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Opportunities for management created by spatial structures : a case study of Finnish reindeer Berkson, James Meyer 1988

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OPPORTUNITIES  FOR MANAGEMENT  BY SPATIAL  CREATED  STRUCTURE:  A CASE STUDY OF FINNISH  REINDEER  By JAMES MEYER BERKSON B.A., The U n i v e r s i t y  A THESIS SUBMITTED  o f C a l i f o r n i a , San Diego  IN PARTIAL FULFILLMENT OF  T H E REQUIREMENTS FOR T H E DEGREE OF MASTER OF SCIENCE  in T H E FACULTY OF GRADUATE STUDIES (Department o f Zoology)  We accept t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d  standard  T H E UNIVERSITY OF BRITISH COLUMBIA January ©  1988  James Meyer B e r K s o n ,  1988  THE  In  presenting  degree  this  at the  thesis  in  partial  fulfilment  University of  British  Columbia,  freely available for reference and study. copying  of  department  this or  publication of  thesis by  for  his  this thesis  or  scholarly her  of  the  I agree  requirements  for  may  representatives.  It  be is  granted  by the  understood  Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  make it  for extensive head  that  for financial gain shall not be allowed without  permission.  advanced  that the Library shall  I further agree that permission  purposes  an  of  my  copying  or  my written  ii  ABSTRACT  This  study  management  when  subdivisions. for  opportunities  population  data  of management  resources,  f o r renewable  are  In p a r t i c u l a r I l o o k  the d e s i g n  monitoring  examines  collected  and  the  by  at p o t e n t i a l  experiments,  resource spatial  applications  t h e d i s t r i b u t i o n of  improvement  of  parameter  estimation. Methods subdivisions to  are for  the  external  to  . Methods could  possible  units.  information  minimize  similar  a r e low o r when  used.  The use of t h e s e  could  notify  managers  external  s i m i l a r l y in the r i s k  of  units.  subdivisions  that  subdivisions  when  stratified  subdivisions  of e x t r e m e l y good  of  experimental  in experimental  to i d e n t i f y  being  Factors  between  could  about  groupings  have r e a c t e d  differences  developed  resources  that  factors  creating  are  provide  monitoring  subdivisions  external  factors  rank  can cause d i f f e r e n c e s  Selecting  past  to  u s e as e x p e r i m e n t a l  the experiment  units.  developed  sampling  as " i n d e x  o r bad y e a r s  is  units" in  a  l a r g e number of s u b d i v i s i o n s . Two  methods  developed  innovative  estimation  subdivided  populations.  estimates  t o be a d j u s t e d  approach  allows  more a c c u r a t e l y  by  techniques A  Walters that  Bayesian  using  be  provide used  with  approach allows  parameter  a known d i s t r i b u t i o n .  Another  similar subdivisions t h a n would  can  (1986)  t o be e s t i m a t e d  be p o s s i b l e i n d i v i d u a l l y .  jointly  Not  a l l renewable  information there  f o r use w i t h  is  little  autocorrelation measurement series  in  steps  reliability  on  experimental identified. are  used The  external pattern  reindeer little could the  units  and  on t h e d a t a  (1986)  of  amounts  of  to  test  p a r t i c u l a r data  sets.  (Rangifer  steps  methods  A the  tarandus  to i l l u s t r a t e  for  the  f o r these  developed.  units  show t h a t located  occuring  subdivisions  biological  carrying  the  where  levels  a p p e a r t o be a p p r o p r i a t e  index  data were  Possible  monitoring  of p a r a m e t e r  are  very  benefit limits  subdivisions  close  are  estimation  by  highly  similar  subdivisions.  regions. and  dependence.  conducting  This  existence  managed  of t h e p o p u l a t i o n  capacities within  the  geographic  of d e n s i t y by  with  t o one a n o t h e r .  caused  within  of any k i n d  potentially  sets  methods.  managers  the t h e s i s  at l e a s t p a r t i a l l y  evidence  most  reliable  s e t as w e l l .  reindeer  bad y e a r s  Data high  for  for  reindeer  using  provide  o r even modest  on t h e i r  data  Walters'  was  extremely  noise,  tested  effects  variation,  throughout  The r e i n d e e r  sets  applications.  introduced  Finnish  when  data  are i n a p p r o p r i a t e is  t a r a n d u s ) are used  methods  the  of t h e methods  Data  methods.  these  common  error  of  resource  The provide  Managers  experiments growth  of  to t e s t  rates  and  iv  TABLE OF CONTENTS ABSTRACT L I S T OF TABLES L I S T OF FIGURES AC KNOWLE G DEMENT S CHAPTER 1. INTRODUCTION Background Experimental Design Monitoring A l l o c a t i o n Parameter E s t i m a t i o n Case S t u d y O r g a n i z a t i o n of t h e T h e s i s CHAPTER 2. IDENTIFYING SUBDIVISIONS WITH SIMILAR RESIDUALS Introduction Simulations Methods Results Discussion Case S t u d y Methods Results Discussion CHAPTER 3. IDENTIFYING SIMILAR SUBDIVISIONS IN THE PRESENCE OF COMPLEXITY Introduction Simulations Methods Results Discussion Case S t u d y CHAPTER 4. OPPORTUNITIES FOR EXPERIMENTAL DESIGN AND MONITORING ALLOCATION Introduction Methods Experimental Design B r u t e f o r c e r a n k i n g of u n i t s f o r s m a l l experiments H i e r a r c h i c a l c l u s t e r i n g t o rank u n i t s f o r l a r g e r experiments Monitoring A l l o c a t i o n Results Experimental Design Monitoring A l l o c a t i o n Discuss ion CHAPTER 5. IMPROVING PARAMETER ESTIMATION OF SUBDIVIDED POPULATIONS Introduction Bayesian Approach  i i V i V i i ix 1 2 5 7 8 9 13 15 15 16 16 21 23 29 29 31 36 37 37 39 39 42 45 48 56 56 58 58 58 59 62 63 63 81 86 90 90 91  V  Methods Results E s t i m a t i o n of common e x t e r n a l Simulations Methods Results Case S t u d y Methods Results Discussion CHAPTER 6. CONCLUDING REMARKS LITERATURE CITED  effects  91 94 99 106 106 107 I l l i l l I l l 112 118 124  vi L I S T OF TABLES Table  1.1  Table  3.1  Table  4.1  Table  4.2  Table  4.3  Table  4.4  Table  4.5  Table  4.6  Table Table Table  4.7 5.1 5.2  Table  5.3  Table  5.4  L i s t of t h e 56 r e i n d e e r management d i s t r i c t s in northern Finland L i s t of r e i n d e e r h e r d s and y e a r s t h a t e x h i b i t t h e 50 l o w e s t s t a n d a r d i z e d r e s i d u a l values Ten c l o s e s t g r o u p i n g s of r e i n d e e r s u b d i v i s i o n s based on r e s i d u a l similarity P e r c e n t a g e of s i m u l a t i o n s r e s u l t i n g i n c o r r e c t i d e n t i f i c a t i o n of quads and p a i r s from c l u s t e r a n a l y s i s T h r e e s e t s of h e r d s u s e d f o r f u r t h e r r a n k i n g of e x p e r i m e n t a l u n i t s Top c a n d i d a t e s f o r e x p e r i m e n t a l units within sets Groups of h e r d s r e m a i n i n g a t t h e 2 0 t h s p l i t f o r e a c h h a l f of t h e t i m e s e r i e s Ten most s i m i l a r p a i r s d u r i n g e a c h h a l f of t h e t i m e s e r i e s P o t e n t i a l key I n d i c a t o r s u b d i v i s i o n s Mean g r o w t h r a t e v a l u e s by h e r d R i c K e r " a " v a l u e s as e s t i m a t e d b e f o r e and a f t e r t h e B a y e s i a n method R e s u l t s of i n d i v i d u a l vs j o i n t estimation simulations E s t i m a t e s of R i c K e r "b" p a r a m e t e r by i n d i v i d u a l and j o i n t p a r a m e t e r analysis  10  53 64 69 76 77 82 83 84 95 100 108  113  vii L I S T OF FIGURES Figure  i . i  Figure  2.1  Figure  2.2a  Figure  2.2b  Figure  2.3  Figure  2.4  Figure  2.5  Figure  2.6  Figure  2.7  Figure  3.1  Figure  3.2  Figure  3.3  Figure  3.4  Figure  3.5  Figure  3.6  Figure  3.7  Figure  4.1  Figure  4.2  Figure  4.3  Figure  4.4  Map of t h e 56 R e i n d e e r management d i s t r i c t s i n Northern Finland P e r f o r m a n c e of 3 s i m i l a r i t y s t a t i s t i c s as i n d i v i d u a l v a r i a t i o n i n c r e a s e d S u r f a c e v i e w of t h e p r o b a b i l i t y of successful pairings versus i n d i v i d u a l and common v a r i a t i o n C o n t o u r v i e w of t h e p r o b a b i l i t y of successful pairings versus i n d i v i d u a l and common v a r i a t i o n Mean c o r r e l a t i o n as an i n d i c a t o r of p r o b a b i l i t y of c o r r e c t p a i r i n g s D i s t r i b u t i o n of c r o s s c o r r e l a t i o n c o e f f i c i e n t s between p a i r s of F i n n i s h r e i n d e e r herds D i s t r i b u t i o n of rank c o r r e l a t i o n v a l u e s c o m p a r i n g t h e f i r s t h a l f of t h e t i m e s e r i e s to t h e second h a l f D i s t r i b u t i o n of rank c o r r e l a t i o n v a l u e s c o m p a r i n g t h e f i r s t h a l f of t h e t i m e s e r i e s t o the e n t i r e time s e r i e s D i s t r i b u t i o n of rank c o r r e l a t i o n v a l u e s c o m p a r i n g t h e s e c o n d h a l f of t h e t i m e s e r i e s t o t h e e n t i r e time s e r i e s E f f e c t of a u t o c o r r e l a t i o n on t h e p r o b a b i l i t y of c o r r e c t p a i r i n g s E f f e c t of measurement e r r o r on t h e p r o b a b i l i t y of c o r r e c t p a i r i n g s E f f e c t of e x t r e m e l y bad y e a r s on t h e p r o b a b i l i t y of c o r r e c t p a i r i n g s D i s t r i b u t i o n of a u t o c o r r e l a t i o n i n r e i n d e e r herd r e s i d u a l s C o m p a r i s o n of r e i n d e e r s t a n d a r d i z e d r e s i d u a l s t o a normal d i s t r i b u t i o n C o m p a r i s o n of r e i n d e e r trimmed, s t a n d a r d i z e d r e s i d u a l s t o a normal distribution Map of r e i n d e e r h e r d s w i t h e x t r e m e l y low r e s i d u a l v a l u e s d u r i n g common y e a r s (3 y e a r s i l l u s t r a t e d ) F i v e of t h e t e n most s i m i l a r p a i r s of subdivisions Amount of t i m e r e q u i r e d t o f i n d t h e 10 top c a n d i d a t e s f o r e x p e r i m e n t a l u n i t s R e s u l t s of d i v i s i v e c l u s t e r i n g of t h e r e i n d e e r herds R e s u l t s of t h e f i r s t 15 s p l i t s of divisive clustering  12 22 24 25 26  32  33  34  35 43 44 46 49 50  52 54 66 68 72 73  vi i i Figure  4.5  Figure  4.6  Figure  4.7  Figure  5.1  Figure  5.2  M a j o r c l u s t e r s of h e r d s r e m a i n i n g a t t h e 15th s p l i t of d i v i s i v e c l u s t e r i n g Best candidate f o r 6 u n i t experiments w i t h i n e a c h of t h e 3 s e t s P o t e n t i a l i n d i c a t o r herds a t the l e v e l of 0.65 shown w i t h t h e h e r d s t h a t t h e y represent D i s t r i b u t i o n of t h e mean g r o w t h r a t e estimated f o r the r e i n d e e r herds R e s u l t of s i m u l a t i o n s c o m p a r i n g individual to joint estimation with h i g h l e v e l of common v a r i a t i o n  74 80 87 98 110  ix ACKNOWLEDGEMENTS Many people provided me with assistance in the completion of this thesis. Dr. Carl Walters introduced the topic to me and helped with my project o r i e n t a t i o n . Dr. Carl Walters and Dr. Don Ludwig assisted me with i t s organization and were v e r y helpful i n discussions. I particularly wish to thank my entire Supervisory Committee; Dr. Carl Walters, Dr. Don Ludwig, Dr. C.S. Holling, Dr. A.R.E. Sinclair, and Dr. N.J. Wilimovsky. Their suggestions and help were always appreciated. John Eadie, Dr. Don Ludwig, Locke Rowe, and Peter Watts had the thankless task of reading an early draft of this thesis. I appreciated a l l of the help provided by the UBC statistics department, particularly the staff of SCARL and Professors Schulzer and Zamar. Brad Anholt, Alistair Blachford, Pete Cahoon, Colin Daniels, John Eadie, Dr. Chris Foote, Dr. Lee Gass, Linda Glennie, Bob Gregory, Gordon Haas, Don Hall, Debbie McClennan, Teresa Patterson, Rob Powell, John Richardson, Don Robinson, Locke Rowe, A r l e n e Tompkins, Andrew T r i t e s , and Peter Watts all honored me with their f r i e n d s h i p by providing helpful comments, needed encouragement, and unwarranted abuse. Financial support was provided by the University of British Columbia in the form of a research assistantship, a university graduate fellowship and a teaching assistantship. Don Hall formatted the equations i n Chapter 5 f o r proper presentation. Finally, I wish to thank my family; Dick, Katie, Ben, Justin, G a r r e t t , and particularly my wife Denise; for always being there when I need them. I would also l i k e to thank Denise f o r proofreading the various d r a f t s of t h i s thesis and for keeping me calm and productive.  1  CHAPTER  1  INTRODUCTION  The wildlife  management  and  basis,  fisheries  with  treated  each  as  a  parameters approach  has  provided the  by  a  following  Fisheries Ricker  (Gulland 1975);  Wildlife  Shaw  1985);  Wildlife  1984,  Wakeley  1982,  Starfield information  and  or  stock  production  Recently,  and  that  i t has  management  been  activities  across  units  may  discussed  1984,  Nelson  1968);  subdivisions  1969,  Resource The  was  and  Robinson Modeling  systematic  first  reviews  in  approach:  Johnson  1984,  has  information  this and  (Bailey  (Cox  literature  utilizing  not  Ecology  1986).  for  Textbooks  Management  Bleloch  from  May  Watt  localized  its  management  approach  have  1983,  as  harvests.  populations.  fields  to  tests)  such  very  herd,  1986),  of  resource  systematic  subdivided  a  variation.  (Walters  policy  the  on  regard  natural  better  recently,  addressed  with  coordination  substantially  resources,  subpopulation,  questioned  experimental  Until not  of  natural  attempted  entity  pattern  that  provide  been  biological  been  (monitoring,  renewable  has  unique  and  suggested  of  1983,  Giles  1969,  and  Bolen  (Grant  1986,  utilization  introduced  by  of  Walters  (1986). This work  of  work  presented  Walters  (1986).  in In  this  thesis  particular,  will new  expand methods  on will  the be  2 introduced  to  allocation.  improve  Examples  estimation  will  experimental  of  be  recent  design  and  improvements  monitoring  in  parameter  presented.  BACKGROUND  A  main  objective  increase  the  commercial  and  wildlife  animals  managers  each  Before  established,  size  increases.  with  the  aid size  population  model  and  a  of  number  population  growth  parameter.  Managers  population  a  an  unknown  rate  is  fit  f o r the  past  for  methods  to  Models  do  difference  observed  in  or  The  rates,  about  understood  predict  or of  a  difference  provide  that  the  best look  parameters. There  predicts a  A  production  constantly  model  between  population  parameters  exactly.  model  The  parameters.  Managers  the  future  size.  present  words,  data  the  better  f o r the  of  be  population  example  data.  can  potential  to  constants,  i n other  values  population.  population  t h e estimation  between  overall  involving  not f i t population  Fisheries number of  be  model  values  population  improve  nature.  can  an  estimate  appropriate,  and  i s to  maximum  harvest  present  of  the most  a  size  equation  a  information  population on  managers  resource.  the  as  relationship  are  is  such  require  based is  their  depleting  policies,  This  population  size  without  between  of  resource  t r y to h a r v e s t  managers  relationship  renewable  value  often  year,  management  of  and  always values  predicted  and  3 observed  value  interested i.e.  in  in  the  is  called  the  a  residual.  information  pattern  of  Managers  unaccounted  natural  for  variation  are  by  also  the  model,  represented  by  the  residuals. With variation,  information managers  develop  harvest  harvest  times.  collect  The  commonly  called  to  new  test  thought  as  case  is of  distinct called  allocation.  on  often or  stocks  1985).  instead over  of  distinct  management  placed  require  develop  units on  a  are  arbitrary  map  of  the  Groundfish  stocks  Management  units  within  subdivisions.  the In  same the  case  as  can  be  their  based  population  of  such  for charge,  management.  factors.  which  want  policies  in  for  in  is  design.  species  population  also  tests  implement  not  often  monitoring  may  These  units  to size.  are  population  proper  and the  order  resources  Managers their  and  population  monitoring  and  but  but  of  in  establish  experiments  Pacific  McFarlane  to  rates.  biologically  grid  managers  They  rates  populations  harvest  providing  distinctions  monitoring  higher  herds  cases,  the  natural  policies.  harvest  about  of  individual  a  for  of  and  policies  portions  of  of  other  data,  monitoring  Managers  other  or  kinds  establish  distribution  policies  testing  to  develop  requiring  priorities.  the  They  parameters  several  policies  available  limited,  population  develop  information,  Resources  on  on  biological  In  many  range  as  Canada are  In  cases in  (Tyler  the and  geographically  range separate  can  also herds  be as  4 well  as  separate  subdivide  the  providing the  population  information  population  Where have  a  With  this,  the  of  at  time,  addition,  patterns within  information  and  or  cycles  one  of  little  managers  will  within  each  improve  the  time  subdivision. the  model,  compared share  each  series  of  Residuals or  effects  between  common  data  collected  a  within  size  be  at  have  a  between  time  residuals these  time  for  this  series  variation.  behavior  cases  time.  some  natural  of  will  over  estimated  on  to  managers  relationship  will  many  can  be  (Box  and  procedures  managers.  coordinated available  subdivision.  among  population  a  series  time  Models  subdivision.  can  New  be  of  f i t to  procedures  can  accuracy  of  parameter  estimation.  residuals  is  available  from  represent external  subdivisions  external  unit,  population  have  data  a  in  is  population  Also,  managers Often  estimates  find  can  use  from  to  one  series  1980),  data  developed  grid,  are  population  of  time  population  be  to  managers  management  subdivisions,  to  information  Chatfield  Where  a  purposes.  size  parameters  providing  1976,  by  population  attempt  Model  residuals  provide  animal  can one  In  discovered  population  i s localized  of  managers  future.  Jenkins  on  series  size  Although  out  f o r management  management  relationship.  laid  subdivisions.  single  population in  regions  variation  to to  effects.  the  model.  identify Policies  not  explained  Residuals  take  by  can  subdivisions which  each  be  that into  5 account more  similarity systematic  populations  in  approach  than  is  EXPERIMENTAL  Managers populations (Holling active  management.  to  learn  a  sizes.  population  sizes  level  active and  the  management  being  practiced.  provide  of  approach  in  can  population With  of  a  subdivided  order  probing,  a  order  grow  this  to  given  a  is  potential  regular  rates  in  broad  at  opportunities  order  range can  annual  Kept  and  information  managers  maximize  their  management  provide  harvest  population  miss  adaptive  information, to  about  recommends  manipulate  will  in  managers  information  management  Managers  how  the  increased  experiments"  population  Without  gain  Adaptive  "probing  subdivisions  DESIGN  using  1978).  among  to  currently  can  by  to  residuals  of set  harvests. a  constant  to  improve  harvests. Another management population  policies is  experimental policies. the  example  units  Managers  Experimental  wolf  in  active a  subdivided,  experimental  fisheries  of  new  policy  policies other  then  policy  i s the  management  while  can  probing  to  tests  have  wildlife  management.  control  project  of  the  can  the  relative  and  experiment. be  units  assess  An  British  testing  the  When  tested  maintain  benefits  and  control  often  been  example Columbia  of  of  on  new the some  control costs  of  baseline. conducted  in  this  is the  Wildlife  Branch  6 (Atkinson  and  compared  between  regions been  Janz  where  made  Janz  regions wolves  based  1986).  on  or  interpret  because  as  would  are  control  treatment  one  can  not  an  factors  There can  use  to  subdivisions interfere  answer  are  subdivisions  or  to  range  of  the  of  not  policy  has  (Hatter  and are  within  difficult  identical  to  each  experiment.  policy  i f the other  to other  Suppose  experiment, and If  control  broad  possible units:  a  two  one one  as  a  as  a  change  and  the  change  is  change  commerce. range  of  criteria  Managers  economically  experimental  in  treatment, due  to  the  in  external  that  managers  way  important  Managers  behaves  so  may  environmental  policy  wish as  wish  to  use  not  to  to  use  conditions  similarly  under  to a  conditions.  thesis units  and  subdivisions.  important  broad  removed  reports  very  policy.  some  are  the  not  certainty  number  a  be  management  which  with  been  experiments  technical  management  experimental  that  selecting  a  select  ensure  This  with  a  are  between  the  these  can  experimental  policy  with  of  laboratory  existing an  between  a for  observed  experimental  experiments  the  have  agencies.  in  used  with is  of  internal  subdivisions be  been  management  many as  have  rates  Overall  experiments  being  with  behavior  of  survival  wolves  results  published  Management  units  where  the  government  they  Ungulate  remain.  Results  unpublished individual  1986).  suggests as  controls  another and  possible treatments  criterion in  for  management  7 experiments: similar  select  responses  differences behaved likely  to  in  effects,  to  behave  subdivisions  similar  effects.  external  be  that  in  subdivisions as  order  in  the in  the  to to  be  have  are  If  of  external  for  treatment  more  pairs  their  identified  and  minimize  past  of  have  that  future. terms  can  control  in  likely  Subdivisions  factors  compared  experiments  are  factors,  similarly  can  management  external  external  similarly to  subdivisions  use  in  units.  MONITORING ALLOCATION  In  many  subdivisions limited  cases  more  monitor.  1977, to  Many  possible  subdivisions  to  want  to  subdivisions. monitored  be  monitored  Subdivisions  concentrating  on or  Tanner  that  (1978)  suggests  where  subdivisions  to  stratified  sampling  monitoring  have  to  single  to  extensively:  accessible  Other  criteria  dynamics  selecting  population units  at  the  Managers important  easily  a  out  select  commercially  whose  plans  effort.  used  only  with  case  the  most  are  some  all  be  intensely.  subdivisions  the  monitoring  the  monitor  of  also  can  monitor  subdivisions on  will  to In  the  select  additional  criteria  most  others.  improved  1967)  precisely  the  understood,  using  receive  wish  permit  must  Murthy  may  than  not  managers  subdivisions  decline.  do  Managers  (Cochran  may  extensively  resources  subdivisions,  managers  may  be  include  are  poorly in  random  steep for  8 intensive  sampling  This  within  thesis  selecting  subdivisions which  of  effects.  external be  past  a  is  good  subdivisions  units".  terms  of  be  Gulland can  with  are  presented  as  for  examples.  understand  of  as  terms  bad  years  subdivisions,  the  can  in  the  future. can  be  The  be  called  compared  representative  index  in  factors  extensively  effects,  use  be data  does  in  subdivisions  units.  from  Walters'  on  not  data  address 1975). by  book  after  may seeing  issue  joint will  using  sample  an  methods  example  1986,  estimation  examine These  the  other  estimation  (1986).  find  by  within  (Grant  Parameter  I  without  gained  collected  using  Walters  be  management  this  subdivisions. by  can  resource  Ricker  managers  implement  management  population  improved  developed in  for  using  1984,  Resource and  in  or  of  external  similarity  literature  May  techniques  to  subdivisions  (1986)  the  of  information  potentially  procedures  response  good  number  pairs  The  1983,  large  more  parameters  Walters  a  select  subdivisions  extremely  for  ESTIMATION  Additional  subdivisions.  other  way  external  identified  to  criterion  monitoring:  monitored  If  PARAMETER  than  for  indicator  their  estimating  this  in  being  "index  can  In  alternative  extensive  similar  identified similarity  an  for  are  provided  strata.  suggests  subdivisions  could  each  of  two  of  techniques data easier their  sets to use.  9 In can  be  one  approach,  adjusted  from  a  assuming  known  Bayesian  to  estimated  individual are  improve  the  second of  estimated  as  year.  parameter  large  to  of  when  a  normal are  jointly,  the  procedure  this  parameters  subdivisions  This  nature"  parameters  from  approach,  as  by  Walters,  random  well  subdivisions  "drawn  estimation  at  estimation  (Walters  randomly  the  group  each  across  According  drawn  a  are  of  were  been  In  for  parameters effects  can  have  distribution.  they  estimates  distribution.  method  assumed  parameter  shared  works  shared  where external  better  external  than effects  1986).  C A S E STUDY  Methods allocation, tested  in  actual  for  and  parameter  this  data  thesis.  study.  from The  management located Reindeer of the  about 56  experimental  estimation  These  methods  design,  will will  be then  monitoring  developed be  used  and on  an  set.  Population tarandus)  improving  data  on  northern  Finland  population purposes  north  of  Raising 366,000  is  (Table the  reindeer  (Rangifer  will  be  subdivided  1.1  and  Kiimiinki  Districts  management  reindeer  is  into  Figure river.  graze  for  1987). on  as 56  the  case  herds  for  l.i). A l l The  responsible  (Anonymous  districts  used  tarandus  herds  are  of  the  Union the  management  Reindeer natural  within pastures.  10  L i s t of the56 reindeer management d i s t r i c t s in Finland  Table l . l  Herd #  Name  Herd #  Name  1  Paistunturi  29  Palojarvi  2  Kaldoaivi  30  Orajarvi  3  Naatamo  31  Kolarin alanen  4  Muddusjarvi  32  JaasKo  5  Vatsari  33  NarKaus  6  Ivalo  34  Niemela  7  Hammastunturi  35  Timis j a r v i  8  Sallivaara  36  Tolva  9  MuotKatunturi  37  Livo  10  NaKKala  36  Isosydanmaa  1i  Kasivarsi  39  Manty j a r v i  i 2  Muonio  40  Kuukas  13  Kyro  41  AlakitKa  14  Kuivasalmi  42  Akanlahti  15  Alakyla  43  Hossa-Irni  44  Kallioluoma  16  Sattasniemi  17  Oraniemi  45  Oivanki  18  Syvajarvi  46  Joki j a r v i  19  Pyhajarvi  47  Taivalkoski  20  Lappi  48  Pudas j a r v i  21  Kemin-Sompio  49  Oijarvi  Herd #  Name  Herd #  Name  22  Sallan pohjoinen  50  Livo  23  Salla  51  Pintamo  24  Hirvasniemi  52  Kilminki  25  Kallio  53  Kollaja  26  Vanttaus  54  Ikonen  27  Poikajarvi  55  Naljanka  28  Lohijarvi  56  Halla  Figure 1.1  Map of the 56 Reindeer management districts In Northern Finland  13 Reindeer and  are  also  own  are  made  with  reindeer  on  and  hundred  improve  will where  develop  from  subdivisions the  with  conditions  families  a  get  base  district  in  concert  are  their  principal  provided  prepared  7500  by  by Dr.  Dr. Timo  include  the  number  of  males,  with  the  number  of  males,  along  Census  from  district  1961  data  through  are  available  1983.  THESIS  develop,  test,  design,  use  will similar necessary  them them  using on  methods  these  Each  Carlo  Finnish  of  methods  allocation,  Monte  the  series  for  use  populations.  introduce time  and  monitoring  subdivided  test  and  data  Data  of  methods,  2  There  harvested.  will  estimation  Chapter  districts.  were  a l l areas  appropriate,  district  populations  experimental  parameter  herding  each  the  reindeer  censused  thesis  from  within  1987).  ORGANIZATION OF T H E  This  of  management  (Anonymous  Finland.  in  means  reindeer  yearlings  a l l years  a  decisions  Management  U.B.C.  calves  by  within  directors  Eight  the  of  and  females, for  of  Rovaniemmo,  females,  of  owning  Walters  Helle,  board  owners. from  living  Management  Union  Data Carl  a  district  feasible.  citizen  reindeer. by  their  where  Finnish  the  income  within  fencing  Any can  kept  and  chapter  simulations  reindeer  to  to  data.  identify  residuals  along  with  methods  to  work  14 effectively. test  Managers  the r e l i a b i l i t y  Chapter real  3  world  absence  of  will  test  complications  judge  the  be  these  complications  of  managers  will  given  suggestions  methods  f o r any  t h e methods to  the  in  further  system.  actual  reliability  and  about given  by The  data  sets  how  to  data set.  adding  many  presence will  appropriateness  or help  of  the  design  and  methods.  Chapter monitoring used  as  select  4  controls  are  Chapter  5  (1986) that  independent  in  Methods  and  subdivisions  Walters  specifically  allocation.  monitoring  shown  looks  at  to  treatments  as  index  experimental  select are  units  subdivisions  developed. for  to  be  Methods  to  more  intensive  developed.  tests to  and uses  improve  many  parameter  cases,  two  of the methods  parameter these  analysis.  estimation. methods  are  developed It  by  will  be  superior  to  15 2  CHAPTER  IDENTIFYING SUBDIVISIONS WITH SIMILAR RESIDUALS.  INTRODUCTION The  preceding  available types  to  possible  monitoring  react be  by  second,  subdivisions  the  where  the  All  identify This  that  account  the  time  residuals  main design,  three  of  subdivisions  effects.  population  comparing  Three  experimental  to  models as  opportunities  estimation.  external  (such  by  are  ability  using  several  populations.  parameter  to  variation  subdivision; among  require  first  of  subdivided  and  similarly  met:  sources  of  mentioned  improvement  allocation,  approaches  that can  managers  of  these  chapter  requirement for  major  size)  within  series  of  are  measured  test  methods  each  residuals  around  the  models. This  chapter  will  similar  subdivisions  methods  will  given  not  suggestions  these methods  methods will  be  in  work on  on  how their  develop  and  terms well to  in Judge  of all  with  cases.  the  individual  demonstrated  external  the  Managers  sets.  Finnish  identify  effects.  potential  data  to  The  will  be  reliability  of  Finally,  reindeer  the  data.  16  SIMULATIONS  Simulation  To  develop  subdivisions must  first  is  with  available  methods  series  used  to  population an  A  example.  in  identifying a  the  Population  year.  size  model  model  population  another  similar  population  population  relate  size  for  residuals,  introduced.  tool  as  test  time  be  the  used  and  from  mathematical year  Methods  size  A  is  is in  simple  a one  model  assumed  to  be  change  in  the  annually.  Suppose following  that  population  size  (i - H  X  is  assumed  to  way:  Nt l,i  = N  +  t | i  t | 1  )  A  where N ^ . i = population size at time t in subdivision i H-t,i  = harvest rate at time t in subdivision i (fraction of  harvested)  Aj = mean population growth r a t e i n subdivision i  Population density by  in  fitting  growth  this  model.  population  (Draper  and  estimated,  a  series  are  Values  data  Smith time  rates  from  1981). of  independent of  each Once  residuals  X^  can  subdivision  of  population  be  estimated  to  parameters can  be  easily  the (X^)  model are  calculated  17 in the following  R  t+i,i  way:  Actual Population  =  Size - Expected Population  = t i,i  - N  where  = the estimated mean growth rate for  N  +  (1 - H  Size  t l i  t > i  ) X  A  subdivision i  A  simulation  of  three  of  residuals.  test  statistics  the  (Ludwig  SoKal  1973)  of  numerical in  time,  of ie.  list  in  terms  fitted  were  the  coefficient were  of  and  account.  the  have This  the the  list  time of  compared  of  same  particular  of  from  time  Magnitude  statistic  is  also  Sneath  and  of  a  of a  a  in over  population tested  correlation  The  statistics  subdivisions. simple  that is  list  compared  rank  pairs  the  the  statistics  the  is  in  responses  coefficient.  association  sign.  were  Three  association,  of  to  analysis  used  terms  population  series.  percentage  series  used  of  1963,  in  residuals  residuals of  been  subjects  in  correlation  between  coefficient  represents  numbers  each  coefficient  calculated The  which  to  a  methods  Sneath  differences  time  1985).  and  work  abilities  frequently  (Sokal  this  of  are  have  are  the  similar  similarity  subjects  of  Identify  Walters  taxonomy  a  model  1981,  For  compare  complicated  measure  which  to  simulations of  Walters to  designed  correctly  Carlo  characteristics.  terms  was  effectiveness  Techniques of  to  Monte  and  field  model  not  named  statistic  two  columns  taken the  into simple  18 matching The  coefficient  ((Zubin  1938,  pair  of  subdivisions  with  coefficient  of  association  would  more  complex  statistic  Sokal the  be  and  Michener  highest  value  considered  to  1958).  for  be  the  the  most  similar. A residuals  is  statistic  the  was  technique  first  as  well  There  is  intermediate of  correlation  (where  is The  A and  Carlo  realistic  chapter.)  year  the  The  t + l ,i  = N  t t i  of  pairs  the  of  factor  residuals  four  are  models  t|i  and  the  non-parametric  rank  statistic,  ranked  from  series)  between  designed  best  model  Xi u>  complexities  from two  1  the to  n  lowest  to  subdivisions  ranks.  subdivisions  )  time  with  this  are  were  worked  of  t ( i  the  coefficient  the  coefficient  subdivision  (1 " H  is  series  to:  N  of  calculate  simulations  series  of This  inverted  statistics  One  time  statistics  four  the  correlation  To  correlation  three  simple  last  from  Monte  of  the  each  the  calculated  these  variety  coefficient. within  series  coefficient.  Magnitudes  association.  residuals  is  a  as  time  signs.  between  coefficient  highest.  1936).  as  compare  correlation  introduced  (Stephenson  considered  n  cross  to  under  was will  to  test  various  which  conditions.  created.  (More  be  in  annually  used  behaved  of  complex  the  next  according  19 with a l l variables defined as above and u>t.,i  =  Two each  noise  components  subdivision  component  each  of  term  randomly  picked  subdivisions.  4.  noise,  1  and  of  year 1  and  being  similar,  were  was  year  noise  was  2  the  shared and  resulted, and  4  for a  two  each  "common" pairs  same  shared by  of  common a  second  definition,  different  to  "individual"  each  of  3  added  an  f o r each  subdivisions design  below.  wt,i  component  component  This 2  noise  randomly  each  and  defined  from  in  3 and  The noise terms were calculated as:  «t, 1 = P e x  w  One  second  component.  subdivisions  year.  Subdivisions  of  common  the  picked  The  as  of  noise  subdivision.  component  term  CV ,1  +  t  *nt ,1-2  +  t > 2  = exp ( y  w ,3  = exp ( y  t ( 3  + Ht.3-4 + K )  W ,4  = exp ( y  t t 4  + l i t , 3-4  t  t  t i 2  + ^1.1-2  *1> +  K ) 2  3  +  * ) 4  where: w  t, l  Vt, 1  :  noise term at time t f o r subdivision i  =  individual noise term at time t f o r sub. 1  •nt.1-2  =  common noise term a t time t f o r subs, i and 2  lt,3-4  =  common noise term a t time t f o r subs. 3 and 4  T  Ki  = correction factor f o r mean to be 1.0 f o r sub. i  All  noise  terms  were  drawn  from  a  normal  distribution  20 with  mean  of  deviations the  were  model  subtracting For the  of  data and  tried.  from  were  tested  t h e highest  were  Parameters values  population rate to  was 1000.  standard varied were  set to Each  from  to  estimate  the  taking  mean the  coefficients  in  One  the  three and 4  hundred of  test  individual  the  data.  i.40, t h e  Mean annual  population  f o r 23  years.  common  size The  standard 0.02.  combination  of  individual  of  was  correct  runs.  100  harvest was s e t  independent each  A l l combinations and  times  common  i n order  to  calculated  by  pairings.  coefficient  a l l possible  100  run  of  annual  deviation  of  correlation of  representative  steps  t h e model  cross  over  of  i and 2, and 3  reindeer  the i n i t i a l  the  probability  average  values  were  set to  ran  0.4,  each  deviations,  A  was  and  standard  then  of t h e model,  combination  model  Finnish  simulation  For  i f the  combinations.  the  0.28, and  0.02  tried.  in  rate  deviation  fitting  subdivision,  simulations  f o r each  f o r the  growth  by  deviations.  used  estimated  standard  expected.  f o r t h e pairs  generated  of  calculated  each  Carlo  see  pairwise  standard  were  the  set of Monte to  combinations  to  large  a l l possible  common  of  Residuals  t h e observed  were  sets  range  earlier  a  statistics  A  described  results  out  0.  pairwise  was  cross  correlation  21 Simulation  The did  a  simulations  better  (subdivision  Job  pair  other  two  amount  of  pairs  of  1  common  and  of the time  the  individual common  deviation nearly  as as  The pairs  held  well  as  proportion the  of  greater  values  of  other  The  rank  the  correlation  individual  variation of  successful  relative  to  individual  factor increased  to  in as  individual  the  2.1  range  of  of t h e of  standard performed  (identifying  depended to  on  common  the  variation  variation. as  There common  both  relative as was  well a  variation  variation. of  variation  identifying  subdivisions  The  ordering  coefficient  pairings  correctly  value  common  pairings  components  variation  The  similar  coefficient.  the  of  important  the  Figure  for a  constant  correlation  successful  the  as  statistics.  reliable.  performed  0.15.  of  did the  identified two  least  of  similar)  Magnitude  pairings  the  probability  increased  correctly  measures  magnitude  than  increased  did t h e other  be  overall  rate  deviation  to  4)  subdivisions  increased.  was  of  coefficient  similar  and  for a  probability  created  correlation  deviations,  for  well.  3  success  than  similarity  standard  pair  coefficient  association  standard  reliabilities  and  variation  more  how  2  the  identifying  The  correlation  shows  that  correctly  The  of  common  showed  statistics.  coefficient  as  Results  sharing  increased  a the  pairs. 2  to  overall  was  an  Successful 1  ratio  of  magnitude  Probability of Successful Pairings  0.05  0.1  0.15  0.25  0.2  0.3  Individual Standard Deviation Correlation  0  Rank Correlation  °  Coeff. of Asso. ro IX)  F i g u r e 2.1  Performance of 3 s i m i l a r i t y s t a t i s t i c s as i n d i v i d u a l v a r i a t i o n i n c r e a s e d  23 of  variation.  subdivisions  sharing  variation  increased  results  of  Figure  2.2a  the  probability in  cases  as  O.i  Similarly, to  the  as  a  a  high  less  i  contour  as  of  a  ranged  Figure as  was of  as  common  variation.  from  most  to  surface  in  of v a r i a t i o n when  decreased  individual  drawing  pairings  cases  of  magnitude  are shown  amount  in  pairings  ratio  overall  successful  where or  2  simulations  and of  a  successful  the  drawing  in  2.2b.  The  high  common  The  as i.O  to as low  variation  was  individual. The indicator cross of  mean of  correlation  the  probability  correlation  correct  correlation overall mean  over  pairings value  mean  of  value.  correlation  identified  of  100  runs  increased individual There  simulations  that  was  have  the  identify  similar  way  Judge  to  the  most  2.3).  The  The  mean  probabililty mean  greatly  occasional  cross  around  the  high  values  of t h e  simulation  runs  which  that  reliable  of  liKely  the  at  similarly  correlation  pairs the  the  good  Results  showed  responded  Unfortunately,  fairly  pairings.  Discussion of S i m u l a t i o n  coefficient  as  varied  in  a  pairings.  increased  runs  were  was  successful  (Figure  coefficient  incorrect  The  coefficient  cross  correlation  identifying  subdivisions  to  did  factors.  always  correctly  not  subdivisions. reliability  external  of  Managers the  need  methods  on  some any  24  •.ot  F i g u r e 2.2a  Surface view of the p r o b a b i l i t y of successful p a i r i n g s versus i n d i v i d u a l a n d common v a r i a t i o n  25  Figure 2.2b  Contour view of the p r o b a b i l i t y of successful pairings versus Individual and common v a r i a t i o n  Probability of Correct Pairings 1.0  0.6 -  -0.1  0  0.1  0.2  0.3  Mean Cross Corr. Coeff. (100 runs) Figure  2.3  Mean c o r r e l a t i o n as an Indicator of p r o b a b i l i t y of correct p a i r i n g s  0.4  27 given  data  set  before  experimentation simulations  of  external  often the  methods  than  the  magnitudes.  the In  should  more  similar.  common  by The  cases  select  The  units  results  judge  or  subdivisions  statistics and  for  of  the  improve  the  with  similar  correct  results  normal  best  of  pairs  simulations to  could  estimate  deviations  found  in  their  the  of  noise  added  signs  The  residuals  and  in  the  correlation  normal  were  added  methods relative  than  not  due  noise.  identical  to  bias  the data  identifying were  in  to most  by  causation. managers  identify common sets  they  were  cannot  always  when  most  use  subdivisions.  individual could  be  variation.  similar  similar and  to  useless  individual  be  similar  created  became  to  that  accurately  managers  advantage  data:  subdivisions  selected  imply  took  of  more  normally  the  correlation  of  small  rather  was  conditions  been  used  1985).  Identification was  noise  distribution.  had  the  pairs  in  the  which  it  coefficient  under  that  because  the  components  (Walters  when  chance  methods  a  noise  variation  these  these  to  produced  available  note  probability  increased  similar  to  estimation  pairs  two  the  work  original  The  In  identify  correlation  from  important  parameter  the  addition,  coefficient  the  to  coefficient  information  came  to  monitor.  information  Only  simulations  is  methods  ways  to  other  available  a l l of  It  to  possible  correlation  distributed. of  these  effects.  The  of  units  provide  reliability  more  or  using  If  standard  roughly  Judge  28 the  reliability  possible.  The  common  data  possible  sets  pair  common  share identify  of  single  kept  that  fed.  analysis,  reliability  a  noise. only  rough As  of  the  By  the  an  individual  should  within  the  of  example, noise,  no  and  level will  way  to  to a  based  As  they  an  the snow  two  if on can  example,  Finland  i n heated  the  gauge  First,  i n northern  increase  of  represent  subdivisions,  a r e kept  each  level  subdivisions  through  up  each  and ice,  barns  distinct  and  groups  the  ratio  and  thereby  increase  coefficient  can a c t  group  of  are  common  method.  mean  indicator  herds  splitting  managers  of  foraging  Real  i n t h e model.  methods.  herds  is  similarity.  between  of  together  improve  their  level  different  should  used  of  from  a  There  to  not  variation:  subdivisions  things  factors  variation  Secondly, as  t h e data  one  different  shares  as  assessments  t h e southern  individual  the  several  i n t h e winter,  a  variation  variation  reliability  supplementally  to  of  is  subdivisions.  common  variation.  a l l of the reindeer  a l l of  before  the  levels  external  the  outside  while  group  common,  increase suppose  can  of  of  of  three  this  design,  shares  of  common  do  by  levels  a l l of  these  can  reliability  known,  group  of common  Managers  managers  of  of  measure  the  many  each  However,  pairs  subdivisions  level  which  between  contain  variation;  some  method. contained,  shared  variation;  common  this  simulations  variation  world  of  of  cross the if a the  correlation amount group expected  of of  common  to individual  subdivisions  mean  cross  contained correlation  29 coefficient  would  increases  relative  correlation  0.  to  As  showed  average  was  amount noise,  should  that a  the  individual  coefficient  simulations on  be  the  good  of  common  noise  the  mean  cross  increase  mean  indicator  cross of  as  well.  correlation  probability  The  coefficient  of  successful  pairings. Thirdly,  the  data  segments  and  segment.  In t h i s  way  occur  over  similarity overall  can  identification  patterns  of  be  methods  managers time.  similarity  broken can  down be  into  tested  can see i f the same Additionally are  due  to  on  can  or  each  patterns  managers one  smaller  two  of  see i f extreme  years.  CASE STUDY  The  reindeer  similarity  of  reindeer  before  the data  improve  parameter  herds  subdivisions  based  can be  used  monitoring  used  as  The on  an  reliability  residuals  with  example  methods  allocation,  must  of of  evaluating identifying  first  be  gauged  4  and 5  i n Chapters  experimental  design,  or  estimation.  Case S t u d y  The  are  subdivisions.  similar  to  data  first based  Methods  step  as  described  on  known  common  earlier external  was  to  factors.  group I  have  30 little in  information  this  could  about  population  not  general  be  the  and  without  completed  geographic  primary  except  units  causes  this  to  (eg.  external  information,  group  far  of  the  north,  noise  this  step  subdivisions central,  into  southern,  etc.). The  second  correlation  The  a l l 56  growth  as  pairs  correlation  The time.  was  designed. 1  simple  periods. other  to  data  set  i i , and  years  herd  For  herds  time.  between  all  of  cross  and  comparing compare was  cross  correlation  12  correlation a l l other  each were  herd, ranked  similarity  patterns  similarity  patterns  the  split  up  through  f o r the two  Cross herd  mean  over  distribution  mean  of  was f i t of  residuals  calculated  cross  pairs  estimates  22  The the  possible  56  of  were  involved test  The  f o r each  each  and  mean  f o r simulations  in  series  the  calculated.  step  earlier.  between  56  subdivisions.  also  through  calculated  all  A  all  earlier  coefficients  third  calculate  resulting  as  of  was  over  years  used  coefficients  coefficient  time  well  correlation  possible  done  model  to  among  subdivisions,  rate  Cross  was  coefficient  subdivisions. to  step  time  into  23.  one  periods  by  as  were one  a l l correlations separately  halves,  Residuals  coefficients herds  two  for  were  had  been  calculated  f o r the two  between the  i t and  two  time  periods. If (pattern  the herds of  had  similarity  shown in  the same  terms  of  structure  common  of  external  similarity effects  31 between  subdivisions)  columns  of  rankings  coefficient  was  rankings  for  the  of  herd.  Each  half  entire  time  series  the  shown  0  only  second 0.16.  value The half 2.6) the  was  of  the  half,  the  the  be  2.7)  varied  over  similarity  produced one  a  for  compared  of  half  of  had  each  to  the  rank  series  to  average  of  produced  an  time  the  time  rank  time  no  series  series.  is  is  is  was among  shown  The and  of  0.27.  relationship in  comparing  overall  0.80.  However,  coefficients  series  correlations the  that  is The  centered  variation.  correlation  shown  average  set.  coefficients  common  the  showed  data  value  distribution  the  the  the  mean  of  analysis  correlation  correlation  The  of  an  cross The  herds  time  half  also  study in  average  0.0.  produced  herd's  coefficients,  apparent  presence  first  the  second  2.4.  the  If  correlation  This  was  two  method.  case  of  cross  distribution  (Figure  very  Figure  would  of  the  similarity  showing  When the  of  of  distribution  same  rank  each  periods.  series  the  distribution  in  above  using  A  the  Results  variation  The  comparing  time  periods,  same,  correlation  the  results  structure  time  the  time  rank  of  both  be  two  56  Case S t u d y  The  would  calculated  distribution  common  during  time  compared  to  herds  was  the  Figure the  series  distribution the  0.59.  overall  average  time  2.5. first  (Figure between series  Number in Range 250  Mean • 0.258 StDev • 0.259  • : • -  200  160  100  I  •  50  -.4  -.2  0  I 0.2  0.4  0.6  0.8  Cross Correlation Coefficient -0.2 means the interval -0.3 < X < -0.2 Figure  2.4  D i s t r i b u t i o n of cross c o r r e l a t i o n c o e f f i c i e n t s between p a i r s of F i n n i s h reindeer herds  CO l\5  Number of Herds in Range  -0.2  -0.1  0  0.1  0.2  0.3  0.4  Rank Correlation Value 0 means -0.1 < X < 0.0 Figure  2.5  D i s t r i b u t i o n of rank correlation comparing the f i r s t half of the series to the second half  values time  0.5  0.6  25  Number of Herds in Range  Mean - 0.80 20  16  10  Rank Correlation Value 0 means -0.1 < X < 0.0 Figure  2.6  D i s t r i b u t i o n of ranK correlation comparing the f i r s t half of the series to the e n t i r e time series  values time  Number of Herds in Range  0.3  0.4  0.5  0.6  0.7  Rank Correlation Values 0 means -0.1 < X < 0.0 Figure  2.7  D i s t r i b u t i o n of r a n k c o r r e l a t i o n values comparing the second h a l f of the t i n e series to the e n t i r e time series  0.8  36  Case S t u d y D i s c u s s i o n  Overall, shared The  the  parameter  mean  both  cross  the  found  in The  between  and  the  use  with  coefficient  is  second  the  half  overall  of  clustering  the  and  reindeer  greater  than  of  time  the  structure  data. 0  and  series  of  similarity  very  different  data.  first  This  conditions,  the  methods  the to  structure  series.  of  similarity  i l years  could  but  policies  among  actions  over  included  in  it  is  the  and  be  12  years  changing  due was  period,  as  last  to  more  time  obviously  the  likely  If  model  was  due  more  herds.  the  to  explainable  the  management  about  management  information factors,  time  environmental  changing  known  this  of  could  rather  be  than  in  residuals. The  easier  process if  and  obvious trying analysis  of  policies by  or  year.  differences to  identifying  additional  management herd  correlation  greatly  the  encourage  estimation  first  contributed  the  results  guess of  information.  the  at  similar  information about  the  Obviously, in  i t with  reindeer  were  principle identifying  causation  would  residuals. data,  herds  In  managers  would  be  available causes  much about  of  noise  by  similarity  based  on  be many may  preferred cases, not  as have  over in  my this  37  CHAPTER 3 IDENTIFYING SIMILAR SUBDIVISIONS IN THE PRESENCE OF COMPLEXITY  INTRODUCTION  In  Chapter  identify of  this  among the  2,  subdivisions statistic  usefulness the  In chapter  of  with  analysis  presence  of  three  residuals,  measurement  presence  or  can  encourage  based  on  similar  of  discourage  exist.  In in  the  of  presence  these  factors two  in  this the in  extremely  will  aid in  will  hinder.  actual  identification  set.  model.  autocorrelation  factors  the  gauge  data  subdivisions  subdivisions,  these  to  factors,  of  similar  variation  offered  simple  to  reliability  particular  very  used  shared  complexities  and  One  were  was The  of  any a  identify  error,  absence or  with  additional  of  of  data  sets  similarity  subdivisions. Autocorrelation  is  suggestions  tested.  identification  The  of  be  amount  additional  to  coefficient residuals.  the  measure  many  ability  the  on  was  the  will  similar  Several the  reality,  years,  correlation  depended  subdivisions.  However,  bad  the  correlated  unmodeled variation  with  occurs  the  components operate  on  noise  when in  involved a  cycle  or  the  noise  in  any  previous  years.  If  in  population  other  nonrandom  given any  of  change pattern  year the or over  38 time,  a u t o c o r r e l a t i o n is present  that  does  not  correctly  environmental variations factors  represent  phenomena,  (Mysak that  previous  i n the residuals  1986),  create  states  for  are  noise  such  as  the  of  to  are  many  in  predators,  model  components.  example  thought  any  oceanographic  occur  i n cycles.  cases  prey,  Some  The  dependent  habitat  on  conditions,  etc. Measurement observed  or  estimated  instead  departs  year.  In  effectively. In normal  from  many In  more  extremely,  population reindeer different extremely  than  limit bad  populations  a  may  by  2, noise not  be  random  factor  are  difficult  to  a  sets  be  would  were  rate  could  subject  A  go  from  data  population  of  There  decrease;  a  a  of  extinct.  to  high.  rate  year. of  census  f r e q u e n t l y , or  maximum  limit.  each  be  drawn  but  i n actual  more  The  good  maximum  population  error  size  size  some  t h e case  years.  in  population  population  terms  biological  the a  actual  in data  triple  on  year,  may  good  is  the  measurement  occur  never  the  populations  This  may  when  actual  of chapter  increase can  not  cases  distribution. years  is  cases  t h e model  Bad  occurs  the  these  sets.  this  error  is  a  in  an  Because  of  skewed  noise  distribution.  In estimate identify  this the  chapter effects  similarity  of  Monte of  Carlo these  simulations complexities  subdivisions.  The  will on  presence  be  used  to  methods  to  or  absence  39 of  these  the  use  elements of  additional for  such  way  an  judge  data  example  complexity  methods.  to  individual  as  of  of  the  This the  sets.  will will  encourage provide  appropriateness  The  reindeer  data  or  managers of  discourage with  these  set  will  will  be  an  methods be  used  methods.  SIMULATIONS  Simulation Methods  The for  the  to  population models  estimate  populations  used  the with  in the  shared  first  was in Chapter  Wi  ,i  time  Vt.i  noise year  + <nj  f i  t  this  i n chapter chapter.  of  2  Simulations  correctly  will  identifying  the basis be  used  pairs  of  effects.  was of  generated  each  in  simulation  the was  following generated  way: as i t  2 :  = exp ( Y i  In  in  used  probability  Autocorrelated noise  model  following to t h a t  = « Yt-1 , i  +  + Ki)  years  a t time  e  n,i  a  term  relating  t - i i s added:  population  size  at  and similarly :  *1t,i-J  = « *nt-i,i-J  +  €  v  t,i-j  where: a  = autocorrelation coefficient  e = square root of ( i - a*) <f>t,i = random normal deviate f o r time t in subdivision i v  t,i-j  = random normal deviate f o r time t f o r subdivision pair i . J  As  a  increased. and of  the a  values,  is  The common  ranged 1000  increased,  the  amount  individual  standard  standard  deviation  from  trials  Measurement  -0.4 were  error  to  0.4.  performed  was  of  deviation was  set  For in  modeled  = t , i * exp N  (e  t f i  t,i  =  to  each  the  Observed population a t time t i n sub. i  to  is O.iO  The  level  combination  using  )  set  0.15.  of  where: N  was  groups  from Chapter 2 with one slight modification:  *t.i  autocorrelation  of  100.  simple  model  41  N  = Actual population a t time t in subdivision i  t i i  calculated as in chapter 2. *t,i  random normal deviate f o r time t  :  for subdivision i For  tests  standard  measurement  deviation  deviation  was  deviation to  of  of  0.45.  was  set  0.10  and  0.15.  The  value  to  each  effects,  set to  t h e measurement  For  error  error  standard  was  the  individual  common  for  then  deviation  the  standard  the  varied  value,  standard from  1000  0.00  simulations  were run.  Simulating two  steps  drawn  number  an  a  a  bad  set to  was a  that  was  year  bad  simulations.  drawn  from  term.  noise  terms  as  This  well  probability  holding  the  as of  low  to  individual  noise  bad  year  standard  of a  of  the  distribution  took  random  bad  the a  year  for that random  bad  noise  year  term f o r  as i n the other  value  place  the  If  was  determine  term  level.  a  required  number  If  noise  and  received  individual  order  the  procedure  a  random  probability  normal  years  years  occurred.  and  the  a  a in  did not occur,  A l l bad  noise  while  than  bad  to t h e p r o b a b i l i t y  predetermined  year  their  The  a  First,  had  occurred  greater  occurring,  year  or equal  year  extremely  distribution  bad  than  of  year.  uniform  less  was  number  simulated  extremely  was  occurring, year  each  from  whether  t h e presence  of  -0.30 f o r  f o r both  common  terms.  was  varied  deviation  0.00 at  t o 0.30, 0.15  and  42 the  common  simulations bad  standard  were  run  deviation  f o r each  at  value  0.10.  of  the  Five  hundred  probability  of a  year.  Simulation  Results  Autocorrelation probability  of  more  had  drastic  increased  the  measurement  correctly  Autocorrelation had  and  a  error  identifying  slight  effects.  effect  The  probability  decreased  similar  while  presence  of  both  subdivisions.  measurement  of  extremely  correctly  the  error  bad  identifying  years  similar  subdivisions.  Autocorrelation, the  chances  Each  point  value  of  correct of  of on  100  0.0  the  plot on  on  of  pattern variation  of  as  a  x axis.  of a  0.25, t h e  f o r other  or  correct  axis  correct  versus an  pairing  probability did not  combinations  pairing  the  With  mean the  lowered  (Figure  3.1).  autocorrelation probability  autocorrelation was  0.83.  dropped  affect of  negative,  to  the  value  With 0.76.  results.  individual  and  of  an The This  common  well.  Measurement correctly  the  the autocorrelation held  positive  represents  the y  the probability  autocorrelation sign  recognizing  runs  pairings  whether  error  identifying  greatly  pairs  (Figure  decreased 3.2).  the  probability  Without  of  measurement  0 . 9  n  a  0 . 8 8  -  0 . 8 6  -  0 . 8 4  -  a o •  0 . 8 2  -  0 . 8  -  0 . 7 8  ~  0 . 7 6  -  0 . 7 4  -  DD  0 . 7 2  -  •  0 . 7  -  • •  a a a  a  • • • CO  • • cn m  •  CD  eq  a a  c •  Q  a•  •  • m  DOC ]  •  a  •  o  a  D  •  • •  on a • an m a on a • a  DO • CO  mn  a  m •  a  a  • a  •  D  •  • • a •  • an  0 . 6 8  -  0 . 6 6  -  0 . 6 4  -  0 . 6 2  -  a a  • •  a •a • •  •  -  0  a T  .  5  -  0  .  3  - 0 . 1  — I — 0 . 1  Mean Autocorrelation Value  Figure 3.1  Effect of autocorrelation on the probab i l i t y of correct p a i r i n g s  0 . 3  —E3B0 . 5  Figure 3.2  Effect of measurement error on the probability of correct p a i r i n g s  45 error,  the probability  standard  deviation  probability  of  probability  common  The  and  from  rose  from of  pairing  measurement  error  pairing  quickly  This  of  As  the  to to  of  0.82. only  to  o f f before  also  held  With 0.10  a  the  0.53.  The  i t leveled  f o r various  at  values  variation.  extremely of  bad  years  correctly  probability  O.i, t h e  of  This  pattern  and  individual  to  the  model  identifying  an  probability  0.6.  common  was  dropped  drop  pattern  probability  0.0 0.5  correct  individual  the  rose  values  to  0.20.  3.3).  a  correct  addition  improved (Figure  of  continued  approximately of  of  extremely  of  a  occurred  pairs  bad  correct for a  year  grouping  variety  of  variation.  Discussion o f S i m u l a t i o n R e s u l t s  The additional estimating that  for  similarity  populations noise  would  not  is  ways  reliable  their  hand,  of  subdivisions  provide that  results  the  managers of  sharing  that  provide  encouraged  simulations  of  this  gauge  chapter  the  little  variation  common  information. large modest  reliable  correlations  f o r populations  to where  of  amounts  results  as  2  This  amounts  provide  reliability  Chapter  have  or even  to  in  subdivisions.  similarity  terms,  use  the  concluded would  chapter  identify extremely  not adds  autocorrelation in  of measurement well.  of  On  similar bad  the  error other  subdivisions years  affect  0.81  Probability of Correct Pairings  0.7  0.6  0.6  0-  0.05  0.1  0.15  0.2  0.26  0.3  Probability of Bad Year Figure  3.3  Effect of extremely bad years on the probability of correct p a i r i n g s  4*  47 groups  of  subdivisions.  The  presence  fundamental  the  autocorrelation  assumptions  coefficient reason  of  to  be  a  implied valid  the correlation presence  world  will  of  not  their  level  autocorrelation effect  simulations have  little  interacts even  with  modest  term.  on  Populations  candidates  acts  for  of  an  the  well  the  in real  autocorrelation and  average  indicator  of t h e  analysis.  levels  to  mask  of  of  From  the  autocorrelation  variation  the with  the  the true  variation, present  of measurement  to  as  small  component  the true  due  in  this  reliability.  error  amounts  undetectable  used  For as  distribution  of  that  perform  level  violates  correlation  similarity.  Subdivisions  The be  terms  standard  not  same  reliability  effect  additional  the  can  appears  Measurement This  of  does  residuals.  on  it  measure  a l l have  noise  the  autocorrelation.  in  overall  for  coefficient  present of  in  error,  additional, inaccurate  correct  in  population by  definition,  the  common  individual, censuses  size.  data.  With  noise  becomes  multiplicative  would  identification  be  of  poor  similar  subdivisions. The increases years  addition the  occur  of  amount in  increases  decreases.  If managers  to  groups  of  and  of  bad  common  individual  variation  common  extremely  the  discover  years  to  variation  subdivisions probability extremely  subdivisions,  the  subdivision present. only,  If bad individual  of  correct  bad  years  prospects  pairs  of  pairings that  are  correctly  48 identifying  similar  subdivisions  are  improved.  CASE STUDY  The the  reindeer  three  data  factors  discussed  Autocorrelation residuals  of  each  the  of  calculating at  ahead  (time  each  distribution small  56  time  shown  of  count  in  of  The  a l l animals  data by  of  t) and  mean  value 3.4.  and  using  the  Autocorrelation  for  was  one  was  -0.21  On  average  should  have  assumed  to  set.  Reindeer  managers  them  into  the  values  is  herding  estimated  between  =  autocorrelation  this  2.  residuals  Figure  error  to each  calculated  coefficient  (time  in  were  Chapter  series  t+i).  Measurement existent  from  step  i n regard  above.  correlation  is  amount  herds  time  =  examined  coefficients  the 56  the  values  set was  be  by  residual time  step  and this  little  the is a  effect.  virtually  counting  are  non-  able  corrals  to  (Anon.  1987). In across first  order herds,  each  herd  in  that  herd.  compared  residuals a  determine  the  normal  was  The  aggregated  distribution  calculated  standard  calculated,  The to  the  residuals,  standardized.  of  then  to  a  normal across  distribution  using  deviation  then  distribution  of  divided  herds. Chi  The  residuals 2,  of  the  residuals  each  residual  into  (Figure residuals  Square  the  chapter  standardized  distribution  a  in  of  test  were  residuals 3.5)  was  for  d i d not (X*  =  all fit  101.4,  Number of Herds in Range 14 i  -  .8  -  .4  0  .4  .8  Autocorrelation Value 0 means -0.1 < X < 0.0 VO  Figure  3.4  Distribution of autocorrelation reindeer herd residuals  in  140  Number in range  -3  -2  -1  0  1  Standardized Residual Value Actual data Figure 3.5  Normal values  Comparison of reindeer standardized residuals to a normal d i s t r i b u t i o n  cn o  51 P  <  The  0.01, df points  7.48,  p  a  the  27).  were  not  <  removed, to  =  .01,  and  the  normal  p  <  points,  (Table  bad  3.1). central  Twenty extremely fifty  three  the  bad  points  year  0  (X*  and  and  of  (X* = then  compared This  (Figure  the  0  were  test.  removal  time  3.6)  the  (X* =  50  lowest  residuals  =  0.068,  suggest  the  of  illustrates the  were  fifty  were  before  regions  of  southern region,  of  than  skewed.  p  <  were 0.75, df  presence  of  data.  affected 3.7  of  restandardized  the  be  points  Chi Square  of  to  mean  lowest  strongly  residuals  northwestern  the  a  With  in the  Figure  and  50  was  mean  results  during  Low  27).  the  years  about  The  using  appeared  restandardization,  years  3.1).  residuals  =  about  Bad  l).  f i t better  the  These  extremely  =  0.05, df  symmetrical 1).  symmetrical  distribution  and  distribution  distribution  distribution  45.95,  =  df  The  regions  lowest  i n twenty associated  population  regions  extremely  evident  year  the  bad  14  in  the  residuals different  with  only  case  study  displaying (from  Table  year  10  in  the  year  6  in  the  northeastern occurred  herds. one  low  years  during  primarily,  range  during  Only  herd  region.  3  in  one  of the  a  unique  year. The  analysis  of  encouraging  f o r the  application  chapters  and  A u t o c o r r e l a t i o n was  was years  4  assumed  to  occurred  5. be in  the  virtually common  of  data  methods  be  looks  very  presented  low, measurement  non-existent,  among  to  set  and  subdivisions.  error  extremely These  in  bad  results  Figure 3 . 6  Comparison of reindeer trimmed, standardized residuals to a normal distribution  53  Table 3. i  L i s t of reindeer herds and years t h a t exhibit the 50 lowest standardized residual values  (i9--) Year  63 2  64 3  66 5  67 6  69 8  71 10  75 14  78 17  80 19  82 21  Herd  51  8  46  3  11  18  8  45  20  13  52  4  12  19  9  27  5  22  10  28  6  23  12  30  7  24  14  31  9  25  15  32  #  48  26  16  29  23  34  32  35 36 37 39 40 41 42 45 50 54 56 Total of herds  # 2  l  2  6  2  20  \bar e  Figure  3.7  «ar 10  «ar 14  Map of reindeer herds with extremely low residual values during common years (3 years illustrated)  55 together  with  correlations are  of  similar  monitoring  results residuals  appropriate  Chapter being  the  for  4 can  the  will aid  of  chapter  to basis  explore the  allocation.  2  identify of  how  processes  suggest  that  similar  upcoming  use  reindeer  of  herds  applications.  subdivisions of  the  identified  experimental  design  as and  56  4  Chapter  OPPORTUNITIES IN EXPERIMENTAL DESIGN AND  MONITORING ALLOCATION  INTRODUCTION  In to  the  preceeding  identify  subdivisions  factors  and  methods  were  addition, effects  to  Judged  the  of  designing  decide  Managers  in  convenience  the  out  that  until  for  coordinating  selecting respond the  will  of  to  recently,  primarily.  information  among  introduces  to  units: external  differences  controls  not  one  range.  The  for to  this  aid  in  allocation.  and  must  treatments.  units  Chapter  no  In  similar  managers  experimental  were  of  methods  monitoring  The  data.  applications  experiments,  selected  there  reindeer  population  and  as  external  methods.  evidence  examine  use  tto&wrimpftii  to  these  the  the  experiments  experimental  chance  will  have  to  identify  practicality  thesis  similarly  of  management  past  of  provided  regions  subdivisions  and  This  applicable  wnwnft  responses  reliability  chapter  designing  methods  similar  data  thesis  This  which  be  within  management  In  the  reindeer  this  information.  to  chapters,  with  estimate  occurring  rest  two  based  one  systematic  on  pointed approaches  subdivisions. possible  Select factors. caused  criterion  subdivisions This by  would the  for which  minimize  experimental  57 policy  occurring  develop  between  methods  residuals,  with  candidates  for  to  In  subsample In  more  heavily  to  criteria  units  at  approach units,  amount  as  could  be  to  developed  to  select  a  will  similarity being  of  obvious  in  the  in that  are  for  in  thesis  bad  each  or one  subdivisions  a  resources. subdivisions  sampling  design  to  is  to  others  as in  managers in  a  be  index  used  index their could  large  Methods  should  A  alternative  select,  way  years  decision.  An  to  more  possible  selecting  1978).  monitored. which  Many  suggests  This  monitor  this  similar  good  to  units.  making  (Tanner  factors.  monitor  obtained.  index  highly  to  some  subdivisions  strata  important  monitoring  monitor  literature  this  many  need  stratified  which  considered  extremely  subdivisions,  in  will  to  monitoring  external  spot  on  involve  limited  wish  decide  within  subdivisions  to  information  the  approach  potentially of  must  random  due may  of  suggested  response  chapter  subdivisions  managers  others  use  systematic  based  decisions  cases  managers  Managers heavily,  allocation  than  the  This  experimentation.  subdivisions  cases  increase  subdivisions  ranking  many  of  other  rank  top  Monitoring tradeoffs.  subdivisions.  number will  as  be index  units. Methods with  the  into  a  developed  reindeer  few  good  in  this  In  this  data. sets  for  chapter case  the  experimentation  will  be  data and  illustrated  cluster  monitoring.  nicely  58 METHODS  Methods f o r E x p e r i m e n t a l Design  To  select  reindeer  herds  residuals  from the  of  for  experiment  for  were the  list  experiment  appropriate  ten  a l l possible  ranked simple  best  of  in  terms  population  experimental  sizes  size  experimental  of  two  2,  3,  pairings  of  of  their  model  of  units 4,  units,  groups  5  chapter  2.  was  of A  created  units.  correlation  were  of  similarity  (herds)  and  cross  herds  units,  For  an  coefficients  ranked.  Brute force ranking of units f o r small experiments  In  comparing  each  experiment,  three  correlation  correlations  between  rank  all  possible  correlations  involved  one  statistic.  average ranking 3  the of  lowest  This  ruled  to  One  the  a  group  in  possible With  the  where  and  each to  2,  groups this  group do  this  two  of  the  a  on  three  high  3  and  rank  correlations  the  3.  To  three  aggregated would  into be  to  conservative  the minimum  found  unit  the  more  were  measure a  be  aggregation  groups  that  2  herds, to  based  a  involved:  3, and  Instead,  was  similarity  and  had  for were  three  method  possibility  1  of  correlations.  pairwise out  1  group  coefficients  groups  way  three  correlations.  the  herds  possible  ranked in  based the  would were  of the  be very  on  group. given high  59 while  the  also  third  ranked As  size  by  to  two,  all  groups)  had  to  all  possible  an  experiment  be  tested.  was  larger most  of  and  ranking  groups  to  the  of  subdivisions.  amount  of  time  56  For  had  to  find  an  experiment herds  be  possible  necessary  the  was  three,  tested.  groups  to  of  (1540  of  subdivisions  would time  number  be  possible  subdivisions  necessary  were  the  size  to  potential  of  above  2  experiment  32,468,436  testing The  increased,  (27,720)  of  and  involving  an  number  compared 20  For  six,  four  increased.  herds  of  size  the  experiments. similar  3  of  value.  groups  tested.  size  decreased,  also  possible  be  Groups  experiment  test  groups  If  low.  correlation  required  groups  of  very  minimum  the  possible  was  For  had  to  to  test  for  even  find  the  determined  and  most  similar  groups  Hierarchical clustering to rank units f o r larger experiments  In highly  order dissimilar  before based  to  testing on  cluster,  groups is  similarity thus  Hierarchical subdivisions  decrease  begun. in  reducing  the  on  Subdivisions and  number  of  analysis  a  numerical  coefficients.  Individuals  to  allowing  clusters  number  subdivisions  residuals  cluster based  of  the  are  groups  then  groups  to  be  ruled  could could  be  ranked  potential  measure,  be  such added  identified  test, out  clustered  within  groups  identifies  continually to  of  each  to  test.  groups  of  as  correlation  or  subtracted  both  at  the  60 course  grain  and  Within exist:  groups  in  correctly method  model  in  "fuse"  best  would  divisive  into  Carlo  to  be  individuals  and  group  would  known  analyses  the  Monte  method  of  steps;  overall  remain.  which  the  of  created  each  correctly 7-8)  similar  the  had  be  finer  simulations  work  used  a  model  common was  identify and  of  best  at  similar. to  The  cluster  common  the  the  the  of  component  of  groups  of v a r i a t i o n  if  pairs  1-2,  the  of  four  to or  three  subdivisions component  four  3-4, as  be  Pairs  5-6, and  well.  The  methods  similar  quads  of  (subdivisions  variation.  clustering  designed  to t h e  had  individual  (subdivisions  see  similar  Eight  group  component to  two.  an  each  was  subdivision  of  having  four  tests  each  instead  eight,  a  Carlo  except  each  group  had  the  2,  with  5-8)  i n Monte  variation  Within  and  each  used  Chapter  variation.  7goal  could  (1-2, 3-4,  designed  t o be  (1-4, 5-8). The  or  the  i960).  types  progressive  subdivisions  worked  components  5-6,  test  two  which  of  individuals  to  (Everitt  herds.  The  within  series  grouping  reindeer  were  a  only  that  model  clusterings  partition  designed  grain  methods,  which  until  were  of  hierarchical  groups  clusterings  8)  t h e fine  agglomerative  into  1-4  at  agglomerative  method  single  link  method.  containing  one  subdivision,  In  used  the were  was  t h e "nearest  beginning, used.  The  eight two  neighbor"  groups, most  each  similar  61 groups  were  group  and  simplify  all  the  similarity groups most  joined.  were  stored). were  divisive  subdivision dissimilar group.  and  the  all  The  the  new who  to  the  this  criterion,  were old  the  each  agglomerative Everitt In  only  the  the  two  was  again  1  group  gained  was  first  When  to no  was  seven.  of  group  on  a  by  new  adding  than  they  the  two  newly  until  8  groups  subdivision. used  most  subdivisions f i t  continued  methods  The  (the  members new  for  correlations  member  repeated  1  all  additional  process  divisive  the  contained  calculated  correlation  additional  similar  which  remaining  the  This  group  using  became  process  the  group,  average  containing  and  one  lowest  group.  of  a l l other  matrix  correlation  and  more  to  values and  until  with  the  group  sub-groups.  existed,  began  within  order  groups,  similarity  new  subdivisions).  subdivision  were  created  8  group  remaining  this  in  highest  new  continued  average  subdivision)  subdivisions  Both  are  reviewed  the in  (1980). each  correlations The  Joined  An  with  the  seven  method  subdivision that  of  between  calculated (the  the  subdivisions.  between  matrix  process  values  were  Of  (containing  The  each  groups  members  This  remained  similarity  similarity  similar  8  other  between  simplified.  all  New  run were  clustering  coefficients.  The  of  the  Monte  calculated  proceeded program  between  using kept  Carlo  the track  simulation, the  eight  matrix of  residuals  how  of many  and  subdivisions. correlation pairs  and  62 quads  were  simulations  correctly  were  deviations.  For  deviations,  the  The  Possible  of  to  was  the  test.  the  groups  two  clusters.  the  The  units  results  were  best  using  produced  three  clustered  using  on  standard  within  cluster  clusters  described  without  possible  were  ranked  units  the  within  of  The  ranked  were  more  simulations.  above.  and  analysis  the  ranked  number  experimental  earlier  The  standard  the  and  the  combined  potential  of  the  tested  method  were  combinations  based  reducing  cluster  process.  times.  were  groups  the  of  100  correlation  results  of  methods  Possible  each  identify  run  greatly  minimum  within  number  subdivisions  clusters,  groups  the  model  in  combination  experimental  several  to  for a  each  reindeer  dependable  using  run  identified  in  top order  among a l l compared  aid  of  to  cluster  analysis. In  order  over  time,  time  series.  half  of  to  see  cluster  the  analysis  The time  if  ten series  identify  coefficients to  a  pre-set  among level.  was  performed  similar  were  possible reindeer A  similarity  most  Methods f o r M o n i t o r i n g  To  the  list  relationships on  pairs  each of  half  herds  changed of for  the each  compared.  Allocation  index herds was  units, were  made  cross  systematically showing:  correlation compared  a l l herds  with  63 a  pairwise  cross  correlation  level;  t h e herds  with  done  for  these  lists  i t was  index  units  i n order  which  correlation  value this  values  easy  value  of  to  greater was  0.65,  pick  out  to represent  than  a  shared.  0.75,  and  t h e herds  t h e maximum  pre-set This  was  0.85.  that  From  could  number  be  of other  herds.  RESULTS  E x p e r i m e n t a l Design  herds  Lists  of  were  produced  Divisive to  clustering  test,  cut  identified herds of  as  limited  down  found  herds  and  the  time  similar  to  the  the  be  of  similar  f o r management  number  of  required groups.  spatially  groupings  the most  highly  experiments.  potential  groupings  substantially, In  adjacent.  reindeer  general, Both  similar  and  similar  the  clusters  herds  changed  time.  Results  of  analysis  herds)  are  56  the  top t e n candidates  Figure  reindeer  the  the  close  containing  candidates  a l l highly  were  over  t h e groups  4.1. to  Most  one  subdivisions  of  made  presented  f o r two  t h e groups  another.  are  (using  The up  of  10  unit  a l l possible in  experiments  a r e made most  various  Table  up  similar  of  groups  4.1.  Five  of  a r e shown i n herds  located  groupings  combinations  of  of  of a  4  small  Table 4.1 Ten closest groupings of reindeer subdivisions based on residual similarity Groups of 2  Rank  Herd  1  Herd 2  Correlation  1  37  42  0.91  2  24  26  0.88  3  37  39  0.87  4  18  27  0.87  5  3  4  0.85  6  36  42  0.85  7  37  50  0. 84  8  24  41  0.84  9  27  32  0.84  10  25  26  0.83  Groups Of 3 Minimum Pairwise Rank  Herd  1  Herd 2  Herd  3  Correlation  1  37  42  50  0.82  2  37  39  42  0.82  3  36  37  42  0.82  4  18  27  32  0.80  Minimum Pairwise Rank  Herd  1  Herd 2  Herd  3  Correlation  5  37  42  56  0.80  6  26  39  42  0,80  7  35  39  42  0.79  8  35  37  42  0.78  9  35  37  39  0.78  10  25  26  39  0.78  Groups of 4 Min . Pairwise Rank  Herd  i  Herd 2  Herd 3  Herd 4  Correlation  1  35  37  39  42  0.78  2  26  37  39  42  0.77  3  36  37  42  50  0.77  4  26  37  42  50  0.76  5  34  37  39  42  0.76  6  26  34  37  42  0.75  7  26  34  39  42  0.75  6  26  34  37  39  0.75  9  26  35  37  42  0.75  10  26  35  39  42  0.75  Figure 4.i  Five of the ten most similar pairs of subdivisions  67 number  of  subdivisions  It  took  groupings 56  a  4.2)  than  the  of  identified  and  the  component  quads  identified Figure  numeric caused  and  4.3  codes.  the  The  The  to  total  of  results.  4.2).  increased,  pairs  As  As  quads  axis  of  lists  divisive all  between  56  herds  herds  of  represents  the  order  the  The  lowest  spikes  lowest  the  occurs  process,  at  several  Figure remained are  split.  each  at  large  4.5 the  related  The  split  55.  within  of 5  split.  The  spike  After  groups  fifteenth  herds.  highest  illustrates  located  split.  the  occurs first  herds  axis at  15  remained  of  the  The  groups  geographic  Y  major for  regions.  splits  the  spike appear  lists  split splits  of  their  represent  height  of  more  by  The  number  As  clustering  procedure.  closely  easily  were  clustering  most  pair  identify.  the  the  pairs  the  by  between  the  more  to  increased,  Both  both  were  difficult  better  pairs.  spikes  of  of  clustering  worked  identify.  pattern  X  of  (Table  was  more  were  possible  a  cases  pattern  increased,  so  all  from  methods  all  variation  shows  herds.  in  difficult  of  than  divisive  became  check  simulations  variation  variation  to  computer.  Carlo  same  more  of  XT  method  of  similar.  subdivisions  the  the  became  reindeer  time  Monte  component  quad  less 20  that  produced  quads  highly  Compaq  the  component  the  a  showed  individual  easily  of  agglomerative  methods  and  total  using  Results performance  are  considerably  from  (Figure  that  the  i , the in  the  (Figure  4.4).  clusters  that  the  most  part  Of  the  five  Log (Minutes needed to find best group)  1130.0 m i n u t e s t  1.9 minutes  -+>  0  o  6  Number in Group 56 Subdivisions Figure 4.2  20 Subdivisions  Amount of time required to f i n d the 10 top candidates f o r experimental units  CTi CO  69  Table 4. 2 Percentage of simulations resulting i n correct identification of quads and pairs using cluster analysis I n d i v i d u a l Standard Deviation  = 0.15  Pair Common Standard Deviation  = 0.05  Quad Common Standard Deviation  = 0.05  Agglomerative  Divisive  */ of Pairs Identified 0  1  '/. of  0 | 48  Quads  1  )  2  Iden.  2  1  1  2  36  4  3  4 0  3  0  i  3 0  7. of Pairs Identified  0  2  0 j 38  0  0  i  0  2  Individual Standard Deviation  | 1  = 0.15  Quad Common Standard Deviation  = 0.05  Agglomerative  z of  2  3 1  14  3  2  4  0  4  2  i  0  0  1  1  0  Divisive  z of Pairs Identified 1  33  2  = 0.05  P a i r Common Standard Deviation  0  1  3  z of Pairs Identified 4  0  1  2  3  4  0  1  0  0  0  0  47  35  1  1  0  0  0  0  34  16  2 1  0  0  0  0  19  0  1  0  0  0  0  49  Quads  1  1  0  0  0  0  Iden.  2 1  0  0  0  0  70 Individual Standard Deviation  = 0.05  Pair Common Standard Deviation  = 0.15  Quad Common Standard Deviation  = 0.10  Agglomerative  Divisive x of Pairs Identified  v- of Pairs Identified 0  1  2  3  4  0  1  2  3  0  1  0  0  0  0  25  0  1  0  0  0  0  26  Quads  1  1  0  0  0  0  35  1  1  0  0  0  0  35  Iden.  2 1  0  0  0  0  40  2 1  0  0  0  0  39  •/. of  Individual Standard Deviation  = 0.05  Pair Common Standard Deviation Quad Common  = 0.05  Standard Deviation  = 0.15  Agglomerative  Divisive  */ of Pairs Identified  z of Pairs Identified  0  1  2  3  4  0  1  0  0  0  0  0  Quads  1  1  0  0  0  0  0  Iden .  2 1  0  2  13  25  60  X of  0  1  2  3  4  0  1  0  0  0  0  0  1  1  0  0  0  0  0  2 1  0  0  6  17  75  71  Individual Standard Deviation  = 0.05  Pair Common Standard Deviation  = 0.10  Quad Common Standard Deviation  = 0.15  Agglomerative  Divisive  of Pairs Identified  '/ of Pairs Identified  0  1  2  3  4  0  x of  0 |  0  0  0  0  0  Quads  1 |  0  0  0  0  3  Iden.  2 1 0  0  0  1  96  0 1 | 2  1  | 0 0  1  0  0  0 0  2  0 0  0  3  4  0  0  0 0  3 97  16 a.  u. o  26  o oc o  36  46  M l l * J4 II 41 J7 M » M » M M t l t l t4 41 40 t l 47 4f St 43 4* J l 4$ 4t It 10 IS 11 17 It 31 14 11 M II 11 14 1 1 X — Set 1 — X X — Set 2 — X X — HERD NUMBERS F i g u r e 4.3  Results of d i v i s i v e c l u s t e r i n g reindeer herds  of  the  4  S 11  I  t  10  tl  Set  • 17  3  M I 44  7 SS St  — X  ro  S4 31 34 It 42 37 M 50 M  * —  M  15 25 23 t i 24 41 40 22 47 4« SI 43 48 51 4S 41 It 10 IS 1» 27 32 31 24 13 X 20 11  Set 1 Figure 4.4  —  X X  —  set 2 Herd Numbers  —  14  1 3 4  X X—  Results of the f i r s t 15 s p l i t s of d i v i s i v e  S 12  1  2 20 21  Set 3 clustering  » 17 It  ( 44  7 SS 52  —  X  Figure 4 . 5  Major clusters of herds remaining at the 15th s p l i t of d i v i s i v e clustering  75 clusters  illustrated  primarily coast, the  in  one  the  The groups factors. for  in  This  herds  4.3)  on  set  consisted  15th  split.  The  well  as  proximity  or  9  Most  range.  The The  were  group  than  of  cluster  occurs  central  western  in  the  center  third  of  sets  similar  2  3,  as  groups  sizes  noted  in  of  and  groups  4,  to  5,  similar  identified  for  3  (Table set,  from  in  other  within  a  sets.  remaining  18  and  (Table the  overall top level  Figure  the the  the  within  higher  to of  overall  units  The  geographic  portion  4.3  the herds  region  the  Figures  the  4.1).  had  of  The  9  clustering  central  (Table  by  of  close  of  in  split.  undivided  northern  6  candidates  15th  produce  and  external  formed  cluster  the  herds  most  southeastern top  the  compared 2,  a  consisted  in  groups  and of  set  are  the  find  groupings  were  remained  in  to  potential  of  to  selecting  terms  are  occur  for  located in  order  similarly  at  that  herds  in  herds  consisted  herds  which  of  evident  herds  set  these  three  the  the  the  occur  find  sets  proximity  The  of  sizes  i , the  20  close  identified  matched  set  in  identified  groups  of  used to  clusters  additional  most  experiment  two  reacted  and  second  range. most  on  clustered  was  Three  the  first  herds,  and  have  information  regions.  reindeer  one  occurs  were  that  alternative  herds.  one  north,  experiments  based  figure,  range.  herds  larger  as  the  reindeer  of  the  southeast,  occurs  reindeer  in  of 4.4.  three  sets  lists  for  4.4).  The  three  sets  population groups of 4.6  the  for  within  similarity shows  for  Table  Set  4.3  i  Three sets of herds used for f u r t h e r ranking of experimental units Set  3  10  1  22  11  2  23  13  3  24  15  4  25  16  5  26  18  6  33  27  7  34  26  8  35  29  9  36  30  12  37  31  14  39  32  17  40  43  20  41  45  21  42  48  38  46  49  44  47  51  52  50  53  55  56 #  of herds  Set  19  54  Total  2  18  77  Table 4.4  Top candidates f o r experimental units within sets  Groups of 2 Minn Herd  St i  St 2  St 3  1  Herd 2  Herd 3  Herd 4  Herd 5  Herd 6  Corr  37  42  0.91  24  26  0.88  37  39  0.87  18  27  0 . 86  27  32  0.84  18  32  0 .80  3  4  0.85  20  21  0.82  4  5  0.72  Groups of 3  St  1  37  42  50  0.82  37  39  42  0.82  36  37  42  0.82  78 St 2  St 3  18  27  32  0.80  27  31  32  0.66  27  29  32  0 .66  1  3  4  0.66  1  4  14  0. 56  6  20  21  0. 55  35  37  39  42  0.78  26  37  39  42  0.77  36  37  42  50  0.77  13  27  31  32  0.60  27  29  31  32  0.60  13  18  27  32  0.59  1  3  4  14  0.53  1  9  12  14  0.51  9  14  20  21  0.48  Groups of 4  St  1  St 2  St 3  79 Groups of 5 St  i  St 2  St 3  26  34  37  39  42  0.75  26  35  37  39  42  0.75  36  37  42  50  56  0.72  18  27  29  31  32  0.58  13  18  27  31  32  0.58  15  18  27  29  32  0.53  1  3  4  9  14  0.48  3  4  9  14  20  0.45  4  6  7  9  20  0.41  26  34  35  37  39  42  0.70  25  26  34  37  39  42  0.70  23  24  25  26  42  50  0.69  10  15  16  18  29  32  0.44  27  28  29  30  31  32  0.38  15  27  28  29  30  32  0.38  1  3  4  9  14  20  0.38  3  4  6  9  14  20  0.37  4  6  7  9  14  20  0.37  Groups of 6 St  1  St 2  St 3  Figure 4.6  81 each  of  the  three  sets,  the  most  similar  herds  for  a  6  unit  patterns  of  experiment. Cluster similarity (Table  between  4.5).  halves,  but  area  compared of  of  the  time  two  were time  to  clusters  10  of  half  of  with  the  the the  second  pairs  the  of  the  were  similar  over the  from  the  from  ranges time  between  similar  the  region  calculated from  the  pairs  of  series.  southeast  herds  two  formed  greatly the  series  the  wider  pairs  while  time  in  entire  varied  of  time  Clusters  similar  series  majority  half  reindeer  using  Herds  time  the  appeared  found  4.6).  the  comprised  series  (Table majority  of  exception.  similar  the  different  groupings  most  comprised  region  halves  the  the  series  first  produced  small  these  half  Lists  the  Several  each  two  analysis  with central  calculated  series.  Monitoring Allocation  As decreased, so  did  the  level  the  number  the  represented  number  (Table  cross  correlation  There  were  of  nine.  being  the  of  potential  of  herds  Nine  coefficient  several If  correlation  4.7).  the  represented,  represent  of  ways goal  to was  four  remaining  index  which  herds  had  greater select to  index 5  required  units  at  one  equal  units  from  maximize  the  number  units  would  herds.  Forty  units and  potentially  least or  index increased  could  than  index  of  be  three  be  pairwise to  0.85.  this  group  of  herds  selected herds  to were  Table 4.5  Tears  Clusters of herds remaining at the 20th split for each half of the time series  1-12  Cluster i  16,  18,  22,  23,  24,  35,  36,  37,  39, 40,  3,  4,  5,  Cluster 2  1,  Cluster 3  12,  15  Cluster 4  38,  48  Cluster 5  19,  31,  Cluster 6  14,  55  Cluster 7  6,  Cluster 8  25,  29,  32,  41, 42,  50,  56  30,  34,  53  13,  20,  21  33, 45,  46,  54  17,  44  Cluster 1  13,  15,  18.  19,  27,  28,  Cluster 2  i .  3,  4,  7.  14,  16  Cluster 3  6,  Tears  27,  9  7,  Cluster 9  26,  13-23  55  Cluster 4  35. 45  Cluster 5  20.  Cluster 6  36 . 40  Cluster 7  31 , 39,  Cluster 6 Cluster 9  21 . 22 , 47  56  8,  10.  12, 4 i , 46  17. 23,  24,  25.  2,  32,  26,  29,  33 , 42,  54  34,  Table 4. 6  Years  Herd  i  Ten most similar pairs during each half of the time series. 1-12  Herd 2  Years  Corr  Herd  1  13-23  Herd 2  Corr  3  4  0.95  27  30  0.95  37  39  0.94  18  30  0.94  36  42  0.94  26  54  0.94  37  50  0.92  23  24  0.93  18  21  0.92  29  54  0.93  40  42  0.91  28  34  0.93  39  50  0.91  18  32  0.93  24  41  0.91  18  27  0.92  33  54  0.91  37  42  0.92  39  40  0.90  30  32  0.92  Table 4. 7  Correlation >  Potential index units  0.85  Index Unit  Herds  Herd #  Represented  3  4  18  27  24  26  37  39  Correlation >  42  0.75  3  4  20  21  26  23  24  25  32  15  18  27  34  19  23  34  35  Correlation >  Index Unit Herd  #  0.65  Herds Represented  4  1  9  17  10  16  £0  21  22  47  32 42  3  5  15  18  27  29  30  31  19  23  24  25  26  34  41  50  51  54  56  35  36  37  39  40  86 involved 0.65.  in  In  at  this  4.7  shows  0.65  and  least case  7  of  the  four the  one  herds  units  would  same  geographic  pairwise  herds  would  potential  that  primarily  relationship  they  represent  at  represent  29  index  units  would  be  other  the  level  others.  at  the  Figure level  representing.  herds  located  of  of  Index  within  the  region.  DISCUSSION  Reindeer ones  that  years.  herds  were The to  The  half  the  subdivisions  of  in  the  chapter  affected  many  of  central 14  and  the time shown  portion 21.  correlation  in  of  the  experimentation  units  results  simulations wishing  that  is  3.1,  an  year  10.  bad  of  chapter  to  select will common  over  the  time.  years  among  reindeer bad  second  occurred  year  half  subdivisions  estimate  highly  time  similarity  The  in  of the  in  years  considering  that  similarity  sensitive  bad  over  years  surprising, to  were  extremely  extremely  among  in  faced  bad  produced  not  used  of  portion  where  are  monitoring have  in  similarity  similarity of  similarity  range  residuals  or  of  series  Table  coefficient  Managers  subdivisions  the  These  series  3,  highest  presence  southeastern  produced of  the  location  time  these  series  the  pattern  the  From  time  by  changing  range.  the  showed  influenced  changing  corresponds first  that  to  among  outliers  as  3. similar  in  most  extreme  subdivisions cases years  want in  to the  for use past  Figure 4. 7  Potential index herds at the level of 0.65 shown with the herds that they represent. (For each shading pattern, index herd shown with shading only in center of area.)  88 and  would  likely  Other  managers  more  interested  subdivisions could  be  series  face  may  not  in  with  calculated of  common be  to  less  emphasis  between  spatial  and scales  similar  subdivisions  each  other.  This  to  the  should  effects excluded, For be this  was  appear  true  of  fewer  show  the  may  not  Many  of  of  consistent  may  find  be  similar  correlations of  the  time  the  time  than  the  overall  second  portions that  are  unless  example,  In  of  known these  information  fed  time  to  no  herds  halfway  should  be  different  longer can in  be  the  through  built  halves  into  will  2.  were  managers  time  into  series.  time  the  model  external should  the  region  the  be  model.  began  series, or  of  important  contain  southern  first  pattern  applicable  built  evident.  The  methods,  that  of  appeared  not  available  be  the  management  were  clustering  near  series  groupings  a  series  or  regionalization  chapter  both  the  when  and  from  longest  factors  i f reindeer  supplementally  using  the  the  but  to  time  of  at  time,  herds  others  showed half  use  degree  but  results  series  picture.  reindeer  important  series,  the  next  Smaller  same  effective  enough  environmental  most  time  with  the  analyzed.  the  the  presumably  However,  are  likely  of  similarity  actions  will  effects  of  logs  Given  the  halves  half  To  or  time.  was  is  roots  future.  and  extremes,  over  series  similarity.  This  square  time  and  during  on  the  extremes  variation.  management  most  actions  year  in  in  residuals.  different  contain  years  interested  year  Environmental  entire  as  extreme  to  either  only  the  89 second  half  of  Before on the  a  level  of  Managers  will  representing whether  in  they  subdivisions  and have  a  can  select  fewer  the  fewer  small want  that  decide  number index  are  not  units as  they  the  The  is  higher  subdivisions subdivisions  closely  closely  can  a  related.  units  and  important  the  level  be be  want  related  decide  an  will  they  representing  must  index  There  whether of  used.  units,  between  this.  the  be  index  represented. doing  to  should  needed  being  desired, units  series  similarity  involved  similarity index  time  managers  subdivisions  tradeoff  as  the  available  represented. index  units  subdivisions large  of  number  or of  90  CHAPTER 5 IMPROVING PARAMETER ESTIMATION OF SUBDIVIDED  POPULATIONS  INTRODUCTION  The in  preceding  monitoring  data  allocation  within  introduce  subdivided The  estimation examples. of  This the  In a  procedures  the  known  first  In  estimates  can  explore data.  the  use  with  can  second  be  to of  a  out  the  reindeer  procedure, for  external  these  two  estimated  parameter  book  of  without  the  factors. on  that  This the  to  using  accuracy  subdivisions  methods  of  assumed  the  to  use  of  belong  to  data.  parameters better  first  estimation  Walters'  examples  collecting  the  number  in  provide  be  improved  responses the  will  procedure,  distribution  approach.  similar  chapter  was  by  parameter  behind laid  improvements  design  (1986)  population  is  potential  experimental Walters  theory  procedures  illustrated  and  subdivisions.  procedures.  two  chapter  a  of  Bayesian parameter  have chapter  Finnish  shown will  reindeer  91  BATESIAN  In been  many  "drawn  simplest assumed This one  cases by  case, to  APPROACH  a  parameters  from  population  been  drawn  would  variance.  parameter  poorly  may  from  distribution  tune  values  nature"  have  overall  parameter  This  estimates  estimated  Known  from  have  a  one  to  normal  be  may  be  distribution.  population can  have  In t h e  subdivisions  overall  each  assumed  distribution.  single  distribution from  be  mean  and  to  fine  used  subdivision,  particularly  parameters.  B a y e s i a n Methods  The 299-302  methods in  parameter by and find 0  i f  f o r this  Walters values  nature"  from  variance the with  (1986). for a  a'p.  most the  If  the  single  result  we  are  that 0$,  distribution  theorem value  described  assume  subdivisions, normal  Bayes  probable  approach  0  can for  on  pages  the were  "drawn  with be  each  actual  mean used  0 to  replicate  being:  fii = Wih + {l-Wi)fi  where subdivision:  Wj  is  a  weighting  associated  with  each  92  is;  and around  0  is  which  a  weighted  nature's  estimate  "sample"  8^  of  the  were  mean  0  drawn:  i  The through o g  can  6.  A  ?  overall an  population  iterative be  new  procedure.  can  A  to  calculate  values  value  of  a»g  be  can  be  trial  used  - »=1  ,3  variance  calculated  estimate of  Wj  calculated  of and as:  m  N  »=i This stop  process  changing With  the with  this  assumed large  average.  or  is  repeated  approach method,  natural variances  Parameter  until  the  successive  estimates  zero.  parameter  estimates  distribution. are  adjusted  estimates  with  are  adjusted  Parameter towards small  the  using  estimates population  variances  change  93 little  if  at  If  the  subdivision over is  population  small  relative  degree  estimation for  population  variances,  the  large  all.  of  managers  To rate be  to  exists.  the  of  If  subdivision  subdivision.  This  with  regard  in  is  to  to  the  variation  population  variances,  each  have  parameter  the  Involved  provide  an  example  earlier  for  distributed.  variances  variance  there  the  is  a  parameter  important  information  experimental  design  and  of  this  the  Using  method,  reindeer the  herds  growth  calculated  the was  rate  earlier,  mean  growth  assumed  to  estimates  and  and  o ,  0  fi  estimated. The  to  range  degree  relative  allocation.  normally  were  large  large  uncertainty  calculated  their  a  is  of  to  monitoring  variance  the  Ricker  model  (Ricker  data  in  reindeer  1954,  order  to  Ricker add  1975) a  was  density  also f i t dependence  parameter.  N  t+l  = t * exp (a - b * s  where Nt_  +1  St a,b  Although the  density  S ) t  = population at time t+1 = population a f t e r harvest at time t = parameters to be  the  carrying  dependence  estimated  capacity ("b")  parameter  would  not  that in  represents theory  be  94 identically rate  of  increase  The Pi  distributed  parameter  Ricker  o- e>  the  Bayesian  Ricker  mean  rate  Original rate  (from  a*p growth (Table growth  rate The  than  the  from  the  Figure  5.1 the  were  identical  used  to  estimate  values  values  had  log of a  growth  of  estimated  little  reason to  the  for 0  0j.  for  effect  on the  annual  growth  a*g  original  of  the  was  one  to  the  annual  i.20 t o of  the  outlier  subdivision  parameter to  the  0.28  value f o r  1.53  annual  value  considerably  each  values  a  from  distribution  updated  equal  of i.33 and  ranged  of  for  the the  rate  mean  Estimates  shows  of  the  value  subdivisions  estimates  this  were  the  but  exception  value  For  the  2.24E-03.  with  1.53.  changed  produced  annual  to  5.1).  be.  estimates  slightly  of  2)  to an  rate  intrinsic  term.  estimates  equal  the  Approach  process  Chapter  corresponding  could  recalculate  "a" parameter  growth  ("a")  to  Results o f B a y e s i a n  The  a l l subdivisions,  "a" parameter  and  f  among  of  larger of  a*g.  estimates, fourth  0^,  decimal  point.  The  Ricker  "a"  parameter  estimates  produced  a  value f o r  •»/  0  equal  7.98E-02.  to These  0.66  and  values  a  value  generated  for estimates  o'g  equal of  0j  to only  Table 5.1 Mean growth r a t e values by herd where Pi was estimated as log ( I 4 ) Herd  B  A  a*g  v  exp (Pi)  1  .31  1.21E-05  1.36  2  .28  3.78E-05  1.32  3  .28  2.28E-05  1.32  4  .23  1.02E-05  1.26  5  .19  1.60E-05  1.21  6  .30  1.82E-05  1.36  7  .27  1.90E-04  1.30  8  .25  1.37E-05  1.28  9  .28  2.53E-05  1.32  10  .25  3.46E-06  1.28  11  .20  1.12E-06  1.22  12  .19  7.34E-06  1.21  13  .27  4.45E-06  1.31  14  .28  4.08E-06  1.31  15  .26  6.03E-07  1.29  16  .31  8.52E-06  1.37  17  .31  1.65E-05  1.36  18  .26  4.20E-06  1.30  19  .32  5.10E-06  1.37  20  .26  4.12E-05  1.29  21  .24  1.10E-05  1.27  96 Herd  exp(P ) A  22  .21  1.15E-05  1.24  23  .24  9.61E-06  1.27  24  .25  1.31E-05  1.28  25  .27  7.51E-06  1.31  26  .21  5.95E-06  1.23  27  .26  3.76E-06  1.30  28  .25  1.10E-06  1.29  29  .25  2.52E-06  1.28  30  .23  9.94E-07  1.26  31  .25  4.36E-06  1.28  32  .24  5.06E-06  1.27  33  .26  3.31E-06  1.30  34  .23  1.93E-06  1.26  35  .22  6.60E-06  1.25  36  .23  5.95E-06  1.26  37  .27  7.07E-06  1.31  38  .33  1.41E-06  1.39  39  .30  2.65E-06  1.34  40  .29  2.07E-06  1.34  41  .19  9.33E-07  1.21  42  .22  1.71E-06  1.25  43  .27  7.85E-07  1.31  44  .27  8.64E-06  1.31  45  .23  3.81E-07  1.25  Herd  P  exp (f^)  A  46  .34  1.27E-06  1.40  47  .36  2.04E-06  1.43  48  .35  1.61E-06  1.43  49  .33  7.74E-07  1.39  50  .31  1.30E-06  1.37  51  .36  4.42E-07  1.43  52  .32  1.88E-06  1.38  53  .28  5.89E-07  1.32  54  .30  2.16E-06  1.35  55  .43  1.02E-05  1.53  56  .30  9.24E-08  1.34  Number of Herds in Range 12 i  1.22 1.24 1.26 1.28  1.3  1.32 1.34 1.36 1.38  1.4  M e a n G r o w t h Rate Value Figure 5.1  D i s t r i b u t i o n of the mean growth rate estimated f o r the reindeer herds  1.42 1.44  1.46  99 slightly 5.2). in  different  Parameter the  with  unrealistic "a"  value  of  3.90,  of  small  ranged for  which  a  Ricker  fit  the  This  use  with  model  to  model  produce  N  values  the  parameters. estimate In share amount is  common of  estimated  large,  external is  a  Ricker  model.  of  parameter  (N) to  the  "b"  For  values  example annual  a  are Ricker  growth  rate  of each  Ricker  value  over  involves  time  between  pages  303-306  in  Walters  The  easiest  way  to f i t  subdivisions  would  subdivision. "a" a  total  procedure  of  could  together.  procedure,  all  subdivisions  are  noise  shared  additional  estimation  by  at  be  When  more  2  N «  N to  assumed  to  step.  The  each  year  time  a l l subdivisions  parameter.  should  each  would  attempt  subdivisions  (noise)  to  and  all  effects  be  This  parameter  estimating  estimation  method  for  external  an  these  estimation  on  parameters  as  Parameter  of  effects  number  joint  common  unchanged.  reindeer.  described  the  Ricker  Walters'  adjusted Parameter  maximum  parameter  individually  A  the  for  were  COMMON EXTERNAL E F F E C T S  values  of  a  (Table  average.  Many  populations. to  estimated  variances  remained  1.36.  impossible  common  for  to  possible  subdivisions.  large  population  corresponds is  Another  (1966)  the  0.10  ESTIMATION OF  estimating  with  reindeer  1.36  originally  variances  from  of  those  estimates  direction  estimates values  than  at  common  accurate.  noise If  is  data  Table 5. 2 Ricker "a" values as estimated before and a f t e r Bayesian process Herd  ^  a«o.  n  e  w  p  A  1  0.85  8.94E-04  0.85  2  1.25  2.34E-03  1.23  3  0.93  1.86E-03  0.93  4  0.77  8.76E-04  0.77  5  0.56  2.56E-04  0.56  6  0.67  5.57E-04  0.67  7  0.63  4.54E-03  0.82  8  0.54  1.10E-03  0.54  9  0.59  5.71E-04  0.59  10  0.70  1.23E-03  0.70  11  0.28  8.37E-05  0.28  12  0.44  5.52E-04  0.44  13  0.63  1.17E-03  0.63  14  0.96  1.60E-03  0.95  15  0.45  8.88E-04  0.46  16  0.69  4.70E-04  0.69  17  0.82  1.47E-03  0.82  18  1.00  2.71E-03  0.99  19  0.89  1.32E-03  0.88  20  1.38  2.20E-03  1.36  21  0.80  1.79E-03  0.80  Herd  pi  new  22  0.55  7.61E-04  0.55  23  0.99  1.39E-03  0.99  24  1.18  2.90E-03  1.16  25  0. 86  4.18E-03  0.85  26  0.77  3.23E-03  0.77  27  0.80  4.03E-03  0.79  28  0.56  1.10E-03  0.56  29  1 . 06  6.45E-04  1 . 06  30  0.68  9.67E-04  0.68  31  0. 99  1.49E-03  0.98  32  0.68  2.36E-03  0.68  33  0.86  1.44E-03  0.85  34  0. 67  3.25E-03  0.67  35  0.51  2.35E-03  0.51  36  0.76  1.23E-02  0. 74  37  0.43  2.61E-03  0.44  38  0.53  9.74E-05  0.53  39  0.43  3.16E-04  0.43  40  0. 56  2.28E-04  0.56  41  0.60  2.63E-03  0 . 60  42  0.26  2 . 19E-03  0.27  43  0.24  8.96E-05  0.24  44  0.47  1.61E-04  0.47  45  0.29  4.39E-05  0.29  Herd  6^  new  e  i  46  0.26  1.79E-04  0.26  47  0.77  3.52E-03  0.76  48  0.30  1.68E-03  0.31  49  0.44  1.78E-05  0.44  50  0.08  1.64E-03  0.10  51  0.14  8.79E-05  0.14  52  0.29  5.01E-05  0.29  53  0.21  1.14E-04  0.21  54  0.44  9.67E-04  0.44  55  0.51  5.70E-05  0.51  56  0.27  1.12E-07  0.27  103 were of  available 2  »  N  procedure. causes make  a  time  series  for  parameters  would  be  increased  deterioration  the  method  the  f i t to  where Hi \,  number in  of  are  years,  estimated to  individual  a  total  using be  performance,  than  this  estimated  which area  will  fitting  large.  parameter  population  T+l  parameters  estimation  accurate  effects  individual  is  data  estimation, from  each  the  following  subdivision:  = the population in sub. l at time t  t  s  less  shared  With equation  T  + The  a  unless  over  i,t-l  :  t  n  e  Population in sub. i at time t-1 a f t e r the harvest  a^ = Ricker a parameter for subdivision i bi w  i,t  The term  = Ricker b parameter f o r subdivision l  Joint  consists  where  normally distributed process e r r o r  =  estimation  of  w  two  i , t  =  procedure  assumes  components:  noise In subdivision i at time t  that  the  noise  104 w^  = common effect shared by all replicates during year t  w»i t = an independent effect due to (  conditions encountered i n sub. i  In matrix  order  to  equation  is  calculate  parameter  values  the  used: .1  A : B *d" .B' M. W • • •  w  :  where b = the b parameter ests.  (1 f o r each subdivision)  w  = the mean noise ests.  ( i f o r each y r : i  A  = diagonal matix having the elements: T  Aw^s,-,-*.)' *=1  B  = a matrix having the following elements:  Bu = {Si,t - §i)  NI = the T x T identity matrix multiplied by the number of subdivisions  d  = an a r r a y having the following elements: T  t=i  T)  following  105 W = an a r r a y having the following elements:  t=i  where  = the arithmetic mean of  over  time f o r subdivision i  y  t>t  = ln(/Y/S,-«_x) itt  y\ = the mean of Yi \ over time f o r t  subdivision i  The estimates of a^ are found by:  fi,- = Y - biSi {  The population 1986). the  The  key  parameters  levels  for  variances  the for  for  setting  subdivisions these  parameters  following way:  where: A and B are matrices as defined above  (BB%=^(5, -5,)(5 t=i  i t  i i t  -5 ) y  are  optimum the are  b  harvest A  (Walters  calculated  in  106  £ E(K,t-a,--$,-s,- -«B)  a  >t  5  a  =  ,  =  l  t  =  t  1  NT - 2N - T + 1  Additional Walters  (196*6).  compared  The  using  estimated Joint  information  using  on  this  Individual  Monte  Carlo  t h e reindeer  procedure  and  Joint  simulations.  data  i s provided  with  both  in  procedures  were  Parameters  were  the individual  and  procedures.  Simulations  Simulation Methods  Data the  Ricker  were  generated  model  with  individual  noise  fit  to  both  estimation amount  of  combinations  subdivisions  as  RicKer  common  of  The  s.d. and  s.d. =  for 6  and  were  data  common  For  each  and  were  then  the  Joint  compared  as the  the  number  run  for  s.d. were  using  common  with  deviation  Simulations  subdivisions.  as  were  standard  0.05;  0.25.  and  and  Simulations  with  The  procedures  common =  2.  increased  subdivisions  and  individually  two  noise  of  parameters  curve  noise:  series  i n Chapter  increased.  individual  individual  Known  added  procedure.  subdivisions  with  a  for a  (s.d.) = run  =  0.05  of two 0.25 with  for  combination  4 of  107 noise  components  simulations  were  At were track  of  point  estimate  whether  kept  track  the  range  of  f o r both  order  to  each  point  subdivisions  0.25,  was  of  the  which amount  the  each  and  100  The  or  value  program or  value.  of  was i n  each  In  estimate,  error.  individual  this  It  estimates.  i t s standard  of  Joint  i t s standard  joint  the  kept  the  value  minus  uncertainty  whether  conditions  parameter  the  by  of  parameter  plus  divided  stored  true  and  relative  was  number  estimate  true  individual  the  a  point  estimate  the  simulations  estimation  stated  100  produced  by  became and  standard In  this  than the  common  are  more  the  Walters  simulations  better  of  to  t h e largest  analyzed. much  point  increased  individual  being  whether  of  parameter  results  created,  The  or  joint  term.  Results  Results  as  subdivisions  parameter.  individual  of  calculated  increased,  t h e "b"  closer  estimate  produced  of  simulation,  was  compare  Simulation  of  each  the  the  error  Joint  of  involving  also  method  number  run.  t h e end  compared  program  and  better noise  of  (1986).  =  the  point  0.05,  5.2  and  Table  5.3.  t h e number variation shows  the  deviation 8  procedure.  procedure The  depended  Individual  =  subdivisions  estimation  estimate  present.  as  standard  Joint  individual  in  common  Figure  common  deviation  the  effective  amount  with  case  listed  method on  the  parameter  108  Table 5. 3  Results of individual vs joint estimation simulations  4 subdivisions , a common = 0.05, o' i n d . = 0 . 2 5 J  z of simulations with closest point estimate  Ind.  Joint  59.5  40.5  57.3  6 2.3  67.0  33.0  Ind.  Joint  55.0  45.0  57.8  59.6  63.0  37.0  z of simulations with true value within range of p t . estimate ± s t a n . e r r o r z of simulations with highest value of p t . estimate  - standard e r r o r  8 subdivisions , a* common = 0.05, a* i n d . = 0.25  z of simulations with closest point estimate z of simulations with t r u e value within range of p t . estimate + s t a n . e r r o r x of simulations with highest value of p t . estimate  - standard e r r o r  109 4 subdivisions , a* common = 0.25, a* i n d . = 0.05  z of simulations with closest point estimate  Ind.  Joint  14.0  86.0  57.5  6 7.8  16.5  83.5  Ind.  Joint  12.0  8 8.0  57.6  6 7.4  12.0  88.0  v. of simulations with true value within range of p t . estimate ± s t a n . e r r o r y- of simulations with highest value of p t . estimate  - standard e r r o r  8 subdivisions , a* common = 0.25, a* i n d . = 0.05  x of simulations with closest point estimate of simulations with t r u e value within range of p t . estimate + s t a n . e r r o r x of simulations with highest value of p t . estimate  - standard e r r o r  90 11  0.0004 Parameter estimate (True — •  Joint Estimation  Figure  5.2  —0.00003) +  Ind. Estimation  Result of simulations comparing i n d i v i d u a l to Joint estimation with high level of common v a r i a t i o n  Ill estimation in  cases  more  produced of  often  little  or  minus  the  highest  common  their  value  with  real  noise.  of  closer True  point  produced  potentially  to  the  parameter  the j o i n t errors.  the  also  could data  of  standard  values  errors  This  estimates  i n the range  plus  standard  point  estimation The  method  estimates the  the  value  values  were  point that  divided  closest  indicate  true  values produced  by  point  their  estimates.  appropriate  method  sets.  Case S t u d y  Case Study Methods  The  Ricker  each  herd  using  the  joint that  number  of  four  common herds  and  intermediate were  both  (Table  within  of  using  a  group  herds  both  estimation  Results  within  the  of  t h e population  individual  variation  values  f i t to  procedure.  number  calculated  was  the  estimation  showed  variation  equation  group  each  of  "b"  parameter  not  both  selected.  the three  sets  from  procedure chapter  decreased  increased.  could  were  from  data  and four  as t h e  Since  common  be  maximized,  Joint  estimates  listed  in  chapter  4.3).  Case Study Results  Values  of  the  from  both  individual  and  112 Joint  estimation  variances  were  variances. reasons:  The  joint the  t h e value  standard  involving  primarily f o r the  13  the  involving higher. sets  of  18  Cluster  was  second  parameter in  herds,  16  most  of  set, i n v o l v i n g  herds,  produced  parameter  variation.  cases the  than  same  the set,  values  were  central  t h e 18  by  first  f o r the  of  two  divided  t h e 20  and  the  for  common  primarily  12  individual  estimate  For  higher,  joint  preferred  shared  greater  northern  their  were  selected  were  The  corresponding  statistic.  analysis  on  the  Joint  southern  primarily  based  the  values  5.4.  estimates  individual  higher, of  than  clusters  error  corresponding  i n Table  a l l less  First,  Second, its  a r e listed  herds,  third  set,  values  were  ranking  of  these  valid,  both  similarity.  Discussion  When  the  Bayesian  method  method  can  subdivisions These used  to  populations. estimation to  appropriate and  provide commpared  t h e estimation improved to  individual  two  improve  parameter  Another procedure  for  each  while  common  one a  methods  would  parameter second  subdivision.  which  involve  is  can be  subdivided  i s estimated  Additional  within  procedures.  for  parameter  the  effects  estimates  estimation  approach  external  estimation  of t h e possible  possible where  of  are  parameter  a r e only  a l l subdivisions,  uniquely  assumptions  a  Joint common  estimated  simulations  are  Table 5.4  Estimates of Ricker "b" parameter by individual and joint parameter analysis  Joint Herd  Estimate  Ind. Estimate  Joint Variance  Ind. Variance  i  .95E-04  83E-04  . 73E-09  . 72E-09  2  .16E-03  16E-03  . 10E-08  . 14E-08  3  .59E-03  60E-03  . 26E-07  .35E-07  4  . 12E-03  13E-03  . 20E-08  . 17E-08  5  .15E-03  15E-03  . 19E-08  . 22E-08  6  .20E-03  17E-03  . 36E-08  .54E-08  7  .26E-03  , 25E-03  .26E-08  . 12E-07  6  . ilE-03  .76E-04  . 17E-08  .24E-08  9  . 89E-04  . 99E-04  . 13E-08  .23E-08  10  .61E-04  .59E-04  .35E-09  .63E-09  11  . 17E-04  . 90E-05  . iiE-09  . 13E-09  12  . 55E-04  .51E-04  .95E-09  .iOE-08  13  .95E-04  . 15E-03  . 17E-08  .37E-08  14  . 15E-03  . 17E-03  .51E-08  . 27E-08  15  .iOE-03  .61E-04  .34E-08  .32E-08  16  . 16E-03  . 13E-03  .ilE-08  . 31E-08  17  . 23E-03  .21E-03  .47E-08  .60E-08  18  . 16E-03  . 19E-03  . 22E-08  . 36E-08  19  . 93E-03  . 81E-03  .35E-07  . 76E-07  20  .14E-03  . 17E-03  .95E-09  . 11E-08  Joint Herd  Estimate  Ind. Estimate  Joint  Ind.  Variance  Variance  21  .46E-04  52E-04  . 35E-09  . 36E-09  22  .49E-04  72E-04  .30E-09  . 13E-08  23  . 10E-03  14E-03  .55E-09  . 14E-08  24  . 26E-03  40E-03  . 33E-08  . 10E-07  25  . 15E-03  . 16E-03  . 16E-08  . 51E-08  26  .41E-03  65E-03  . 26E-07  .78E-07  27  . 87E-04  , 12E-03  . 20E-08  . 37E-08  28  . 34E-03  25E-03  . 22E-07  . 23E-07  29  . 17E-03  . 21E-03  .21E-08  . 17E-08  30  .49E-03  .42E-03  . 32E-07  . 28E-07  31  .58E-03  .58E-03  . 16E-07  . 23E-07  32  . 31E-03  .45E-03  . 23E-07  .53E-07  33  . 26E-03  . 34E-03  . 63E-08  . 11E-07  34  .39E-03  .30E-03  . 16E-07  .27E-07  35  . 33E-03  . 19E-03  .68E-08  .23E-07  36  . 35E-03  . 30E-03  . 12E-07  . 38E-07  37  .28E-03  . 10E-03  .69E-08  . 25E-07  38  . 23E-03  . 15E-03  . 19E-07  .67E-08  39  .42E-03  . 15E-03  . 13E-07  . 27E-07  40  .43E-03  .25E-03  .97E-08  . 15E-07  41  .26E-03  .26E-03  . 19E-07  .22E-07  42  .22E-03  . 18E-04  , 12E-07  .22E-07  43  .82E-05  . 17E-04  . 17E-08  . 16E-08  Joint Herd  Estimate  Ind. Estimate  Joint  Ind.  Variance  Variance  44  . 17E-03  14E-03  .59E-08  .61E-08  45  . 71E-04  31E-04  . 26E-08  . 16E-08  46  . 12E-03  IiE-03  .15E-07  . 23E-07  47  . 34E-03  23E-03  . 12E-07  . 19E-07  48  •41E-04  28E-04  .65E-08  . 93E-08  49  . 18E-03  ,IiE-03  •52E-08  .43E-08  50  . iiE-03  ,14E-03  . iiE-07  . 15E-07  51  .74E-04  , 92E-04  . 36E-08  . 13E-08  52  . 28E-04  ,57E-04  . 19E-07  .98E-08  53  . 58E-04  ,76E-04  . ilE-07  .84E-08  54  . 80E-03  .35E-03  . 91E-07  .20E-06  55  . 13E-03  .91E-04  .64E-08  .75E-08  56  .57E-04  .36E-04  . 15E-08  . 18E-09  116 needed  to  better  methods.  Additional  methods  a r e used  As herds  variances  the  rates  were  of  of  the  surplus  population  each  higher  population  provide  greater  RicKer  model  the  reindeer  to  density  dependence  some  curve cases,  population  carrying  growth  rate  This  was  the  parameters  population order a  to  values  true  of  were  up  herds,  achieved  different  increased  of  the southern  much  faster  through  biased  curve  upwards  see i f  order  to  meaning  when  i f any,  set.  Values  f o r the  of  curve  fits.  negative the  herds.  In  after  cases  at  possible  points. the  The  a  grew.  other  increased  data  In  showing  population  biologically  even  smaller  little,  that  the a c t u a l  Kept  very  as  than  due to  to in  little  was  that  Higher  year.  Kinds  term  Managers  with  be  data  very  harvesting the  experiment  because  were  Managers  herds  each  approach,  possibly  by  having  this  a  out  herds  of  reindeer  constant.  to  can  produced  levels level  want  capacity  many  these  level.  size  ended  several  the  rates  Managers  in  when  herds  population  for harvest  existed  show  within  constant  rates  surpluses  other  Bayesian  animals.  a  and  Finnish  inactive  may  growth  gained  the  rates  these  year.  rates  applied  RicKer  at  these  the  i n t h e southern  populations  growth  The  Keep  found  be  With  overall  to  of  methods,  growth  reproductively  animals  will  managed.  the  attempted  culling  most  to  uses  sets.  these  t h e mean  relative  growth  data  by  highly  around  probably  other  out  very  the  information  on  pointed  were  small  understand  low in  RicKer  Bayesian  117 modifications. Carrying maximum  number  Managers  of  informative Managers to  of  may  the  wish  capacities  to  increased  to  4.  of  Adjoining  similar  their  of  policies.  Management  evidenced  by  south.  high  order  be  carrying  to  little  capacities. populations  provide  informative  This  and  year.  their  data.  estimated  the  each  provided  allowing  population  would  populations  allow to  be  potential.  management found  subdivisions  environmental  have  by  determining  harvested  estimate  in  maximum  similarity  the  to  in  be  herds  decrease)  carrying  factor can  experiment  their  structure  that  needed  to  (and/or  actions  a  reindeer  within  The  are  animals  variation  increase  variation  capacities  by  will  should the  have  conditions policies growth  stongly  methods similar and  were  similar  rate  policy  influence  used  residuals  similar within of  in  the  chapter due  to  management regions  managers  in  as the  118  CHAPTER 6  CONCLUDING REMARKS  ways  The  research  that  resource  presented managers  subdivided  populations.  coordinating  policies  management.  This  Walters  (1986).  results  of  each  thesis  improve  The  chapter,  chapter  discusses  the  upon I  briefly  and  of  approach  is  the  some  management  systematic  subdivisions  expands  this  this  could  among  thesis  In  in  new  to  ideas  resource  introduced  review  discuss  for  the  by  primary  future  research  directions. The sharing  simulations similar  under  proper  common  other  statistics  coefficient amounts  was  of  of  was  good  to  information,  managers  can  effectiveness  of  Results reindeer correlation  of  data was  the  be  being  correctly  more  contained greater  on  begin  to on  study common than  high  than  cross  data  Chapter  variation.  a The  of  the  correlation  of  gage  of  have  either  the  variation.  zero.  levels  identified.  average  their  identified  variation  roughly  in  subdivisions  correctly  mean  individual  methods case  reliable  The  that  sharing  individual  indicator  common  these  can  to  tested.  a  showed  Subdivisions  relative  coefficient  2  effects  conditions.  probability  correlation  Chapter  external  variation  greater  in  relative  With the  this  expected  sets. 2  showed The  that  mean  Unfortunately,  the cross no  119 information the  population  grouped  the  time  absence  of  which  between  to  error  identifying  among  subdivisions,  subdivisions  half  subdivisions The  and  showed  in  of  to  pattern  the  be of  second  the  gaging  half  the  residuals  effectiveness  to  probability extremely  present  correctly  the  subdivisions.  If  were  or  extremely  hindered  the  subdivisions.  presence  and  similar  decreased  ability  the  autocorrelation  level  low  that error,  identify  similar  by  3  managers  any  reindeer  autocorrelation, low  and  residuals  within  regions  results  encourage  of bad  in  common  identify  similar  herds  displayed  little,  if  occurred of  the  any,  within  the  the  of  levels  measurement  population use  low  same  year  range.  the  of error.  to  herds  A l l three  methods  to  of  identify  subdivisions.  Chapter treatment  4  units  monitoring. management the  the  within  increased.  The  on  variation  factors.  first  levels  at  represented  similar  aid High  years  these  of  measurement  correctly  Measurement  located  the  i n Chapter  could  methods.  Extremely  enabled  external  autocorrelation,  residuals  correctly  causes  series.  of  ability  the  have  similar  simulations  these  on  would  to  changed  The  low  available  according  similarity of  was  introduced  methods  f o r management  In  order  experiments,  minimum  to  select  experiments  select  a l l possible  correlation  to  the  and best  groups  coefficient  control  index  and  units  candidates  for for  were  ranked  based  within  each  group.  120 Groups  with  the  similar  external  management  effects  was  by  order  correlation  then  divided  can  act  the  as  level  identified  of  the  located  larger  groups  were  found  low  The  residuals  idea  of  external  effects  addition  to  experiments, not  be  effects  series  for  minimizes  data  additional  experimental  and  chance could  of  units.  For by  cause  information  to  these aid  fields  not  similar  methods in  the  shared  types  other  the  3).  on  many  studies  of  experienced  chapter  are  were methods.  Many  differences  ecological  region,  new  many  factors  herds  units  benefit  with  listed  represented.  based  treatments  that  be  units  In  be  that  (from  pairs  fewer  the  regions  year  units  the  another.  experimental  Selecting  available  level  within  of  represented.  index  management.  treatments. is  one  same  controls  the  to  be  can  using  potentially  manipulation  and  be  the  selecting  identical.  controls  in  resource  units  experimental  to  would  herds  can  to  and  number  subdivision  desired,  units  for  groups.  level  herds  most  hierarchical  the  all  herds  the  candidates  reduce  pre-set  fewer  close  best  dissimilar  and  the  reindeer  were  to  units,  a  shared  experiments  similarity  and  herds  the  highly  index  experimental  for  way  units  Similar  extremely  a  above  units  be  large  out  index  index  Possible  as  select  into  higher  For  values  correlation  would  ruling  to  with  The  and  suggested  groups  In  minimum  experiments.  clustering possible  highest  and  in of can  external than  the  between  the  where  time  provide  an  selection  of  121 Chapter improved In  It  this  no  book  on  series  data.  cases  the  The  little  confirmed  by  parameters. estimates  very  year  level. rate  as  managers  the  mean  of  each  can  rate  The  mean  growth  the  herds  allowed  to  vary  and  was  also  rate  Some as  may  wish of  This mean  high  the  around  at  a on  as  the  they  herds  growth  each  herd.  increase  population  southern to  to experiment their  constant  subdivisions  the  herds. animals  mean  should  total  1.4  variance  surplus  within  among of  parameter  relatively  size  rate  individual  the  the  growth  population  harvest  increase  rates  productivity  most  harvests,  growth 5).  in  overall  population  to  that  i s dependent  their  and/or  mean  chapter  productive  increase  total  little  managed  changed  populations  harvested the  to  reindeer  time  the  variances  appear  improve  growth  (from  maintained less  to  herd.  greatly  of  as  method  these  reindeer  not  around  use  was  the  data  provided  the  addition, There  (1986).  sample  methods  variation.  the  the  on  In  highly  the  Walters  to  within  were  because  than  keep  number  For  so  methods f o r  f o r managers  results.  variances  managers  well  the  sets.  Bayesian  larger  and  The  of  were  little  Reindeer each  low  The  much  examples  informative  the  by  used  i t easier  sizes  the  described  dependence  herds  of  not  interesting  density  two  were  data  population  provided  so  own  some  of  the  make  of  as  methods  that  their  evidence  was  the  will  produced  examples  estimation  i s hoped  chapter  methods herds  provided  parameter  Waiters*  sets. in  5  by  1.5.  size  varies herds  Managers  t o see i f they changing  their  122 composition. with  By  additional  herd  can  young  be  population attempt  size  to  population  safe  levels  in  allocation  and  of  data  sets the it  areas  combined  5,  along  managers  may  do  Although  many  experimental Chapter which  4 may  experimental select out units  units of  can  environment.  be  increased  of  the  New design are  and  and  collected  independently  for  Walters'  book  (1986),  joint  parameter  trying and  tests,  the  policy.  fisheries  units  methods  chance  more  Chapter  more  closely  are  for  Wildlife  monitor  the  of and  4  region,  each  region  entire  population.  suggest  selected  estimation. conduct  arbitrarily.  effects  fisheries  managers  introduces represent  that  experimental  either  the  by  observing  intensively  in  wildlife  managers  selecting  were  monitoring  presented  fisheries  been  benefits  methods  with by  to  coordinated  for  introduces  will  experiments  analyzed  wildlife  improve  Many has  the  and  policy  to  data  if  adopting  methods  analyzed  better  convenience. that  the  either  Chapter  by  estimation.  where is  recent  the  the  examples  experimental  of  for  capacity.  subdivisions.  of  rates  of  potentially  managers  animals  determine  capacity  provided  examples  data  growth  carrying  population  exist  mean  to  fertile  management  then  the  resource  the  less  order  carrying  has  parameter  is  In  could  below  among  the  older,  increased,  the  size  to  area  be  thesis  presented  or  can  Just  available policies  females  establish  This  the  increased.  The  yet  replacing  at  methods others.  units of  the often  random to There  or  select are,  123 however, that  limits  many  to  data  The  new  important  fact:  itself  methods  as  and  implemented  additional  testing.  studies.  such  as  in  will These  the  this  to  be  design  will  well.  learn  of  and  will  All  a the  provide  experiments  of  a one  only the  of  this  great  deal  field  New  others  on not  advantage  in  is  based  information as  out  methods.  is  populations.  methods  point  populations  take  created  3  these  introduced  real  and  area  provides  Researchers  on  for  subdivisions  are  2  subdivided  attempt  methods  techniques  fields  other  thesis  estimation  of  subdivision  information.  concepts  actually  study  about  Chapter  inappropriate  Research  Each  this  methods.  are  field.  but  in  additional  sets  systematic  relatively  about  these  and  parameter will  uses in  undergo in  other  ecological  124  L i t e r a t u r e Cited Anonymous. 1987. 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