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Comparison of linearly and nonlinearly statistically downscaled atmospheric variables in terms of future… Gaitan Ospina, Carlos Felipe 2013

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Comparison*of*linearly*and*nonlinearly*statistically*downscaled*atmospheric*variables*in*terms*of*future*climate*indices*and*daily*variability** by**Carlos*Felipe*Gaitan*Ospina**B.Eng.*Pontificia*Universidad*Javeriana,*2006*M.Sc.*Pontificia*Universidad*Javeriana,*2008** A*THESIS*SUBMITTED*IN*PARTIAL*FULFILLMENT*OF**THE*REQUIREMENTS*FOR*THE*DEGREE*OF**DOCTOR&OF&PHILOSOPHY&&in*The*Faculty*of*Graduate**and*Postdoctoral*Studies*(Atmospheric*Science)** THE*UNIVERSITY*OF*BRITISH*COLUMBIA*(Vancouver)*October*2013*?*Carlos*Felipe*Gaitan*Ospina*2013* ii Abstract *Statistical*downscaling*(SD)*of*global*climate*model*output*assumes*that*the*SD*skills*in* present* climate* are* retained* in* future* climate* (i.e.* timeYinvariant).* To* check* this*assumption,* I* used* regional* climate*model* output* as* pseudoYobservations* to* verify*the* downscaled*models?* performance* in* terms* of* both* daily* variability* and* climate*indices* for*historical*(1971Y2000)*and*future*(2041Y2070)*periods.*The*variables*of*interest* are* daily* maximum* and* minimum* temperatures,* daily* precipitation*occurrences*and*amounts,*and*surface*wind*speed.**In*particular,*a*variety*of*nonlinear*statistical/machine*learning*models*(e.g.*Bayesian*neural* network* (BNN),* adaptive* regression* sufficiently* smooth* polynomials,* and*classification* and* regression* trees* (CART))* and* multiple* linear* regression* models*were* used* to* downscale* the* Canadian* Global* Climate* Model* 3.1* output* using* the*Canadian*Regional*Climate*Model*4.2*output*as*pseudoYobservations.*The*regions*of*interest*are*southern*Ontario*and*Quebec,*Canada,*for*temperature*and*precipitation,*and*Haida*Guaii,*British*Columbia,*Canada,*for*surface*wind*speed.****The* results* indicate* that* choosing* the* best* model* based* on* the* historical* period*performance*could*result*in*having*one*of*the*worst*models*for*the*future*period.*In*particular,*when*downscaling*temperatures,*using*SD*models*with*greater*ability*to*model*complicated*relations,*by*having*either*nonlinear*capability*or*additional*nonY iii temperature*predictors,*seemed*to*alleviate*the*drop*in*performance*found*in*future*climate*conditions.*When*downscaling*precipitation*occurrences,*nonlinear*methods*outperformed*their*linear*counterparts*in*terms*of*the*Peirce*skill*score*and*the*skill*did* not* diminish* for* future* climate.* On* the* other* hand,* when* downscaling*precipitation*amounts,*the*model*performances*deteriorated*in*future*climate,*and*a*BNN*model*had*the*best*future*performance*in*terms*of*daily*variability,*even*though*the*model?s*performance*varied*widely*among*individual*climate*indices.*Finally,*the*Wind*INDices*for*the*evaluation*of*EXtremes*(WINDEX)*were*introduced,*and*it*was*shown* that* a* BNN* model* and* a* probabilistic* model* were* the* best* in* simulating*pseudoYobserved* surface* wind* speed* daily* variability* and* the* WINDEX* climate*indices,*respectively.** & iv Preface *This* thesis* results* from* a* collection* of* four* journal* manuscripts* on* statistical*downscaling* (Chapters*2,*3,*4*and*Appendix*A).*All* the*manuscripts*were*prepared*for*publication,*or*published*in*peerYreviewed*scientific*journals.**Versions*of*Chapter*2*and*Chapter*3*were*submitted* independently* for*publication,*with*authors*Carlos*F.*Gaitan*(first*author),*William*W.*Hsieh,*and*Alex*J.*Cannon.*A*version*of*Chapter*4*with*authors*Carlos*F.*Gaitan*and*Alex*J.*Cannon*was*published*in* Renewable* Energy,* and* a* version* of* Appendix* A*was* submitted* for* publication,*with* authors* Carlos* F.* Gaitan* (first* author),* William*W.* Hsieh,* Alex* J.* Cannon* and*Philippe*Gachon.**Chapter* 2* is* based* on* the* results* presented* at* the* 92nd* Annual* American*Meteorological* Society* (AMS)*meeting,* where* I* was* awarded* a* Commendable* Oral*Presentation*Award.*Chapter*3*elaborates*on*the*results*presented*at*the*93rd*Annual*AMS* meeting,* where* my* oral* presentation* was* distinguished* with* an* Honorable*Mention,* and*Appendix*A* is*based*on* the* results*presented*at* the*91st*Annual*AMS*meeting*where* I* received*a*First*Place*Student*Paper*Award.*Similarly,*Appendix*A*complements* the* results* presented* at* the* 2011* World* Climate* Research* Program*(WCRP)* Open* Science* Conference,* where* I* received* a* World* Meteorological*Organization*travel*award.* v *I*was*responsible*for*the*implementation*and*analysis*of*all*the*experiments,*and*the*majority* of* the* writing.* Professors*William*W.* Hsieh* and* Alex* J.* Cannon* provided*extensive* feedback* for* Bayesian* neural* networks* and* statistical* downscaling*methods.* They* also* provided* numerous* editorial* comments,* helped* me* with* the*figures* and* were* always* kind* to* share* their* invaluable* comments* prior* to* the*submission*of*the*documents.*Prof.*Cannon*provided*the*data*used*in*Chapters*2,*3,*and*4.*Finally,*Prof.*Philippe*Gachon*assisted*me*with*the*Appendix*A,*by*providing*editorial*comments*and*scientific*feedback,*and*Prof.*Van*Nguyen*as*leader*of*a*multiYuniversity* team* supported* by* a* Special* Research* Opportunity* project* grant* was*responsible*for*the*selection*of*the*study*region*and*the*ten*observational*sites*used*in*Chapters*2,*3*and*in*the*Appendix*A.**I*am*authorized*by*Elsevier*to*use*all*the*text,*tables*and*figures*included*in*Chapter*4*(Evaluation*of*historical*and*future*statistically*downscaled*pseudoYobserved*surface*wind*speeds*in*terms*of*annual*climate*indices*and*daily*variability).*** *! vi!Table&of&Contents!ABSTRACT............................................................................................................................(ii(PREFACE...............................................................................................................................(iv(TABLE(OF(CONTENTS......................................................................................................((vi(LIST(OF(TABLES...................................................................................................................(x(LIST(OF(FIGURES................................................................................................................(xi(ACKNOWLEDGEMENTS..................................................................................................(xvi(DEDICATION(......................................................................................................................(xiii(1! INTRODUCTION.....................................................................................................1!1.1! Motivation.............................................................................................................1!! 1.1.1!Study!areas.................................................................................................10!! 1.1.2!Models!used...............................................................................................16!1.2! Background........................................................................................................18!! 1.2.1!Model!evaluation,!verification!and!validation.........................................27!1.3! Thesis!structure................................................................................................28!2( COMPARISON(OF(STATISTICAL(DOWNSCALING(METHODS(FOR(FUTURE(WEATHER(AND(CLIMATE:(SURFACE(TEMPERATURE(IN(SOUTHERN(ONTARIO(AND(QUEBEC,(CANADA.(................................................................................................32(2.1! Introduction.......................................................................................................32!2.2! Study!area,!predictors!and!predictands.......................................................36!! vii!2.3! Methods..............................................................................................................38!2.3.1! Stepwise!multiple!linear!regression....................................................!38!2.3.2! Bayesian!neural!networks!(BNN)(........................................................39!2.4! Daily!maximum!and!minimum!surface!temperature!downscaling.........40!2.5! Model!evaluation...............................................................................................40!2.6! Conclusion!and!discussion..............................................................................49!3( COMPARISON( OF( STATISTICALLY( DOWNSCALED( PRECIPITATION( IN(TERMS(OF(FUTURE(CLIMATE(INDICES(AND(DAILY(VARIABILITY(FOR(SOUTHERN(ONTARIO(AND(QUEBEC,(CANADA.(......................................................................................52(3.1! Introduction.......................................................................................................52!3.2! Precipitation!statistical!downscaling(..........................................................!57!3.2.1! Predictors,!predictands!and!study!area.................................................59!3.2.2! Model!evaluation....................................................................................!61!3.3! Data!methods!....................................................................................................!62!3.3.1! Discriminant!analysis!classification......................................................63!3.3.2! Na?vePBayes!classifier............................................................................!63!3.3.3! kPnearest!neighbor!classifier.................................................................!64!3.3.4! Decision!trees!and!decision!trees!ensemble........................................!65!3.3.5! Stepwise!multiple!linear!regression!(SWLR)(......................................!66!3.3.6! Artificial!neural!networks!for!classification!and!regression................66!3.3.7! Adaptive!regression!sufficiently!smooth!polynomials!!(ARES)(.........!68!! viii!3.4! Results.................................................................................................................68!3.4.1! Classification!and!regression!models....................................................69!3.4.2! Regression!models!conditioned!on!the!occurrence!model..................73!3.5! Conclusions!and!recommendations!..............................................................88!4( EVALUATION( OF( HISTORICAL( AND( FUTURE( STATISTICALLY(DOWNSCALED( PSEUDODOBSERVED( SURFACE( WIND( SPEEDS( IN( TERMS( OF(ANNUAL(CLIMATE(INDICES(AND(DAILY(VARIABILITY.................................................91(4.1! Introduction......................................................................................................!91!4.2! Material!and!methods.....................................................................................!95!4.2.1! Study!area!and!data...............................................................................!95!4.3! Evaluation!method.........................................................................................100!4.3.1! Daily!variability!evaluation.................................................................!100!4.3.2! Climate!of!extremes!evaluation:!The!WINDEX!indices......................101!4.4! Theory!and!calculation..................................................................................103!4.4.1! Downscaling!models............................................................................!103!4.4.2! Regression!downscaling......................................................................!106!4.5! Downscaling!results!and!discussion..........................................................!107!4.5.1! Evaluation!results!for!historical!pseudoPobservations!.....................107!4.5.2! Evaluation!results!in!future!data........................................................!112!4.6! Conclusions!and!recommendations!...........................................................!118!5( CONCLUSION(...............................................................................................................122(! ix!5.1! Summary..........................................................................................................126!5.2! Future!work.....................................................................................................134!REFERENCES(............................................................................................................................136(APPENDICES(............................................................................................................................151(APPENDIX( A:( EVALUATION( OF( LINEAR( AND( NONLINEAR( DOWNSCALING(METHODS( IN( TERMS( OF( WEATHER( AND( CLIMATE( INDICES:( SURFACE(TEMPERATURE(IN(SOUTHERN(ONTARIO(AND(QUEBEC,(CANADA.........................151(A.1.! Introduction.....................................................................................................151!A.2.! Datasets............................................................................................................157!A.2.1.Predictors................................................................................................157!A.2.2.Predictands..............................................................................................160!A.3.! Regression!based!downscaling!models......................................................161!A.3.1.!Linear!regression!(LR)(..........................................................................162!A.3.2.!Multi!layer!perceptron!(MLP)!BNN.......................................................163!A.4.! Daily!maximum!and!minimum!temperature!downscaling....................164!A.4.1.!Model!evaluation.....................................................................................165!A.5.! Results...............................................................................................................168!A.6.! Summary!and!discussion..............................................................................174! x List&of&Tables&*Table*1.*Examples*of*recent*downscaling*studies*(2005Y2012).*.............................................*6*Table*2.*STARDEX*temperatureYrelated*annual*indices*used*in*this*study.*....................*35*Table*3.*Models*and*their*characteristics.*.......................................................................................*38*Table*4.*Extreme*precipitation*indices*used*in*this*study.*The*sdii*index*is*defined*as*the* annual* sum* of* the* daily* precipitation* amount* on*wet* days* divided* by* the*total*number*of*wet*days*per*year*(Karl*et*al.*1999).*The*ETCCDI*uses*monthly*Rx1day*and*Rx5day*instead*of*yearly*maximum*values*(Zhang*et*al.*2011).*.........*53*Table* 5.* Summary* results* for* the* climate* indices.* ?Y?* represents* a* positive* answer*(yes)*to*the*header,*while*a*blank*represents*a*negative*answer*(no).*...................*87*Table*6.*Model*descriptions*and*predictor*sets*used.*Each*CGCM*grid*point*provides*u*and*v*wind*components*(U*and*V*respectively),*and*surface*air*pressure*(P).*....*99*Table*7.*WINDEX*annual*indices.*.....................................................................................................*103*Table*8.**Model*MAEs,*correlation*coefficients*and*WAIs*for*the*historical*and*future*periods.*..............................................................................................................................................*116*Table* 9.* Models'* average* ranking* based* on* daily* variability* and* WINDEX* indices*performance.*...................................................................................................................................*118*Table*10.*STARDEX*Temperature*related*annual*indices*used*in*this*study.*..............*156*Table*11.*Predictor*variables*available*for*the*NCEP/NCAR*reanalysis.*........................*159*Table*12.*Linear*and*nonlinear*models*used.*.............................................................................*170* xi List&of&Figures&Figure*1.*Soil*categories.*Southern*Quebec*(Agriculture_Canada*2013).*..........................*13*Figure*2.*North*American*map.*Squares*indicate*the*location*of*the*selected*weather*stations;* area* in* red* indicate* the* northern* yearYround* (approximate)* range* of*the*Northern*Cardinal*YCardinalis*cardinalis*(Canadian*Geographic,*2009).*.........*14*Figure* 3.* Haida* Guaii,* British* Columbia.* The* West* Moresby* NOMAD* buoy* is*represented*by*a*circle.*.................................................................................................................*16*Figure*4.*Schematic*diagram*illustrating*the*types*of*extrapolation*used*by*different*downscaling*methods.**The*area*in*gray*represent*the*historical*period*domain,*where* Ysim=* f(XGCM).* The* area* in* white* represents* conditions* outside* the*training/historical*period.*............................................................................................................*24*Figure* 5.* Historical* period* (1971Y2000)* model* comparison,* with* the* daily*temperature*MAE*plotted*againt*the*unified*STARDEX*index*IOA.*The*BNN*model*results,*averaged*over*the*10*stations,*are*plotted*with*solid*symbols,*and*their*MLR* counterparts*with* open* symbols.* Triangles* represent* the*models*with* all*predictors*considered*(BNNall,*and*LRall),* circles,* those*with*only* temperature*predictors*(BNNT*and*LRT),*and*squares,* those*with*PCs*as*predictors*(BNNPC*and* LRPC).* For* consistency* with* the* MAE* performance* function,* error* bars*display*the*mean*absolute*deviations*(MAD)*computed*from*the*10*stations.*....*44*Figure* 6.* Model* comparison* in* the* future* period* (2041Y2070),* with* the* daily*temperature* MAE* versus* the* unified* STARDEX* index* IOA.* The* future* period* xii results*are*shown*with*MAD*error*bars,*while*the*historical*period*results*(from*Figure*5)*are*plotted*without*error*bars,* and*black* lines* connect* the*historical*and*the*future*climate*results.*....................................................................................................*46*Figure*7.**Euclidean*distance*to*the*perfect*model*point*(1,1)*in*the*2YD*IOA*space*for*weather* and* climate,* with* smaller* distances* indicating* better* models.* Darker*bars* show* the* historical* period* results* and* lighter* colored* ones,* the* future*period*results.*....................................................................................................................................*48*Figure*8.* *A*comparison*of*the*mean*values*of*the*downscaled*daily*(left)*maximum*and* (right)* minimum* temperatures* with* the* mean* of* the* CRCM* pseudoYobservations* for* historical* (20C3M)* and* future* (A2)* climates.* * The* thick*horizontal*lines*mark*the*mean*values*from*the*pseudoYobservations.*..................*48*Figure*9.*North*American*map.**Canadian*provinces*of*Quebec*and*Ontario*are*shown*in* dark* gray.* Squares* indicate* the* location* of* the* selected* weather* stations;*circles*indicate*the*location*of*the*closest*CGCM3*grid*points.*....................................*58*Figure*10.*Precipitation*occurrence*models?*average*Peirce*skill*score*(PSS).*Lighter*and* darker* coloured* bars* represent* the* historical* * (20C3M)* and* future* (A2)*periods,* respectively.* Error* bars* represent* mean* absolute* deviations* (MAD)*from*the*ten*stations.*......................................................................................................................*71*Figure*11.*Precipitation*amount*models?*average*MAEs.*Lighter*and*darker*coloured*bars* represent* the* historical* * (20C3M)* and* future* (A2)* periods,* respectively.*Error*bars*represent*the*MAD.*...................................................................................................*72* xiii Figure* 12.* Empirical* cumulative* distributions* of* the* statistically* downscaled* time*series*for*Lennoxville,*Quebec,*with*and*without*using*variance*inflation*(dashed*and*dotYdashed,*respectively)*versus*the*CRCM*pseudoYobservations*(solid).*The*SD*model*is*ANNYR.*.........................................................................................................................*73*Figure*13.*Average*IOA*differences*between*climate* indices*from*the*future*and*the*historical* periods* from* ANNYF* (left),* ARESYF* (middle)* and* SWLR* (right),* with*error*bars*indicating*the*MAD.*...................................................................................................*78*Figure*14.* IOAs* (averaged*over* ten*stations)*of* the*precipitationYrelated* indices* for*the*historical*(20C3M)*and*future*periods*(A2)*for*ANNYF*(top),*ARESYF*(middle)*and* SWLRYF* (bottom),* with* precipitation* occurrences* calculated* using* ANNYC,*and*error*bars*indicating*the*MADs.*........................................................................................*79*Figure*15.*Number*of*KS*tests*(out*of*a*maximum*of*ten)*not*rejected*at*a*significance*level*0.05.*The*null*hypothesis*is*that*each*simulated*climate*index*comes*from*the*same*continuous*distribution*as*the*corresponding*climate*index*calculated*from* the* CRCM* pseudoYobservations.* * Figure* shows:* ANNYF* (top),* ARESYF*(middle)*and*SWLRYF*(bottom).*................................................................................................*80*Figure* 16.* Empirical* cumulative* distributions* for* all* models* during* the* historical*(left)*and*future*periods*(right).*................................................................................................*85*Figure* 17.* Historical* (20C3M)* and* future* (A2)* empirical* cumulative* distribution*functions* (ECDF)* of* precipitation* from*ANNYF* (for* precipitation* days* from* the*ANNYC*classification*model)*versus*the*CRCM*pseudoYobservations.*Darker*lines*show* the* SDS.* Dashed* lines* show* the* future* ECDF.* The* ECDFs* are* from* the* xiv pseudoYobserved* and* statistically* downscaled* time* series* closest* to* Ottawa,*Canada.*..................................................................................................................................................*86*Figure*18.*Study*area:*Haida*Guaii,*British*Columbia,*Canada.*Red*markers*represent*CGCM* grid* points.* The* red* diamond* corresponds* to* the* closest* grid* point,* or*pseudo*station*5*(ST5).*The*buoy*is*represented*with*a*blue*circle.*.........................*96*Figure*19.*Statistical*downscaling*diagram.*The*upper*section*represents*the*coarse*resolution*CGCM*and*the*lower*one*the*CRCM*pseudo.*..................................................*97*Figure*20.*Observed*wind*speed*quantiles*(msY1)*versus*RCM*simulated*wind*speed*quantiles* (msY1).* * Red* line* joins* the* 25th* and* 75th* percentiles* of* a* normal*distribution*line.*...............................................................................................................................*98*Figure* 21.* Statistically* downscaled* wind* speed* MAEs* versus* the* WINDEX* indices*average*IOA*for*the*historical*period.*..................................................................................*109*Figure*22.*WINDEX*indices*IOAs*for*the*historical*period.*..................................................*111*Figure*23.*WINDEX*CO*index.*The*solid*black*lines*show*ANNall*(top),*LRall*(center)*and* CDFt* (bottom),* and* the* dashed* red* lines* the* pseudoYobservations.*Horizontal*axes* correspond* to* the*30*years*historical*period* (1970Y1999).*The*vertical*axes*show*the*total*number*of*days*per*year*with*wind*speed*above*the*cutYout*speed.*..................................................................................................................................*113*Figure* 24.* Statistically* downscaled* wind* speeds* MAEs* versus* the*WINDEX* indices*average*IOA*for*the*A2*scenario*run.*....................................................................................*114*Figure*25.*WINDEX*indices*IOAs*for*the*A2*scenario*run.*....................................................*116* xv Figure* 26.* Selected* weather* stations* located* over* southern* Qu?bec* and* Ontario*(Canada)* with* altitude* (in* m)* of* this* area* given* in* color* scale.* Black* lines*correspond*to*the*CGCM3*grid.*...............................................................................................*162*Figure*27.*TAV*MAE*vs.*USI*(average*IOA*of*the*STARDEX*indices).*The*multiYstation*means*from*the*nonlinear*BNN*models*are*plotted*with*solid*symbols,*and*those*from*the*LR*models*with*open*symbols.**The*four*different*sets*of*predictors*are*identified* by* the* shape* of* the* symbols,* with* diamonds* for* the* case* using* all*predictors,* circles,* the* four* temperatures,* stars,* the* four* thicknesses,* and*squares,* the* 3* PCs.* Horizontal* and* vertical* error* bars* represent* the* mean*absolute*deviation**(MAD).*.......................................................................................................*171*Figure*28.*Models?*STARDEX* indices* IOAs.*Nonlinear*methods*are*represented*with*dark*bars*and*the*linear*methods*with*lighter*coloured*bars.**Error*bars*indicate*the*mean*absolute*deviations*(MAD)*of*individual*stations.*.....................................*173*Figure*29.*QuantileYquantile*plots.**(a)*Mean*BNNall*versus*mean*TMAX*observed,*(b)*Mean* LRall* versus*mean*TMAX*observed,* (c)*Mean*BNNall* versus*mean*TMIN*observed,* and* (d)* Mean* LRall* versus* mean* TMIN* observed.* The* red* line*corresponds*to*a*perfect*agreement*between*model*and*observations.*..............*176* * & xvi Acknowledgements **First*and*foremost,*I*want*to*thank*the*support*and*collaboration*of*Prof.*William*W.*Hsieh,* Prof.* Alex* J.* Cannon,* and* the* Climate* Prediction* Group* members* for* their*invaluable*contribution*and*feedback.*Additionally,*I*want*to*thank*Prof.*Roland*Stull*and*the*Geophysical*Disaster*Computational*Fluid*Dynamics*Centre*team*for*inviting*me*to*their*seminars*and*their*involvement*in*my*project.*Similarly,*I*want*to*thank*Prof.*Ian*McKendry*for*giving*me*the*opportunity*to*assist*him*with*his*?Atmospheric*Environments?* course,* and* to* Professors* Marwan* Hassan* and* David* Ley* for* their*trust*and*confidence,*and*for*permitting*me*to*lecture*?Climate*and*Vegetation?*and*?Water*and*Landscapes?*at*the*Department*of*Geography;*also*I*am*most*grateful*for*Prof.*Douw*Steyn?s*advice*and*constant*support.****I*am*thankful*for*the*support*from*a*Special*Research*Opportunity*grant*awarded*by*the* Canadian* Natural* Sciences* and* Engineering* Research* Council,* and* to* Prof.* Van*Nguyen*at*McGill*University*for*leading*the*multiYuniversity*project,*and*for*inviting*me* to* collaborate*with* Dr.* Andrew*Harding* and*Milka* Radojevic* at*McGill.* I* thank*Prof.* Philippe*Gachon* (Universit?* du*Qu?bec* ?*Montr?al* and* Environment* Canada)*for*his*comments*and*help*with*the*Appendix*A,*and*for*his*assistance*during*my*visit*to*Montreal,*and*to*Andrew*and*Milka*for*their*support*and*hospitality.** xvii I*also*wish*to*acknowledge*the*Data*Access*Integration*(DAI)*Team*for*providing*the*data* and* technical* support.* The* DAI* Portal* (http://loki.qc.ec.gc.ca/DAI/)* is* made*possible*through*collaboration*among*the*Global*Environmental*and*Climate*Change*Centre*(GEC3),*the*Adaptation*and*Impacts*Research*Division*(AIRD)*of*Environment*Canada,*and*the*Drought*Research*Initiative*(DRI).***Lastly,*I*am*most*thankful*to*Keith*Dixon,*Berrien*Moore,*Renee*McPherson*and*the*people*at*the*South*Central*Climate*Science*Center*and*at*NOAA?s*Geophysical*Fluid*Dynamics*Lab*for*their*support*in*2013.* * xviii Dedication ** To*Elvira,*Luis*Orlando*&*D?sir?e**  1 Chapter 1 1 INTRODUCTION 1.1 Motivation As*mentioned* by* Coiffier* (2011)* in* his* book* ?Fundamentals* of* Numerical*Weather*Prediction?,* numerical* models* have* now* become* essential* tools* in* environmental*science,* particularly* in* weather* forecasting* and* climate* prediction.* * These* models*when*developed*and*applied*to*generate*future*global*climate*projections*are*known*as*global*climate*models*(GCMs).*GCMs*aim*to*dynamically*solve*the*complexities*of*the* interactions* affecting* the* climate* system* (Timbal* et* al.* 2008),* but* as*acknowledged*by*several*authors*(e.g.*Denis*et*al.* (2002);*Wilby*(1998);*Wilby*and*Wigley*(1997a);*Fowler*and*Wilby*(2007);*Haylock*et*al.*(2006);*Friederichs*(2010);*Schoof*and*Pryor*(2001);*Huth*(1999))*their*resolution*is*often*too*coarse*for*direct*use* in* regional* climate* change* impact* studies* (Warner* 2011),* and* doubling* their*resolution* generally* implies* 24* * times* the* number* of* computations* (Coiffier* 2011).**Because* of* the* final* users?* need* for* fineYscale* information* at* lower* computational*cost,* various* statistical* techniques* and* higher* resolution* regional* climate* models*(RCMs)*have*been*developed*for*downscaling*GCM*simulations*to*regional*and*local*scales*(Denis*et*al.*2002).*However,*the*RCMs*can*also*be*computationally*intensive*and* their* spatial* resolution* (from* 5* to* 50* km)* does* not* always* provide* the*information* required* by* regional* climate* change* impact* studies* (Vrac* and* Naveau*  2 2007).*Moreover,*the*RCMs*depend*on*boundary*conditions*prescribed*by*the*GCMs*and* on* time* invariant* subYgrid* parameterizations* schemes* (Khalili* et* al.* 2013).* On*the* other* hand,* the* statistical* downscaling* techniques* derived* from* the* local*forecasting*model*output*statistics*(MOS)*method*(e.g.*quantile*matching*approaches*like* the* CDFt* method* (Michelangeli* et* al.* 2009)),* or* from* the* numerical* weather*prediction?s*perfect*prog*postYprocessing*technique*(Marzban*et*al.*2006),*establish*a*synchronous*(valid*at*a*given*time)*statistical*relation*between*the*coarse*resolution*predictors* (i.e.* GCMs* or* the* global* reanalysis* data* sets)* and* the* smallYscale*predictand(s)* representing* the* climate* of* a* region* (Warner* 2011).* * In* general,* the*statistical* downscaling* methods* are* computationally* inexpensive* and* easier* to*implement* than* the* RCMs* but* rely* on* the* time* invariance* of* the* statistical*relationships.* Therefore,* both* the* RCMs* and* the* statistical* downscaling* methods*depend*on*the*credibility*of*the*coarser*scale*GCM.**As*aforesaid,*statistical*downscaling*techniques,*like*the*ones*used*in*this*thesis,*are*often* needed* to* generate* finer* scale* projections* of* variables* affected* by* local* scale*processes*not* resolved*by* the* coarser* resolution*GCMs.*However,* in* the*absence*of*future*observations,*statistical*downscaling*studies*rely*on*historical*data*to*validate*their*models*and*assume*that*these*historical*simulation*skills*will*be*retained*in*the*future*(Wilby*et*al.*1998),*as*they*consider*that*the*statistical*relationship(s)*between*the* coarse* resolution* global* climate* model* predictors* and* the* finer* scale*  3 predictand(s)* remain* constant* over* time* (i.e.* are* timeYinvariant).* Additionally,* as*mentioned* by* Charles* et* al.* (1999),* the* evaluation* of* downscaling* models* using*historical* data* does* not* guarantee* the* models* will* be* valid* under* future* climate*conditions.* The* present* dissertation* tackles* this* generally* overlooked* assumption,*and*verifies*if*downscaling*relationships*are*time*invariant,*when*downscaling*daily*maximum*and*minimum*temperatures*(TMAX*and*TMIN,*respectively),*precipitation*occurrences*and*amounts,*and*surface*wind*speeds.***As* mentioned* in* Cannon?s* doctoral* dissertation* (2008a),* most* * statistical*downscaling* models* have* been* developed* for* observations* at* a* single* station.*Nevertheless,*it*is*common*to*find*studies*focusing*on*specific*geographical*areas,*like*watersheds* and* islands,* political* divisions* (e.g.* countries,* states,* provinces)* and*broader* geographic* areas* like* continents* (see* Table* 1).* To* the* date,* there* are* no*studies*showing*statistically*downscaled*global*results*for*any*variable,*and*few*had*downscaled* broader* regions* of* North* America* using* gridded* data* as* observations.**For*example*Maurer*and*Hidalgo*(2008)*produced*continuous*gridded*time*series*of*precipitation*and*surface* temperature*at*1/8*degree*resolution*(approximately*140*km2*per*grid* cell)*over*western*U.S.*using* two*statistical*downscaling*methods*and*the*NCEP/NCAR*reanalysis*data.**  4 Even* though* the* lack* of* global* downscaling* studies* compromises* the* ability* to*generalize* the* downscaling* models?* results,* it* is* important* to* recall* that* the*downscaling*studies*are*formulated*to*obtain*information*at*a*finerYscale/local*level,*thus* the* downscaled* results* and* the* downscaling* models* should* be* considered*relevant* for* their* specific* region* of* interest.* For* example,* it* is* incorrect* to* assume*that*because*a*downscaling*model*was* skillful* in* southern*Ontario* the* same*model*will* be* skillful* in* Ellesmere* Island,* Nunavut.* This* could* happen* for* a* variety* of*reasons* including*incomplete*understanding*of*the*physical*processes* involved*(e.g.*seaYice*interaction),*deficiencies*with*the*parameterizations*(e.g.*poor*representation*of*seaYice*albedo),*different*relevant*predictors*and*the*GCM?s* inability*to*represent*observed*local*conditions.**Regression*models*used*to*downscale*can*represent*linear*or*nonlinear*relationships*between* predictands* and* largeYscale* predictors* (Fowler* et* al.* 2007).* In* particular,*nonlinear* regression*models* like* artificial* neural* networks* (ANN)* have* been* used*extensively* for* statistical* downscaling* (e.g.,* Wilby* and*Wigley* (1997a);* Crane* and*Hewitson* (1998b);* Weichert* and* Burger* (1998);* Schoof* and* Pryor* (2001);* Dunn*(2004);*Cannon*(2008b);*Huth*et*al.*(2008)).*Artificial*neural*networks,*are*inspired*by* the* human* brain* and* its* network* of* about* 86* billion* interconnecting* building*blocks* called* neurons* (Azevedo* et* al.* 2009),* and* represent* the* first* wave* of*breakthrough*in*machine*learning*(Hsieh*2009).*By*the*mid*1990?s*many*connections*  5 between* the* artificial* neural* networks* and* statistical* physics,* probabilistic* models*and* statistics* became*well* known* (Rasmussen* and*Williams* 2006).* In* general,* the*nonlinear*models*can*give*similar* results* to*multiple* linear* regression*downscaling*methods*for*temperature*and*precipitation*(Schoof*and*Pryor*2001),*and*are*capable*of* outscoring* linear* models* when* relationships* are* nonlinear* and/or* interactive*(Tang*et*al.*2000).*****Statistical* downscaling* experiments* for* future* climate* rely*on* several* suppositions,*one* of* them* being* the* timeYinvariance* assumption,* as* one* cannot* know* the* true*change* in* the* variable* of* interest,* or* cannot* validate* the*models*with* data* not* yet*observed.* In*particular,*Wilby* (1997a)* studied* the* relationship*between* circulation*indices* and* local* precipitation* in* the* United* Kingdom,* and* concluded* that* the*empirical* relationships* used* could* not* be* assumed* to* be* timeYinvariant.* The*most*common* approach* used* to* investigate* model* stationarity* involves* splitting* the*historical* period* data* in* two* folds,* where* the* first* fold* is* used* for* training* the*downscaling* model* and* the* second* fold* is* used* for* validation.* * If* the* data* in* the*second* fold* belongs* to* a* ?warmer?* period* than* the* data* used* in* the* first* fold,* the*stationarity*assumption*can*be*tested*in*a*slowly*warming*climate*context*(Benestad*et*al.*2008b).*However*as*the*observed*changes*in*mean*temperature*during*the*past*50*years*are*much*milder*than*the*projected*temperature*change*for*the*21st*century,*this*approach*is*not*recommended.**  6 Table*1.*Examples*of*recent*downscaling*studies*(2005Y2012). ********************Author* Variable* Year* Region* Number*of*stations*A.*Bardossy*et*al.! Precip.! 2005! Essen,*Germany! 1!F.*Wetterhal*et*al.! Precip.! 2006! Central*Sweden! 7!C.*M.*Sordo*et*al.! Wind*speed! 2007! Spain! 11!A.*Busuioc*et*al.! Precip.! 2007! EmiliaYRomagna*region,*Italy! 41!M.*Hessami*et*al.! Tmax! 2007! Eastern*Canada! 10!D.I.*Jeong*et*al.! Precip! 2008! Southern*Quebec,*Canada! 9!R.*Huth*et*al.! Temp.! 2008! Europe! 8!A.*Cannon! Precip.! 2008! British*Columbia,*Canada! 10!P.A.*Michelangeli*et*al.! Wind*speed! 2009! France! 26!A.*Anandhi*et*al.* Tmax,*Tmin* 2009* Malaprabha*reservoir,*India* 18*A.*Anandhi*et*al.* Precip.* 2009* 11*R.E.*Benestad* Precip.* 2010* Oslo,*Norway* 1*A.*Bardossy*and*G.*G.*S.*Pegram* Precip.* 2009* BadenYWurtemberg.*Germany* 32*SYT.*Chen*et*al.* Precip.* 2010* ShihYMen*Reservoir,*Taiwan* 10*D.*Fasbender*and*T.*B.*M.*J.*Ouarda* Tmax,*Tmin* 2010* Gulf*of*St.*Laurence,*Canada* 12*C.L.*Curry*et*al.* Wind*speed* 2011* British*Columbia* 20*D.*I.*Jeong*et*al.* Tmax,*Tmin,*precip.* 2012* Southern*Ontario,*Canada* 4*E.P.*Maurer*and*H.G.*Hidalgo* Temp.*and*precip.* 2012* Western*U.S.* Gridded*data*C.S.*Cheng*et*al.* Wind*gusts* 2012* Ontario* 14*G.*Burger*et*al.* Tmax,*Tmin*Precip.* 2012* British*Columbia,*Canada* 6*  7 For*example,*Barrow*et*al.*(2004)*indicate*that*by*the*2080s*the*winters*in*southern*and* western* Canada* will* likely* be* between* 6?C* and* 8?C* warmer* than* at* present,*following*the*SRES*A2*scenario.*This*projected*warming*is*unprecedented*in*the*20th*century*historical*record.***In*contrast,*Vrac*et*al.?s*method*(2007)*uses*regional*climate*model*(RCM)*outputs*as*pseudoYobservations*to*estimate*model*performance*in*the*context*of*future*climate*projections* by* replacing* historical* and* future* observations*with*model* simulations*from*the*RCM,*nested*within*the*domain*of*a*coarse*resolution*GCM.*The*method*was*originally* used* to* ?identify* strengths/weaknesses* of* a* nonhomogeneous* stochastic*weather*typing*method?*and*was*exclusively*applied*to*downscale*daily*precipitation*distributions* for* different* stations* in* Illinois,* USA.* Their* work* also* compared* the*downscaled*and*the*observed*time*series*in*terms*of*10th,*25th,*50th,*75th,*90th*and*99th*quantiles.*In*this*thesis*I*extended*Vrac*et*al.?s*approach*and*compared*not*only*one*but* several* combinations* of* methods* and* predictors* to* downscale* not* only* daily*precipitation,* but* also* daily* maximum* and* minimum* temperature,* and* daily* wind*speeds.*Additionally,*I*further*extended*the*evaluation*of*the*downscaled*time*series*by* comparing* different* climate* indices* with* those* obtained* from* the* pseudoYobservations.* To* check* this* assumption,* I* used* regional* climate* model* output* as*pseudoYobservations,* following* Vrac* et* al.* (2007b),* to* verify* the* statistical*downscaling* models?* performance* in* terms* of* both* daily* variability* and* climate*  8 indices* for* historical* (1971Y2000)* and* future* (2041Y2070)* periods.* * Both* periods*correspond* to* the* time* windows* used* by* the* North* American* Regional* Climate*Change*Assessment*Program*(NARCAPP).**Specifically,* nonlinear* and* linear* regression* models* were* used* to* downscale* the*Canadian* Global* Climate* Model* 3.1* (CGCM3)* output* using* the* Canadian* Regional*Climate* Model* 4.2* (CRCM)* outputs* as* pseudoYobservations.* * The* aforementioned*climate*models*were*chosen*to*be*consistent*with*the*models*used*by*Environment*Canada* to* produce* climatological* maps* from* the* CRCM* and* CGCM* over* North*America* for* the* current* and* future* periods* (DAI_Team* 2009),* and* with* the* North*American* Regional* Climate* Change* Assessment* Program* (NARCCAP)* as* I* used* the*fourth*member*of*the*CGCM3*and*the*SRES*A2*emission*scenario*(IPCC*2000).***Nevertheless,* the*reader*must*be*aware*of*several* limitations*when*downscaling* to*gridded*data,*including:*1)*the*gridded*predictand*represents*an*area*average*instead*of* point*measurements;* 2)* the* variance* of* a* variable* averaged* over* a* large* area* is*expected*to*be*smaller*than*the*variance*of*the*same*variable*at*a*particular*weather*station/point,*and*3)*the*wet*spells*calculated*from*the*gridded*data*likely*last*longer*than* the* observed* ones,* as* CRCM* overpredicts* the* number* of* days* with* nonYzero*precipitation.* Nonetheless,* according* to* Bourdages* and* Huard* (2010)* the*precipitation*values*generated*by*the*CRCM*4.2*(driven*by*the*CGCM3)*are*very*close*  9 to* the* observation* datasets,* while* the* simulated* temperatures* are* lower* than* the*observed*values.*Additionally,*when*using*RCM*output*as*pseudoYobservations,*one*should*be*aware* that* since*RCMs*simulate* climate*over*a* specified*area*of* interest,*they* require* nesting* information*which* describes* the* evolution* of* the* atmospheric*circulation*at* their* lateral*boundaries* (Music*and*Sykes*2011),* thus*are*affected*by*the*driving*GCM*uncertainties.**Finally,* it* is* worth* mentioning* that* although* the* study* of* the* timeYinvariance*assumption*is*the*main*interest*of*this*thesis,*it*is*not*the*main*cause*of*uncertainty*in*climate*change*regional*projections.*Many*studies*have*shown*that*the*driving*GCM*may*significantly* impact* the*simulated*regional*climate*(e.g.*Rowell*2006;*D?qu?*et*al.,*2007;*De*Elia*and*Cot?,*2010),*and*the*GCMs?* internal*variability* is*estimated*to*account*for*at*least*half*of*the*interYmodel*spread*in*projected*climate*trends*during*2005?2060* in* the* CMIP3* multiYmodel* ensemble* (Deser* et* al.* 2010).* Current*international*efforts*to*quantify*the*GCMs*uncertainty*include*the*generation*of*larger*ensembles* of* GCM* simulations* * (from* diverse* institutions),* following* different*emission*scenarios* (now*called*representative*concentration*pathways* (RCP)).*This*task*requires*significant*economic,*human*and*computational*resources,*although*the*probability* that* any* single* emissions* path* will* occur* as* described* in* the* Special*Report*on*Emissions*Scenarios*Y*SRES*(IPCC*2000)*is*highly*uncertain.**  10 1.1.1 Study areas *The* study* regions* are* southern* Ontario* and* Quebec* (for* precipitation* and*temperatures),* and* Haida* Guaii,* British* Columbia* (for* the* surface* wind* speed).*Assessing*if*the*statistically*downscaled*relationships*are*timeYinvariant*in*southern*Ontario*and*Quebec*is*key*to*gain*confidence*on*recent*downscaling*studies*focusing*on*the*area*(e.g.*Fasbender*and*Ouarda*(2010),* Jeong*et*al.*(2012a))*as*their* future*climate* projections* were* based* on* time* invariant* relationships.* * Additionally,* the*downscaled* results* could* potentially* benefit* 1/7th* of* the* Canadian* population*(Statistics_Canada*2012)*as*in*this*thesis,*I*also*calculate*the*skill*of*different*climate*indices* with* high* societal* impact* like* heat* wave* duration,* number* of* frost* days,*number* of* wet* days* and* number* of* dry* days,* among*many* others.* * Other* sectors*interested* in* downscaled* information* include:* agriculture,* health* and* ecology* (e.g.*Bergant*et*al.*(2002)).**The* selection* of* Haida* Guaii* as* an* area* of* interest* to* downscale* wind* speeds* is*motivated*by*the*recent*interest*from*the*Naikun*Wind*Development*Inc.*to*develop*Canada?s*first*offshore*wind*energy*project,*between*Haida*Gwaii*and*Prince*Rupert.*Personally,* it* is*of*uttermost* importance*to*determine* if* the*statistical*relationships*based*on*historical*data*will*hold*in*the*future,*as*Cheng*et*al.*(2012)*already*warned*about*the*possible*impacts*of*climate*change*on*wind.*  11 1.1.1.1 Southern Ontario and Quebec *The* region?s* climate* is* continental,* distributed* from*moderate* in* the* south* to* subY*polar*in*northern*areas,*with*cold*winters*and*hot?mild*summers*modulated*by*the*presence*of*water*masses*as*the*Gulf*of*St.*Lawrence*(Fasbender*and*Ouarda*2010).*The* study*area* includes* important* socioYeconomic* centers* like*Ottawa,* the*nation?s*capital,* and*Montreal* in* Quebec;* the* cities* have* a* combined* population* (including*metropolitan*areas)*of*5.1*million*people*(Statistics_Canada*2012).*The*study*region*has* soils* of* the* Gleysolic* and* Brunisolic* orders,* some* of* the* best* soils* in* southern*Quebec*(Figure*1).*For*details*see*Agriculture*Canada*(1974)*soil*classification.*According*to*the*City*of*Ottawa*(2012),*the*city?s*the*rural*economy*contributes*over*$1* billion* CND* to* the* Canadian* gross* domestic* product* (GDP).* The* Ottawa*agricultural* sector* represents* close* to* 300,000* acres* of* land* farmed* by*more* than*1,300* agricultural* operations,* employing* approximately* 10,000* people.* Rural*economic*activity* includes*areas* that*might*be*affected*by* future*climate*change*as*agriculture,* forestry,* tourism,* manufacturing,* services,* and* transportation.* For*example,* a* warming* climate* will* likely* increase* environmental* stresses,* and* may*result* in* less* resilient* ecosystems* that* are* unable* to* combat* invasive* species*(Hellmann*et*al.*2008).*According*to*the*Ontario*Invasive*Species*Action*Plan*(2012),*this*potential*effect*on*the*ecosystems*is*worrisome*as*a*warming*climate*will*likely*increase* the*rate*of*new*species* invasions*and*may*promote* the*spread*of*alreadyYestablished*species.**  12 The*success*of*commercial*scale*agriculture*has*always*been*dependent*of*the*local*weather*conditions,*for*example*in*2012*warmer*than*usual*temperatures*in*March*induced*an*early*bloom*from*different*fruit*harvests*across*Ontario,*Quebec*and*the*northeastern*U.S,*which*was*vulnerable*to*subsequent*deep*frosts,*causing*Ontario?s*apple*growers*to*loose*about*80*per*cent*of*their*crops*(Leung*2012).**Similarly,* as* the* ecosystems* link* plants,* animals* and* their* interactions* with* the*physical* environment,* a* changing* climate* affects* the* ecosystems* because*temperature* and* moisture* are* among* the* physical* limiting* factors* on* species*distributions* (Christopherson* et* al.* 2013).* For* example,* the* Northern* Cardinal*(Cardinalis* cardinalis)* does* not* migrate* and* is* common* in* the* southeastern* U.S.;*however,* in*recent*decades*its*northern*yearYround*range*moved*north*through*the*United*States*and*southeast*Canada*(Figure*2).*Overall* the* selected* weather* stations* belong* to* a* region* with* the* best* soils* in*southern* Quebec,* are* located* inside* the* Northern* Cardinal?s* northern* yearYround*range,* and* include* important* socioYeconomic* cities* like*Montreal* and* Ottawa,* thus*the* results* derived* from* the* evaluation* of* the* stationarity* assumption* over* this*region*may*be*taken*into*consideration*by*Ontario?s*agricultural*sector,*by*biologists*interested*in*obtaining*local*scale*temperature*and*precipitation*information*to*drive*their* ecological* models,* and* by* different* decision* makers* involved* with* the*development*and*execution*of*long*term*city*plans,*among*many*others.*  13 !Figure!1.!Soil!categories.!Southern!Ontario!and!Quebec!(Agriculture!Canada!2013).!  14 !Figure!2.!North!American!map.!Squares!indicate!the!location!of!the!selected!weather!stations;! area! in! red! indicate! the! northern! year=round! (approximate)! range! of! the!Northern!Cardinal!=Cardinalis!cardinalis!(Canadian!Geographic,!2009).!!!1.1.1.2 Haida Gwaii !Haida!Gwaii! is!an!archipelago!of!over!150!islands! located!on!the!northwest!coast!of!British!Columbia,!Canada.!!Formerly!known!as!the!Queen!Charlotte!Islands,!they!are!nestled! below! the! Alaskan! Panhandle! and! separated! from! the!mainland! by! Hecate!Strait,! which! has! some! of! the! strongest! and!most! consistent!winds! in! Canada.! The!islands!constitute!the!most!westerly!landforms!of!Canada!!(Figure'3).!!!!Nowadays,! there! is! interest! from! the! Naikun! Wind! Development! Inc.! to! develop!Canada?s!first!offshore!wind!energy!project,!between!Haida!Gwaii!and!Prince!Rupert.!!!  15 I!selected!this!area!because:!? It!is!of!prime!economic!interest!for!British!Columbia,!!!? It!has!strong!and!consistent!winds,!!? It!is!relevant!for!the!study!of!wind!speed!future!projections,!and!? The!West!Moresby!NOMAD!buoy!(52?31'12"!N!132?41'23"!W)!is!located!near!the!area,!and!can!be!used!to!verify! if! the!CRCM!output!and!the!observations!belong!to!the!same!probability!distribution!family,!and!to!find!the!CRCM!bias!versus!the!observations.!!Pseudo=observed! daily! maximum! wind! speed! from! the! 1970=1999! period!obtained!from!a!CRCM!grid!cell!located!near!the!buoy!=!located!west!of!Frederick!Island,!opposite!to!the!Hecate!Strait!=!was!used!as!predictand.!The!maximum!and!minimum! values! on! record! are! 22.5! and! 0.9! meters! per! second,! respectively.!Wind!speeds!between!these!values!will!allow!the!determination!of!the!number!of!days!a!hypothetical!wind!turbine!would!not!operate!continuously.!For!example,!it!is!possible! to!calculate! the!number!of!days!with!wind!speeds!above! the!cut=out!speed!(i.e.!when!the!turbine?s!blades!are!stopped!so!the!turbine!is!not!damaged!by!wind!speeds!above!the!design!specifications),! the!number!of!days!where!the!winds!are!below!the!cut=in!speed!(i.e.!lowest!wind!speed!needed!by!the!blades!to!turn,! and! by! the! turbine! to! start! producing! energy),! and! other! indices! (see!Chapter!4).!  16 !Figure!3.!Haida!Guaii,!British!Columbia.!The!West!Moresby!NOMAD!buoy!is!represented!by!a!circle.!!1.1.2 Models used Radi?! and! Clarke! (2011)! evaluated! the! performance! of! 22! AOGCMs,! from! the!Intergovernmental! Panel! on! Climate! Change! (IPCC),! in! simulating! the!mean! annual!cycle! and! interannual! variability! of! 6! selected! variables! (precipitation,! sea! level!pressure,!geopotential!height!at!850!hPa!and!500!hPa,!specific!humidity!at!850!hPa,!and!air! temperature!at!850!hPa)!over!North!America!during!1980=1999,!and! found!that!the!five!top!performing!models!were!ECHAM5MPI=OM,!HadCM3,!CGCM3.1(T47),!MIROC3.2(medres),!and!CGCM3.1(T63),!hence!the!CGCM!is!a!good!model!to!provide!predictors!for!statistical!downscaling.!!  17 CGCM3.1! outputs! have! been!used! as! potential! predictors! in! the! past! by!Khan! et! al.!(2006)! and! Dibike! and! Coulibaly! (2006),! among! others,! for! downscaling!temperatures! in! northern! Quebec,! Canada;! and! by! Jeong! et! al.! (2012a)! for!downscaling!temperatures!and!precipitation!in!southern!Quebec,!Canada.!The! potential! predictors! I! have! available! from! the! CGCM3.1! and! the! NCEP/NCAR!reanalysis!=!through!Environment!Canada?s!Data!Access!and!Integration!Portal!(2008)!=! include! daily! values! of! 25! variables! comprising! of! temperature,! humidity,! surface!pressure,! as! well! as! upper! air! measures! of! wind! speed! and! direction,! vorticity,!divergence,! and! geopotential! height.! However,! I! did! not! use! all! of! them,! since,! in!downscaling! applications! it! is! important! to! use! predictors! that! can! be! modelled!correctly!by!the!AOGCMs!and!that!are!able!to!incorporate!the!forcing!responsible!for!the! climate! change! signal,! and! not! all! the! available! predictors! have! these!characteristics.! In! particular,! vorticity! at! 500! hPa,! 850! hPa! and! 1000hPa,! wind!direction! at! 500! hPa! and! the! zonal! component! of! the! wind! velocity! at! 1000! hPa!showed! the! largest! root! mean! squared! error! (RMSE)! differences! between! the!reanalysis!and!the!CGCM3.1;!T2!also!showed!some!biases!during!winter!(Jeong!et!al.!2012a).!As! I! am! interested! in! downscaling! temperatures! and! precipitation! in! southern!Ontario!and!Quebec,!Canada,!and!I!have!available!the!same!reanalysis!data!that!Jeong!et!al.!(2012b)!interpolated!to!the!AOGCM!(CGCM3.1)!grid,! it! is!natural!for!me!to!use!  18 predictors!recommended!by!Jeong!et!al.! (2012a).! ! In!particular,! Jeong!et!al.! (2012a)!mentioned! that! their! predictor! sets! ??! should! be! relevant! to! project! anticipated!predictand! variables! because! they! include! sensitive! predictors! for! future! climate!signal!and!variability!such!as!temperature!at!2!m!(T2),!specific!humidities!at!500!hPa,!850! hPa! and! 1000! hPa,! and! 500! hPa! and! ! 850hPa! geopotential! heights?,! they! also!mentioned!that!?specific!humidity,!geopotential!height!at!different!levels!and!T2,!were!important! explainable! predictors! for! the! daily! temperatures?.! The! relationship!between! geopotential! thickness! and! temperature! can! be! appreciated! from! the!hydrostatic! equation,! and! the! relationship! between! the! coarser! temperatures! or!precipitation! outputs! and! the! local! variables! is! straightforward! as! the! coarse!resolution! output! can! be! viewed! as! a! smoother! version! of! the! local! variable! of!interest.!!1.2 Background A!global!climate!model! is!a!numerical!representation!of! the!main!chemical,!physical!and!biological!components!of!the!global!climate!system.!GCMs!can!be!used!to!simulate!historical! climates! and! project! future! climates! under! different! emission! scenarios!resulting! from! different! assumptions! about! socio=economical! trends! (IPCC! 2000).!Nevertheless,! the! GCM's! low! resolution! prevents! it! from! resolving! small=scale!dynamical! processes,! local! orographic! effects! and! other! regional! physiographical!features.! Thus! accurate! regional! estimates! of! observed! climate! are! unlikely! to! be!  19 produced.!On!the!other!hand,!meteorological!variables!at!weather!station!scale!or!at!a!higher! resolution! than! the!one!provided!by! the!GCMs!are!often!needed.!To!address!this!need,!downscaling!techniques!can!be!used!to!generate!finer!scale!projections!of!near! surface! climatologies! (Salameh! et! al.! 2008).! Similarly,! downscaling! techniques!can! be! used! to! understand! the! underlying! relationships! between! the! coarse!resolution!predictors!and!surface!observations.!!!There! are! two! main! downscaling! approaches:! dynamical,! and! statistical.! The!dynamical!technique!is!based!on!extracting!regional!scale!information!using!regional!climate!models!(RCMs).!RCMs!use!as!lateral!boundary!conditions!information!from!a!coarser! resolution! GCM! model! (Mearns! et! al.! 2003;! Laprise! 2008).! Sea! surface!temperature! (SST),! sea! ice,! greenhouse! gas! (GHG)! and! aerosol! forcing,! as! well! as!initial! soil! conditions,! are! also! more! often! provided! by! the! driving! GCM.! ! The!statistical! approach! is! based! on! finding! statistical! relationships! between! the!predictors!(i.e.!atmospheric!variables!from!coarse=resolution!model!outputs)!and!the!finer=scale!predicted!variables! (predictands)!required!by! the!climate!change! impact!studies,! with! time=invariance! of! the! predictor=predictand! relationships! assumed!(Haylock!et!al.!2006).!!!Statistical!downscaling!models!can!be!divided!in!three!general!categories:!regression!models,! weather! typing/classification! schemes! and! weather! generators.! The!regression!models! represent! linear! or! nonlinear! relationships!between!predictands!  20 and!large=scale!predictors!(Fowler!et!al.!2007).!Within!this!category,!the!most!popular!methods! include! multiple! linear! regression! (Murphy! 1999),! canonical! correlation!Analysis!(CCA)!(von!Storch!et!al.!1993),!artificial!neural!networks!(ANN)!(Crane!and!Hewitson! 1998a)! and! singular! value! decomposition! (SVD).! ! An! early! reference! to!regression! based! downscaling,! could! be! found! looking! at! the!work! of!Wigley! et! al.!(1990)! who! regressed! local! temperature! and! precipitation! observations! on! the!spatial! area! averages! of! those! two! variables,! mean! sea=level! pressure,! 700! hPa!geopotential! heights! and! other! predictor! variables.! He! found! significant!spatiotemporal!variations!in!model!performance!near!mountain!and!coastal!regions.!!Weather!typing/classification!schemes!basically!select!the!date!in!the!training!period!when!the!situation!most!closely!resembled!the!day!for!which!the!prediction!is!made!(Benestad!et!al.!2008a;!Benestad!et!al.!2008b).!In!other!words,!the!model!searches!a!database!describing!all!weather/climate!events! in!the!past,!and!local!measurements!of!the!predictand.!A!major!drawback!is!that!it!cannot!predict!magnitudes!outside!the!range! of! the! historical! data,! since! the! predicted! values! are! taken! from! past!observations.!!!Classification! methods! use! a! set! of!features!or!parameters!to! characterize! a! given!object! (assuming! the! features?! relevance! to! the! problem).! The! circulation=based!approach! to! downscaling! remains! particularly! appealing! because! it! is! founded! on!  21 sensible! physical! linkages! between! the! large! scale! climate! and! local! scale! weather!(Gangopadhyay! and! Clark! 2005).! An! example! is! the! statistical! and! physical!dependence! of! daily! precipitation! variations! on! time! series! of! circulation! changes!(Wilby!and!Wigley!1997b).!!!The! third! category,!weather!generators,! replicate! the! statistical! attributes!of! a! local!climate!variable!(mean!and!variance)!but!not!observed!sequences!of!events!(Wilby!et!al.! 2004).! The! principal! issue! involving! the! application! of! stochastic! weather!generators!to!future!climates!has!been!the!challenge!of!adjusting!the!parameters!in!a!physically!realistic!and!internally!consistent!way!(Wilby!and!Wigley!1997b).!!!Comparisons! between! downscaling! methods! are! usually! carried! out! in! terms! of!correlations!(Weichert!and!Burger!1998;!Bertacchi!Uvo!et!al.!2001;!Cheng!et!al.!2008;!Chu! et! al.! 2010;! Souvignet! and! Heinrich! 2011),! root! mean! square! errors! (RMSE)!(Huth!1999;!Kostopoulou!et!al.!2006;!Fasbender!and!Ouarda!2010),!or!similar!metrics!(Cawley! et! al.! 2007)! between! the! daily! downscaled! and! observed! values.!Characteristics!such!as!extreme!values,!probability!distributions!and!spatio=temporal!structures!were!infrequently!treated!(Huth!et!al.!2008),!but!recently!there!has!been!a!substantive! increase! in! the! number! of! studies! considering! extremes! (Busuioc! et! al.!2008;! Michelangeli! et! al.! 2009;! Cannon! 2010;! Foresti! et! al.! 2010;! Mannshardt=  22 Shamseldin! et! al.! 2010;! B?rger! et! al.! 2012),! as! these! studies! are! essential! for!understanding!how!models!replicate!the!full!range!of!observed!characteristics.!!!In! contrast,! fewer! authors! have! been! interested! in! studying! the! stationarity!assumption,!as!the!model!relationships!are!by!definition!stationary!or!time=invariant.!!Notable!examples!of!research!done!in!this!area!include!(in!chronological!order)!i)!the!sensitivity! analysis!proposed!by!Wilby! (1994)!where! the!historical! record! could!be!fragmented! into!warm/cold!or!wet/dry!years,!with! the!statistical!models! trained! in!one! fragment! (usually! the! coldest! one,! if! interested! in! future! projections)! and!validated! in! the! other! (usually! the! warmest,! assuming! that! the! local! mean!temperature!will!increase!in!the!future),!ii)!the!validation!methodology!from!Vrac!et!al.! (2007)!which! inspired! this! thesis,! iii)! the! evaluation! of! bias! correction!methods!using! different! conditions! from! those! used! for!model! calibration! (Teutschbein! and!Seibert! 2012),! and! iv)! the! ?Perfect! Model?! ! or! Big=Brother! experimental! design!!(Dixon!et!al.!2013)!where!the!observations!used!for!training!the!downscaling!model!are! substituted! by! high! resolution! GCM! output! (~25! km! grid! spacing)! and! the!predictors! are!derived! from!a! coarsened! (200!km!grid! spacing)!version!of! the!high!resolution! GCM,! thus! allowing! the! calculation! (and! comparison)! of! historical! and!future!downscaling!skills.!!  23 In! this! thesis,! I! evaluated! linear! and! nonlinear! downscaling!models! using! different!sets!of!predictors!in!terms!of!their!ability!to!reproduce!day=by=day!variability,!and!in!terms!of! their! ability! to! reproduce! climate! indices.! I! aimed! to! i)!determine! if! linear!and! nonlinear! models! have! similar! skills! simulating! daily! variability! and! climate!indices! in!the!historical!period,! to! ii)!determine! if! linear!and!nonlinear!models!have!similar!skills!simulating!daily!variability!and!climate!indices!in!the!future!period,!iii)!determine!if!one!has!to!trade!off!daily!variability!for!climate!indices!simulation!skills,!and! iv)! describe! a! validation! methodology! comparing! these! skills.! ! In! particular,! I!examined! if! the! linear! and! nonlinear! models! have! similar! skills! simulating! daily!variability! and! climate! indices! in! the! historical! and! future! periods.! This! inquiry! is!relevant!to!all!the!statistical!downscaling!methods!using!different!forms!of!regression,!as! the! ANN! models! extrapolate! very! differently! from! traditional! multiple! linear!regression! methods,! with! ANN! using! bounded! hyperbolic! tangent! or! sigmoidal!functions! (Figure! 4),! in! contrast! to! the! unbounded! behavior! of! the! polynomial! and!linear!extrapolations!(Hsieh!2009).!!  24 !Figure!4.!Schematic!diagram!illustrating!the!types!of!extrapolation!used!by!different!downscaling! methods.! ! The! area! in! gray! represent! the! historical! period! domain,!where! Ysim=! f(XGCM).! The! area! in! white! represents! conditions! outside! the!training/historical!period.!!The!regression!models!used!to!downscale!represent!linear!or!nonlinear!relationships!between!the!finer=scale!predictands!and!the!coarse!resolution!predictors!(Fowler!et!al.!2007)!and!have!the!undesirable!characteristic!of!under!predicting!the!variance.!To!deal!with! this! problem!one! can! (i)! add! noise! to! the! statistically! downscaled! series,!thus!breaking!the!temporal!correlation!of!the!data!(Huth!et!al.!2001)!or,!(ii)!inflate!the!downscaled! time! series! as! proposed! by! Karl! et! al.! (1990)! assuming! that! all! local!variability!can!be! traced!back! to! large=scale!variability.!The!present!study!opted! for!the! latter! approach,! as! some! of! the! climate! indices! used! (e.g.! heat! wave! duration,!  25 growing!season!length)!needed!the!temporal!correlation!to!be!kept,!nevertheless!I!am!aware!of!the!limitations!and!caveats!of!the!variance!inflation!approach!as!mentioned!by!von!Storch!(1999).!!As! mentioned! earlier,! nonlinear! regression! models! like! ANN,! using! multi=layer!perceptrons!(MLP)!have!been!used!extensively!for!statistical!downscaling.!In!general!MLP!downscaling!models!give!similar!results!compared!to!multiple!linear!regression!downscaling! methods! for! temperature! and! precipitation! (Schoof! and! Pryor! 2001),!and!are!capable!of!outscoring!linear!models!when!relationships!are!nonlinear!and/or!interactive! (Tang! et! al.! 2000).! Nevertheless,! there! is! no! consensus! on! their!performance! versus! linear! models! (Jeong! et! al.! 2012b).! ! For! example! when!downscaling! temperatures! over! Europe,! Huth! et! al.! (2008)! concluded! that! the!nonlinear!methods!did!not!bring!an! improvement!over! linear!methods.! In! contrast,!Miksovsky!and!Raidl!(2005),!concluded!that!the!nonlinear!techniques!outperformed!linear!regression!in!the!majority!of!cases,!when!downscaling!daily!temperatures!from!25!European!stations.!!!!In! general,! ANN! models! allow! the! predictors! and! predictands! to! have! linear! or!nonlinear!relationships;!however!these!models!have!several!limitations!including!the!subjectivity! in! the! choice!of!model! architecture! and! training! algorithm! (Jeong! et! al.!2012b),!and!more!noticeably!the!possibility!of!being!trapped!at!local!minima!(Hsieh!  26 2009).!To!overcome! the! last! limitation,!an!ensemble!modeling!strategy! is!preferred!(Cannon! and! Whitfield! 2002;! Miksovsky! and! Raidl! 2005).! Typically,! in! ensemble!modeling! the! generalization! error! of! the! final! predictive! model! is! reduced! by!averaging!the!outputs!from!a!number!of!ensemble!members.!Breiman!(1996)!found!that! the! improvements! in!performance! tend! to! level!out!after!adding!more! than!25!models!to!the!ensemble.!!This!research! is!part!of! the!"Probabilistic!assessment!of!regional!changes! in!climate!variability! and! extremes"! project! funded! by! the! Natural! Sciences! and! Engineering!Research! Council! of! Canada! through! a! Special! Research! Opportunity! (NSERC=SRO)!grant.! The! project! benefits! from! the! partnership! between! the! Canadian! Climate!Analysis! Group! and! the! European! ENSEMBLES! project! (an! European! Community!major! initiative! funded! by! the! European! Commission),! and! aims! to! develop! high=resolution! climate! change! information! with! the! CGCM3! and! downscaling!methodologies.!!!For! comparison! purposes,! all! the! group!members! are! analyzing! a! common! area! of!interest,! and! employing! a! multi=station! modeling! framework! for! assessing! the!changes!in!precipitation,!daily!minimum!and!maximum!temperatures!in!the!southern!Ontario!and!Quebec!area.!Besides!the!contributions!presented!in!this!thesis,!statistical!downscaling! models! developed! by! the! author! were! also! used! in! two! other!  27 manuscripts! prepared! by! other! project! members:! i)! ?A! common! comparison! for!statistical!downscaling!over!southern!Quebec?!by!Harding!et!al.!!(in!preparation),!and!ii)! ?Statistically! downscaled! climate! change! scenarios! for! southern! Quebec?! by!Harding!et!al.!(in!preparation).!!Overall,! I! expect! that! knowing! the! methods?! performance! differences! between! the!historical!and!future!time!periods!will!contribute!to!the!discussion!about!the!level!of!confidence!one! should!attribute! to! the!downscaled! time! series! commonly!used!and!needed! by! professional! groups! interested! in! using! downscaled! information! like:!hydrologists,!engineers,!ecologists,!biologists,!land!conservation!officers,!agronomists,!insurance! analysts,! epidemiologists,! municipal! planners! and! policy! makers,! among!many!others.!!1.2.1 Model evaluation, verification and validation !Verification! methods! allow! us! to! compare! model! outputs! with! real! observed!situations! (Coiffier! 2011);! nevertheless! the! verification! and! validation! of! natural!systems! is! impossible! because! the! model! results! are! nonunique,! and! because! the!Earth! is! an! open! system! (Oreskes! et! al.! 1994).! Our! incomplete! knowledge! of! the!natural!system!thus!precludes!us! from!confirming!the!models.!Model!evaluation,!on!the!other!hand,!does!not!require!real!observed!situations,!as!it!is!possible!to!evaluate!a!model!in!terms!of!its!number!of!outliers,!its!sequence!of!events,!autocorrelation,!and!  28 statistical! moments,! to! name! a! few!metrics.! For! example,! when! downscaling! daily!mean! temperatures! in! Quebec,! a! candidate! model! can! erroneously! simulate!maximum! temperatures! below! the! minimum! temperature,! or! a! precipitation!downscaling!model! can! simulate! negative! precipitation! amounts.! In! both! cases! the!observations!were!not!needed!to!assess!the!models?!performance.!!In!that!context,!the!models!used! in!Chapters!2,!3!and!4!were!evaluated!and!compared!with!RCM!model!data! (used! as! pseudo=observations),! thus! strictly! speaking! were! not! validated.!Therefore,!Vrac!et!al.!(2007a)!terminology!is!inaccurate.!!1.3 Thesis structure This! thesis! results! from! a! collection! of! four! journal! manuscripts! on! statistical!downscaling.! ! In! Chapters! 2,! 3,! and! 4,! the! time! invariance! assumption! is! tested! by!comparing! historical! and! future! regional! climate!model! outputs! versus! statistically!downscaled!time!series;!while! in!Appendix!A,! I!downscaled!the!daily!maximum!and!minimum! temperatures! gathered! at! ten! weather! stations! in! southern! Ontario! and!Quebec! using! the! NCEP/NCAR! reanalysis! (Kistler! et! al.! 2001;! Kalnay! et! al.! 1996)!outputs!as!predictors.!Additionally!I!introduced!an!evaluation!method!for!the!models?!performance!simulating!daily!variability!and!climate! indices,!and! for!comparing! the!performance!of!different!potential!predictors.!!  29 In!Chapter!2,!daily!maximum!and!minimum!temperatures!over!southern!Ontario!and!Quebec,! Canada,! were! downscaled! and! different! linear! and! nonlinear! statistical!downscaling! methods! were! compared! in! terms! of! daily! variability! mean! absolute!error! and! in! terms! of! indices! of! agreement! (Willmott! et! al.! 2012)!when! simulating!climate! indices.! In! Chapter! 3,! different! classification! and! regression!methods! were!compared!when!downscaling!precipitation!occurrences!and!amounts!over! the!same!observational!sites!used!in!Chapter!2.!!The! experimental! setup! for! these! two! chapters! follows! Vrac! et! al.! (2007b)! and!involves! using! the! Canadian! regional! climate! model! (CRCM)! outputs! as! pseudo=observations! to! estimate! model! performance! in! the! context! of! future! climate!projections! by! replacing! historical! and! future! observations!with!model! simulations!from! the! CRCM,! nested!within! the! domain! of! the! CGCM3.! In! particular,! I! validated!statistically! downscaled! daily! precipitation! time! series! in! terms! of! the! Peirce! skill!score,!mean! absolute! errors,! and! climate! indices;!while! the! downscaled! TMAX! and!TMIN! time! series! were! validated! in! terms! of! mean! absolute! errors! and! indices! of!agreement! for! the! climate! indices.! The! predictors! were! obtained! from! the! CGCM3!20C3M! (1971=2000)! and! A2! (2041=2070)! simulations,! and! pseudo=observation!outputs!from!the!CRCM!4.2!forced!with!the!CGCM3!boundary!conditions!were!used!as!predictands.!!  30 In!Chapter!4,!present!and!future!daily!surface!wind!speed!at!a!hypothetical!offshore!wind! energy! plant! northwest! of! the! Canadian! archipelago! of!Haida!Guaii! (formerly!known! as! Queen! Charlotte! Islands)! was! downscaled! using! different! linear! and!nonlinear! methods.! As! in! Chapters! 2! and! 3,! I! employ! a! pseudo=observation!downscaling!verification!approach,!which!allows!one!to!estimate!model!performance!in! the! context! of! future! climate! projections! by! replacing! historical! and! future!observations!with!model!simulations!from!the!CRCM!nested!within!the!domain!of!the!CGCM3.! The! new! evaluation! methodology! compares! historical! and! future! pseudo=observations!in!terms!of!both!downscaled!daily!variability!and!annual!climate!indices!characterized! by! the! proposed! Wind! INDices! for! the! evaluation! of! EXtremes!(WINDEX)!(Gaitan!and!Cannon!2013).!!!Finally,!Appendix!A!lays!the!groundwork!for!Chapter!2!as!it!uses!reanalysis!not!GCM!data!and!observed!daily! temperatures! instead!of!pseudo=observations.!Additionally,!as! downscaling! techniques! can! be! used! to! understand! the! underlying! relationships!between! the! coarse! resolution!predictors!and! surface!observations,!daily!maximum!and! minimum! temperatures! over! southern! Ontario! and! Quebec,! Canada,! were!downscaled!and!different! linear!and!nonlinear!methods!were!compared! in! terms!of!mean!absolute! errors! and! indices!of! agreement.! ! The! study!period!was!1961=2000.!This!time!window!is!consistent!with!the!CMIP3!model!simulations!(Meehl!et!al.!2007)!and!with! the! recent!work! from! Jeong! et! al.! (2012a)! exploring! the! use! of! CGCM3.1!  31 predictors!for!daily!temperature!(and!precipitation)!downscaling!in!southern!Quebec,!Canada.!!Appendix! A! constitutes! one! of! the! first! deliverables! of! the! aforementioned! inter=comparison! project,! and! its! results! were! presented! at! the! 2011! ?Probabilistic!Assessment! of! Regional! Changes! in! Climate! Variability! and! Extremes?!workshop! at!McGill! University! in! Montreal,! Canada.! Specifically,! atmospheric! reanalysis! outputs!were! downscaled! using! linear! regression! and! artificial! neural! networks! (ANN)!ensembles!to!obtain!daily!station!values!of!maximum!and!minimum!temperature!for!ten!weather!stations.!!!As!Appendix!A!lays!the!basis!for!Chapter!2,!a!natural!order!would!be!to!read!Appendix!A!before!Chapter!2.!!!! !  32 Chapter 2 2 COMPARISON OF STATISTICAL DOWNSCALING METHODS FOR FUTURE WEATHER AND CLIMATE: SURFACE TEMPERATURE IN SOUTHERN ONTARIO AND QUEBEC, CANADA !!2.1 Introduction Atmosphere=ocean!global!climate!models!(AOGCM)!like!the!Canadian!Global!Climate!Model! 3.1! (CGCM)! are! commonly! used! to! assess! the! possible! changes! of! different!components!of! the!Earth! system!under!possible! climate! change!emission! scenarios,!where!each!scenario!represents!an!alternative!future!(IPCC!2000).!Although!AOGCMs!are! being! used! as! learning! tools! for! complex! systems! and! for! policy!making! (IPCC!2004),!there!is!often!the!need!to!downscale!their!coarse!spatial!resolution!outputs!to!local! or! weather! station! scale,! as! the! AOGCMs! are! unable! to! resolve! small=scale!dynamical! features! (Salameh! et! al.! 2008).! Additionally,! local! scale! meteorological!variables! are! often! needed! by! hydrologists,! engineers,! and! other! professionals! to!assess!possible! impacts!and!adaptation!responses,!and!to!design!bridges,!drainages,!and!irrigation!structures.!!  33 Downscaling! techniques!can!be!divided! in! two!categories,!dynamical!and!statistical.!Statistical! downscaling! (SD)! techniques,! like! the! ones! used! in! the! present! study,!establish! statistical! relationships! between! the! coarse! resolution! predictors!representing!the!climate!system!(e.g.!AOGCM!or!reanalysis!products)!and!finer!local!scale! observations.! In! contrast,! dynamical! downscaling! techniques! use! regional!climate!models! (RCM)!with!boundary! conditions!provided!by! the! coarse! resolution!global!model!(Laprise!2008)!to!extract!finer!resolution!local!estimates.!!In!traditional!statistical!downscaling,!observations!are!needed!to!evaluate!the!model?s!performance;!therefore!one!cannot!evaluate!the!model?s!future!performance,!as!there!are!no!future!observations.!Thus,!one!has!to!assume!that!present!simulation!skills!will!be!maintained! in! the! future! (Wilby! et! al.! 1998).! However,! by! using! the! finer=scale!RCM! outputs! for! future! climates! as! pseudo=observations! (Vrac! et! al.! 2007b),! this!limitation! can!be!partially! addressed.! I!will! follow! the!Vrac!et! al.! approach! in!using!RCM!outputs!as!pseudo=observations.!!Similarly,! an! increased! interest! in! understanding! how! the! models! replicate! the!observed! variability! has! encouraged! novel! studies! on! extreme! values,! probability!distributions!and!spatio=temporal!relationships!between!predictor(s)!and!predictand!(Michelangeli! et! al.! 2009;!Mannshardt=Shamseldin! et! al.! 2010;! Foresti! et! al.! 2010).!This! contrasts! with! traditional! statistical! downscaling! studies! using! only! daily!  34 variability/weather! metrics! like! root! mean! squared! error! (RMSE)! or! correlation!coefficients! (Chu! et! al.! 2010;! Souvignet! and!Heinrich! 2011;! Fasbender! and! Ouarda!2010;!Cheng!et!al.!2008).!!In! this! chapter,! I! used!multiple! linear! regression! (MLR)!and!multi=layer!perceptron!(MLP)! Bayesian! neural! networks! (BNN),! a! nonlinear! regression! method,! to!statistically!downscale!the!coarse!resolution!Canadian!General!Circulation!Model!3.1!outputs! (predictors)! to! a! finer! scale.! The! Canadian! RCM! 4.2! daily! maximum! and!minimum! temperatures! were! used! as! pseudo=observations! (predictands)! for! the!southern!Quebec!and!Ontario!region.!!!!Although! neural! networks! have! been! used! extensively! on! statistical! downscaling!studies! (e.g.Wilby! et! al.! (1998);!Weichert! and! Burger! (1998);! Dibike! and! Coulibaly!(2006);!Cannon!and!Whitfield!(2002);!Cannon!(2008b)),!and!Schoof!and!Pryor!(2001)!showed!that!MLP!and!MLR!methods!provide!comparable!results!when!downscaling!temperature,!there!are!still!contrasting!results!when!comparing!linear!and!nonlinear!statistical! downscaling! methods.! For! example,! Huth! et! al.! (2008)! showed! that!nonlinear! methods! did! not! outperform! multiple! linear! regression! methods! when!downscaling! temperatures!over!Europe,!while!Miksovsky!and!Raidl! (2005)! showed!neural!network!models!outscoring!their!linear!counterparts.!!  35 Additionally,! the! historical! (1971=2000)! and! future! (2041=2070)! pseudo=observations! were! compared! against! the! statistically! downscaled! series! for! both!periods! in! terms! of! (i)! mean! absolute! errors! (MAE)! to! evaluate! the! models?!performance! in! simulating!daily!variability,! and! (ii)! indices!of!agreement!calculated!from! the! annual! STARDEX! climate! indices! (Goodess! 2005)! used! to! evaluate! the!performance! in! simulating! the! climate! of! extreme! weather! (e.g.! Tomozeiu! et! al.!(2006)).!Specifically,!I!used!six!annual!climate!indices,!namely,!the!90th!percentile!of!the!daily!maximum!temperature,!10th!percentile!of!the!daily!minimum!temperature,!the! intra=annual! temperature! range! calculated! from! the! previous! two! indices,! the!number!of! frost!days,!the!growing!season!length!and!the!heat!wave!duration!(Table!2).!A!study!by!Khaliq!et!al.!(2007)!dealt!with!the!heat!spell!occurrences!in!Montreal,!Canada.! The! ability! of!MLR! and! BNN!models! to! downscale! historical! climate!when!driven!by!reanalysis!fields!was!evaluated!in!Appendix!A.!! Table!2.!STARDEX!temperature=related!annual!indices!used!in!this!study.!! Acronym Temperature Indices T10 10th percentile of TMIN  T90 90th percentile of TMAX  IATR Intra-Annual extreme Temperature Range (T90 - T10)  FD Number of Frost Days (with TMIN < 0?C)  GSL Growing Season Length (period between TMEAN > 5?C for more than 5 days and TMEAN < 5?C for more than 5 days) HWDI Heat Wave Duration Index (maximum period of consecutive days with TMAX exceeding the climatological T90)    36 This! chapter! presents! for! the! first! time! the! historical! and! future! statistical!downscaling! performance! in! simulating! weather! and! climate! of! extremes! for! the!southern! Ontario! and! Quebec! region,! and! aims! to! provide! some! insight! into! the!effectiveness! of! the! time! invariance! assumption! applied! to! linear! and! nonlinear!methods!when!downscaling!daily!maximum!and!minimum!temperatures.!Specifically,!the! results! indicate! that! skills! simulating! present! weather! and/or! climate! do! not!necessarily!imply!similar!skills!in!future!climates.!!!2.2 Study area, predictors and predictands To!evaluate!the!models?!performance!simulating!weather!and!climate!indices!for!the!historical!period!(1971=2000)! I!used!the!CGCM!3.1!scenario!20C3M,!which!assumes!greenhouse! gas! emissions! increasing! as! observed! through! the! past! century.!Additionally,! the! Canadian! RCM! (CRCM),! ran! using! the! CGCM! output! as! boundary!conditions,!was!used! to!obtain! the!pseudo=observations.!Future!performances!were!evaluated!using!the!CGCM!and!CRCM!model!outputs!following!the!Special!Report!on!Emissions! Scenarios! (SRES)! future! climate! scenario!A2.! Predictors! and! predictands!were! obtained! from! the! North! American! Regional! Climate! Change! Assessment!Program!dataset!(Mearns!et!al.!2007).!!The! study! area! includes! ten! CRCM! grid! points! in! southern! Ontario! and! Quebec!(Figure'9).!I!extracted!daily!maximum!and!minimum!surface!temperatures!from!the!  37 CRCM! grid! points! and! used! them! as! pseudo=observations.! ! For! predictors,! I! used!CGCM!model!outputs!from!the!nine!grid!points!nearest!to!the!study!region.!!From! each! grid! point,! six! predictors,! namely! the! daily! values! of! maximum! and!minimum! temperatures,! near=surface! u! and! v! wind! components,! specific! humidity,!and! sea! level! pressure! were! extracted! for! the! historical! (1971=2000)! and! future!(2041=2070)!time!periods.!!!The!historical!period!uses!the!atmospheric!component!of!CGCM!3.1!20C3M!transient!run,!while!the!future!period!uses!the!SRES!A2!scenario!(IPCC!2000)!forced!with!the!CGCM! 3.1! T47! run! number! 4.! This! scenario! assumes! a! very! heterogeneous! world,!with!a!high!population!growth!and!less!concern!for!rapid!economic!development.!!!With! 54! predictors! (i.e.! 9! grid! points! x! 6! predictors/point),! I! constructed! three!different! predictor! sets:! (i)! ?all?! uses! stepwise! regression! (Wilks! 2011;! Darlington!1990)! to! select! predictors! from! the! 54! available! predictors,! (ii)! ?T?! uses! stepwise!selected!predictors!from!the!9!CGCM!outputs!for!temperature!(either!daily!maximum!or! minimum),! and! (iii)! ?PC?! uses! stepwise! selected! predictors! from! the! leading!principal! components! (PC)! of! the! standardized! anomalies,! ! where! each! PC?s!contribution!exceeds!0.1%!of!the!variance!(with!20=29!PCs!above!this!cutoff).!I!kept!the! same! PC! structure! between! periods! by! applying! principal! component! analysis!(PCA)!to!an!extended!dataset!containing!both!historical!and!future!predictors!(Imbert!  38 and!Benestad! 2005).! ! I! also! tested! linear! and! nonlinear!models! using! the! stepwise!selection! procedure! on! the! PCs! obtained! after! first! using! Preisendorfer?s! n=rule!(1988)! to! select! the! significant! PCs! from! the! 54! principal! components,! but! these!models!were! outscored! by! the!models! using! the! set! of! predictors! used! in! (iii).! For!consistency,!the!nonlinear!SD!models!used!the!same!predictors!selected!for!the!linear!models.!Table!3!lists!the!three!linear!and!three!nonlinear!model!runs.!! Table!3.!Models!and!their!characteristics.!!Model ID Type Method Predictors used LRall linear MLR stepwise selected from 54 CGCM predictors LRT linear MLR stepwise selected from 9 CGCM temperatures LRPC linear MLR stepwise selected from CGCM leading PCs BNNall nonlinear BNN same as in LRall BNNT nonlinear BNN same as in LRT BNNPC nonlinear BNN same as in LRPC !  2.3 Methods 2.3.1 Stepwise multiple linear regression  This!type!of!screening!procedure!selects!from!a!pool!of!potential!predictors!a!subset!of! predictors! chosen! after! implementing! forward! selection! and/or! backward!elimination! (Wilks! 2011;! Darlington! 1990)! based! on! the! probabilities! that! the!random!variable!would!assume!a!value!greater! than!or!equal! to! the!observed!value!  39 strictly!by!chance!(p=values).!The!method!involves!creating!different!MLR!models!and!comparing!them!with!and!without!a!potential!predictor,!assuming!that!the!predictor!to!be!added/removed!has!a!zero!regression!coefficient!(Hill!and!Lewicki!2006).!I!kept!MATLAB?s!default!p=values!for!a!predictor!to!be!added!and!removed!!at!5!%!and!10!%,!respectively.!!2.3.2 Bayesian neural networks Bayesian! neural! networks! (MacKay! 2003;! Bishop! 2006)! belong! to! the! multi=layer!perceptron! (MLP)! architecture! family,! and! use! Bayesian! regularization! to! prevent!overfitting.!MLP!neural! networks!were! trained! to!minimize! the!RMSE!performance!function!between!the!dependent!variable?s!observed!and!predicted!values.!!Regarding!the!MLP!architecture,! I! used!one! input! layer,! one!hidden! layer! and!an!output! layer!with! one! neuron! (i.e.! single! output).! ! In! addition,! I! used! the! hyperbolic! tangent!function!to!map!from!the! input! layer!to!the!hidden! layer!(with!30!hidden!neurons),!and! a! linear! function! to! map! from! the! hidden! layer! to! the! output! neuron.! The!MATLAB!?trainbr?!code!for!was!used!for!the!BNN!models.!Due!to!the!presence!of!local!minima! during! the! nonlinear! optimization! process,! BNN! models! initialized! with!different! random! initial!weights! tended! to!behave! somewhat!differently,!hence! it! is!common! to! build! an! ensemble! of! BNN! models! and! ensemble! average! the! model!outputs.!   40 2.4 Daily maximum and minimum surface temperature downscaling After!removing!the!climatological!seasonal!cycle!from!the!predictors!and!the!pseudo=observations! to! get! the! daily! anomalies,! the! historical! and! future! anomalies! were!standardized!using! the!historical! variance! as! reference.!The!next! three! steps! in! the!downscaling!procedure!were:! (i)! train! the!model!using! the!historical! synoptic=scale!circulation! data! from! CGCM! as! predictors! and! CRCM! pseudo=observations! as!predictands;! (ii)! use! cross=validation! (Bishop! 2006)! ! to! compute! the! model?s!validation! error! on! independent! data;! and! (iii)! after! training! and! validating! the! SD!model! for! the! historical! period,! use! future! scenario! simulations! by! CGCM!as!model!inputs!to!downscale!future!climates.!!!2.5 Model evaluation The!evaluation!procedure!involves!validating!the!downscaled!results!in!terms!of!daily!variability!(i.e.!weather)!and!in!terms!of!climate!indices,!for!the!historical!and!future!periods.!To!get!the!daily!variability!error!I!calculated!the!mean!absolute!error!(MAE)!between!the!pseudo=observations!and!the!statistically!downscaled!data!for!the!daily!mean!temperature.!MAE!was!used!instead!of!the!root!mean!squared!error!(RMSE)!as!it!is!a!more!natural!measure!of!the!average!error!(Willmott!and!Matsuura!2005).!!For!the!climate!evaluation!procedure,!I!used!the!downscaled!maximum!and!minimum!temperatures!to!calculate!six!temperature=related!annual!climate!indices!used!by!the!  41 STARDEX! project! (Goodess! 2005)! (Table' 2).! Next,! I! calculated! the! revised! index! of!agreement! (IOA)! (Willmott! et! al.! 2012)! between! each! downscaled! and! observed!index,!to!evaluate!the!ability!of!the!SD!models!to!replicate!the!climate!of!extremes!as!quantified!by!the!STARDEX!indices.!!Willmott?s!revised!IOA!statistic!is!defined!by! IOA = 1 - [?i ?Pi - Oi ?]  /  [2 ?i ? Oi - ? ?], when  ?i ?Pi - Oi ? ? 2 ?i ? Oi - ? ?, or IOA = [2 ?i ? Oi - ? ?] / [?i ?Pi - Oi ?]  - 1, when  ?i ?Pi - Oi ? > 2 ?i ? Oi - ? ?,     (1)  where Pi and Oi are! the! downscaled! and! observed! values,! respectively,! and ? the!observed!mean.!In!general,!a!larger!IOA!value!indicates!better!forecast!performance,!and! if! IOA=! 0,! it! signifies! that! the! sum! of! the! magnitudes! of! the! errors! ! (i.e.                   ?i ?Pi - Oi ?) and! the! sum! of! the! perfect=model=deviation! and! observed=deviation!magnitudes (i.e. ?i (? Oi - ? ?+? Oi - ? ?) = 2 ?i ? Oi - ? ?) are!equivalent!(Willmott!et!al.!2012).! As!there!are!6!STARDEX!indices,!there!are!6!IOAs,!so!I!averaged!the!six!IOA!values!to!yield!the!unified!STARDEX!index!(USI)!IOA!(Appendix!A)!to!characterize!the!model?s!ability!to!reproduce!the!selected!group!of!extreme!climate!indices.!!!!  42 !I! used! cross=validation! to! calculate! the! validation! error! of! the! entire! dataset.! The!procedure! involved!dividing! the!data! record! into! four! sections,! training! the!models!with! three! sections,! and! validating! (verifying)! the! predictions! over! the! remaining!section.!This!procedure!was!repeated!with!a!different!section!selected!for!validation,!until! all! four! sections!had!been!used! for! validation.!Next,! as! ensemble! averaging! of!models! can! yield! an! expected! error! less! than! or! equal! to! the! expected! error! of! an!individual!model!(Hsieh!2009),!I!calculated!an!ensemble!average.!For!each!validation!section,!50!BNN!models!were!built!by!starting!from!random!initial!weights,!and!the!15! best! models! (in! terms! of! MAE! over! the! training! data)! were! selected! for! the!ensemble.!As! regression!approaches!under=predict! the!variance! (Karl!et!al.!1990),! I!multiplied! the! downscaled! outputs! by! a! scale! factor,! i.e.! the! ratio! of! the! pseudo=observed!standard!deviation!to!that!from!the!SD!model.!Finally,! for!future!climate,! I!ensemble=averaged! the! forecasts! from! the! 60! best! ensemble!members! (i.e.! 15! best!members!for!each!section!x!4!sections).!A!similar!procedure!was!used!for!MLR,!except!that!no!ensemble!construction!was!needed.!!Figures!5!and!6!show!the!models?!skill!simulating!weather!and!climate!indices!on!the!historical!and! future!periods,!respectively.!Particularly! the!ordinate!shows!the!daily!variability! errors! in! terms! of! MAE,! and! the! abscissa! shows! the! models?! unified!STARDEX! index! IOA.! ! A! model! good! in! simulating! daily! weather! will! have! low!ordinate!values,!while!a!model!good!in!simulating!the!climate!of!extremes!will!have!a!  43 high! abscissa! value.! Thus! the! ideal! model,! good! in! both! weather! and! climate!simulation,!would!lie!in!the!lower!right!hand!corner!of!the!plot.!!The! historical! period! comparison! (Figure! 5)! shows! that! although! the! LRT! model!(MLR!using!only! temperature!predictors)!marginally!outscored! the!other!models! in!terms!of!climate!simulation,!and!the!LRall!model!(MLR!with!all!predictors!available)!marginally! outscored! the! others! in! terms! of! weather! simulation,! all! the! models!attained! similar! IOA! and! MAE! values.! This! underperformance! of! the! BNN! models!contrasts!with!previous! results! (Appendix!A)! obtained!when!downscaling! from! the!NCEP/NCAR! reanalysis! (1961=2000)! (Kalnay! et! al.! 1996)! to! observed! data! from!weather! stations,! where! a! BNN! model! using! a! set! of! stepwise! selected! predictors!slightly!outperformed!all!MLR!models!in!terms!of!climate!and!weather.!!This!could!be!partially!caused!by!the!discrepancies!between!the!CRCM!and!observed!data!(e.g.!the!relations!between! the! reanalysis!data!and! the!observations!may!be!more!nonlinear!than!those!between!the!CGCM!and!CRCM!data),!or!by!using!different!predictors!in!the!two!studies.!!!  44 !Figure! 5.! Historical! period! (1971=2000)! model! comparison,! with! the! daily!temperature! MAE! plotted! againt! the! unified! STARDEX! index! IOA.! The! BNN!model!results,!averaged!over!the!10!stations,!are!plotted!with!solid!symbols,!and!their!MLR!counterparts!with!open!symbols.!Triangles!represent!the!models!with!all!predictors!considered! (BNNall,! and! LRall),! circles,! those! with! only! temperature! predictors!(BNNT!and!LRT),!and!squares,!those!with!PCs!as!predictors!(BNNPC!and!LRPC).!For!consistency! with! the! MAE! performance! function,! error! bars! display! the! mean!absolute!deviations!(MAD)!computed!from!the!10!stations.!!!But!how!do!these!models!perform!in!the!future!period?!Figure!6!compiles!the!2041=2070!downscaled!results!under!the!SRES!A2!emission!scenario,!and!compares!them!with!the!historical!results!shown!in!Figure!5.!  45 The! results! show! that! when! evaluating! the! model's! future! performance,! a! higher!score!simulating!the!past,!does!not!guarantee!a!similar!future!score.!Moreover,!all!the!methods! decreased! their! performance! in! terms! of!MAEs.! Similarly,! the! two!models!using! only! the! temperature! predictors! decreased! their! weather! and! their! climate!downscaling!performance! the!most! among! the! six!models.!This! could!be! caused!by!the! difference! between! models'! complexity,! as! the! remaining! four! models! used! a!higher! number! of! predictors,! hence! were! able! to! capture! more! complicated!interactions!between!atmospheric!variables.!!While!previous!studies!have!noted!that!using!additional!predictors!besides!temperature!benefit! the!statistically!downscaled!historical!temperature!outputs!(Huth!1999,!2003),!the!new!finding!here!is!that!using!additional! non=temperature! predictors! seems! to! alleviate! the! performance! drop! in!future!climate!too!(for!the!southern!Ontario!and!Quebec!region).!!Among! the! six!models,! the! nonlinear! BNNPC!method! proved! to! be! the!most! stable!over!time!in!terms!of!MAE!and!IOA!changes,!presenting!similar!climate!and!weather!downscaling!performance!between!the!two!periods.!In!the!future!climate,!the!BNNPC!has!marginally! the!best!climate!downscaling!performance,!while!BNNall,!marginally!the!best!?weather?!performance.!In!contrast,! in!the!historical!period,!marginally,!the!best!performer!in!climate!indices!is!LRT,!and!in!?weather?!is!LRall,!both!linear!models.!Again,! analogous! to! the! situation! involving! additional! non=temperature! predictors,!  46 the!ability!to!model!more!complicated!relations!with!nonlinear!BNN!models!seem!to!alleviate!the!performance!drop!in!future!climate.!!!Figure! 6.! Model! comparison! in! the! future! period! (2041=2070),! with! the! daily!temperature!MAE!versus! the!unified!STARDEX! index! IOA.!The! future!period!results!are!shown!with!MAD!error!bars,!while! the!historical!period!results! (from!Figure!5)!are!plotted!without!error!bars,!and!black! lines!connect! the!historical!and!the! future!climate!results.!!!!  47 To! determine! which! model! was! the! best! in! terms! of! both! weather! and! climate!downscaling,!I!calculated!the!daily!mean!temperature!IOA.!This!daily!variability!IOA!and!the!unified!STARDEX!index!IOA!allow!us!to!characterize!a!model?s!performance!as!a!point!in!a!2=D!space,!where!a!perfect!model!occupies!the!point!(1,!1),!as!both!IOAs!take!on!the!perfect!score!of!unity.!The!Euclidean!distance!between!the!perfect!model!point!of!(1,!1)!and!an!actual!model?s!point!in!this!2=D!space!gives!a!single!measure!of!the! model?s! performance! in! terms! of! both! weather! and! climate,! with! a! smaller!Euclidean! distance! indicating! better! model! performance.! Figure! 7,! showing! the!Euclidean! distance! to! the! perfect! model! for! both! the! historical! and! future! periods!confirms! that! the! largest! change! in! performance! going! from! the! historical! to! the!future!period!occurred!in!models!using!only!temperature!predictors!(BNNT!and!LRT),!and! shows! BNNPC! as! the! best! model! in! simulating! both! climate! indices! and! daily!variability!for!the!future!climate.!Finally,!although!the!models!were!able!to!respond!to!the! CGCM!predicted!warming! trend! and!were! able! to! replicate! the! historical!mean!temperature,! I! found!that!all! the!models!overpredicted!the! future!mean!of! the!daily!maximum! temperature! (Figure!8! (left))! and!even!more! severely! for! the!mean!daily!minimum!temperature!!(Figure!8!(right)).!!  48 !Figure!7.!!Euclidean!distance!to!the!perfect!model!point!(1,1)!in!the!2=D!IOA!space!for!weather! and! climate,! with! smaller! distances! indicating! better!models.! Darker! bars!show!the!historical!period!results!and!lighter!colored!ones,!the!future!period!results.!!!Figure!8.! !A!comparison!of!the!mean!values!of!the!downscaled!daily!(left)!maximum!and!(right)!minimum!temperatures!with!the!mean!of!the!CRCM!pseudo=observations!for!historical!(20C3M)!and!future!(A2)!climates.!!The!thick!horizontal!lines!mark!the!mean!values!from!the!pseudo=observations.!  49 2.6 Conclusion and discussion Here! I! present! for! the! first! time! the! historical! and! future! performances! simulating!weather! and! climate! of! extremes! of! two! popular! statistical! downscaling! methods:!stepwise! multiple! linear! regression! and! nonlinear! Bayesian! neural! network! using!three! different! predictor! sets,! for! southern! Ontario! and! Quebec.! ! The! statistically!downscaled!data!were!compared!against!a!regional!climate!model?s!daily!maximum!and!minimum!temperatures,!used!here!as!past!pseudo=observations!and!proxies!for!future!conditions.!!A! major! limitation! of! statistical! downscaling! methods! is! the! assumption! that! the!present! day! statistical! relationships! between! the! coarse! resolution! atmospheric!predictors!and!the!predictand(s)!will!remain!constant!in!the!future,!thus!valid!under!possible!climate!change!scenarios!(Wilby!et!al.!1998).!Here!I!showed!that!not!all!SD!models!satisfy! the!assumption,!and!that! the!model!skills!simulating!daily!variability!and! climate! indices! may! vary! from! present! to! future! climate! depending! on! the!predictor! set! used.! Specifically! I! showed! that! the! best! model! simulating! both!historical!?weather?!and!climate!indices!in!Figure!7!(i.e.!LRT)!was!not!the!best!one!in!the! future! period! (in! fact! this! model! was! the! worst).! This! finding! has! significant!repercussions! ==! as! one! of! the! statistical! downscaling! paradigms! is! to! assume! that!present!simulation!skills!will!be!kept!in!the!future,!a!decision!maker,!by!assuming!that!  50 validating! the!model! in! the!past!would!suffice,! could!end!up!selecting!a!model!with!poor!future!performance.!!!!Previous!studies!found!using!circulation!and!temperature!variables!superior!to!using!only! single!predictors!when!downscaling! temperature!or!precipitation! (Wilby!et! al.!1998;! Huth! 1999,! 2002,! 2003;! Gachon! 2008,! 2005;! Hessami! et! al.! 2008).! ! Here! I!further!noted!that!SD!models!with!additional!non=temperature!predictors!seemed!to!suffer! less! performance! deterioration!when! shifting! from!present! to! future! climate!than! models! with! only! temperature! predictors.! Moreover,! nonlinear! BNN! models!seemed! to! deteriorate! less! than!MLR!models!when! shifting! from! present! to! future!climate.!Hence!using!models!with!greater!ability! to!model!complicated!relations,!by!having!either!nonlinear!capability!or!additional!non=temperature!predictors,!seemed!to!alleviate!the!drop!in!performance!found!in!future!climate!conditions.!!Although!the!performance!differences!between!the!six!models!are!small,!the!BNNPC!model!is!generally!best!in!terms!of!both!?weather?!and!climate!indices.!This!contrasts!with!my!previous!study!of!temperature!downscaling!to!observed!data!(Appendix!A),!where! the! BNNall! model! was! the! best,! in! both! weather! and! climate.! This! may! be!explained! by! the! difference! between! observed! data! and!RCM!data,!where! the! RCM!data!had!less!variability!than!the!observations.!The!diminished!local!variability/signal!in!the!RCM!data!could!enhance!the!approach!using!PCs!as!predictors!(as!in!the!BNNPC!  51 model),! since! PCs! are! best! for! capturing! larger! scale! signals.! The! diminished! local!variability/signal!could!also!lessen!the!advantage!of!nonlinear!SD!models!over!linear!models,! as! greater! advantage! of! the! nonlinear! BNN! models! over! MLR! models! in!climate! downscaling! was! found! in! the! study! using! real! observed! data! than! in! the!present!chapter!using!RCM!data.!!!Finally,! in! terms! of! climate! performance! during! the! historical! period,! the! models!using!only!temperature!predictors!(LRT!and!BNNT)!performed!well!here!!(Figure!5),!but!in!the!study!using!observed!data!were!considerably!poorer!than!the!models!using!more!predictors!(LRall!and!BNNall).!Again,!this!difference!may!be!due!to!the!fact!that!the! relationship! between! the! reanalysis! data! and! the! observed! data! is! more!complicated!than!that!between!the!GCM!data!and!the!RCM!data.!!To!close,!I!recommend!future!studies!to!consider!the!influence!of!spatial=correlation!on!the!effective!number!of!weather!stations.! !Here!I!used!10!neighbouring!RCM!grid!points! as! pseudo=observations! but! the! effect! of! spatial=correlation! considerably!reduced!the!effective!number!of!uncorrelated!stations.!In!Chapters!3!and!4,!I!evaluate!the! effectiveness! of! the! SD! methods! on! non=normally! distributed! datasets,! i.e.!precipitation!and!wind!speed!datasets.!!!! !  52 Chapter(3!3 COMPARISON+ OF+ STATISTICALLY+ DOWNSCALED+PRECIPITATION+IN+TERMS+OF+FUTURE+CLIMATE+INDICES+AND+DAILY+VARIABILITY+FOR+SOUTHERN+ONTARIO+AND+QUEBEC,+CANADA.+!!3.1 Introduction++In! the! absence! of! future! observations,! statistical! downscaling! studies! rely! on!historical! data! to! validate! their!models! and! assume! that! these!historical! simulation!skills! will! be! retained! in! the! future! (Wilby! et! al.! 1998),! as! they! consider! that! the!statistical!relationship(s)!between!the!coarse!resolution!global!climate!model!(GCM)!predictors!and!the!finer!scale!predictand(s)!remain!constant!over!time.!!The!present!study!tackles!this!generally!overlooked!assumption,!and!verifies!if!the!performances!of!different!linear!and!nonlinear!regression!and!classification!downscaling!models!for!precipitation!occurrences!and!amounts!in!southern!Ontario!and!Quebec,!Canada,!are!time!invariant.!!Additionally,! as! climate! change! studies! have!been! shifting! their! focus! from!average!behavior! (e.g.! Busuioc! et! al.! (2008)! and! Huth! (1999))! to! the! effects! on! extremes!(B?rger! et! al.! 2012),! and!as! the!performance!on!observations! cannot!persist! as! the!  53 only! criterion! of!model?s! quality! (Huth! 2003),! and! because! climate! extreme! events!have! always! been! associated! with! impacts! on! socioeconomic! and! natural! systems!(Zhang!et!al.!2011),!I!opted!to!use!the!CLIMDEX!climate!precipitation!indices!(Table!4)! recommended! by! the! Expert! Team! on! Climate! Change! Detection! and! Indices!(ETCCDI)! sponsored! by! the! World! Meteorological! Organization?s! Commission! of!Climatology!and!the!Climate!Variability!and!Predictability!(CLIVAR)!project!(Peterson!2005),!and!tested!if!the!models?!skill!simulating!them!was!time=invariant!too.!!!Table!4!Extreme!precipitation!indices!used!in!this!study.!The!sdii! index!is!defined!as!the! annual! sum!of! the! daily! precipitation! amount! on!wet! days! divided! by! the! total!number!of!wet!days!per!year!(Karl!et!al.!1999).!The!ETCCDI!uses!monthly!Rx1day!and!Rx5day!instead!of!yearly!maximum!values!(Zhang!et!al.!2011).!!Index!number! Index!acronym!! Index!definition! Units!1! Rx1day! Monthly!maximum!1!day!precipitation+ mm!2! Rx5day! Monthly! maximum! consecutive! 5! day!precipitation+ mm!3! sdii! Simple!daily!(precipitation)!intensity!index+ mm!day!=1!4! r10mm! Annual!count!of!days!when!precipitation!>!10!mm.+ days!5! r20mm! Annual!count!of!days!when!precipitation!>!20!mm.+ days!6! r30mm7 Annual!count!of!days!when!precipitation!>!30!mm! days!7! cdry7 Maximum!length!of!dry!spell! days!8! cwet! Maximum!length!of!wet!spell!+ days!9! r95! Annual! total! precipitation! from! days! >! 95th!!percentile+ mm!10! r99! Annual! total! precipitation! from! days! >! 99th!!percentile! mm!11! prcptot! Annual!total!precipitation!on!wet!days.! mm!!  54 In! general,! downscaling! methods! are! divided! in! two! categories:! dynamical!downscaling!and!statistical!downscaling.!Statistical!downscaling!methods!(SDM),!like!the! ones! presented! in! this! study! aim! to! find! time=invariant! statistical! relationships!between! the!coarse!resolution!predictors!and! the! finer!scale!predictand! (Wilby!and!Wigley!1997a),!and!represent!a!computationally!inexpensive!alternative!to!dynamical!downscaling.! In! particular,! Schmidli! et! al.! (2007)! compared! statistically! and!dynamically! downscaled! daily! precipitation! outputs! over! the! European! Alps! and!showed! that! most! statistically! and! dynamically! downscaled! precipitation! statistics!were!similar.!!Habitually,! linear! and! nonlinear! regression! methods! are! used! to! statistically!downscale!different!atmospheric!variables! like! temperature,!precipitation!and!wind!speeds.! Just! to! mention! a! few! examples,! Jeong! et! al.! (2012b)! recommended! using!annual!multiple! linear!regressions!(i.e.!one!model! for!all!months)!over!monthly!(i.e.!one!model!per!month)!models!when!downscaling!temperature!and!precipitation!over!Canada,! because! the! annual! models! ?showed! as! good! performance! as! monthly!multiple! linear! regression! ones! in! spite! of! its! mathematical! simplicity?.! Similarly,!when! downscaling! monthly! precipitation! over! Spain,! Trigo! and! Palutikof! (2001)!indicated! that! linear! or! slightly! nonlinear! artificial! neural! network! (ANN)! models!outperformed!more!complex!ANN!models.!Tomassetti!et!al.!(2009)!used!an!ANN!and!terrain! information!to!downscale!hourly!precipitation!over!central! Italy.!Schoof!and!  55 Pryor! (2001)! compared! different! precipitation! downscaling! methods! over! the!midwest! region! of! the! United! States.! Chen! et! al.! (2010)! used! discriminant!classification!and!support!vector!machines!(SVM)!to!obtain!precipitation!occurrences!and!amounts!in!Taiwan.!!Classical!statistical!downscaling!studies!assume!that!present!simulation!skills!will!be!retained! on! the! future,! as! it! is! not! possible! to! validate! the! downscaled! time! series!versus!data!not!yet!observed.!Vrac!et!al.!(2007b)!proposed!the!use!of!regional!climate!model!(RCM)!outputs!as!pseudo=observations!or!proxies!of!future!climate,!where!the!RCM!receives!boundary!conditions! from!a!coarse! resolution!GCM.!The!method! thus!allows! downscaling! and! validating! the! statistically! downscaled! series! (SDS)! in! the!future,!as!future!RCM!outputs!are!available.!+Here! I! extended! Vrac! et! al.?s!methodology! and! validated! the! statistical! downscaled!series!not!only! in!terms!of!daily!variability!but!also! in!terms!of!annual!and!monthly!climate!precipitation! indices,! and!validated!different! linear!and!nonlinear! statistical!downscaling!methods!under!the!Special7Report7on7Emissions7Scenarios!(SRES)!20C3M!and!A2!scenarios!(more!on!the!scenarios!in!Section!3.2).!In!particular,!I!evaluated!the!models?! performance! downscaling! the! coarse! resolution! Canadian! Global! Climate!Model! (CGCM3)! 3.1! to! a! finer! scale! obtained! using! the! Canadian! Regional! Climate!Model! (CRCM)! 4.2! daily! precipitation! outputs! as! pseudo=observations,! in! terms! of!  56 annual!or!monthly!climate! indices!and!daily!variability.!Specifically,! I!used!different!linear!and!nonlinear!classification!and!regression!techniques!(e.g.!adaptive!regression!sufficiently!smooth!polynomials,!neural!networks,!classification!and!regression!trees)!to! obtain! the! statistically! downscaled! series! and! compared! them! with! historical!(1971=2000)!and!future!(2041=2070)!pseudo=observations.!Although! a! variety! of! studies! have! compared! the! performance! of! different!downscaling! methods! (e.g.! Wilby! and! Wigley! (1997a),! Schoof! and! Pryor! (2001),!Harpham!and!Wilby!(2005),!Dibike!and!Coulibaly!(2006),!Fr?as!et!al.!(2006),!Haylock!et! al.! (2006),!Khan!et! al.! (2006),! and!more! recently!B?rger!et! al.! (2012)),!my!work!differs! from! those! previous! studies! as! it! compares! the! statistically! downscaled!outputs!with!historical!and!future!RCM!outputs.!!!As!in!B?rger!et!al.!(2012)!my!choice!of!methods!is!in!part!dictated!by!opportunity.!The!methods!used!span!a!wide!range!of!approaches!and!most!of!them!are!relatively!advanced!and!not!available!to!the!average!user.!This! work! is! part! of! the! "Probabilistic! assessment! of! regional! changes! in! climate!variability! and! extremes"! project! funded! by! the! Natural! Sciences! and! Engineering!Research! Council! of! Canada! through! a! Special! Research! Opportunity! (NSERC=SRO)!grant.!The!project!aims! to!develop!high=resolution!climate!change! information!with!the! Canadian! GCMs! and! different! downscaling! methodologies.! For! comparison!purposes,!a!common!area!of!interest!is!being!analyzed!by!all!the!group!members,!and!  57 a! multi=station! modeling! framework! is! employed! for! assessing! the! changes! in!precipitation,! minimum! and! maximum! temperatures! in! ten! weather! stations! near!southern!Ontario!and!Quebec,!Canada.!The!study!area!(Figure!9)!includes!important!socio=economic! centers! like! Ottawa,! the! nation?s! capital,! and! Montreal! in! Quebec.!According! to! Statistics! Canada! (2012)! the! cities! have! a! combined! population!(including!their!metropolitan!areas)!of!5!.1!million!people.!!Overall,! planning! agencies,! stakeholders! and! hydrological! modelers,! among! many!others,!will!benefit!from!knowing!the!differences!between!the!methods?!performance!simulating! extreme! indices,! classifying! precipitation/no=precipitation! days! and!determining!precipitation!amounts!over!a!given!period,!as!these!differences!provide!valuable! information! regarding! the! level! of! confidence! I! should! attribute! to! the!downscaled!climate!projections.!!3.2 Precipitation+statistical+downscaling+!First!I!removed!the!climatological!seasonal!mean!from!the!predictors!and!the!pseudo=observations!to!get!the!anomalies!and!then!standardized!them!to!unit!variance.!!!!  58 !Figure!9.!North!American!map.!!Canadian!provinces!of!Quebec!and!Ontario!are!shown!in!dark!gray.!Squares!indicate!the!location! of! the! selected! weather! stations;! circles! indicate! the! location! of! the! closest! CGCM3! grid! points.  59 Afterwards,+I+used+the+following+downscaling+procedure:++I. Train+ the+ model+ using+ the+ historical+ CGCM3+ data+ as+ predictors+ and+ CRCM+pseudo?observations+as+predictands;+II. Compute+ the+ model?s+ validation+ error+ on+ independent+ historical+ data+ using+cross?validation+(Bishop+2006);+and++III. Use+ the+CGCM3+ future+A2+ scenario+ simulation+ as+model+ inputs+ to+downscale+for+the+possible+future+climate.++!3.2.1 Predictors,!predictands!and!study!area!To+evaluate+the+models?+performance+simulating+daily+variability+and+climate+indices+for+ the+ historical+ period+ (1971?2000)+ I+ used+ the+ CGCM3+ 20C3M+ scenario;+ this+scenario+ assumes+ greenhouse+ gases+ emissions+ increasing+ as+ observed+ through+ the+past+century.+Additionally,+I+used+the+CRCM+4.2+using+the+CGCM3+output+as+boundary+conditions+ to+ obtain+ precipitation+ pseudo?observations.+ Future+ performances+ were+evaluated+using+the+CGCM3+and+CRCM+model+outputs+following+the+Special(Report(on(Emissions(Scenarios+(SRES)+A2+scenario+(IPCC+2000).++The+data+were+provided+by+the+North+American+Regional+Climate+Change+Assessment+Program+(Mearns+et+al.+2007).+From+each+grid+point+daily+values+of+maximum+and+minimum+temperatures,+zonal+and+meridional+ wind+ components+ (u,v),+ sea+ level+ pressure+ and+ precipitation+ were+extracted+ for+ the+historical+ (1971?2000)+and+ future+ (2041?2070)+ time+periods,+with+the+ future+ period+ using+ the+ A2+ scenario+ from+ the+ CGCM3+ T47+ run+ number+ 4.+ This+  60 scenario+ assumes+ a+ very+ heterogeneous+world,+with+ a+ high+ population+ growth+ and+less+ concern+ for+ rapid+ economic+ development.+ The+ statistical+ downscaling+ models+used+daily+precipitation+pseudo?observations+for+the+same+periods.++The+study+area+for+the+inter?comparison+project+(and+this+study)+includes+ten+weather+stations+near+southern+Ontario+and+Quebec+(Figure+9).++I+extracted+daily+precipitation+outputs+ from+ the+ nearest+ CRCM+ points+ to+ each+ weather+ station+ and+ used+ them+ as+pseudo?observations.+ The+ CGCM3+ predictors+ were+ obtained+ from+ nine+ grid+ points+near+the+study+region.+Using+a+maximum+of+54+potential+predictors+(i.e.+9+CGCM3+grid+points+times+6+predictors+per+grid+point),+I+downscaled+precipitation+occurrences+and+amounts+at+ each+CRCM+pseudo?observational+ site.+ + In+particular,+when+downscaling+temperature+and+precipitation+in+southern+Quebec,+Jeong+et+al.+(2012a)+found+that+the+mean+sea+ level+pressure,+ the+specific+humidity+and+the+u+and+v+ components+showed+potential+ as+ predictors+ for+ daily+ precipitation.+ In+ addition,+ as+ the+ predictors+ should+carry+the+climate+change+signal,+I+ included+sensitive+predictors+between+periods+like+daily+maximum+and+minimum+temperatures.++In+general,+ the+region+ is+affected+by+the+presence+of+ the+Gulf+of+St.+Lawrence+and+ its+estuary,+which+ tend+ to+ reduce+ the+ continental+ conditions+of+ its+ climate+ (Jeong+ et+ al.+2012a);+ particularly,+ Farnham+ and+ Drummondville,+ located+ closer+ to+ the+ Gulf+(northeast)+are+the+wettest,+and+Ottawa,+located+southwest+from+the+other+sites,+is+the+  61 driest.+ On+ average,+ during+ the+ historical+ period,+ there+ were+ 4930+ days+ with+precipitation+out+of+a+total+of+10918+days.++3.2.2 Model!evaluation!I+ validated+ the+ downscaled+ results+ in+ terms+ of+ daily+ variability+ and+ annual+precipitation+ indices+ for+ the+ historical+ and+ future+ periods.+ The+ daily+ variability+evaluation+involved+calculating+the+mean+absolute+errors+(MAE)+between+the+pseudo?observations+and+the+statistically+downscaled+series,+as+MAE+is+a+better+score+than+the+mean+ squared+ error+ (MSE)+ (Willmott+ and+Matsuura+ 2005).+ The+ climate+ evaluation+procedure+involved+calculating+the+refined+Index+of+Agreement+(Willmott+et+al.+2012)+(IOA),+between+the+downscaled+and+pseudo?observed+annual+climate+indices+in+Table+4.+In+general,+ the+IOA+is+bounded+between+?1+and+1+and+a+larger+IOA+value+indicates+better+forecast+performance.+The+precipitation+indices+in+Table+4+were+defined+by+the+Expert+ Team+ on+ Climate+ Change+ Detection+ Monitoring+ and+ Indices+ (ETCCDI)+sponsored+ by+ the+World+ Meteorological+ Organization?s+ Commission+ of+ Climatology+and+ the+Climate+Variability+ and+Predictability+ (CLIVAR)+project+ (Peterson+2005).+ In+particular,+I+used+the+pcic_climdex(v.(0.491(R+package+(Bronaugh+2012)+to+calculate+the+ETCCDI+ precipitation+ climate+ indices+ (hereafter+ CLIMDEX+ indices),+ and+ then+calculated+ and+ compared+ the+multistation+ average+ IOAs.+ In+ particular,+ the+ cwet+ and+cdry+ indices+depend+on+precipitation+occurrence,+the+remaining+nine+indices+depend+on+ the+ precipitation+ amounts+ and+ the+ sdii+ and+ prcptot+ indices+ are+ considered+  62 aggregated+ indices+ as+ they+ depend+ on+ both+ the+ precipitation+ occurrences+ and+ the+amounts.++The+ validation+ error+ of+ the+ entire+ dataset+ was+ obtained+ using+ cross?validation.+Specifically,+ I+ divided+ the+ historical+ data+ (1971?2000)+ into+ four+ adjacent+ sections,+used+three+sections+to+train+the+model+and+the+remaining+section+to+test+the+model+on+independent+ data.+ I+ repeated+ the+ procedure+ four+ times,+ until+ all+ four+ sections+were+used+to+test+predictions.+As+regression+under?predicts+the+variance+(Karl+et+al.+1990)+I+multiplied+ the+ downscaled+ outputs+ by+ the+ ratio+ of+ the+ pseudo?observed+ standard+deviation+to+the+downscaled+standard+deviation+from+the+training+data.+3.3 Data!methods!+Precipitation+ occurrences+ were+ calculated+ using+ different+ linear+ and+ nonlinear+methods+ such+ as+ artificial+ neural+ networks,+ k?nearest+ neighbors+ (an+ analogue+method),+classification+trees,+classification+tree+ensembles,+discriminant+classification,+and+ na?ve?Bayes+ classifiers.+ The+ model+ outputs+ were+ then+ compared+ to+ the+persistence+forecast+using+the+Peirce+skill+score+(PSS)+(Wilks+2011).+++In+general,+classification+methods+take+an+input+vector+x+and+assign+it+to+only+one+of+the+ discrete+ output+ classes+ available,+ with+ the+ input+ space+ divided+ into+ decision+regions+ by+ boundaries+ known+ as+ decision+ surfaces+ (Bishop+ 2006).+ + In+ particular,+ I+classified+ the+ pseudo?observed+ precipitation+ series+ in+ two+ categories:+ precipitation+  63 and+no+precipitation,+with+precipitation+days+exceeding+the+1+mm+day?1+threshold,+as+in+Iizumi+et+al.+(2011),+and+compared+the+downscaled+results+versus+the+persistence+forecast+in+terms+of+PSS.+++On+the+other+hand,+precipitation+amounts+were+calculated+using:+a)+stepwise+multiple+linear+ regression+ (SWLR),+ b)+ artificial+ neural+ networks+ for+ regression+ (ANN?R),+ c)+regression+ trees+ ensembles+ using+ bagging+ (TreeBagg),+ and+ d)+ adaptive+ regression+sufficiently+smooth+polynomials+(ARES).+The+regression+target+data+corresponded+to+the+transformed+precipitation+amounts+using+the+Yeo?Johnson+extension+of+ the+Box?Cox+transform+(Wilks+2011)+to+make+the+predictand+distribution+more+Gaussian.+++3.3.1 Discriminant!analysis!classification!I+ created+ a+ two?classes+ linear+ discriminant+ analysis+ classifier+ to+ separate+ the+precipitation+ and+ no?precipitation+ classes.+ In+ general,+ this+ classifier+ results+ from+ a+linear+equation+of+the+form:++ y((x)+=+wTx++wo++ (2).+Equation+ 2+ creates+ a+ boundary+ between+ the+ two+ classes+ using+ the+ training+ data+ to+estimate+the+weight+vector+(w)+and+bias+(wo).+A+vector+x+is+assigned+to+the+first+class+if+y+ (x)+?+0+or+ to+ the+second+class+otherwise+ (Hastie+et+al.+2009).+ +My+ implementation+used+Matlab??s+classify+function.+(3.3.2 Na?ve@Bayes!classifier!The+na?ve?Bayes+classifier+first+estimates+the+parameters+of+a+probability+distribution+assuming+conditional+independence+of+the+training+sample+predictors+xk,+given+a+class+  64 G(=(j.+Then,+on+the+prediction+step+the+method+computes+the+posterior+probabilities+of+an+ unseen+ test+ sample+ belonging+ to+ each+ class,+ and+ then+ assigns+ the+ sample+ to+ the+class+with+the+largest+posterior+probability+(Wilks+2011).+++The+ na?ve?Bayes+ classifier+ has+ remained+ popular+ over+ the+ years+ and+ often+outperforms+more+ sophisticated+ alternatives+ (Hastie+ et+ al.+ 2009).+ See+Wilks+ (2011)+Chapter+14+or+Hastie+et+al.+(2009)+Chapter+6+for+details.+Specifically,+I+used+Matlab??s+Statistics+Toolbox?+NaiveBayes.fit+function+to+classify+the+data.+3.3.3 k@nearest!neighbor!classifier!The+ k?nearest+ neighbor+ (KNN)+ classifier+ is+ one+ of+ the+ most+ robust+ and+ useful+classifiers+(Hastie+et+al.+2009).+The+classification+uses+the+majority+vote+among+an+odd+number+of+neighbors+to+avoid+ties.+The+method+takes+the+k+nearest+neighbors+of+the+new+inputs+and+predicts+the+new+output+based+on+the+most+frequent+outcome,+0+or+1,+among+these+neighbors+(Tebaldi+and+Knutti+2007).+See+Hastie+et+al.+(2009)+Chapter+13+for+details.++I+ used+ Matlab??s+ Bioinformatics+ Toolbox?+ knnclassify+ with+ the+ Euclidean+ distance+function,+and+varied+the+number+of+neighbors+to+find+the+best+classification+model.+ +I+found+ that+ during+ training+ a+ KNN+ classifier+ using+ 45+ neighbors+ presented+ the+maximum+ average+ number+ of+ correct+ forecasts+ when+ compared+ to+ the+ pseudo?observations.++  65 3.3.4 Decision!trees!and!decision!trees!ensemble!!Two+main+types+of+decision+trees+can+be+used+in+climate+downscaling:+a)+classification+trees+ and+ b)+ regression+ trees;+ although+ the+ term+ classification+ and+ regression+ tree+(CART)+ refers+ to+ both+ types+ of+ trees.+ In+ particular,+ I+ created+ classification+ trees+ for+predicting+ the+ precipitation+ occurrences,+ and+ regression+ trees+ for+ predicting+ the+precipitation+amounts+as+a+function+of+the+coarse+resolution+predictors.+++To+ model+ the+ precipitation+ occurrence+ process+ I+ used+ bootstrap+ aggregation+(bagging)+ of+ 500+ classification+ trees+ (TreeEnsemble),+ where+ every+ tree+ in+ the+ensemble+ is+ grown+ on+ an+ independently+ drawn+ bootstrap+ replica+ of+ the+ input+ data+(Breiman+1996)+and+compared+the+ensemble+output+(with+classification+determined+by+the+majority+vote)+against++a+model+using+a+single+classification+tree+(ClassTree).+I+used+ the+ classregtree+ function+ included+ in+ Matlab??s+ Statistic+ Toolbox?+ to+ obtain+precipitation+occurrences.++To+model+precipitation+amounts,+I+used+bagging+on+500+regression+trees+(TreeBagg)+(Breiman+ 1996).+ + In+ general,+ bagging+ allows+ us+ to+ obtain+ maximum+ likelihood+estimates+ of+ standard+ errors+ and+ other+ quantities+where+ no+ formulas+ are+ available+(Hastie+ et+ al.+ 2009).+ My+ implementation+ used+ the+ TreeBagger+ function+ included+ in+Matlab??s+Statistics+Toolbox?.++  66 3.3.5 Stepwise!multiple!linear!regression!(SWLR)!This+ multiple+ linear+ regression+ approach+ uses+ forward+ selection+ and/or+ backward+elimination+ to+ select+ a+ predictor+ subset+ from+ an+ original+ pool+ containing+ all+ the+available+predictors+(Wilks+2011;+Darlington+1990).++The+ratio+of+sample+variances+is+computed+ to+ test+ models+ with+ and+ without+ a+ potential+ predictor,+ assuming+ a+ null+hypothesis+ that+ the+ predictor+ to+ be+ added+ or+ removed+ has+ a+ zero+ regression+coefficient+(Hill+and+Lewicki+2006).++In+particular,+I+used+the+stepwisefit+function+included+in+Matlab??s+Statistics+Toolbox?+to+implement+the+stepwise+multiple+linear+regression+(SWLR),+and+kept+its+default+p9values+for+adding+and+removing+a+variable+at+0.05+and+0.1,+respectively.++3.3.6 Artificial!neural!networks!for!classification!and!regression!I+used+the+feed?forward+multilayer+perceptron+(MLP)+ANN+with+one+input+layer,+one+hidden+ layer+ and+ an+ output+ layer+ to+ map+ the+ nonlinear+ relationships+ between+ the+coarse+resolution+CGCM3+predictors+and+the+dependent+output+variables.+In+general,+MLP+neural+networks+aim+to+minimize+the+MSE+between+the+predicted+and+observed+value+ of+ the+ dependent+ variable(s)+ (Hsieh+ 2009).+ In+ particular,+ I+ used+ ANN+ with+Bayesian+ regularization+ (i.e.+ Bayesian+ neural+ networks)+ to+ prevent+ overfitting+(MacKay+1992).+ANN+was+used+ for+nonlinear+ classification+ (ANN?C)+of+precipitation+and+ non?precipitation+ days,+ and+ to+ obtain+ downscaled+ precipitation+ amounts+ by+nonlinear+regression+(ANN?R).+To+implement+the+classification+and+regression+models+  67 I+used+the+patternet+and+trainbr+functions+respectively+in+Matlab??s+Neural+Networks+Toolbox?.+++One+of+the+advantages+of+Bayesian+neural+networks+over+traditional+neural+networks+is+ that+ they+can+estimate+the+optimal+weight+penalty+parameter+without+ the+need+of+validation+data+(Hsieh+2009),+nevertheless+their+solutions+can+still+be+trapped+in+local+minima+during+the+nonlinear+optimization+(Bishop+2006),+hence+an+ensemble+of+runs+with+random+initial+weights+is+commonly+used.+++Recall+ in+the+cross?validation,+ the+data+record+was+divided+into+four+sections,+and+at+each+ iteration+of+ the+computation+ loop+only+three+sections+were+available+ for+model+training.+With+ANN+I+also+determined+the+optimal+model+architecture+by+varying+the+number+of+hidden+neurons+from+1+to+30.+Only+two+thirds+of+the+available+training+data+was+actually+used+for+model+training,+while+the+remaining+third+was+used+to+compute+the+mean+absolute+error+(MAE)+to+allow+the+selection+of+the+best+model+architecture.+With+ the+ optimal+ number+ of+ hidden+ neurons+ thus+ determined,+ I+ next+ ran+ the+ ANN+model+ (using+ all+ the+ training+ data)+ 10+ times+ with+ random+ initial+ weights+ and+ensemble+ averaged+ their+ output.+ In+ general,+ ensemble?averaging+ decreases+ the+expected+error+of+the+ANN+(Hsieh+2009).+This+process+was+repeated+four+times+for+the+four+ sections+used+ in+ the+ cross?validation+process.+For+ the+21st+ century+outputs,+ the+  68 ensemble?averaging+was+performed+over+40+ensemble+members+(i.e.+10+members+for+each+section+x+4+sections).++3.3.7 Adaptive!regression!sufficiently!smooth!polynomials!(ARES)!Based+on+ the+multivariate+adaptive+regression+splines+ (MARS)+ technique+(Friedman+1991),+ the+ goal+ of+ ARES+ is+ to+ obtain+ the+ best+ piecewise?linear+ or+ piecewise?cubic+regression+models.+According+to+Hastie+et+al.+(2009)+this+technique+can+be+viewed+as+generalization+of+ stepwise+ linear+ regression+or+ a+modification+of+ the+CART+method.+The+model?s+algorithm+uses+ forward+selection+and+backward+elimination+ to+prevent+overfitting+ and+ can+ include+ automatically+ generated+ nonlinearities+ and+ interaction+terms.+I+used+piecewise?cubic+models+and+the+ARESLab+toolbox+(Jekabsons+2010)+for+Matlab?+to+estimate+the+precipitation+amounts.++3.4 Results!The+present+section+shows+the+results+from+two+different+points+of+view:+(A)+separate+analysis+ of+ the+ classification+ and+ the+ regression+ models,+ and+ (B)+ analysis+ of+ the+regression+models+conditioned+on+a+classification+model+output.+The+main+difference+between+the+two+is+that+in+the+former,+the+precipitation+amounts+are+not+affected+by+the+skill+of+the+precipitation+occurrence+models+(i.e.+only+uses+CRCM+and+CGCM3+data+from+ precipitation+ days+ according+ to+ CRCM),+ while+ in+ the+ latter+ the+ precipitation+amounts+ were+ evaluated+ on+ the+ precipitation+ days+ predicted+ by+ a+ selected+classification+model.+Setup+B+is+consistent+with+operational+reality,+as+the+user+selects+a+downscaling+model+based+on+historical+results+and+then+uses+ it+ to+generate+ future+  69 predictions;+ while+ setup+ A+ allows+ us+ to+ evaluate+ if+ the+ statistical+ relationships+between+the+large+scale+predictors+and+the+precipitation+amounts+are+time?invariant.+3.4.1 Classification!and!regression!models!The+classification+analysis+(Figure+10)+shows+that+complicated+nonlinear+models+(e.g.+ANN?C+ and+ TreeEnsemble)+ outperformed+ linear+ models+ (Discriminant+ and+ na?ve?Bayes)+ and+ simpler+ nonlinear+ models+ (KNN+ and+ ClassTree)+ in+ terms+ of+ the+ PSS+(averaged+ over+ the+ ten+ stations),+ with+ all+ the+ models+ outscoring+ the+ persistence+forecast;+additionally,+ the+models+did+not+show+a+decrease+ in+performance+ in+ future+climate.+ This+ indicates+ that+ the+ relationships+ between+ the+ coarse+ resolution+predictors+and+the+binary+predictand+(indicating+precipitation+or+no+precipitation)+for+most+of+the+models+(and+for+this+region)+are+unchanged+into+the+21st+century.+Under+these+conditions,+one+could+argue+ that+ the+precipitation+occurrence+process+ is+ time?invariant+for+the+region+of+southern+Ontario+and+Quebec.++Also+note+the+improvement+of+the+PSS+when+using+an+ensemble+of+classification+trees+(TreeEnsemble)+instead+of+a+single+tree+(ClassTree).++On+the+other+hand,+the+regression+results+(Figure+11)+under+setup+A+show+that+all+the+models+ degraded+ their+ performance+ in+ terms+ of+ the+ MAE,+ when+ compared+ to+ the+historical+period.+ In+general,+ the+ future+MAE+ is+~1+mm+day?1+higher+ than+within+ the+historical+period.+Changing+predictor?predictand+relationships+between+the+historical+and+ future+ periods+ could+ cause+ this+ change+ in+ skill.+ An+ alternative+ hypothesis+  70 suggesting+ that+ this+ change+ was+ mainly+ caused+ by+ the+ extrapolation+ error+ was+rejected+ as+ the+ regression+ tree+ model+ (used+ in+ TreeBagg)+ is+ known+ for+ having+ no+extrapolation+ capacity+ (Hastie+ et+ al.+ 2009).+ Figure+ 11+ also+ shows+ the+ MAEs+ of+ the+variance+inflated+SDS+required+to+calculate+the+CLIMDEX+climate+indices.+++The+ differences+ in+ the+ performance+ among+ the+ regression+ models+ were+ generally+small+ between+ ARES,+ ANN?R+ and+ SWLR+ (for+ the+ historical+ period),+ with+ the+ three+models+outscoring+TreeBagg.+For+the+future+period,+ANN?R+was+the+best+model,+while+ARES+and+SWLR+marginally+outscored+TreeBagg.++In+Figure+12+I+compare+the+empirical+cumulative+distribution+functions+(ECDFs)+of+the+CRCM+precipitation+output+ (black)+with+ two+SD+time+series+ for+Lennoxville,+Quebec,+the+ first+ one+ (dot?dashed)+ corresponds+ to+ the+ raw+ ANN?R+model+ output,+ while+ the+second+ one+ (dashed)+ shows+ the+ variance+ inflated+ ANN?R+model+ output.+ The+ figure+also+ shows+ a+ knock?on+ effect+ of+ variance+ inflation:+ for+ the+ last+ three+ deciles+ (daily+precipitation+ >+ 8+ mm)+ the+ SDS+ using+ variance+ inflation+ is+ closer+ to+ the+ pseudo?observed+ quantiles+ than+ the+ SDS+ without+ it,+ but+ when+ the+ pseudo?observed+precipitation+ amounts+ are+ less+ than+ 8+mm+ day?1+ the+ ANN?R+ raw+ output+ provides+ a+better+fit+than+the+variance+inflated+time+series.++  71 !Figure!10!Precipitation!occurrence!models?!average!Peirce!skill!score!(PSS).!Lighter!and!darker!coloured!bars!represent!the!historical!!(20C3M)!and!future!(A2)!periods,!respectively.!Error!bars!represent!mean!absolute!deviations!(MAD)!from!the!ten!stations.!  72 !!Figure! 11.! Precipitation! amount! models?! average! MAEs.! Lighter! and! darker! coloured! bars! represent! the! historical!!(20C3M)!and!future!(A2)!periods,!respectively.!Error!bars!represent!the!MAD.!  73 !Figure! 12! Empirical! cumulative! distributions! of! the! statistically! downscaled! time!series!for!Lennoxville,!Quebec,!with!and!without!using!variance!inflation!(dashed!and!dot@dashed,! respectively)! versus! the! CRCM! pseudo@observations! (solid).! The! SD!model!is!ANN@R.!!3.4.2 Regression-models-conditioned-on-the-occurrence-model-!As!three!out!of!four!regression!models!performed!equivalently!(i.e.!ARES,!ANN@R!and!SWLR)!and!outscored!the!TreeBagg!model!on!the!historical!period! in!terms!of!MAE!(see! Figure! 11),! setup! B! deals! with! the! performance! of! these! three! models! when!conditioned!on!precipitation!days!simulated!by!ANN@C,!one!of!the!best!classification!models.!Hereafter!I!will!refer!to!these!final!combined!models!as!ARES@F,!ANN@F!and!SWLR@F.! Overall,! for! the! historical! and! future! periods,! I! calculated! the! eleven!  74 CLIMDEX!climate!indices!from!the!SDS!and!compared!them!with!the!CLIMDEX!indices!generated!from!the!historical!and!future!CRCM!pseudo@observations! in!terms!of! the!refined!IOA!(Willmott!et!al.!2012).!!!To! show! if! the!models?! IOAs! calculated! from! the! CLIMDEX! indices! skills!were! time!invariant!I!opted!to!plot!the!average!IOA!differences!for!each!climate!index!between!the!future!and!historical!periods!!(Figure!13),!where!positive!IOA!differences!indicate!higher! future! skills! and! negative! differences! indicate! higher! historical! skills.! ! If! the!time! invariance! assumption! holds! one! should! expect! the! model! differences! to! be!centered!on!zero.!!!!In!particular,! I! see! that! for! the! classification! related! indices! (i.e.! cwet! and!cdry)! the!differences! are!marginally! positive! (taking! into! consideration! the!MAD! error! bars)!and!their!performance!is!the!same!for!the!three!final!models!as!anticipated!?!cwet!and!cdry!depemd!on!ANN@C@.!On!the!other!hand,! for!the!regression!related!indices,!I!see!that!the!r20mm!presents!the!only!positive!differences!for!ANN@F!and!SWLR@F,!while!the! r99! differences! are! slightly! positive! for! the!ARES@F!model! and! negative! for! the!ANN@F!and!SWLR@F!models,!and!the!r95!differences!are!only!negative!for!the!SWLR@F!model.! Also! it! is!worth!noting! that! for! all! the!models,! the! two!monthly! indices! (i.e.!Rx1day! and! Rx5day)! degraded! their! skill,! and! that! ARES@F! and! SWLR@F! showed!negative!IOA!differences!when!analyzing!the!prcptot3index.!!  75 Overall! it!could!be!agued!that!for!this!study!case,! five!out!of!nine!regression!related!indices! (i.e.!prcptot,! r95,! r10mm,! r30mm! and! sdii)! had! time! invariant! performances!when!using!ANN@F,!while! three! (i.e.! r95,! r20mm! and! sdii)! and! two! (i.e.! r10mm! and!sdii)!of!the!nine!indices!had!time!invariant!skills!for!the!ARES@F!and!SWLR@F!models,!respectively.! ! Alternatively,! I! can! count! the! number! of! significant! negative! IOA!differences! in! Figure! 9,! which! occurred! for! three! (ANN@F),! five! (ARES@F)! and! six!(SWLR@F)!indices,!indicating!ANN@F!to!be!best!in!avoiding!performance!deterioration!in!future!climate.!!Although! testing! the! stationarity! assumption! is! important,! having! time@invariant!performances!does!not!suffice,!as! it! is!possible!to!obtain!models!with!no!skill! in!the!historical! and! future! periods.! For! this! reason,! I! complemented! the! analysis! by!showing! the! IOAs! for! each! period! and! model! (Figure! 14),! and! determined! if! each!CLIMDEX!climate!index!time!series!(i.e.!obtained!after!calculating!one!value!per!year,!for!the!nine!annual!indices,!or!one!value!per!month!for!the!two!monthly!indices)!from!the!downscaled!and!the!pseudo@observed!time!series!belong!to!the!same!distribution!(Figure!15).!!The! IOA! is! bounded! between! 1! and! @1,! where! +1! indicates! complete! agreement!between! the! observations! and! the! simulations,! and! @1! indicates! complete!disagreement!(see!Chapter!2!for!more!details).!Figure!14!shows!the!IOAs!derived!by!  76 comparing!the!CLIMDEX!climate!indices!obtained!from!the!pseudo@observations!with!the! ones! calculated! from! the! statistically! downscaled! time! series.! The! indices!were!calculated!for!each!model!and!for!the!historical!and!future!time!periods.!!Focusing!the!analysis!on!the! indices!with!time!varying!IOA!from!Figure!13!I!see!that! the!positive!difference!between!the!future!and!historical!cdry!IOA!was!caused!by!the!classification!model! improving! its! performance! from! near! zero! IOA! in! the! historical! period! to!positive!IOA!in!the!future.!The!occurrence!model!(ANN@C)!overpredicted!the!number!of!no@precipitation!days!(F(psds)=0.61!versus!F(pobs)=0.56).!!Regarding! the! models?! performance,! when! studying! ANN@F! I! see! that! the! r20mm!positive!difference!(Figure!13)!was!caused!by!the!non@skillful!historical!period!results!(negative! IOA)!and!the!slightly!positive!or!close! to!zero! IOA! in! the! future.!Similarly,!when!considering!the!ANN@F!indices!that!lowered!their!IOA!between!periods!(i.e.!r99,!Rx1day! and!Rx5day),! all! three! indices! had! positive! IOA! in! the! historical! and! future!periods,!so!this!statistical!downscaling!model!was!actually!adequate!for!these!indices.!!!Figure! 14! shows! that! ANN@F! has! difficulties!modeling! the! r10mm,! r20mm! and! to! a!lesser!extent!sdii.!Nevertheless,!the!complications!modeling!sdii!were!expected!as!this!index! depends! not! only! on! the! correctness! of! the! classification! model,! but! also!depends! on! the! ability! of! the!ANN@R!model! to! obtain! the! total! yearly! precipitation.!Overall!a!complex!nonlinear!model! like!ANN@C!helped!ANN@R!to!predict!most!of!the!  77 CLIMDEX! indices.! ! In!particular,!seven!out!of!eleven! indices!obtained!positive!mean!IOAs!in!the!historical!period,!and!eight!out!of!eleven!indices!in!the!future!period.!!The!figure! shows! that! ANN@F! obtained! higher! historical! skills! when!modeling! r99! than!when!simulating!the!less!extreme!r95! index,!and!also!shows!the!model!increased!its!r95!IOA!between!periods.!For!the!future!comparison,!the!model!slightly!improved!its!r10mm! and! r20mm! predictions! (relative! to! the! historical! period),! but! for! the! two!monthly! indices! (Rx1day! and! Rx5day)! the! IOAs! degraded! when! compared! to! the!20CM3!values.!!!For! ARES@F,! prcptot! and! sdii! showed! negative! IOA! on! both! historical! and! future!periods! and! r10mm! presented! the! biggest! drop! in! terms! of! IOA! between! periods!(Figure!9),!changing! from!a!positive!historical!value!to!a!negative! IOA! in! the! future.!Finally,!when!looking!at!the!SWLR@F!model,! I!noticed!that!three!indices!(i.e.!prcptot,!r99!and!sdii)!had!negative!IOAs!in!the!historical!and!future!periods,!while!r95!and!the!two!monthly!indices!had!negative!IOAs!in!the!future.!!On!the!other!hand,!the!historical!and! future! IOAs! from! r10mm! and! r20mm! were! positive.! ! Given! these! results! I!recommend! using! the! methods! with! caution,! as! positive! historical! IOAs! do! not!necessarily!imply!positive!IOAs!in!the!future!(at!least!for!this!particular!region).!!!  78 !Figure!13!Average!IOA!differences!between!climate!indices!from!the!future!and!the!historical!periods!from!ANN=F!(left),!ARES=F!(middle)!and!SWLR!(right),!with!error!bars!indicating!the!MAD.!  79 !Figure!14! IOAs! (averaged!over! ten! stations)!of! the!precipitation=related! indices! for! the!historical! (20C3M)!and! future!periods!(A2)! for!ANN=F!(top),!ARES=F!(middle)!and!SWLR=F!(bottom),!with!precipitation!occurrences!calculated!using!ANN=C,!and!error!bars!indicating!the!MADs.!!  80 !Figure!15!Number!of!KS!tests!(out!of!a!maximum!of!ten)!not!rejected!at!a!significance!level!0.05.!The!null!hypothesis!is!that! each! simulated! climate! index! comes! from! the! same! continuous! distribution! as! the! corresponding! climate! index!calculated!from!the!CRCM!pseudo=observations.!!Figure!shows:!ANN=F!(top),!ARES=F!(middle)!and!SWLR=F!(bottom).!!  81 I"had"shown"which"indices"IOAs"were"or"were"not"time2invariant"and"which"indices"had" positive" or" negative" IOAs"when" derived" from" the" ANN2F," ARES2F" and" SWLR2F"models;"but"what"about" the" indices?"probability"distributions?"Do" the"observed"and"simulated" indices" belong" to" the" same" distribution?" To" determine" so" I" performed"Kolmogorov2Smirnov"(KS)"tests"with"a"significance"level"?"="0.05,"and"assumed"a"null"hypothesis" (Ho)" where" the" observed" and"model" calculated" indices" belonged" to" the"same"distribution." This" procedure"was" conducted" for" each" index" at" each" of" the" ten"pseudo2observational"sites"and"for"all"the"final"models."Afterwards,"I"aggregated"the"number"of"Ho"rejections"per" index"per"method,"and"plotted" the"number"of" tests"not"rejected" at" the" selected" significance" level" for" the" historical" period" and" the" future"period"(Figure"15).""The" results" show" that" the" occurrence"model" (ANN2C)" responsible" for" the" cwet" and"cdry"indices"is"appropriate"since"for"most"of"the"ten"sites,"the"observed"and"simulated"cwet"and"cdry"distributions"were"not"different"from"each"other"at"the"5"%"significance"level."This"is"not"unexpected"as"both"indices"had"positive"IOA"for"both"periods"and"the"IOA"did"not"degrade"with"time."Regarding"the"other"nine"indices,"for"the"ANN2F"model"only"prcptot"and"r99"had"similar"distributions"to"observations" for"the"historical"and"future" periods," while" the" model" derived" r95" and" r30mm" distributions" were"surprisingly" similar" to" the" future" pseudo2observed" distributions" despite" the" high"rejection"rate"from"their"KS"tests"for"the"historical"period."  82 The" ARES2F" model" produced" r99" and" r95" distributions" that" were" similar" to" the"observed"ones"for"most"of"the"pseudo2observational"sites"during"the"historical"period,"but"for"the"other"indices"only"a"minority"of"sites"did"not"reject"the"null"hypothesis,"and"in" general" the" number" of" rejections" was" higher" for" the" future" period" than" for" the"historical"one."For"the"SWLR2F"model"during"the"historical"period,"the"r20mm"and"r99"indices"had"few"rejections"by"the"KS"tests,"with"r20mm"being"the"index"with"the"least"number" of" historical" and" future" rejections." Similarly," when" considering" only" the"indices"related"to"precipitation"amounts"(i.e."excluding"cwet"and"cdry)"r20mm"was"the"only"one"with"less"than"five"rejections"on"the"future."As"with"the"ARES2F"model,"all"the"other"indices"showed"an"increased"number"of"rejections"in"the"future"period.""Based"on"the"IOA"and"KS"test"results,"it"seems"best"to"use"ANN2F"for"prcptot"and"r99,"ARES2F" for"r95" (with" reservations)," and"SWLR2F" for"r20mm" in" this"particular" study"region.""""Regarding" the" differences" between" the" pseudo2observed" time" series" and" the" SDS," I"plotted" the"historical" and" future" empirical" cumulative" distributions" from" the"CRCM"and" from" the" statistical" downscaling"models" (Figure" 16)." The" figure" shows" that" in"general" the"models" overpredict" precipitation" amounts" between" 2" and" 6"mm" day21."The"figure"also"shows"an"increase"in"the"number"of"projected"rainy"days"rain" in"the"  83 future"period"than"the"historical"period,"and"a"better" future"agreement"between"the"CRCM"distribution"and"the"one"from"the"ANN2F"distribution.""For"high"precipitation"values"(e.g."between"25240"mm"day21),"Figure"12"shows"that"the"ANN2F"model"gave" fewer"high"precipitation"values" than"the"CRCM,"whereas"ARES2F"and"SWLR2F"gave"more"high"values"than"the"CRCM."This"difference"may"partly"explain"why" the" results" from" ARES2F" and" SWLR2F" were" generally" more" similar" than" the"results"from"ANN2F"in"Figures"10"and"11."The"mathematical"structures"of"the"models"are"very"different," the"ANN"being"a" sum"of"hyperbolic" tangents" functions," the"ARES"being" piecewise" cubic" polynomials" and" the" SWLR," a" linear" function." For" extreme"values"of" the"predictors," the"asymptotic"behavior"of" the"model" functions"comes" into"play."As"the"tanh"function"is"bounded,"the"ANN"is"bounded"asymptotically,"while"the"SWLR"and"ARES"are"unbounded,"which"is"consistent"with"Figure"12"where"the"ANN2F"gave"fewer"high"precipitation"values"than"the"other"two"models."In"future"climate,"the"predictors"may"shift"to"give"more"extreme"values,"rendering"the"asymptotic"behavior"of"the"models"even"more"relevant.""In" order" to" further" fathom" this" difference"between" the"historical" and" future"ANN2F"ECDFs,"I"opted"to"show"the"modeled"and"observed"upper"quantiles"(F(p)">"0.90)"for"a"pseudo2observational"point"near"Ottawa,"Canada"(Figure"17)."The" figure"shows"that"for"the"historical"(and"future)"simulations,"the"ANN2F"models"have"biggest"difference"  84 versus"the"CRCM"pseudo2observations"near"20"mm"day21,"converging"gradually"on"the"upper"end"of"the"distribution,"which"explains"the"higher"skills"when"simulating"r99."In"general," the" model" underpredicts" values" above" the" 90th" pseudo2observed" quantile,"even"though"the"future"maximum"precipitation"from"the"ANN2F"model"(83"mm"day21)"exceeds"the"one"from"the"pseudo2observations"(65"mm"day21)."""To" conclude," Table" 5" summarizes" the" results" from" Figure" 13," Figure" 14" and" the"evaluation" of" field" significance" tests" on" the" results" from" Figure" 15." " Specifically," I"evaluated" the" joint" statistical" significance"of"multiple"null"hypothesis" tests" (i.e." field"significance)" (Wilks" 2006)" by" comparing" the"minimum"p2value" (from" the" KS" tests)"from"each"group"of"K" local"tests"versus"?"="0.05."If"the"minimum"p2values"from"each"group"of"(K"="10)"local"tests"was"smaller"than"or"equal"than"pw"(from"equation"3)"then"the" global" null" hypothesis" was" rejected" at" ?" =" 0.05," indicating" field" significance."Equation"(3)"is"the"basis"of"the"Walker"test"(Wilks"2006)""" pw"="1"?"(12?)1/K"""""(3).""The" table" shows" if" the" final" models" (i.e." ANN2F," ARES2F" and" SWLR2F)" were"appropriate"for"modeling"the"different"CLIMDEX"indices"on"three"properties:"1)"IOA"time2invariance," 2)" positive" IOAs," and" 3)" historical" and" future" rejection" of" field"significance.""  85 !Figure!16!Empirical!cumulative!distributions!for!all!models!during!the!historical!(left)!and!future!periods!(right).!  86 !Figure! 17! Historical! (20C3M)! and! future! (A2)! empirical! cumulative! distribution!functions!(ECDF)!of!precipitation!from!ANNCF!(for!precipitation!days!from!the!ANNCC!classification!model)! versus! the! CRCM! pseudoCobservations.! Darker! lines! show! the!SDS.!Dashed! lines! show! the! future!ECDF.!The!ECDFs!are! from! the!pseudoCobserved!and!statistically!downscaled!time!series!closest!to!Ottawa,!Canada.!!Overall! I! see! that! the! occurrence! model! responsible! for! the! cdry! and! cwet! indices!values!is!appropriate!for!generating!rainy!days!if!one!does!not!take!into!consideration!the! timeCvariance! of! their! IOAs,! although! it! could! be! argued! that! a! model! whose!biggest!flaw!is!having!higher!future!IOAs!does!not!represent!a!major!concern!to!the!downscaling!community.!!On!the!other!hand,!it!is!evident!that!no!model!outscored!the!others!in!all!categories,!and!that!no!model!was!appropriate!for!modeling!all! indices.!!  87 In! particular,! ANNCF! and! ARESCF!were! best! for!modeling! r95! and! SWLRCF!was! the!best! for! modeling! r10mm! (at! least! for! this! particular! region).! For! the! remaining!indices,!no!model!was!able!to!pass!on!all!three!properties.!Overall,!in!Table!5!with!33!entries,!ANNCF!had! the!highest!number!of!positive! results! (16),! versus!13! from! the!ARESCF!model!and!10!from!the!SWLRCF.!Table! 5! Summary! results! for! the! climate! indices.! ?Y?! represents! a! positive! answer!(yes)!to!the!header,!while!a!blank!represents!a!negative!answer!(no).!!Index& Historical&IOA&=&Future&IOA&& Positive&historical&and&future&IOA&& Historical&and&future&global&KS&tests&not&rejected&&ANN?F& ARES?F& SWLR?F& ANN?F& ARES?F& SWLR?F& ANN?F& ARES?F& SWLR?F&prcptot& Y& & & Y& & & & & &r99& & & & Y& Y& & Y& Y& Y&r95& Y& Y& & Y& Y& & Y& Y& &r10mm& Y& & Y& & & Y& & & Y&r20mm& & Y& & & Y& Y& & & Y&r30mm& Y& & & Y& & & & & &cdry& & & & Y& Y& Y& & & &cwet& & & & Y& Y& Y& Y& Y& Y&sdii& Y& Y& Y& & & & & & &Rx1day& & & & Y& Y& & & & &Rx5day& & & & Y& Y& & & & && & & & & & & & & &Total& 5& 3& 2& 8& 7& 4& 3& 3& 4&  88 3.5 Conclusions,and,recommendations,,The! present! work! contributed! to! the! understanding! of! the! generally! overlooked!assumption!of!time!invariance,!inherent!to!the!statistical!downscaling!studies!(i.e.!the!statistical! relationship(s)! between! the! predictors! and! the! predictand(s)! remain!constant!over!time).!Especially,!the!results!suggest!that!future!statistical!downscaling!analyses!should!assume!no!stationarity,!and!only!use!classical!approaches!when!the!time! invariance! of! the! relationship! between! predictors! and! predictand(s)! can! be!verified.!!In! particular,! when! analyzing! precipitation! occurrences,! I! found! that! complex!nonlinear!models!like!artificial!neural!networks!and!ensembles!of!classification!trees!outperformed!linear!models!and!simpler!nonlinear!models!in!terms!of!the!Peirce!skill!score,! and! that! the!models?! performance!did!not! deteriorate! in! future! climate.! This!suggests!that!the!linear!discriminant!classification!method!used!by!Chen!et!al.!(2010)!to! obtain! precipitation! occurrences! might! not! be! the! best! (at! least! for! southern!Ontario!and!Quebec).!On!the!other!hand,!when!downscaling!precipitation!amounts!I!found!that!all!the!regression!models!showed!important!MAE!differences!between!the!historical!and!future!periods!(i.e.! the! future!MAE!is!~1!mm!dayC1!higher!than! in!the!historical!period).!A!similar!conclusion!can!be!drawn!when!comparing!the!CLIMDEX!climate! indices! in!terms!of! IOAs,!as!different! indices!changed!between!periods!from!negative! to! positive,! or! marginally! increased! or! decreased! their! IOA! values.! From!  89 these! results,! one! could! argue! that! timeCinvariance! holds! for! the! precipitation!occurrence!process!but!not!for!the!precipitation!amount!in!the!southern!Ontario!and!Quebec!region.!!In!the!simulation!of!the!indices,!as!the!models!were!trained!on!daily!precipitation!and!not!on!the!indices!themselves,!the!indices?!performance!was!?more!independent?!and!?less! prone! to! artificial! skill! from! overfitting?! as! B?rger! et! al.! (2012)!mentioned! in!their!recent!paper.!In!particular,!the!occurrence!model!responsible!for!the!cwet!index!produced!simulated!distributions!that!were!not!different!from!observed!ones!at!the!5!%!significance!level!for!most!sites,!had!positive!IOA!during!both!periods!and!the!IOA!did!not!degrade!in!the!future!period.!Overall,!ANNCF!generally!showed!better!results!than!ARESCF!and!SWLRCF!when!simulating!most!of! the!historical!and!future!climate!indices!(Table!5).!!Future!studies!should!be!made!around!the!Globe!with!different!climatic!regions!and!predictands/predictors! in! order! to! determine! for! which! cases! the! stationarity!assumption!holds!(e.g.!precipitation!occurrence)!!or! does! not! hold! (e.g.! precipitation!amounts,! climate! indices),! and!whether!nonstationary!downscaling! approaches,! e.g.!Kallache!et!al.!(2011)!should!be!used.!Similarly,!future!studies!should!complement!the!training/calibration! phase,! common! to! most! empirical! downscaling! methods! (i.e.!estimate! the! function! parameters! using! reanalysis! predictors! and! finer! scale! data)!  90 (B?rger! et! al.! 2012),! by! verifying! the! time! invariance! assumption.! Only! then,! it! is!recommended!to!use!the!methods!in!a!climate!change!context.!!In! the!end,! I! aspire! that!knowing! the!differences!between! the!models?!performance!between!periods!will!provide!valuable!information!regarding!the!level!of!confidence!I!could!attribute! to! the!downscaled!climate!projections,! and!will! foster!discussion!on!the! advantages! and! limitations! of! different! regression! and! classification! models!commonly!used!by!the!downscaling!community.!! !  91 Chapter(4!4 EVALUATION OF HISTORICAL AND FUTURE STATISTICALLY DOWNSCALED PSEUDO-OBSERVED SURFACE WIND SPEEDS IN TERMS OF ANNUAL CLIMATE INDICES AND DAILY VARIABILITY. !4.1 Introduction Traditional!evaluations!of!wind!farm!viability!and!wind!energy!potential!rely!on!good!quality!historical!data.! It! is! surprising! that!some!studies!suggest!multimillionCdollar!investments! based! on! a! limited! number! of! observations,! recorded! extreme! events,!and! assumptions! of! stationarity.! Given! evidence! that! wind! resources! may! be!susceptible! to! climate! change,! more! surprising! is! the! limited! role! of! future! wind!projections!in!these!assessments!(Pryor!and!Barthelmie!2010).!!!In!the!past!decades,!Global!Climate!Models!(GCMs)!have!been!developed!to!represent!the!climate!system!and!used!to!simulate!the!actual!climate!and!project!future!climates!under! different! greenhouse! gas! emission! and! concentration! scenarios! (Emissions!Scenarios.!A!Special!Report!of!Working!Group!III!of!the!Intergovernmental!Panel!on!Climate!Change!!2000).!Although!their!accuracy!at!simulating!largeCscale!patterns!has!greatly! improved!since!the!first!generation!of!models,! their!spatial!resolution!is!still!  92 not! sufficient! to! resolve! local!orographic! effects! and! small! scale!physical!processes.!This!has!motivated! the!use!of!downscaling!methods,!both!dynamical!and!statistical,!for!bridging!the!gap!in!scales!between!GCMs!and!station!observations!(Wigley!et!al.!1990).!!!As! surface!wind! speed! variability! cannot! be! resolved! by! current! generation! GCMs,!downscaling!techniques!are!needed!to!generate!finer!scale!projections!of!near!surface!wind!climatologies!(Salameh!et!al.!2008).!Statistical!downscaling!techniques,!like!the!ones!used!in!this!work,!find!empirical!relationships!between!largeCscale!atmospheric!circulation! predictors! and! the! localCscale! variable! required! by! the! climate! change!impact!study!over!a!historical!period!(Huth!1999).! !Once! the!empirical! relationship!has! been! identified! from! observations! and! represented! by! a! statistical! model,!downscaled! projections! are! obtained! by! applying! the! model! to! GCM! outputs.! In!particular,! these! techniques!are!useful! for! the!wind!energy!sector,!as,!depending!on!the!region!under!consideration,!climate!changes!might!positively!or!negatively!impact!wind!energy!development!(Pryor!and!Barthelmie!2010).!!!!In!this!context,!future!wind!projections!are!becoming!increasingly!more!important!for!governments! and! stakeholders! and! relevant! studies! are! starting! to! emerge.! For!instance,! a! project! was! recently! completed! to! identify! downscaling! methods! that!could! be! used! to! assess! possible! changes! in! wind! statistics! in! British! Columbia,!  93 Canada! (van! der! Kamp! et! al.! 2010).! Pryor! and! Schoof! (2010)! concluded! that,! for!Europe,! the! influence!of! the!emission!scenario!on!projections!of!wind!climatologies!was! small! compared! to! natural! variability! and! other! sources! of! uncertainty! in! the!downscaled!projections.!Additionally,!Pryor!and!Barthelmie!(2010)!studied!the!wind!resources!over!northern!Europe,! concluding! that! there!was! insufficient! evidence! to!suggest! that! climate! changes! could! jeopardize! the! exploitation! of! wind! energy.!However,!they!also!suggested!a!need!to!improve!the!confidence!in!their!projections.!!Classical!statistical!downscaling!experiments!setups!are!unable!to!infer!which!model!performs!best!in!a!future!climate!change!scenario,!as!one!cannot!know!what!the!true!change! in! the! variable! of! interest! will! be! due! to! the! fact! that! the! future! is! as! yet!unobserved.!Additionally,! the!ability!of!models!to!reproduce!historical!climatologies!does! not! imply! they! will! be! able! to! accurately! simulate! future! climate! conditions!(Wilby! and! Wigley! 1997a).! Furthermore,! as! changes! in! the! intensity! and/or!frequency! of! extreme! events! are! one! of! the!major! concerns!with! regard! to! climate!change!(Clausen!et!al.!2010),!!the!study!of!characteristics!such!as!extreme!values!and!probability! distributions! is! becoming! essential! for! understanding! how! models!replicate! the! full! range! of! local! scale! characteristics.! ! Comparisons! between!downscaling!methods!have,!however,!traditionally!been!carried!out!only!in!terms!of!correlations! (Chu! et! al.! 2010),! average! errors! (Fasbender! and! Ouarda! 2010),! or!  94 similar!metrics! (Fowler! et! al.! 2007)! between! the! observations! and! the! downscaled!values.!!To!partially!address!some!of!these!limitations,!Vrac!et!al.!(2007b)!proposed!a!general!validation! method! which! uses! Regional! Climate! Model! (RCM)! outputs! as! pseudoCobservations! in!place!of! observed!historical!weather! station!data.!As!RCMs!outputs!are! driven! by! boundary! conditions! from! GCM! simulations,! dynamicallyCconsistent,!higherCresolution! climate! series! can! be! obtained! for! both! historical! periods! and!future! climate! change! projections.! As! a! result,! the! pseudoCobservation! approach!means! that! statistical! downscaling! models! can! be! validated! in! terms! of! their!performance!in!a!future!climate!change!context.!!!As!the!correct!simulation!of!both!weather!and!climate!conditions!is!required!for!the!assessment!of!wind!energy!potential,! the!present!work!extends! the!aforementioned!approach! and! introduces! a! new! wind! downscaling! evaluation! methodology! which!compares! historical! and! future! pseudoCobservations! with! statistically! downscaled!values!in!terms!of!daily!variability!(e.g.!weather)!as!characterized!by!mean!absolute!error!(MAE),!and!in!terms!of!annual!climate!indices,!measured!here!by!the!proposed!WINDEX!(WIND!evaluation!of!EXtremes)!indices,!using!the!index!of!agreement!(IOA)!(Willmott!1981).!!!  95 Local!scale!pseudoCobservations!for!the!historical!(1970C1999,!20C3M!scenario)!and!future! periods! (2040C2069,! SRES!A2! scenario)!were! taken! from!Canadian!Regional!Climate! Model! (CRCM)! 4.2! simulations! using! boundary! conditions! supplied! by! the!Canadian! Global! Climate! Model! (CGCM)! 3.1! T47! run! number! 4.! Three! statistical!downscaling! methods,! one! linear! and! two! nonlinear,! each! using! different! sets! of!predictors,!were!trained!on!the!CGCM/CRCM!data.!The! first!method,!multiple! linear!regression! (LR),! is! based! on! a! standard! stepwise! linear!model;! the! second!method!uses!feedCforward!artificial!neural!networks!with!Bayesian!regularization!(ANN),!and!the! last! method! is! the! cumulative! distribution! function! transform! method! (CDFt)!introduced!by!Michelangeli!et!al!(2009).!!The!study!region!is!Haida!Guaii!in!northwest!British!Columbia,!Canada,!the!proposed!location!of!Canada?s!first!offshore!wind!energy!project.!Besides!providing!renewable!energy,!the!project!aims!to!displace!approximately!450,000!tonnes!of!greenhouse!gas!(GHG)!emissions!per!year!in!natural!gas!(Naikun!Inc.!2010).!! 4.2 Material and methods 4.2.1 Study area and data Haida!Gwaii! is!an!archipelago!of!over!150!islands! located!on!the!northwest!coast!of!British!Columbia,!Canada.!!Formerly!known!as!the!Queen!Charlotte!Islands,!they!are!nestled! below! the! Alaskan! Panhandle! and! separated! from! the!mainland! by! Hecate!  96 Strait.!The!islands!constitute!the!most!westerly!point!of!Canada!(Figure&18).!The!area!is!close! to! the! NaiKun! Offshore! Wind! Energy! Project,! a! proposed! 396Cmegawatt!offshore!wind!energy!plan,!located!in!Hecate!Strait,!between!Haida!Gwaii!and!Prince!Rupert.! This! area! has! some! of! the! strongest! and!most! consistent! winds! in! Canada!(Frequently! Asked! Questions! ! 2010).! ! The! hypothetical! turbine! (represented! by! a!circle!in!Figure&18)!is!located!west!of!Frederick!Island,!opposite!Hecate!Strait.!  Figure!18.!Study!area:!Haida!Guaii,!British!Columbia,!Canada.!Red!markers!represent!CGCM!grid!points.!The!red!diamond!corresponds!to!the!closest!grid!point,!or!pseudo!station!5!(ST5).!The!buoy!is!represented!with!a!blue!circle.!!  97 PseudoCobserved! daily! maximum! wind! speeds! (msC1)! from! the! 1970C1999! period!obtained!from!a!CRCM!grid!point!located!near!a!buoy!at!54?!N,!134.25?!W!were!used!as!predictand.!Predictors!were!obtained!from!9!CGCM!3.1!grid!points!near!the!buoy.!From!each!grid!point!daily!values!of!surface!pressure!(P)!(Pa)!and!zonal/meridional!surface! winds! (u,! v)! (m/s)! were! extracted! for! historical! (1970C1999)! and! future!(2040C2069)!time!periods!(Figure&19).!The!historical!RCM!simulation!is!forced!by!the!CGCM!3.1!20C3M!transient!simulation,!and!the!future!simulation!by!the!CGCM3.1!T47!SRES!A2!scenario.! Figure!19.!Statistical!downscaling!diagram.!The!upper!section!represents!the!coarse!resolution!CGCM!and!the!lower!one!the!CRCM!pseudo.!   98 The! historical! RCM! pseudoCobservations! were! compared! with! observations! from!1990C2011!of!the!West!Moresby!NOMAD!buoy!(52?31'12"!N!132?41'23"!W)!from!the!Canadian!Department!of!Fisheries!and!Oceans!(DFO)!to!verify!if!both!belonged!to!the!same!probability!distribution!family!and!to!assess!the!bias!of!the!RCM!simulation.!It!was!found!that!both!datasets!can!be!adequately!fit!by!Weibull!distributions,!and!that!the!CRCM!has!a!positive!mean!bias!of!2.35!m/s.!A!twoCsample!KolmogorovCSmirnov!test! rejected! the! hypothesis! that! the! pseudoCobservations! and! the! buoy! data! were!from!the!same!continuous!distribution!at!the!5!%!significance!level.!Visual!inspection!using!a!quantileCquantile!plot!!(Figure&20)!shows!that!the!main!differences!occur!on!the!lower!tail!of!the!distribution!as!less!days!with!zero!wind!speeds!are!simulated!by!the!RCM!than!are!observed!at!the!buoy.!! Figure!20.!Observed!wind!speed!quantiles!(msC1)!versus!RCM!simulated!wind!speed!quantiles!(msC1).!!Red!line!joins!the!25th!and!75th!percentiles!of!a!normal!distribution!line.!!   99 From! the! original! 27! predictors! (3! predictors! per! CGCM! grid! point),! 3! different!subsets!were!analyzed!(Table&6).!The!smallest!subset! includes! just!one!predictor,! the!scalar!wind!speed! from! the!CGCM!grid!point!nearest! to! the!buoy;! the!next! smallest!includes! the! P,! u,! and! v! predictors! from! the! nearest! CGCM! grid! point;! the! most!complicated!uses!all!27!predictors!from!the!nearest!9!CGCM!points.! Table!6.!Model!descriptions!and!predictor!sets!used.!Each!CGCM!grid!point!provides!u!and!v!wind!components!(U!and!V!respectively),!and!surface!air!pressure!(P).!!! ID! Type! Number!of!grid!points!used! Included!variables! Maximum!number!of!predictors!ANNall6 Nonlinear! 9! u,v!and!P! 27!ANNST56 1!(ST5)! u,v!and!P! 3!ANNPC6 9! Leading!principal!components! 5!LRall6 Linear! 9! u,v!and!P! 27!LRST56 1!(ST5)! u,v!and!P! 3!LRPC6 9! Leading!principal!components! 5!CDFt6 Probabilistic! 1! Wind!speed! 1! !The! description! of! the! predictor! sets! is! as! follows:! ALL! refers! to! all! 27! predictors!available! from! the! 9! grid! points,! ST5! refers! to! the! 3! predictors! from! the! buoy's!nearest! CGCM! point.! Finally,! PC! refers! to! the! leading! 5! combined! principal!components! of! the!ALL! subset! for! historical! and! future! CGCM!outputs! (Imbert! and!Benestad!2005).!Use! of! PCA! reduces!predictor!dimensionality! and! ensures! that! the!predictors!are!uncorrelated.!!  100 As! previous! studies! in! the! area! showed! that! downscaling! by! linear! regression! and!PCA!produced!mixed!results!(van!der!Kamp!et!al.!2010),!3!LR!models,!3!ANN!models,!and! a! probabilistic! downscaling! model! (CDFt)! were! used! to! downscale! daily!maximum!wind! speeds.! Identical! atmospheric! circulation! predictors!were! used! for!the! LR! and!ANN!models,!while! only!wind! speed! from! the! nearest! CGCM!point!was!used!with!the!CDFt!model.!! 4.3 Evaluation method !Inspired! by! the! work! of! Vrac! et! al.! (2007b)! on! validating! statistical! downscaling!methods! under! future! climate! change,! and! assuming! the! high! resolution! CRCM! 4.2!can!adequately!reconstruct!the!observed!wind!climatology!when!driven!by!the!CGCM!3.1! 20C3M! transient! run,! I! propose! an! evaluation! method! in! which! downscaling!models!are!assessed!in!terms!of!their!ability!to!reproduce!daily!variability!and!annual!climate!extremes!represented!by!the!WINDEX!indices.!Present!and!future!simulation!skills!will!be!shown!and!compared,!as!the!ability!to!reproduce!historical!climatologies!does! not! imply! an! ability! to! simulate! future! climate! conditions! (Wilby! and!Wigley!1997a).!!4.3.1 Daily variability evaluation Models! were! trained! and! evaluated! using! 6Cfold! crossCvalidation! on! data! from! the!1970C1999!historical!period!(Bishop!2006).!Data!were!split!into!six!contiguous!5Cyear!  101 segments.!Models!were!trained!on!4!of!the!6!segments,!validated!on!one!of!them,!and!model!predictions!made!on!the!remaining!test!segment.!This!procedure!was!repeated!until! predictions! had! been!made! on! all! years! of! data.!Model! performance! statistics!were!then!computed!on!the!concatenated!test!segments.!!To!determine!how!well! the!models! represented!daily! variability! over! the!historical!period,!MAEs!between!downscaled!daily!winds!and!CRCM!pseudoCobservations!were!calculated.! Evaluation! in! terms! of! annual! climate! indices! involves! using! the! daily!downscaled! series! to! calculate! the!WINDEX! indices! shown! in! Table&7! for! each! year.!The!index!of!agreement!(IOA)!between!each!of!the!annual!downscaled!indices!and!the!annual!observed!indices!! IOA = 1 ? [?i ?fi ? gi ??]  /  [?i (? fi - ? ? + ?gi - ??) ?] ,  (4)  was!calculated,!where fi and gi are!the!downscaled!and!observed!values,!respectively, ? the!observed!mean,!and ? can!be!1!or!2!(Willmott!1981). I use ? = 1 as! it! is!more!appropriate!and!consistent!with!the!MAE.!Values!of!IOA!range!between!0!and!1!with!larger!values!indicating!better!forecast!performance.  4.3.2 Climate of extremes evaluation: The WINDEX indices The!design!and!operation!of!a!wind!farm!requires!one!to!consider!different!threshold!wind!speeds!relevant!to!turbine!operation,!for!example!the!cutCout!speed!(maximum!  102 speed! of! operation),! rated! speed! (minimum! wind! speed! that! generates! maximum!power),!and!the!cutCin!speed!(minimum!wind!speed!required!to!generate!power),!as!well! as! factors! such! as! the! number! of! days! that! wind! turbines! cannot! operate!continuously! or! the! number! of! days! that! turbines! can! generate! maximum! power.!Similarly,! the! analysis! of! extreme! events! is! relevant! for! turbine! designers,! and! the!study! of! wind! frequencies! and! magnitudes! is! relevant! for! wind! farm! operation!(Clausen!et!al.!2010).!!!For!these!reasons!the!WINDEX!indices!shown!in!Table&7!are!proposed!to!facilitate!the!evaluation! of! downscaling! models! in! terms! that! are! relevant! to! the! wind! energy!sector.!W10!and!W90!provide!information!about!the!magnitude!of!the!extreme!wind!events;! R! is! important! to! determine! the! number! of! days! generating! the!maximum!power! per! year;! CI! and! CO! determine! the! nonCoperating! number! of! days! per! year!caused!by!low!and!high!wind!speeds;!and!NODI!is!defined!as!the!total!number!of!nonCoperating!days!per!year.!The!first!two!indices!are!defined!in!terms!of!quantiles!such!that!fixed!thresholds!are!not!needed;!an!extreme!wind!event!is!therefore!site!specific.!The! other! indices! are!wind! turbine! specific! and! depend! on! their! power! generation!curve!characteristics.!!For!the!present!study,!3!msC1,!12!msC1!and!20!msC1!are!used!in!the!WINDEX!indices!as!the!cutCin!(CI),!rated!(R),!and!cutCout!(CO)!wind!speeds!of!a!prototypical!wind!turbine.!I!acknowledge!that!the!turbines!are!designed!to!stop!their!operation!while!the!wind!  103 speeds!are!above!their!specifications,!and!that!the!daily!maximum!wind!speeds!used!in!this!study!are!unlikely!to!be!sustained!for!24Chours.!Nevertheless,!for!the!purpose!of!this!research!I!consider!the!daily!maximum!speeds!characteristic!of!that!day.!!!! Table!7.!WINDEX!annual!indices.!! , Wind!Indices! Abbreviation!!1! 10th!percentile!of!!wind! W10!2! 90th!percentile!of!wind!!! W90!3! Number!of!days!!with!wind!speeds!>!RATED!speed!! R!4! Number!of!!days!with!wind!speeds!<!CUT!IN!speed! CI!5! Number!of!days!with!wind!speeds!>!CUT!OUT!speed! CO!6! NonCoperating!duration!index!(Total!number!of!nonCoperating!days)! NODI!!4.4 Theory,and,calculation,4.4.1 Downscaling,models,In!addition!to!the!CDFCtransform!method!introduced!by!Michelangeli!et!al.!(2009),!the!present! work! includes! two! different! regressionCbased! downscaling! methods! (i)!multiple! linear! regression! (LR)! based! on! stepwise! variable! selection;! and! (ii)!Bayesian! neural! networks! (ANN).! In! general,! regression! models! represent! time!invariant! linear! or! nonlinear! relationships! between! predictors! and! predictands!(Fowler!et!al.!2007).!  104 4.4.1.1 Linear,regression,(LR),Based!on! the! subsets! defined! in!Table&6,! downscaling!methods!using!multiple! linear!regression! and! stepwise! (SW)! variable! selection! were! implemented! to! find! linear!relationships!between!predictors!and!predictands.!!!In!general,!SW!regression!is!a!systematic!method!for!adding!and!removing!predictors!from!a!multiple!linear!regression!model!(Darlington!1990).!An!initial!model!is!created!at!the!first!iteration!and!then!the!pCvalue!of!an!FCstatistic!is!computed!to!test!models!with!and!without!a!potential!predictor.!For!both!cases!the!null!hypothesis!is!that!the!predictor!to!be!added!or!removed!has!a!zero!coefficient!(Hill!and!Lewicki!2006).!The!maximum! pCvalue! for! a! predictor! to! be! added! and! the! minimum! pCvalue! for! a!predictor!to!be!removed!were!set!to!0.05!and!0.10!respectively.!!4.4.1.2 Multi,layer,perceptron,(MLP),ANN,MultiClayer! perceptron! (MLP)! ANNs! can! be! used! to! find! nonlinear!predictor/predictand! relationships.! The! goal! of! the! ANN! is! to! minimize! a! leastCsquares! cost! function! between! predicted! and! observed! values! of! the! predictand!variable.!A!MLP!ANN!is!composed!of!an!input!layer,!any!number!of!hidden!layers,!and!an!output!layer!of!neurons.!MLP!ANNs!are!reported!to!give!similar!results!compared!to! multiple! regression! downscaling! methods! for! temperature! and! precipitation!  105 (Schoof! and! Pryor! 2001),! and! have! been! used! by! GarciaCBustamante! and! others!(GarciaCBustamante!et!al.!2004)!for!wind!speed!downscaling.!!In!the!current!study,!Bayesian!regularization!was!used!to!train!the!ANN!models.!This!regularization!was! introduced!by!MacKay! (1992),! and! gives! an! estimate! of! optimal!weight! penalty! parameters!without! the! need! for! validation! data! (Hsieh! 2009).! For!more! information! on! Bayesian! neural! networks! see! Bishop! (Bishop! 2006)! and!MacKay!(2003).!!A!model!ensemble!was!created!and!its!average!considered!to!be!the!final!ANN!output.!Ensemble!averaging!provides,!in!general,!average!expected!error!that!is!less!than!or!equal!to!those!from!the!individual!models!in!the!ensemble!(Hsieh!2009).!!In!this!case,!all!the!potential!ensemble!models!have!one!hidden!layer!with!a!different!number!of!hidden! neurons,!with! the! hyperbolic! tangent! activation! function!mapping! from! the!inputs!to!the!hidden!layer,!and!a!linear!function!mapping!from!the!hidden!layer!to!the!output! layer.! The!best! network! architectures!per! decade!were! chosen!using! a! twoCstep! procedure.! First,! I! selected! the! best! 20!models! based! on! their!mean! absolute!error!(MAE)!between!the!observations!and!the!downscaled!sets,!then!I!selected!from!this! subset! the! best! 10! models! based! on! their! correlation! coefficient! between! the!observed! and! downscaled! values.! This! procedure!was! repeated! on! each! validation!dataset,!as!outlined!in!subsection!3.!  106 4.4.1.3 CDFFtransform,method,The!CDFCtransform! (CDFt)!method,! as! explained!by!Michelangeli! et! al.! (2009)! is! an!extension! of! the! quantilesCmatching! approach! (Panofsky! and! Brier! 1958).! The!method! assumes! the! existence! of! a! nonlinear! transformation! T! that! allows! us! to!translate!a!GCM!predictor?s!cumulative!distribution!function!(CDF)!into!the!CDF!of!the!local!scale!variable!or!predictand.!For!the!present!study,!downscaled!time!series!were!generated!using!the!CDFt!implementation!described!in!Vrac!and!Michelangeli!(2009)!with!CGCM!wind!speeds!at!the!nearest!grid!point!as!the!lone!predictor.!4.4.2 Regression downscaling The!climatological!seasonal!mean!was!removed!from!the!predictors!(Table&6)!and!the!predictands,! and! standardized! anomalies! were! used! as! inputs! to! the! linear! and!nonlinear!regression!models.!The!downscaling!procedure!consists!of! three!different!steps:! 1)! train! the! regression! model! using! the! historical! synopticCscale! circulation!data!as!predictors!and!surface!observation!data!as!predictands;!2)!validate!the!model!from!step!1!using!independent!data;!and!3)!enter!GCM!predictors!from!future!climate!change! scenarios! into! the! trained! and! validated! regression! models! to! obtain!downscaled!wind!projections.!!!  107 4.5 Downscaling results and discussion 4.5.1 Evaluation results for historical pseudo-observations I! assessed! the!performance! in! terms!of!daily!variability!by!comparing!modeled!and!RCM!pseudoCobserved!wind!speed!anomalies!using!MAEs!over!the!1970C1999!period.!!Performance! in! terms! of! annual! climate! indices! was! measured! by! IOAs! between!modeled!and!observed!WINDEX!indices!for!the!same!period.!For!the!climate!indices,!the! climatological! seasonal! cycle!was! added!back! to! the!downscaled! series!prior! to!assessment!of!model!performance.!As!an!aggregated!measure!of!model!performance!for! the! annual! indices,! an! average! performance! statistic,! the!mean! IOA! over! the! 6!WINDEX! indices,! from!now!on! referred! to! as! the!WINDEX! average! IOA! (WAI),!was!also!calculated.!!Figure!21,!which!plots!daily!MAEs!against! annual!WAIs,! shows! that!ANNall!has! the!best!overall!performance!simulating!daily!variability!(minimum!MAE),!while!CDFt!has!the! best! performance! simulating! annual! climate! indices! ! (maximum! WAI).! In! the!latter! case,! note! that! CDFt! was! also! the! second! worst! model! in! terms! of! daily!variability.!Regarding!the!relative!performance!by!predictor!set!(see!Table&6),!models!using!all!predictors!presented!the!lowest!MAE!errors,!followed!by!those!that!used!PCs!and!ST5.!!  108 When!analysing!the!WAI,!the!ANN!models!generally!outperformed!the!corresponding!LR!models!(with!the!exception!of!ST5).! !In!comparison!to!LRPC,!ANNPC!models!that!consider! nonlinear! relationships! between! the! leading! principal! components!improved! the! simulation! of! the! historical! period! climate! indices.! However,! as!mentioned! earlier,! the! probabilistic! CDFt! method! using! only! one! predictor!outperformed! all! other! models.! This! behaviour! can! be! explained! by! the! method's!consistency!in!simulating!all!6!indices,!whereas!the!ANN!model!showed!reduced!skill!in!simulating!CI,!CO!and!NODI.!That!said,!when!analysing!the!remaining!indices,! the!ANN!methods!performed!better!than!CDFt!for!the!W90,!W10!and!R!indices,!while!the!LR!methods!outperformed!the!other!models!in!simulating!the!CO!index.!!As!no!model!beats! the!others! in!simulating!all! indices,! it! is!evident! that! the!user!has! to!choose!a!model! based! on! their! specific! needs.! Furthermore,! this! may! entail! a! tradeCoff! of!"weather"!for!"climate"!simulation!skill.!   109   Figure! 21.! Statistically! downscaled! wind! speed! MAEs! versus! the! WINDEX! indices!average!IOA!for!the!historical!period.! !To! illustrate! in!more! detail,! Figure&22! compares! each! index! IOA! independently.! The!W90! analysis! shows!ANNall! and!ANNPC! outperforming! the! other!models!with! IOA!values!0.2!units!higher! than! the! fourth!best!model! (ANNST5),! and!more! than! twice!that! of! the! LR! IOAs.! A! similar! pattern! is! shown! for! W10,! although! the! gap! in!performance!between!the!nonlinear!and! linear!models! is!smaller! than! for!W90.!For!  110 the!linear!models,!IOAs!for!W90!and!W10!are!consistently!close!to!0.2!regardless!of!the!predictors!used.!This!implies!that!any!improvements!in!skill!above!this!value!are!caused!by!nonlinearities.!!When!analyzing! the!number!of! days!with!wind! speeds! above! the! cutCin! speed! (CI),!CDFt!clearly!outperformed!the!other!models.!The!ANN!based!models!performed!less!well! on! this! index! relative! to! W90! and! W10.! This! shows! the! ANN! models! have!difficulty! characterizing! the! number! of! events! under! lower! thresholds.! Similar!behaviour!was!shown!by!the!LR!methods,!although!ANNall!and!ANNPC!outscored!the!linear!methods.!!Regarding! the! total! number! of! days! per! year!with! speeds! above! the! cutCout! speed!(CO)! (Figure& 23),! the! ANN! models! consistently! overestimated! the! number! of!exceedances!during!the!historical!period.!Good!performance!by!the!CDFt!method!can!be!attributed!to!the!fact!that!the!modeled!index!values!were!in!the!same!range!as!the!observations!with!no!consistent!under/over!prediction.!In!terms!of!IOA,!the!3!linear!models! outperformed! the! other!models,! despite! a!marked! tendency! towards!under!prediction.! ! The! larger! positive! bias! for! the! ANN!models! and,! in! the! case! of! CDFt,!phase!errors!resulted!in!lower!IOA!values!relative!to!the!linear!models.!   111 &Figure!22!WINDEX!indices!IOAs!for!the!historical!period.!!!For!the!number!of!days!above!the!rated!speed!(R),!the!LR!models!performed!worst,!while! the! ANN! and! CDFt! models! performed! better! than! their! linear! counterparts.!!When!looking!at!the!total!number!of!nonCoperating!days!(NODI),!model!performance!is!comparable!to!that!for!the!CI!index.!This!behaviour!is!expected!as!NODI!is!simply!the!sum!of!CI!and!CO!and!the!climatological!frequency!of!CI!days!far!exceeds!that!of!CO!days!!(~10!and!~2!days!per!year!respectively).!!  112 4.5.2 Evaluation,results,in,future,data,Following!the!historical!evaluation,!models!were!also!validated!against!future!CRCM!pseudoCobservations.! In! this! case,! the!A2! simulation! for! the!2040C2069!period!was!used! as! the! observational! reference.! Results,! which! are! shown! in! Figure&24,! suggest!that! model! performance! in! the! future! climate! scenario! was! similar! to! that! for! the!historical!training!period.!Only!one!model,!ANNPC,!exhibited!a!reduction!in!WINDEX!IOA! between! the! historical! and! future! periods.! The! assumption! that! statistical!downscaling! relationships! are! stationary,! a! necessary! condition! for! the! validity! of!climate!projections!using!statistical!downscaling!techniques,!generally!holds!up!well.!!In!terms!of!specific!results,!again,!ANNall!outperformed!the!other!models!in!terms!of!daily! variability! while! CDFt! did! the! same! in! terms! of! the! WINDEX! indices.!Additionally,! there! is!not!a!clear!advantage!of! the!ANNPC!and!ANNST5!models!over!the!corresponding!LR!models.!  113 !Figure'23'WINDEX'CO'index.'The'solid'black'lines'show'ANNall'(top),'LRall'(center)'and'CDFt'(bottom),'and'the'dashed'red' lines' the' pseudoIobservations.' Horizontal' axes' correspond' to' the' 30' years' historical' period' (1970I1999).' The'vertical' axes' show' the' total' number' of' days' per' year' with' wind' speed' above' the' cutIout' speed.  114 !Figure' 24' Statistically' downscaled' wind' speeds' MAEs' versus' the' WINDEX' indices'average'IOA'for'the'A2'scenario'run.'''Figure!25'shows'the'WINDEX'IOAs'for'the'future'period.'For'W90'and'R,'ANNall'was'the'best'model,'followed'by'CDFt,'the'ANNST5,'the'LR'models,'and'ANNPC.'With'the'exception' of' CDFt' and'ANNPC,'models' show' a' better' performance' simulating'W90'than'W10.' Considerable' differences' between' the' LR' and' ANN'models' were' found,'especially' for' W10.' For' CI,' as' observed' in' the' historical' period,' the' CDFt' model'performed' better' than' the' other' 6' models,' performance' of' the' ANN' models' was'uneven,'with'the'LR'models'outscoring'ANNST5.'  115 Regarding'CO,' the'CDFt'model'presented' the'biggest'drop' in'skill' relative' to' the'historical'period,'while'performance'of'ANNPC'was'comparable'to'that'obtained'by'the'linear'methods.''When'analyzing'the'R'index,'the'CDFt,'ANNall'and'ANNST5'models'outperformed'the' 3' linear' models.' ANNall' obtained' the' highest' IOA' (0.65).' It' is' worth'mentioning' that' this'model' outscored' the' second' best' by'more' than' 0.15' units,'and'its'linear'counterpart'by'0.40'units.'As'in'the'historical'period,'the'CDFt'model'performance' was' close' to' 0.50.' Finally,' regarding' the' NODI' index,' as' in' the'historical'period,'model'performance'was'comparable'to'CI.''Tables' 8' and' 9' show' model' performance' statistics' and' rankings' respectively.'Table'8'includes'''WINDEX'average'IOA'and'the'daily'MAE'information'presented'previously,'but'also'includes'correlation'coefficients'calculated'from'the'daily'time'series.' Correlation' was' calculated' as' an' indicator' of' the' strength' of' linear'dependence'exclusive'of'systematic'biases.''''''''''  116 'Table' 8.' ' Model' MAEs,' correlation' coefficients' and'WAIs' for' the' historical' and'future'periods.'Model' Model' Historical'A2' A2'21ST'century'Number' ID' MAE' Correlation'Coefficient'WINDEX'Avearge'IOA'(WAI)' MAE'Correlation'Coefficient'WINDEX'Average'IOA'(WAI)'1' ANNall' 1.68' 0.83' 0.40' 1.64' 0.84' 0.45'2' ANNST5' 2.17' 0.75' 0.27' 2.20' 0.76' 0.24'3' ANNPC' 2.08' 0.77' 0.44' 2.12' 0.79' 0.26'4' LRall' 1.94' 0.73' 0.29' 1.92' 0.75' 0.27'5' LRST5' 2.13' 0.67' 0.26' 2.11' 0.69' 0.25'6' LRPC' 2.08' 0.69' 0.29' 2.06' 0.71' 0.26'7' CDFt' 2.17' 0.69' 0.48' 1.68' 0.71' 0.51'''!Figure'25'WINDEX'indices'IOAs'for'the'A2'scenario'run.'''  117 The'results'show'that'while'the'MAEs'and'correlation'coefficients'were'consistent'between' periods,' the'WINDEX' IOAs' indicate' that' the' models' using' the' leading'principal'components'as'predictors'drop' their'skill' for' the' future'period.'This' is'particularly'the'case'for'ANNPC.''Table'9'shows'the'average'ranking'of'models'over'both'the'historical'and'future'periods.'The'final'results'indicate'that'ANNall' is'the'best'model'in'terms'of'daily'variability'MAEs'and'correlation'coefficients,'and'is'only'outscored'by'CDFt'when'modeling'the'WINDEX'indices.'In'general,'most'models'traded'annual'climate'for'daily' weather' simulation' skill,' or' vice' versa,' and' the' nonlinear' models'outperformed'the'linear'models.'This'last'fact'indicates'that'wind'farm'design'can'potentially'be'improved'by'accounting'for'nonlinear'downscaling'relationships.''In'general,'ANNall'outscored' the'other'models'when'considering'historical'daily'variability'(MAE'and'correlation'coefficient),'while'CDFt'obtained'higher'WINDEX'IOAs' than'ANNall' for' the'historical' and' future'periods.'On' the'other'hand,' even'though'CDFt'outscored'models'2'to'6'in'terms'of'future'period'MAE,'skill'in'terms'of' the' correlation' coefficient'was' low,' indicating' relatively' unbiased' predictions'but'large'phase'errors.'''''  118 Table'9.'Models''average'ranking'based'on'daily'variability'and'WINDEX' indices'performance.' Model' Model'ID' Average'ranking'Number' MAE' Correlation''Coefficient' WINDEX''IOA'1' ANNall' 1' 1' 2'2' ANNST5' 7' 3' 6'3' ANNPC' 5' 2' 3'4' LRall' 3' 4' 4'5' LRST5' 6' 7' 7'6' LRPC' 4' 5' 5'7' CDFt' 2' 6' 1'''Furthermore,'it'is'shown'that'the'models'using'the'leading'principal'components'performed' well' during' the' historical' period,' but' their' climate' simulation' skills'dropped'significantly'for'the'future'period.'The'results'show'that'calculating'the'principal'components'from'a'unified'matrix'containing'historical'and'future'GCM'outputs'does'not'guarantee'the'success'of'the'model'downscaling'future'data,'at'least'in'terms'of'the'WINDEX'indices.''4.6 Conclusions and recommendations As'classical'statistical'downscaling'experiments'are'unable' to' infer'which'model'performs'better' in'a' future'climate'change'scenario,' the' true'change' in' terms'of'daily' and' climate' like' variability' of' a' variable' of' interest' cannot' be' known.' The'present'study'addresses'those'challenges'by'extending'the'approach'of'Vrac'et'al.'  119 (2007)'by'comparing'present'and'future'daily'variability'in'terms'of'MAE'and'the'ability'of'models'to'replicate'historical'and'future'climate'indices,'using'IOAs.'For'the' study' of' these' features,' the' WINDEX' indices' were' introduced.' This' set' of'indices' include' variables' of' interest' when' operating' wind' plants,' such' as' the'number' of' days' below' the' cut\in' speed' (CI),' number' of' days' above' the' cut\out'speed'(CO),'total'number'of'non\operating'days'(NODI),'days'operating'above'the'rated' speed' (R),' and' 10th' and' 90th' percentiles' of' the' daily' wind' distributions'(W10'and'W90,'respectively).'The'evaluation'methodology'also'suggests'that'the'final'users'should'select'downscaling'models'based'on' their'particular'needs,'as'the' best'models' for' representing' day\by\day' variability' need' not' necessarily' be'the'best'at'simulating'climate'variables'such'as'W90'or'R.'For'the'present'study,'3'ms\1,'12'ms\1'and'20'ms\1'represented'the'cut\in,'rated,'and'cut\out'velocities'of'a'prototypical'wind'turbine.''The' results' presented' in' this' study' were' derived' for' Haida' Guaii,' Canada.'Nevertheless,' the' proposed' evaluation' procedure' can' be' extended' to' other'regions' and' other' linear' and' nonlinear' models' commonly' used' for' wind'downscaling.' ' Furthermore,' the'present' study' corroborated' the' flexibility' of' the'ANN'models'when'used'to'statistically'downscale'synoptic'scale'variables'to'local'(weather' station)' scale,' and' compared' them' with' traditional' linear' regression'models' and' a' probabilistic' model' using' the' CDF\transform' method.' With' the'exception' of' ANNPC,' the' ANN' downscaling' techniques' reliably' simulated'historical' and' future'wind' climatologies' under' the' A2' emission' scenario.' MAEs'  120 and'IOAs'were'consistent'between'the'two'periods.'Overall,'ANNall'obtained'the'best' results' simulating' daily' variability,' and' CDFt' obtained' the' best' results'simulating'climate'variability'for'the'historical'and'future'periods.'The'remaining'models'traded'"climate"'for'"weather"'performance'or'vice'versa.''Even'though'the'superiority'of'the'ANN'over'the'linear'models'was'expected'given'the' non\linearity' of' the' process,' other' nonlinear'models' such' as' support' vector'machines' (Cristianini' and' Shawe\Taylor' 2000),' or' temporal' neural' networks'(Dibike'and'Coulibaly'2006)'could'potentially'outscore'the'three'nonlinear'models'implemented'here'in'terms'of'both'daily'variability'and'climate'indices.'Similarly,'because' of' the' underlying' distribution' of' the' wind,' preprocessing' techniques'involving' a' more' refined' nonlinear' transformation' of' the' predictand' (e.g.' Box\Cox)' could' potentially' reduce' the' biases' generated' by' possible' violations' of' the'normality'of'the'error'assumption'(Acock'2010).''Concerning' the' predictor' selection,' the' models' using' all' the' predictors'outperformed'the'ones'using'the'leading'principal'components,'or'the'variables'of'the' closest' grid'point' to' the'buoy,' and' considerable'differences' simulating'daily'variability'and'the'WINDEX'indices'were'found.'This'corroborates'the'fact'that'a'proper' predictor' selection' procedure' is' as' relevant' as' the' method' used' to'downscale.' Regarding' the' nonlinear' model' using' the' leading' principal'components'as'predictors,'even'though'the'results'show'that'their'MAEs'and'their'correlation' coefficients' were' consistent' between' periods,' their' WINDEX' IOAs'  121 indicate'that'the'model's'ability'to'replicate'future'pseudo\observations'was'very'limited.''In' general,' the' linear' methods' over' predicted' the' lower' values' and' under'predicted'the'higher'percentiles;'therefore,'the'models'were'not'able'to'reproduce'correctly'most'of'the'climate'like'indices.'Nonlinear'ANN'models'marginally'under'predicted'W10,'and'slightly'over'predicted'W90,'CO'and'NODI,'while'showing'an'above' average' performance' simulating' the' number' of' days' operating' at' rated'speed.'On'the'other'hand,'the'CDFt'method'matched'the'multiyear'average'of'W10'and'W90'perfectly'due'to'its'quantile\matching'origin.'No'systematic'over/under\predictions'were'found'when'CDFt'was'used'to'simulate'the'6'indices.''' '  122 Chapter(5'5 CONCLUSION 'In' this' thesis'my'aim'was' to'build'an'understanding'of'how' the' time\invariance'assumption,' common' to' all' statistical' downscaling' methods' affects' the' model'outputs' from'different' statistical'downscaling'methods.' 'More' specifically,' I'was'interested' in' knowing' if' the' skills' of' different' climate' indices' (i.e.' STARDEX'indices,'CLIMDEX'indices,'and'WINDEX'indices)'varied'with'time,'and'to'know'if'present'simulation'skills'would'be'kept'in'the'future.'''The' main' goal' was' to' develop' different' linear' and' nonlinear' regression' and'classification' methods' and' to' use' them' to' statistically' downscale' the' coarse'resolution'Canadian'GCM'3.1'outputs'into'a'finer'local'scale.'Particularly,'by'using'the' Canadian'RCM'4.2' output' as' pseudo\observations' I'was' able' to' validate' the'statistically'downscaled'time'series'against'not'only'the'historical'period'outputs'but'also'against'the'CRCM'future'projections.'The'downscaled'variables'were'daily'precipitation,'wind'speed,'and'maximum'and'minimum'temperatures.''The' methodology' herein' is' certainly' not' the' only' means' of' downscaling'statistically' the' aforementioned' climate' variables.' A' plethora' of' methods' for'downscaling' temperature,' precipitation' and' to' a' lesser' extent'wind' speed' have'been' developed' in' the' last' few' decades.' As'mentioned' by' Vimont' et' al.' (2010)'  123 these' methods' include' different' regression' models,' weather' generators' (e.g.'Markov'models),'weather'classification'techniques'and'analogue'methods.''The'present'work' contributed' to' the'understanding'of' the' generally'overlooked'assumption' of' time' invariance,' inherent' to'most' statistical' downscaling' studies.'Time\invariance' assumes' that' the' present' day' statistical' relationships' between'the' coarse' resolution' atmospheric' predictors' and' the' predictand(s)'will' remain'constant'in'the'future,'thus'valid'under'possible'climate'change'scenarios'(Wilby'et' al.' 1998).' Especially,' the' results' suggest' that' future' statistical' downscaling'analyses' should' assume' non\stationary' relationships,' and' only' use' classical'approaches'when'the'time'invariance'of'the'relationship'between'predictors'and'predictand(s)'can'be'verified.''The'results'shown'in'this'thesis'are'for'models'trained'with'the'coarse'resolution'Canadian' GCM' 3.1' and' the' Canadian' RCM' 4.2' as' pseudo\observations;' it' is'unknown' if' the' results' are' robust' against' the' use' of' alternative' GCMs' as' it' is'unlikely' to' obtain' similar' results' when' using' outputs' from' different' coarser'resolution' GCMs' as' predictors.' Similarly,' as' differing' degrees' of' future' climate'change' may' affect' not' only' temperature' but' also' precipitation' and' extremes'(Tebaldi'et'al.'2006)'it'is'uncertain'how'the'models'could'behave'when'driven'by'different'climate'change'scenarios'(e.g.'A1B);'but'as'mentioned'in'the'latest'report'of' the'National' Fish,'Wildlife'&'Plants'Climate'Adaptation'Partnership' (National'Fish,'Wildlife'&'Plants'Climate'Adaptation'Strategy''2012),'?this'uncertainty'is'not'  124 a' reason' for' inaction,' but' rather' a' reason' for' prudent' action:' using' the' best'available'information'while'striving'to'improve'our'understanding'over'time?.'''Consequently,'as'the'driving'GCM'is'the'main'contributor'of'uncertainty'in'terms'of' the' simulated' climate' (Music' and' Sykes' 2011),' and' because' the' driving' GCM'may'significantly'impact'the'simulated'regional'climate'(e.g.'Rowell'2006;'D?qu?'et' al.,' 2007;' De' Elia' and' Cot?,' 2010),' ' there' is' a' need' for' an' improved'understanding' of' sub\grid' processes' in' GCMs' and' for' comparative' studies'with'appropriate' downscaling' models.' Nevertheless,' quoting' Maurer' and' Hidalgo'(2008),' ' ?the'choice'of'most'appropriate'downscaling' technique'depends' in'part'on' the'variables,'seasons,'and'regions'of' interest?,' thus'determining'appropriate'downscaling' models' for' a' study' region' (as' done' in' this' study)' is' of' uttermost'importance.''As' mentioned' earlier,' the' present' study' downscaled' CGCM' outputs' to' gridded'data,' represented' by' the' CRCM' outputs.' Previous' studies' downscaling' coarse'resolution'outputs'to'gridded'scale'include'Maurer'and'Hidalgo'(2008),'Vrac'et'al.'(2007b)' and' more' recently' Maurer' et' al.' (2013).' However,' there' are' several'caveats'when'downscaling' to'gridded'data,' including:'1)' the'gridded'predictand'represents'an'area'average'not'point'measurements;'2)'the'variance'of'a'variable'averaged'over'a'large'area'is'expected'to'be'smaller'than'the'variance'of'the'same'variable' at' a' particular' weather' station/point,' and' 3)' the' wet' spells' calculated'from' the' gridded' data' likely' last' longer' than' the' observed' ones.' Nonetheless,'  125 according'to'Bourdages'and'Huard'(2010)'the'precipitation,'values'generated'by'the'CRCM'4.2' (driven'by' the'CGCM3)'are'very'close' to' the'observation'datasets,'although'the'simulated'temperatures'are'lower'than'the'observed'values.'''As'classical'statistical'downscaling'experiments'are'unable' to' infer'which'model'performs'better' in'a' future'climate'change'scenario,' the' true'change' in' terms'of'daily' and' climate' like' variability' of' a' variable' of' interest' cannot' be' known.'Additionally,' previous' studies' of' the' stationarity' assumption' usually' divide' the'historical'period'data'in'two'folds,'and'then'use'a'?colder?'fold'(i.e.'the'fold'with'lower' mean' temperatures)' for' training' the' downscaling' model' and' the' second'fold' for'validation,' thus' testing' the'stationarity'assumption' in'a' slowly'warming'climate' context' (Benestad' et' al.' 2008b).' However' as' the' observed' changes' in'mean'temperature'during'the'past'50'years'(i.e.'historical'period)'are'milder'than'the' projected' temperature' change' for' the' 21st' century,' this' approach' is' not'recommended.' In'particular,'as'Barrow'et'al.' (2004)' indicated'that'by'the'2080s'the'winters' in' southern'and'western'Canada'will' likely'be'between'6?C'and'8?C'warmer' than' at' present' (following' the' Special' Report' on' Emissions' Scenarios'(SRES)'A2'scenario'(IPCC'2000)),'I'consider'this'classical'approach'not'optimal'for'the'study'region.'The'present'study,'although'limited'by'the'use'of'gridded'data'as'pseudo\observations,' allows' the' comparison' of' statistically' and' dynamically'downscaled'time\series'even'when'the'projected'temperature'changes'are'larger'than'the'ones'observed'during'the'training'period.''  126 Here'I'provide'a'summary'of'the'main'findings'from'each'of'the'three'research'chapters'and'discuss'possible'alternatives'for'future'work.''5.1 Summary When'downscaling' daily'maximum' and'minimum' temperatures,' and' comparing'them'to'the'CRCM'pseudo\observations'(Chapter'2)'I'found'that'choosing'the'best'model' based' on' performance' in' the' historical' period' could' result' in' having'contrasting' results' for' the' future' period' (i.e.' breaking' the' time' invariance'assumption).' 'This'result'partially'corroborates'Charles'et'al.' (1999)'hypothesis,'as'the'evaluation'of'a'downscaling'technique'using'historical'data'does'not'imply'it' will' be' equally' valid' under' future' climate' conditions.' In' particular,' using' SD'models' with' greater' ability' to' model' complicated' relations,' by' having' either'nonlinear'capability'or'additional'non\temperature'predictors,'seemed'to'alleviate'the'drop'in'performance'found'in'future'climate'conditions.'''Additionally,' the'present' study'shows' for' the' first' time' the'historical'and' future'performances' simulating' weather' and' climate' of' extremes' of' two' popular'statistical' downscaling' methods:' stepwise' multiple' linear' regression' and'nonlinear' Bayesian' neural' network' using' three' different' predictor' sets,' for'southern'Ontario'and'Quebec.' 'The'statistically'downscaled'data'were'compared'against' a' regional' climate'model?s' daily'maximum' and'minimum' temperatures,'used'here'as'past'pseudo\observations'and'proxies'for'future'conditions.''  127 Specifically,'it'is'shown'in'Chapter'2'that'not'all'SD'models'satisfy'the'stationarity'assumption,' and' that' the' model' skills' simulating' daily' variability' and' climate'indices'may'vary' from'present' to' future' climate'depending'on' the'predictor' set'used.'Moreover,' the'best'model'simulating'both'historical'?weather?'and'climate'indices'(i.e.'LRT)'was'not'the'best'one'in'the'future'period'(in'fact'this'model'was'the' worst).' This' finding' has' significant' repercussions,' as' one' of' the' statistical'downscaling'paradigms'is'to'assume'that'present'simulation'skills'will'be'kept'in'the' future,' a' decision'maker,' by' assuming' that' validating' the'model' in' the' past'would'suffice,'could'end'up'selecting'a'model'with'poor'future'performance.'''Previous' studies' found' using' circulation' and' temperature' variables' superior' to'using' only' single' predictors'when' downscaling' temperature' (Wilby' et' al.' 1998;'Huth'1999,'2002,'2003;'Gachon'2008,'2005;'Hessami'et'al.'2008).''Here'I'further'noted' that' when' downscaling' temperature' SD' models' with' additional' non\temperature' predictors' seemed' to' suffer' less' performance' deterioration' when'shifting' from' present' to' future' climate' than' models' with' only' temperature'predictors.'Moreover,'nonlinear'BNN'models'seemed'to'deteriorate'less'than'MLR'models'when'shifting'from'present'to'future'climate.'Hence,'as'mentioned'earlier,'using'models'with'greater'ability'to'model'complicated'relations,'by'having'either'nonlinear'capability'or'additional'non\temperature'predictors,'seemed'to'alleviate'the'drop'in'performance'found'in'future'climate'conditions.''  128 Although'the'performance'differences'between'the'six'models'used'in'Chapter'2'are'small,'the'BNNPC'model'is'generally'best'in'terms'of'both'weather'and'climate'indices.'This'contrasts'Appendix'A?s'results,'where'the'BNNall'model'was'the'best,'in' both'weather' and' climate.' This'may' be' explained' by' the' difference' between'observed' data' and' RCM'data,'where' the' RCM'data' had' less' variability' than' the'observations.' The' diminished' local' variability/signal' in' the' RCM' data' could'enhance'the'approach'using'PCs'as'predictors'(as'in'the'BNNPC'model),'since'PCs'are'best'for'capturing'larger'scale'signals.'The'diminished'local'variability/signal'could' also' lessen' the' advantage' of' nonlinear' SD'models' over' linear' models,' as'greater' advantage' of' the' nonlinear' BNN' models' over' MLR' models' in' climate'downscaling'was'found'in'Appendix'A'(using'real'observed'data)'than'in'Chapter'2'(using'RCM'data).'''Finally,' in'terms'of'climate'performance'during'the'historical'period,' the'models'using'only'temperature'predictors'(LRT'and'BNNT)'in'Chapter'2'performed'well,'in'stark'contrast'with'Appendix'A,'where'those'models'were'considerably'poorer'than'the'ones'using'more'predictors'(LRall'and'BNNall).'Again,'this'difference'may'be' due' to' the' fact' that' the' relationship' between' the' reanalysis' data' and' the'observed'data'is'more'complicated'than'that'between'the'GCM'data'and'the'RCM'data.''To' close' the' analysis' of' Chapter' 2,' future' studies' are' recommended' to' consider'the' influence' of' spatial\correlation' on' the' effective' number' of'weather' stations.''  129 Here' I' used' 10' neighbouring' RCM' grid' points' as' pseudo\observations' but' the'effect' of' spatial\correlation' considerably' reduced' the' effective' number' of'uncorrelated'stations'''When' downscaling' precipitation' occurrences' (Chapter' 3),' I' found' that' complex'nonlinear'models' like' artificial' neural' networks' and' ensembles' of' classification'trees'outperformed' linear'models' and' simpler'nonlinear'models' in' terms'of' the'Peirce'skill'score,'and'that'the'models?'performance'did'not'deteriorate'in'future'climate.'On'the'other'hand,'when'downscaling'precipitation'amounts'I'found'that'all' the' regression' models' showed' important' MAE' differences' between' the'historical'and'future'periods'(i.e.'the'future'MAE'is'~1'mm'day\1'higher'than'in'the'historical' period).' A' similar' conclusion' can' be' drawn' when' comparing' the'CLIMDEX'climate'indices'in'terms'of'IOAs,'as'different'indices'presented'changes'between'periods'from'negative'to'zero,'or'marginally'increased'or'decreased'their'IOA' values.' Under' these' conditions,' one' could' argue' that' the' precipitation'occurrence'process'is'time\invariant'for'the'southern'Ontario'and'Quebec'region,'but' that' the' predictor\predictand' relationships' related' to' the' precipitation'amounts'vary'with'time.' 'Overall,' it' is'worth'noting'that'smaller\scale'dynamical'processes'not'captured'at'the'scale'of'GCM'fields'may'influence'projected'changes'in'precipitation'over'this'region.''The' precipitation' occurrences' results' suggest' that' the' linear' discriminant'classification' method' used' by' Chen' et' al.' (2010)' ' might' not' be' the' most'  130 appropriate' (at' least' for' southern'Ontario'and'Quebec).' In' the' simulation'of' the'indices,'as'the'models'were'trained'on'daily'precipitation'and'not'on'the'indices'themselves,'the'indices?'performance'was'?more'independent?'and'?less'prone'to'artificial' skill' from'overfitting?'as'B?rger'et'al.' (2012)'mentioned' in' their'recent'paper.' In' particular,' the' occurrence' model' responsible' for' the' cwet' index'produced' simulated'distributions' that'were'not'different' from'observed'ones'at'the'5'%'significance'level'for'most'sites,'had'positive'IOA'during'both'periods'and'the' IOA' did' not' degrade' in' the' future' period.' Overall,' ANN\F' generally' showed'better' results' than' ARES\F' and' SWLR\F'when' simulating'most' of' the' historical'and'future'climate'indices.''When' statistically' downscaling' wind' speeds' from' Haida' Guaii' (Chapter' 4),' an'artificial' neural' network'model' using' predictors' from' the' 9' nearest' CGCM' grid'points'performed'best'in'terms'of'daily'variability,'as'indicated'by'mean'absolute'errors,' for' the' historical' and' future' periods,' whereas' the' best' performance' in'terms'of'annual'extremes,'as'indicated'by'indices'of'agreement'for'WINDEX,'was'obtained'by'a'variant'of'the'probabilistic'quantile\matching'method.''This'result'is'consistent'with'Brinkmann' (2002)'as' the'model'using'only' local'predictors'was'outscored'by'the'more'complex'models'using'information'from'other'grid'points.'The' results' also' suggest' that' the' best'models' for' representing' day\by\day'wind'variability'need'not'necessarily'be'the'best'at'simulating'specific' indices'of'wind'extremes' like' number' of' days' below' the' cut\in' wind' speed' or' number' of' days'above'the'cut\out'wind'speed.'Regarding'the'development'of'the'WINDEX'indices,'  131 this'set'of'indices'includes'variables'of'interest'when'operating'wind'plants,'such'as'the'number'of'days'below'the'cut\in'speed'(CI),'number'of'days'above'the'cut\out'speed'(CO),'total'number'of'non\operating'days'(NODI),'days'operating'above'the'rated'speed'(R),'and'10th'and'90th'percentiles'of'the'daily'wind'distributions'(W10'and'W90,'respectively).'''Furthermore,' the' present' study' corroborated' the' flexibility' of' the' ANN'models'when'used'to'statistically'downscale'coarse'resolution'variables'to'local'scale,'as'generally'the'ANN'downscaling'techniques'reliably'simulated'historical'and'future'wind' climatologies' under' the' A2' emission' scenario.' MAEs' and' IOAs' were'consistent' between' the' two' periods.' Overall,' ANNall' obtained' the' best' results'simulating'daily'variability,'and'CDFt'obtained'the'best'results'simulating'climate'variability' for' the' historical' and' future' periods.' The' remaining' models' traded'"climate"'for'"weather"'performance'or'vice'versa.'''Even'though'the'superiority'of'the'ANN'over'the'linear'models'was'expected'given'the' non\linearity' of' the' process,' other' nonlinear'models' such' as' support' vector'machines' (Cristianini' and' Shawe\Taylor' 2000),' or' temporal' neural' networks'(Dibike'and'Coulibaly'2006)'could'potentially'outscore'the'three'nonlinear'models'implemented'here'in'terms'of'both'daily'variability'and'climate'indices.'Similarly,'because' of' the' underlying' distribution' of' the' wind,' preprocessing' techniques'involving' a' more' refined' nonlinear' transformation' of' the' predictand' (e.g.' Box\  132 Cox)' could' potentially' reduce' the' biases' generated' by' possible' violations' of' the'normality'of'the'error'assumption'(Acock'2010).''Concerning' the' predictor' selection,' the' models' using' all' the' predictors'outperformed'the'ones'using'the'leading'principal'components,'or'the'variables'of'the' closest' grid'point' to' the'buoy,' and' considerable'differences' simulating'daily'variability'and'the'WINDEX'indices'were'found.'This'corroborates'the'fact'that'a'proper' predictor' selection' procedure' is' as' relevant' as' the' method' used' to'downscale'(Huth'2003).' In'general,' the' linear'methods'over\predicted'the' lower'values'and'under\predicted'the'higher'percentiles;'therefore,'the'models'were'not'able' to' reproduce' correctly' most' of' the' climate' like' indices.' Nonlinear' ANN'models'marginally'under'predicted'W10,'and'slightly'over'predicted'W90,'CO'and'NODI,' while' showing' an' above' average' performance' simulating' the' number' of'days'operating'at'rated'speed.'On'the'other'hand,' the'CDFt'method'matched'the'multiyear'average'of'W10'and'W90'perfectly'due'to'its'quantile\matching'origin.'No' systematic' over/under\predictions' were' found' when' CDFt' was' used' to'simulate'the'6'indices.''In' the' end,' it' is' important' to' note' that' the' statistical' downscaling'models'were'judged' relative' to' future' CRCM' model' outputs' and' not' relative' to' real'observations.'Nevertheless,'Vrac'et'al.'(2007)'considered'it'??unlikely'that'the'SD'model'would'work'well'when'fitted'to'the'data,'when'it'does'not'work'well'when'fitted'to'RCM'outputs,'since'the'GCM'and'the'RCM'are'more'closely'linked'than'the'  133 GCM' and' the' actual' climatology?.' ' Also' I' aspire' that' knowing' the' differences'between' the' models?' performance' between' periods' will' provide' valuable'information' regarding' the' level' of' confidence' one' should' attribute' to' the'downscaled'climate'projections,'and'will'foster'discussion'on'the'advantages'and'limitations'of'different'regression'and'classification'models'commonly'used'by'the'downscaling'community.'''In' Appendix' A,' I' downscaled' temperature' using' predictors' from' NCEP/NCAR'reanalysis' and' observed' temperatures' from' ten' meteorological' stations' in'southern'Ontario'and'Quebec;'the'results'show'that'the'nonlinear'models'usually'outperform'their'linear'counterparts'in'daily'variability'and,'to'a'greater'extent,'in'annual' climate' variability.' ' In' particular,' the' best' model' simulating' daily'variability' and' climate' indices'was' a' Bayesian' neural' network' (BNN)' ensemble'using' a' combination' of' surface' and' upper' level' predictors,' followed' by' a' BNN'ensemble' using' the' 3' leading' principal' components' from' the' NCEP/NCAR'reanalysis' predictors.' Finally,' I' showed' that' the' three' climate' indices,' T90,' T10'and'IATR,'showed'higher'skills'than'those'for'the'growing'season'length,'number'of'frost'days'and'the'heat'wave'duration.'''The' results' in' the'Appendix' could'be'better' than' Jeong' et' al.' (2012b)'because' I'included'10'weather'stations'located'in'eastern'Canada'(as'Hessami'et'al.'(2008)'did),'and'the'downscaled'time'series'from'these'stations'showed'a'relative'better'performance'than'those'obtained'in'the'continental'climate'area'of'Saskatchewan'  134 &'Manitoba,'also'used'by'Jeong'et'al.'(2012b).'Weichert'and'Burger'(1998),'Schoof'and'Pryor'(2001),'and'Easterling'(1999)'have'also'reported'a'performance'similar'to'that'shown'by'Jeong'et'al.'(2012b).''5.2 Future work An'interesting'direction'for'future'work'is'to'extend'this'evaluation'methodology'to' other' GCM\RCM' combinations' and' to' use' finer' RCM' models' as' pseudo\observations' as' the' CRCM' model' used' was' not' able' to' resolve' certain' local'processes' occurring' at' much' finer' scales.' A' promising' alternative' to' this'evaluation' methodology' using' RCM' pseudo\observations' is' to' use' a' so\called'?perfect\model?'approach'(see'Dixon'et'al.'(2013)'for'details).'This'approach'uses'very' high\resolution' GCM' outputs' as' pseudo\observations' to' train' and' validate'the' statistical' downscaling' methods.' As' with' the' methodology' used' in' this'manuscript,' the' ?perfect\model?' approach' allows' the' comparison' of' future'statistically'downscaled'time'series'with'the'outputs'generated'by'the'dynamical'high' resolution' GCM;' thereby' allowing' the' time\invariance' assumption' to' be'tested.'''On' the' other' hand,' because' the' NSERC\SRO' inter\comparison' modeling'framework' used' only' ten' stations' located' in' southern' Ontario' and' Quebec,' it'would' be' useful' to' obtain' statistically' downscaled' outputs' of' precipitation' and'temperature'at'other'sites'located'over'a'wider'range'of'climate'zones,'and'to'test'the'time'invariance'assumption.''  135 'Future' studies' should'be'made'around' the'Globe'with'different' climatic' regions'and'predictands/predictors'in'order'to'determine'for'which'cases'the'stationarity'assumption'holds'(e.g.'precipitation'occurrence)''or' does' not' hold' (e.g.'precipitation'amounts,' climate' indices),' and'whether'nonstationary'downscaling'approaches,' e.g.' Kallache' et' al.' 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Nevertheless,' the'AOGCM's' low' resolution' prevents' them' from' resolving,' small\scale' dynamical'features,' local' orographic' effects' and' other' regional' physiographical' features.'Thus'accurate'local'estimates'of'observed'climate'are'unlikely'to'be'produced.'On'the'other'hand,'meteorological'variables'at'weather'station'scale'are'often'needed'as'the'energy'sector,'hydrological'and'actuarial'sciences,'engineering'studies,'and'the' impacts' and' adaptation' community' among' others,' regularly' use' local' scale'variables.'To'address'this'need,'downscaling'techniques'can'be'used'to'generate'finer' scale' projections' of' near' surface' climatologies' (Salameh' et' al.' 2008).'  152 Similarly,' downscaling' techniques' can' be' used' to' understand' the' underlying'relationships'between'the'coarse'resolution'predictors'and'surface'observations.'''There' are' two' main' downscaling' approaches:' dynamical' and' statistical.' The'dynamical' technique' is' based' on' extracting' regional' scale' information' using'regional' climate' models' (RCMs).' RCMs' use' as' lateral' boundary' conditions'information'from'a'coarser'resolution'model'(Mearns'et'al.'2003;'Laprise'2008).'Sea'surface'temperature'(SST),'sea'ice,'greenhouse'gas'(GHG)'and'aerosol'forcing,'as' well' as' initial' soil' conditions,' are' also' more' often' provided' by' the' driving'AOGCM.' The' statistical' approach' is' based' on' finding' statistical' relationships'between' predictors' (i.e.' atmospheric' variables' from' coarse\resolution' model'outputs)'and'the'predicted'local'variables'required'by'the'climate'change'impact'studies.'This'appendix'will'deal'with'statistical'methods.'''Usually' comparisons' between' downscaling'methods' are' carried' out' in' terms' of'correlations' (Weichert' and'Burger'1998;'Bertacchi'Uvo' et' al.' 2001;'Cheng' et' al.'2008;' Chu' et' al.' 2010;' Souvignet' and' Heinrich' 2011),' root'mean' square' errors'(RMSE)' (Huth' 1999;' Kostopoulou' et' al.' 2006;' Fasbender' and' Ouarda' 2010),' or'similar'metrics'(Cawley'et'al.'2007)'between'the'daily'downscaled'and'observed'values.' Characteristics' such' as' extreme' values,' probability' distributions' and'spatio\temporal' structures' were' infrequently' treated' (Huth' et' al.' 2008),' but'recently' there' has' been' a' substantive' increase' in' the' number' of' studies'considering'extremes'(Busuioc'et'al.'2008;'Michelangeli'et'al.'2009;'Cannon'2010;'  153 Foresti' et' al.' 2010;' Mannshardt\Shamseldin' et' al.' 2010),' as' these' studies' are'essential'for'understanding'how'models'replicate'the'full'range'of'weather'station'characteristics.''This' research' is' part' of' the' "Probabilistic' assessment' of' regional' changes' in'climate' variability' and' extremes"' project' funded' by' the' Natural' Sciences' and'Engineering'Research'Council'of'Canada'through'a'Special'Research'Opportunity'(NSERC\SRO)' grant.' The' project' benefits' from' the' partnership' between' the'Canadian' Climate' Analysis' Group' and' the' European' ENSEMBLES' project' (an'European'Community'major'initiative'funded'by'the'European'Commission),'and'aims' to' develop' high\resolution' climate' change' information' with' the' Canadian'AOGCMs'and'downscaling'methodologies.'For'comparison'purposes,'all'the'group'members'are'analyzing'a'common'area'of'interest,'and'employing'a'multi\station'modeling' framework' for' assessing' the' changes' in' daily' precipitation,'minimum'and'maximum'temperatures'in'the'southern'Ontario'and'Quebec'area.''Although,' many' recent' studies' have' compared' the' performance' of' different'downscaling'methods'(Wilby'and'Wigley'1997a;'Schoof'and'Pryor'2001;'Harpham'and'Wilby'2005;'Dibike'and'Coulibaly'2006;'Fr?as'et'al.'2006;'Haylock'et'al.'2006;'Khan' et' al.' 2006),' those' derived' from' the' Statistical' and' Regional' dynamical'Downscaling'of'Extremes'for'European'regions'(STARDEX)'project'were'some'of'the'first'to'rigorously'and'systematically'compare'different'downscaling'methods'(Fowler' et' al.' 2007).' The' STARDEX'project'was' part' of' the' ENSEMBLES'project'  154 and'developed'extreme'climate'indices'to'compare'the'changes'in'frequency'and'intensity' of' extreme' weather' events' in' current' and' future' climates' (Goodess'2005).'''Since' our'project' derived' from'a'partnership'with' the'ENSEMBLES'project,' it' is'natural' for'us'to'use'the'STARDEX'indices'to'focus'on'an'agreed,'standard'set'of'daily' temperature' extremes.' The' project' framework' incorporates' observational'data'provided'by'Environment'Canada,'and'the'NCEP/NCAR'(National'Centers'for'Environmental'Prediction'and'National'Center'for'Atmospheric'Research)'(Kalnay'et'al.'1996)'reanalysis'product'interpolated'to'the'Canadian'Global'Climate'Model'(CGCM)' 3.1' grid' to' generate' and' validate' daily' maximum' and' minimum'temperature'projections.'The'study'period'is'1961\2000.''In'particular,'the'present'work'uses'different'predictor'selection'methodologies'in'conjunction' with' linear' (multiple' linear' regression)' and' nonlinear' [multi\layer'perceptron'(MLP)'Bayesian'neural'network'(BNN)]'models'to'obtain'statistically'downscaled' daily' maximum' temperature' (TMAX)' and' minimum' temperature'(TMIN)'of'10'weather' stations' located' in' southern'Quebec'and'Ontario,'Canada.'Additionally,' this' study' provides' a' comparison' carried' out' not' only' in' terms' of'mean'absolute'error' (MAE)'between' the'downscaled'and'observed'daily'values,'but'also' in' terms'of' the' refined' index'of'agreement' (IOA)' (Willmott'et'al.'2012)'between'annual'extreme'climate' indices.' In'particular,' I'will' focus'on'six'climate'indices' used'by' the' STARDEX'project:' the' 90th' percentile' of' the' daily'maximum'  155 temperature'(TMAX),'10th'percentile'of' the'daily'minimum'temperature'(TMIN),'number'of'frost'days,'heat'wave'duration,'growing'season'length'and'intra\annual'temperature'range'(Table'10).''''Nonlinear' regression' models' like' artificial' neural' networks' (ANN)' using' multi\layer' perceptrons' (MLP)' have' been' used' extensively' for' statistical' downscaling'(SD)' (e.g.,' (Wilby' and'Wigley' 1997a;' Crane' and'Hewitson' 1998b;'Weichert' and'Burger' 1998;' Schoof' and' Pryor' 2001;' Dunn' 2004;' Cannon' 2008b;' Huth' et' al.'2008)).' In' general' MLP' downscaling' models' give' similar' results' compared' to'multiple' linear' regression' downscaling' methods' for' temperature' and'precipitation' (Schoof' and' Pryor' 2001),' and' are' capable' of' outscoring' linear'models'when' relationships' are' nonlinear' and/or' interactive' (Tang' et' al.' 2000).'Nevertheless,' there' is' no' consensus' on' their' performance' versus' linear'models'(Jeong'et'al.'2011b).''For'example,'when'downscaling'temperatures'over'Europe,'Huth' et' al.' (2008)' concluded' that' the' nonlinear' methods' did' not' bring' an'improvement' over' linear' methods.' In' contrast,' Miksovsky' and' Raidl' (2005)'concluded' that' the' nonlinear' techniques' outperformed' linear' regression' in' the'majority' of' cases,' when' downscaling' daily' temperatures' from' 25' European'stations.'''''''  156 Table'10'STARDEX'Temperature'related'annual'indices'used'in'this'study.'' ' Temperature'Indices' Abbreviation''1' 10th'percentile'of'TMIN'' T10'2' 90th'percentile'of'TMAX'' T90'3' Intra\Annual'extreme'Temperature'Range'(T90'\T10)'' IATR'4' Number'of'Frost'Days'(TMIN'<'0?C)'' FD'5' Growing'Season'Length'(period'between'TMEAN'>'5?C'for'more'than'5'days'and'TMEAN<5?C'for'more'than'5'days)'GSL'6' Heat'Wave'Duration'Index'(maximum'period'of'consecutive'days'with'TMAX'exceeding'the'climatological'T90)''HWDI''To'overcome'some'of' the' limitations'single'ANN'model'outputs'encounter'when'being'applied'to'problems'in'hydrology,'an'ensemble'modeling'strategy'was'used'(Cannon'and'Whitfield'2002;'Miksovsky'and'Raidl'2005).'Typically,' in'ensemble'modeling' the' generalization' error' of' the' final' predictive'model' is' controlled' by'combining' outputs' from' a' number' of' ensemble' members.' However,' Breiman'(1996)'found'that'the'improvements'in'performance'tend'to'level'out'after'adding'more'than'25'models'to'the'ensemble.''In' this' chapter,' I' evaluated' linear' and'nonlinear' downscaling'models' using' four'different' sets' of' predictors' in' terms' of' their' ability' to' reproduce' day\by\day'temperature'variability,'and'in'terms'of'their'ability'to'reproduce'the'6'STARDEX'climate' indices.' ' In' particular,' nonlinear' Bayesian' neural' networks' ensembles'were'implemented'and'compared'against'linear'models.'I'aimed'to'i)'determine'if'  157 linear'and'nonlinear'models'using'different'sets'of'predictors'have'similar'skills'simulating'daily'variability'and'climate'indices,'ii)'determine'if'one'has'to'trade'off'daily' variability' for' climate' indices' simulation' skills,' and' iii)' describe' an'evaluation'methodology'comparing'these'skills.''A.2. Datasets A.2.1. Predictors Consistent' with' our' project' framework,' I' used' a' standard' set' of' 25' predictors'from' the' NCEP/NCAR' reanalysis' interpolated' to' a' T47' Gaussian' grid'(approximately' 3.75?' lat.' by' 3.75?' lon.,' i.e.' the' grid' resolution' of' the' Canadian'AOGCM,'CGCM3.1,'Flato'et%al,'2000)'over'the'current'climate'period'(1961\2000).'All'NCEP/NCAR'reanalysis'data'have'been'averaged' from'6'hourly' to'daily'data'before' being' linearly' interpolated' from' the' 2.5?' x' 2.5?' grid' to' the'T47'Gaussian'grid'(see'further'details'in'DAI'(2008)).''''As' the' ultimate' objective' of' downscaling' is' to' generate,' based' on' AOGCM'predictors,' reliable' future' scenarios' (for' which' I' do' not' yet' have' observed' or'reanalysis'climate'data),'the're\gridding'procedure'of'the'NCEP/NCAR'reanalysis'to'the'CGCM3.1'grid'was'done'to'promote'consistency'between'further'work'using'CGCM3' predictors' (see' Chapters' 2' and' 3)' and' the' downscaling'methods' in' this'appendix.' Similar' sets' of' predictors' have' been' used' in' the' past' by' Khan' et' al.'(2006)' and' Dibike' and' Coulibaly' (2006),' among' others,' for' downscaling'  158 temperatures' in' Northern' Quebec,' Canada;' and' most' recently' by' Jeong' et' al.'(2011a)' for' analyzing' the' performance' of' the' CGCM' predictors' over' southern'Quebec.''The'predictors'shown'in'Table!11'include'daily'values'of'25'variables'comprising'of'temperature,'humidity,'surface'pressure,'as'well'as'upper'air'measures'of'wind'speed'and'direction,'vorticity,'divergence,'and'geopotential'height.'However,'I'did'not' use' all' of' them,' since' in' downscaling' applications' it' is' important' to' use'predictors'that'can'be'modeled'correctly'by'the'AOGCMs'and'that'incorporate'the'forcing' responsible' for' the' climate' change' signal,' and' not' all' the' available'predictors' have' these' characteristics.' Therefore,' I' opted' to' use' the' predictor'subset'recommended'by'Jeong'et'al.'(2011a).' 'From'these'predictors'I'calculated'the'500'hPa' \' 850hPa' thickness' at' each' grid'point' and'used' them'as'predictors'instead'of'the'aforementioned'geopotential'heights.''Overall,'from'the'four'nearest'gridpoints' I'obtained'specific'humidities'at'500'hPa,'850'hPa'and'1000'hPa,' the'500'hPa\850'hPa'thicknesses,'and'the'2'm'temperatures.'These'20'predictors'(i.e.'4'grid'points'x'5'predictors/point)'constitute'our'final'predictor'set.''''''''  159 Table'11'Predictor'variables'available'for'the'NCEP/NCAR'reanalysis.'' Predictor variable Predictor No. Mean'sea'level'pressure' 1 26 51 76 1000'hPa'wind'speed' 2 27 52 77 1000'hPa'U'component' 3 28 53 78 1000'hPa'V'component' 4 29 54 79 1000'hPa'vorticity'(Pa's\1)' 5 30 55 80 1000'hPa'wind'direction' 6 31 56 81 1000'hPa'divergence'(s\1)' 7 32 57 82 500'hPa'wind'speed' 8 33 58 83 500'hPa'U'component' 9 34 59 84 500'hPa'V'component' 10 35 60 85 500'hPa'vorticity'(Pa's\1)' 11 36 61 86 500'hPa'geopotential' 12 37 62 87 500'hPa'wind'direction' 13 38 63 88 500'hPa'divergence'(s\1)' 14 39 64 89 850'hPa'wind'speed' 15 40 65 90 850'hPa'U'component' 16 41 66 91 850'hPa'V'component' 17 42 67 92 850'hPa'vorticity'(Pa's\1)' 18 43 68 93 850'hPa'geopotential' 19 44 69 94 850'hPa'wind'direction' 20 45 70 95 850'hPa'divergence'(s\1)' 21 46 71 96 500'hPa'specific'humidity' 22 47 72 97 850'hPa'specific'humidity' 23 48 73 98 1000'hPa'specific'humidity' 24 49 74 99 Temperature'at'2'm' 25 50 75 100 CGCM 3.1 / NCEP Grids 77x12y 77x13y 78x12y 78x13y  'From'the'final'group'of'predictors'I'formed'four'predictor'sets'using:'(a)'stepwise'(SW)' selection,' (b)' principal' component' analysis' (PCA),' (c)' reanalysis'temperatures'only,'and'(d)'reanalysis'thicknesses'only.'For' (a),' the' SW' selection' procedure' involves' forward/backward' stepwise'selection' with' an' F\test' stopping' criterium' (see' Section' A.3.1' for' details).' This'  160 selection' method' reduced' the' total' number' of' predictors' used' from' 20' to' an'average'of'16'and'17'for'TMAX'and'TMIN,'respectively. For'(b),'I'used'principal'component'analysis'for'data'reduction.'Specifically,'I'used'the' first' 3' PCs' of' the' standardized' anomalies' data' as' predictors.' The' optimal'number' of' PCs' was' determined' following' the' N' selection' rule' (Preisendorfer'1988).' For' this'dominant\variance' rule' I' calculated' the'principal' components'of'1000' Monte' Carlo' simulations' (with' similar' mean' and' variance' to' the' actual'predictors)' and' compared' them' to' the' principal' components' obtained' from' the'actual' predictors.' The' rule' suggests' selecting' the' number' of' PCs' before' the'intersection' of' the' 95%' significance' interval' from' the'Monte' Carlo' experiments'with'the'observed'PCs.'In'addition,'even'though'the'present'work'focuses'on'the'historical'period,'to'guarantee'the'same'PC'structure'on'the'historical'and'future'periods,'PCA'was'applied'to'a'matrix'comprising'the'reanalysis'and'the'CGCM'3.1'SRES'A2'21st'century'outputs'as'suggested'by'Imbert'and'Benestad'(2005).'''For'(c),'as'in'direct'downscaling'studies,'only'the'2'm'air'temperature'fields'from'the'four'nearest'NCEP/NCAR'reanalysis'grid'points'(the'25th,'50th,'75th'and'100th'variables'from'Table'11)'were'used'as'predictors.'Similarly,'(d)'uses'only'the'four'thicknesses'from'the'nearest'grid'points.''A.2.2. Predictands I' used'as'predictands'observed'daily'maximum'and'minimum' temperature'data'from'10'weather'stations'of'Environment'Canada'(EC)'located'in'Southern'Quebec'  161 and' Ontario,' Canada.' I' obtained' and' retrieved' data' from' 1961\2000' from' the'National' Climate' Data' and' Information' Archive' operated' and'maintained' by' EC'using'the'Data'Access'and'Integration'(DAI)'portal'(http://loki.qc.ec.gc.ca/DAI/).'Figure'26'shows'the'station'names'and'their'location'across'the'study'area.'A.3. Regression based downscaling models The' regression' models' represent' linear' or' nonlinear' relationships' between'predictands' and' large' scale' predictors' (Fowler' et' al.' 2007).' As' with' other' SD'methods,' regression' approaches' have' the' undesirable' characteristic' of' under'predicting'the'variance.''To' deal'with' this' problem' one' can' (i)' add' noise' to' the' statistically' downscaled'series,'thus'breaking'the'temporal'correlation'of'the'data'(Huth'et'al.'2001),'or'(ii)'inflate'the'downscaled'time'series'as'proposed'by'Karl'et'al.'(1990).'The'present'study' opted' for' the' latter' approach,' as' some' of' the' STARDEX' indices' (e.g.' Heat'Wave'Duration'and'Growing'Season'Length)'need'the'temporal'correlation'to'be'kept.' Another' limitation' of' all' regression' techniques' is' the' assumption' of' time\invariance'in'the'predictand\predictor'relationships'(Haylock'et'al.'2006).''''Two' different' regression\based' downscaling'methods'were' used' in' the' present'work,' namely' (i)' multiple' linear' regression' (LR)' based' on' stepwise' variable'selection,'and'(ii)'Bayesian'neural'networks'(BNN).'''  162 ''Figure'26.'Selected'weather'stations'located'over'southern'Qu?bec'and'Ontario'(Canada)'with'altitude'(in'm)'of'this'area'given'in'color'scale.'Black'lines'correspond'to'the'CGCM3'grid.  'A.3.1. Linear regression (LR) I'considered'four'linear'downscaling'models'with'all'of'them'using'stepwise'linear'regression' to' find' linear' relationships' between' predictors' and' predictands.' The'first' model' starts' with' the' final' predictor' set' with' 20' predictors' and' performs'stepwise' selection'of'predictors.'The' final'number'of'predictors' selected' for' the'different'weather'stations'varied'from'13'to'17'for'TMAX'and'from'14'to'17'for'TMIN.'The'second'model'performs'stepwise'selection'on'the'first'3'PCs,'while'the'  163 third' and' fourth' models' use' stepwise' regression' on' the' four' temperature'predictors,'and'the'four'thicknesses'predictors,'respectively.'''Stepwise'techniques'for'regression'analysis'are'described'in'Darlington'(1990).'In'general,' stepwise' regression' is' a' systematic' method' for' adding' and' removing'predictors'from'a'multiple'linear'regression'model'(see'its'use'in'the'recent'MLR'method'developed'by'Hessami'et'al.'(2008)).'An'initial'model'is'created'at'the'first'iteration,'then'the%p'value,'is'computed'to'test'models'with'and'without'a'potential'predictor.'The'null'hypothesis'is'that'the'predictor'to'be'added'or'removed'has'a'zero'regression'coefficient'(Hill'and'Lewicki'2006).'A.3.2. Multi layer perceptron (MLP) Bayesian neural networks (BNN) MLP' neural' networks' are' used' to' map' nonlinear' relationships' between' input'variables' and' dependent' output' variables.' The' goal' of' the' neural' network' is' to'minimize'the'mean'squared'error'(MSE)'or'other'performance'function'between'the'predicted'and'observed'value'of'the'dependent'variable.'An'MLP'is'composed'of'an'input'layer,'a'number'of'hidden'layers'and'an'output'layer'of'neurons'(Hsieh'2009).'''Although' the'weights'of'an'MLP'are'similar' to'non\linear'regression'coefficients'(Crane'and'Hewitson'1998b),'it'is'generally'not'fruitful'trying'to'find'meaningful'interpretations' from' the' large' number' of' weights' (Hsieh' 2009).' The' main'differences'between'LR'and'MLP'are'that'while'LR'has'a'closed'form'solution,'MLP'  164 uses'an'iterative'process,'and'while'LR'assumes'a'functional'form,'MLP'allows'the'data'to'define'the'functional'form'(Schoof'and'Pryor'2001).'This'study'will'use'the'Matlab?' neural' network' toolbox' code' ?trainbr?,' an' MLP' BNN' model' with'automated' regularization.' The' BNN' model' was' introduced' by' MacKay' (1992),'which' uses' Bayes' theorem' to' estimate' the' optimal' weight' penalty' parameter'without'the'need'of'validation'data'(Hsieh'2009).'''Similar'to'the'LR'approach,'four'models'were'created'using'ensemble'BNNs.'The'models'used'the'same'sets'of'predictors'employed'by'LR'models,'one'hidden'layer'with'different'number'of'hidden'neurons'for'each'ensemble'member,'and'a'single'output'neuron'(predictand).'For'more'information'on'Bayesian'neural'networks,'see'Bishop'(2006)'and'MacKay'(2003).''To'create'a'model'ensemble' for'BNN,' several'methods'can'be'used'all' involving'elements' of' uncertainty' either' in' the' data' or' in' the' model' itself' (Tebaldi' and'Knutti' 2007).' In' this' case,' all' the' potential' ensemble' models' have' one' hidden'layer,'with'the'hyperbolic'tangent'activation'function'mapping'from'the'inputs'to'the'hidden'layer,'and'a'linear'activation'function'mapping'from'the'hidden'layer'to'the'output'layer.''A.4. Daily!maximum!and!minimum!temperature!downscaling!The' downscaling' procedure' consists' of' three' different' steps:' 1)' Train' the'statistical'downscaling'model'using' the'historical' synoptic\scale' circulation'data'  165 as' predictors' and' surface' observation' data' as' predictands;' 2)' Compute' the'model?s' validation' error' using' independent' data;' 3)' Once' the' statistically'downscaled' model' has' been' trained' for' the' control' period' and' validated,' the'scenario' simulations' can' be' used' as'model' inputs' to' downscale' future' possible'climates.'Step'3'will'not'be'done'in'this'appendix,'as'I'will'focus'on'the'historical'1961\2000'period'analysis.'''The' climatological' seasonal' mean' was' removed' from' the' predictors' and' the'predictands,' yielding' anomalies' from' the' mean' datasets,' and' the' predictor'anomalies'were'then'standardized'to'unit'variance.''A.4.1. Model evaluation The'models'were' evaluated' using' cross' validation' (Bishop'2006),'with' the' data'split'into'four'contiguous'10\year'segments.'Models'were'trained'on'three'of'the'four'decades,'and'predictions'were'made'and'validated'on'the'remaining'decade.'This'procedure'was'repeated'until'predictions'had'been'validated'on'all'years'of'data.'In'general,'cross\validation'is'a'technique'that'allows'the'entire'dataset'to'be'used' for' validation,' so' the' validation' error' can'be' computed' for' the'whole' data'record.''For BNN, the optimal number of hidden neurons was determined as follows: given three decades of training data, the model was first trained using two decades of data, then the optimal number of hidden neurons was chosen (from 1-50) based on model   166 performance in the third decade of data. With the number of hidden neurons now determined, an ensemble of 20 BNN models were trained with random initial weights on three decades of data, and the 10 best ensemble members were selected based on the MAE between the observed and model values in the training data.  'The' cross\validation' procedure' was' implemented' for' each' of' the' linear' and'nonlinear' models' in' Table' 12.' For' the' control' period' in' the' case' of' the' linear'models,' each' segment' will' have' only' one' downscaled' data' set' (D);' for' the'nonlinear'models' each' segment'will' have'm' downscaled' values,'where'm' is' the'number' of' ensemble'members.' The'm' model' values' were' averaged' to' give' the'ensemble'mean,'considered'to'be'the'downscaled'data'set'(D).'The'D'data'sets'for'the'four'10\year'segments'were'assembled'to'give'predictions'over'40'years.'As'regression' approaches' under' predict' the' variance,' the' predicted' values' were'?inflated?'so' that' the'predicted'and'observed'variances'match'(Karl'et'al.'1990).''This'was'accomplished'by'multiplying'the'vector'with'predictions'by'the'ratio'of'the' observed' and' predicted' standard' deviations.' Henceforth' MAE' refers' to' the'MAE'of'the'inflated'results'unless'otherwise'specified.''For'the'day\by\day'variability'or'?weather?'evaluation'procedure'the'downscaled'TMAX'and'TMIN'were'compared'with'the'observations'for'each'weather'station,'and'then'the'MAEs'between'the'datasets'were'calculated.'Then'for'every'weather'station,' the'downscaled'TMAX'MAE'and'TMIN'MAE'were'averaged'to'obtain'the'mean'temperature'MAE.'This'value'will'be'used'to'determine'how'well'the'models'  167 represented'the'daily'variability'over'the'control'period.'The'MAE'was'selected'as'performance' function' following' the' recommendation' of'Willmott' and'Matsuura'(2005),'as' they' found' it' to'be'a'more'natural'measure'of' the'average'error' than'the'RMSE.''The' climate' evaluation' procedure' involves' using' the' downscaled' values' to'calculate'the'six'STARDEX'temperature'related'indices'shown'in'Table!10.''The'IOA'between' the'downscaled' indices' and' the'observed' indices' allows'us' to' evaluate'the' ability' of' the' downscaled' models' to' replicate' the' climate' extremes' as'quantified'by'the'STARDEX'indices.'The'(refined)'IOA'is'defined'by' IOA = 1 - [?i ?Pi - Oi ?]  /  [2 ?i ? Oi - ? ?], when  ?i ?Pi - Oi ? ?  2 ?i ? Oi - ? ?, or IOA = [2 ?i ? Oi - ? ?] / [?i ?Pi - Oi ?]  - 1, when  ?i ?Pi - Oi ?  > 2 ?i ? Oi - ? ?    (A.1)  where Pi and Oi are' the' downscaled' and' observed' values,' respectively, ? the'observed'mean.'The'IOA'statistic'inversely'follows'a'scaling'(1/2)'of'the'average\error'and'deviation'measures'(Willmott'et'al.'2012).'A'larger'IOA'value'indicates'better'forecast'performance. '  168 Finally,' a' Unified' STARDEX' Index' (USI)' characterizing' the' model?s' ability' to'reproduce'a'group'of'extreme'climate'indices'was'defined'as'the'mean'of'the'six'IOAs'for'the'STARDEX'indices,'i.e.'' USI IOA = (T10 IOA + T90 IOA + IATR IOA + FD IOA +GSL IOA + HWDI IOA)/6 . (A.2) 'The'USI'was'calculated'for'each'weather'station.'A.5. Results  First,' for' every' station' and'model,' the' statistically' downscaled'TMIN' and'TMAX'series' were' compared' to' the' observed' TMAX' and' TMIN,' and' their' MAEs' were'calculated.'The'two'MAEs'were'then'averaged,'and'the'multi\station'mean'MAEs'were' calculated' for' each'model' (TAV'MAE)' and'were' plotted' against' the'multi\station'mean'USI' Index'of'Agreement' in'Figure'27.'This' figure'allows'the'user'or'decision'maker'to'choose'a'model'based'on'his'specific'needs,'e.g.'trading'weather'simulation'skill'for'extreme'climate'simulation'skill.''''As'the'accuracy'of'the'SD'temperatures'(i.e.'variance'inflated'anomalies'plus'the'seasonal'cycle)'is'quantified'in'terms'of'MAEs,'the'calculated'errors'may'be'biased'because' the' variance' inflation' leads' to' greater' inaccuracy' of' the' MAE.'Nevertheless,' it' is' worth' mentioning' that' when' analyzing' the' MAE' of' the'uninflated'anomalies'the'relative'performance'of'the'four'main'predictor'selection'methods' remained' unchanged.' However,' as' the' SD' temperatures,' not' the' SD'  169 temperature' anomalies' are'needed' to' calculate' the' STARDEX' indices,' I' opted' to'plot'the'MAE'of'the'inflated'anomalies'along'the'ordinate.'Figure'23'compares'the'performance'of'the'eight'models'used'(Table'12).'''The' results' show' that' among' the' methods' studied,' an' ensemble' of' Bayesian'neural' networks' using' stepwise' selection' from' the' final' set' of' predictors'presented'the'best'results'simulating'day\by\day'variability'(2.49'?C'in'MAE)'and'the'highest'USI'(0.64).'A'linear'model'using'the'same'set'of'predictors'presented'a'larger'MAE'and'a'smaller'USI.'A'cluster'of'five'models'(BNNPC,'BNNT,'LRPC,'LRT'and' LRall)' was' found' on' the' lower' central' region' of' the' figure,' representing'intermediate'MAE'and'USI'between'0.52'and'0.62.'The'clustered'nonlinear'models'marginally' outscored' the' linear' ones' in' terms' of' daily' variability' and' when'simulating' climate' of' extremes,' but' generally' their' differences' were' not'significant.'The'only'two'exceptions'were'between'BNNPC'and'LRPC'(in'terms'of'USI)' and' between' BNNHD' and' LRHD' (in' terms' of' daily' variability).' When'comparing' the' different' linear' models' in' terms' of' the' Bayesian' Information'Criterion' (BIC),' LRall' obtained' the' lower' BIC,' proving' to' be' the' best' predictive'model.'I'did'not'compare'the'nonlinear'models'in'terms'of'BIC'as'determining'the'effective' number' of' parameters' of' the' BNN' models' is' complicated' as' I' used'Bayesian'regularization.''''  170 'Table'12'Linear'and'nonlinear'models'used.''Model Number Model type Regression method Predictors used Model ID 1 Linear LR SW selection from specific humidity at 500 hPa, 850 hPa, and 1000 hPa, 850-500 hPa thickness, and 2 m temperatures from 4 NCEP/NCAR grid points LRall 2 Linear LR First 3 PCs LRPC 3 Linear LR SW selected from 4 NCEP/NCAR temperatures LRT 4 Linear LR SW selected from 4 NCEP/NCAR 850-500  hPa thicknesses LRHD 5 Nonlinear BNN Same as in LRall BNNall 6 Nonlinear BNN Same as in LRPC BNNPC 7 Nonlinear BNN Same as in LRT BNNT 8 Nonlinear BNN Same as LRHD BNNHD     171  Figure' 27.' TAV'MAE' vs.' USI' (average' IOA' of' the' STARDEX' indices).' The'multi\station'means'from'the'nonlinear'BNN'models'are'plotted'with'solid'symbols,'and'those'from'the'LR'models'with'open'symbols.''The'four'different'sets'of'predictors'are' identified'by' the' shape'of' the' symbols,'with'diamonds' for' the' case'using'all'predictors,'circles,'the'four'temperatures,'stars,'the'four'thicknesses,'and'squares,'the' 3' PCs.' Horizontal' and' vertical' error' bars' represent' the' mean' absolute'deviation''(MAD).'''Figure' 28' shows' the' models'' STARDEX' indices' IOAs.' In' general,' the' models''performance' simulating' the' first' three' indices' (T90,' T10,' IATR)' is' marginally'higher' than'simulating' the'remaining'ones'(FD,'GSL,'and'HWDI),'although'many'are' not' even' significantly' higher' when' taking' into' account' the' MADs.' This'behaviour' implies' a' lesser' success' in' simulating' series' of' events,' like' the' ones'  172 needed'to'calculate'the'annual'growing'season'length,'heat'wave'duration,'and'the'number'of'frost'days.''Regarding' the' IOA' of' individual' indices,' for' T90,' I' found' important' differences'between'BNNall'and'BNNHD,'and'between'BNNall'and'all'four'linear'models.''For'T10,' BNNall' clearly' outscored' BNNHD,' LRHD,' and' LRPC.' For' IATR,' the'models''behaviour' is' similar' to' the'previous' two' indices' (specially'T10),' as' this' index' is'calculated'from'the'differences'between'T90'and'T10.'For'FD,'the'LRHD'and'LRPC'models'were'clearly'outperformed'by'BNNall.''For'the'GSL,'BNNall'outscored'BNNHD'and'LRHD,'although'its'IOA'was'lower'than'for' the' previous' four' indices.' This' suggests' that' the' GSL' analysis' may' require'other'surface'and'circulation'variables'to'be'modeled'correctly.'When'comparing'the'magnitude'of'the'six'indices'IOA,'it'is'evident'that'I'need'to'take'into'account'other'variables'to'improve'the'models''performance'simulating'them.'This'means'for'example'including'surface'conditions'such'as'frost'and'thaw'conditions'of'the'soil,' or' potential' snow' cover' on' the' ground,' as' these' induce' some' nonlinear'interactions' with' the' overlying' air' temperature' behaviour,' especially' at' the'beginning'of'the'growing'season.''   173  Figure 28. Models? STARDEX indices IOAs. Nonlinear methods are represented with dark bars and the linear methods with lighter coloured bars.  Error bars indicate the mean absolute deviations (MAD) of individual stations.   For'HWDI,' although' the'nonlinear'BNNall,'BNNT'and'BNNPC'models'marginally'outscored'their'linear'counterparts,'the'performances'of'LRHD'and'BNNHD'were'notable.' This' suggests' that' the' heat' wave' duration' modeling' partially' benefits'from' the' use' of' the' 500' hPa' ?' 850' hPa' thicknesses,' as' there' is' a' relationship'between'heat'waves'and'upper\air'high'pressure'(Alexander'and'Arblaster'2009).'The' large' MAD' for' BNNall' indicated' important' differences' between' individual'stations'values.'This'suggests'that'the'users'should'select'a'model'based'on'their'particular' needs' and' location' of' interest,' or' use'models'with' smaller' deviations'  174 from'the'mean'(LRHD'and'BNNHD),'but'be'aware'of'their'limitations'in'modeling'the'other'five'indices.''Overall' BNNall' was' the' best' model' simulating' daily' variability' and' 5' out' of' 6'climate' indices' (Figures' 23' and' 24),' while' LRHD' was' the' best' simulating' the'remaining' index' (HWDI).' When' comparing' the' two' best' models,' BNNall' and'BNNPC,' I' note' that' the' advantage' of' the' first'model' over' the' latter' one' is' very'small'for'climate'indices,'but'is'larger'for'daily'variability.'A.6. Summary and discussion The' present' study' compared' four' different' sets' of' predictors' and' two' different'statistical'downscaling'models,'one'linear'and'one'nonlinear,'not'only'in'terms'of'their' ability' to' reproduce' daily' temperature' variability,' but' also' their' ability' to'simulate' six' climate' indices.' The' methods?' performance' was' evaluated' using'cross\validation' and'quantified' in' terms'of'MAEs,' for' the'day\by\day'variability,'and'Unified'STARDEX'Index'IOAs,'for'the'climate'of'extremes.'In'general,'I'showed'that'the'advantage'of'nonlinear'models'over'linear'models'is'marginal'for'climate'extreme' indices'and' for'day\to\day'variability' (taking' into'account' the'MAD).' In'particular,' ' an' ensemble' of' Bayesian' neural' networks' using' stepwise' selection'from'the'20'NCEP/NCAR'predictors'outscored'the'rest'of'the'models'in'terms'of'daily'variability'and'in'terms'of'climate'of'extreme'indices.'''  175 The'disparity'on'performance'simulating'climate'of'extremes'between'linear'and'nonlinear'models' can'be'partially' attributed' to' their'differences'when'modeling'lower'and'higher'quantiles'(Figure'29),'and'to'the'ability'of'the'nonlinear'models'to' capture' the' minimum' temperature' behaviour' below' zero' Celsius.' When'modeling' daily' variability' using' only' four' temperatures' as' predictors,' the'marginal'differences'between'the'linear'and'nonlinear'methods'can'be'caused'by'approximately' linear' relationships'between'predictors'and'predictands' (Huth'et'al.'2008).''For'the'daily'variability,'all'four'nonlinear'methods'produced'smaller'mean'MAEs'than'their'linear'counterparts,'although'their'differences'were'less'than'two'times'their' MADs.' ' The' result' is' consistent' with' Miksovsky' and' Raidl' (2005),' as' the'linear' regression' and' the' nonlinear' models' analyzed' in' this' study' presented'comparable'MAEs.'''Several'implications'can'be'derived'from'the'results,'the'first'is'that'the'statistical'downscaling'models'using'a'limited'subset'of'coarse'resolution'predictors'did'not'reproduce' climate' of' extremes' and'weather\like' variability' as' accurately' as' the'models'using'more'predictors.'This'is'in'agreement'(for'the'daily'variability'case)'with'other'studies'(Wilby'et'al.'(1998),'Huth'(2002);'(Huth'2003),'Gachon'(2005))'which' have' shown' that' the' use' of' combined' predictors' (circulation' and'temperature)' is' superior' to' that' of' any' single' predictor' when' downscaling'temperature' and/or' precipitation' (Hessami' et' al.' 2008).' Regarding' the' models'  176 using'only'thicknesses'(LRHD'and'BNNHD),'as'their'ability'to'model'most'of'the'climate'indices'is'very'limited'and'their'MAEs'are'more'than'1.5'degrees'Celsius'higher'than'the' leading'model'(BNNall),' I'do'not'recommended'their'use,'unless'the'user'is'interested'in'modeling'the'HWDI.'Similarly,'I'might'need'to'incorporate'other'predictor'variables'such'as'vorticity,'divergence'and'wind'speed/direction'to'model'the'last'three'indices'correctly'(GSL,'FD,'and'HWDI).'I'encourage'future'studies' to' test' such' models' where/when' the' aforementioned' variables' are'successfully'reproduced'by'the'AOGCMs.''''Figure'29'Quantile\quantile'plots.''(a)'Mean'BNNall'versus'mean'TMAX'observed,'(b)'Mean'LRall'versus'mean'TMAX'observed,'(c)'Mean'BNNall'versus'mean'TMIN'observed,' and' (d)' Mean' LRall' versus' mean' TMIN' observed.' The' red' line'corresponds'to'a'perfect'agreement'between'model'and'observations.''I'recommend'the'evaluation'procedure'to'be'used'in'climate'downscaling'studies,'as' the' traditional' way' of' determining' the' models' skill' is' not' robust' enough' to'  177 establish'the'models?'behaviour'simulating'weather'variability'and'climate'related'indices.' On' the' other' hand,' I' recommend' the' final' users' to' select' the' nonlinear'BNNall'model'as'its'performance'modeling'daily'variability,'and'to'a'lesser'extent'climate'of'extremes'was'superior.''Finally,' as' good' agreement' in' the' present' climate' does' not' guarantee' good'performance'simulating'future'events,'Chapters'2,'3'and'4'deal'with'validating'the'models' performance' using' RCM' outputs' as' present' and' future' ?pseudo\observations?,' following'Vrac' et' al.' (2007b).' Furthermore,' it' is' recommended' to'study'other'predictor'selection'methods,'as'it'has'been'shown'that'the'predictors'used' can' cause' significant' performance' differences.' Further' work' should' also'explore' the' seasonal' effects' of' the' suggested' weather' and' climate' evaluation'procedure' on' the' downscaling'model' performance,' to' better' evaluate' nonlinear'versus' predominantly' linear' interactions' linked' with' the' air' temperature'behaviour.'

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