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 Â© 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: Â«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 â€¢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 Â° 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 â€¢.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 â€¢ 0.258 StDev â€¢ 0.259 â€¢ : â€¢ - 200 160 100 I â€¢ 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 = Â« 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 = Â« *nt-i,i-J + â‚¬ 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 â€¢ 0 . 8 2 - 0 . 8 - 0 . 7 8 ~ 0 . 7 6 - 0 . 7 4 - DD 0 . 7 2 - â€¢ 0 . 7 - â€¢ â€¢ a a a a â€¢ â€¢ â€¢ CO â€¢ â€¢ cn m â€¢ CD eq a a c â€¢ Q aâ€¢ â€¢ â€¢ m DOC ] â€¢ a â€¢ o a D â€¢ â€¢ â€¢ on a â€¢ an m a on a â€¢ a DO â€¢ CO mn a m â€¢ a a â€¢ a â€¢ D â€¢ â€¢ â€¢ a â€¢ â€¢ an 0 . 6 8 - 0 . 6 6 - 0 . 6 4 - 0 . 6 2 - a a â€¢ â€¢ a â€¢a â€¢ â€¢ â€¢ - 0 a T . 5 - 0 . 3 - 0 . 1 â€” I â€” 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 â€”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 Â«ar 10 Â«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 â€¢/. 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 Â» M Â» 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 â€” Set 1 â€” X X â€” Set 2 â€” X X â€” 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 â€¢ 17 3 M I 44 7 SS St â€” X ro S4 31 34 It 42 37 M 50 M * â€” M 15 25 23 t i 24 41 40 22 47 4Â« SI 43 48 51 4S 41 It 10 IS 1Â» 27 32 31 24 13 X 20 11 Set 1 Figure 4.4 â€” X X â€” set 2 Herd Numbers â€” 14 1 3 4 X Xâ€” 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 Â» 17 It ( 44 7 SS 52 â€” 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 Â£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 aÂ»g be can be trial used - Â»=1 ,3 variance calculated estimate of Wj calculated of and as: m N Â»=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 â€¢Â»/ 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 Â« 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 ^ aÂ«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 Â» 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 wÂ»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 â€¢ â€¢ â€¢ 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 - Â§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,-Â«_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 Â£ E(K,t-a,--$,-s,- -Â«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 Â± 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 Â± 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 â€” â€¢ Joint Estimation Figure 5.2 â€”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 â€¢41E-04 28E-04 .65E-08 . 93E-08 49 . 18E-03 ,IiE-03 â€¢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. Suomen P o r o t a l o u s . The R e i n d e e r I n d u s t r y of F i n l a n d . Paliskuntain Yhdistys. Rovaniemi, Finland. 5 pages. A t k i n s o n , K. and D.W. J a n z . 1986. E f f e c t of w o l f c o n t r o l on B l a c k - T a i l e d d e e r i n t h e N i m p k s i h V a l l e y on V a n c o u v e r Island. W i l d l i f e W o r k i n g R e p o r t Number WR-19. Ministry of t h e E n v i r o n m e n t . W i l d l i f e Branch. Nanaimo B.C. 31 pages. B a i l e y , J.A. 1984. P r i n c i p l e s of W i l d l i f e Management. W i l e y and Sons. New Y o r k . 373 pages. Box, G.E.P. and G.M. J e n k i n s . 1976. Time S e r i e s f o r e c a s t i n g and c o n t r o l . Revised edition. San F r a n c i s c o . 575 pages. John Analyis: Holden-Day. C h a t f i e l d , C. 1980. The A n a l y s i s of Time S e r i e s : An Introduction. Second e d i t i o n . Chapman and H a l l . London. 268 pages. C o c h r a n , W.G. 1977. S a m p l i n g t e c h n i q u e s . Third J o h n W i l e y and Sons. New Y o r k . 426 pages. Cox, Edition. G.W. ( e d . ) 1969. R e a d i n g s i n C o n s e r v a t i o n E c o l o g y . Meredith Corporation. New Y o r k . 595 pages. Draper, N.R. and H. S m i t h . 1981. A p p l i e d R e g r e s s i o n Analysis. Second E d i t i o n . J o h n W i l e y and Sons. York. 709 p a g e s . New E v e r i t t , B. 1980. C l u s t e r a n a l y s i s . Second E d i t i o n . Halsted Press. John W i l e y and Sons. New Y o r k . 136 pages. Giles, R.H. J r . ( e d . ) 1969. W i l d l i f e Management T e c h n i q u e s . Third Edition. The W i l d l i f e S o c i e t y . W a s h i n g t o n , D.C. 632 pages. Grant, W.E. 1966. Systems a n a l y s i s and s i m u l a t i o n i n w i l d l i f e and f i s h e r i e s s c i e n c e s . J o h n W i l e y and Sons. New Y o r k . 338 pages. G u l l a n d , J.A. 1983. F i s h S t o c k A s s e s s m e n t . Sons. Chichester. 223 pages. John W i l e y and 125 H a t t e r , I. and D. J a n z . 1986. R a t i o n a l e f o r wolf c o n t r o l i n the management of t h e V a n c o u v e r I s l a n d p r e d a t o r - u n g u l a t e system. W i l d l i f e B u l l e t i n No. B. M i n i s t r y of t h e Environment. W i l d l i f e Branch. V i c t o r i a , B.C. 33 pages. H o l l i n g , C . S . ( e d . ) 1978. A d a p t i v e environmental assessment and management. Wiley. C h i c h e s t e r , New Y o r k . 377 pages. Ludwig, D. and C . J . W a l t e r s . 1981. Measurement e r r o r s and u n c e r t a i n t y i n p a r a m e t e r e s t i m a t e s f o r s t o c k and recruitment. Can J . F i s h . A q u a t . S c i . 38:711-720. May, R.M. (ed.) 1984. Springer-Verlag. Exploitation Berlin. 370 Murthy, M.N. 1967. Sampling t h e o r y Publishing Society. Calcutta, Mysak, of M a r i n e pages. and methods. Statistical India. 684 pages. L.A. 1986. E l Nino, i n t e r a n n u a l f i s h e r i e s i n the n o r t h e a s t P a c i f i c Aqua. Sci. 43(2)1464-497. N e l s o n , L.A. and D.L. J o h n s o n ( e d s . ) Techniques. American F i s h e r i e s Maryland. 468 pages. Communities. variability Ocean. Can and J. Fish 1983. Fisheries Society. Bethesda, R i c k e r , W.E. 1954. S t o c k and r e c r u i t m e n t . B o a r d . Can. 11:559-623. J. Fish Res. R i c k e r , W.E. 1975. C o m p u t a t i o n and i n t e r p r e t a t i o n of b i o l o g i c a l s t a t i s t i c s of f i s h p o p u l a t i o n s . Bulletin F i s h Res. B o a r d Can. No. 191. 382 pages. of R o b i n s o n , W.L. and E.G. B o l e n . 1984. W i l d l i f e E c o l o g y and Management. M a c m i l l a n . New Y o r k . 478 pages. Shaw, J.H. 1985. McGraw-Hill. I n t r o d u c t i o n to W i l d l i f e New Y o r k . 316 pages. Management. Sneath, P.H.A. and R.R. S o k a l . 1973. N u m e r i c a l Taxonomy: the p r i n c i p l e s and p r a c t i c e of n u m e r i c a l classification. Freeman and Co. San F r a n c i s c o . 573 pages. Sokal, R.R. and C D . M i c h e n e r . 1958. A s t a t i s t i c a l method for evaluating systematic r e l a t i o n s h i p s . U n i v . Kansas S c i . B u i 1. 38: 1409-1438. 126 Sokal, R.R. and P.H.A. S n e a t h . 1963. P r i n c i p l e s of N u m e r i c a l Taxonomy. W.H. Freeman and Company. San F r a n c i s c o . 359 pages. S t a r f i e l d , A.M. and A.L. B l e l o c h . 1986. B u i l d i n g Models f o r C o n s e r v a t i o n and W i l d l i f e Management. Macmillan. New York. 253 pages. S t e p h e n s o n , W. 1936. The i n v e r t e d J. Psychol. 26:344-361. Tanner, J . T . 1978. The U n i v e r s i t y pages. Tyler, factor technique. Brit. G u i d e t o t h e S t u d y of A n i m a l P o p u l a t i o n s . of T e n n e s s e e P r e s s . Knoxville. 186 A.V. and G.A. M c F a r l a n e ( e d s ) . 1985. Groundfish s t o c k a s s e s s m e n t s f o r t h e w e s t c o a s t of Canada and recoommended y i e l d o p t i o n s f o r 1985. Canadian M a n u s c r i p t R e p o r t of F i s h and Aqua. S c i . No. 1813. 353 pages. Wakeley, J . S . ( e d . ) 1982. W i l d l i f e Population Ecology. The Pennsylvania State U n i v e r s i t y Press. U n i v e r s i t y Park. 385 pages. Walters, C.J. 1985. Bias i n the e s t i m a t i o n of f u n c t i o n a l r e l a t i o n s h i p s from time s e r i e s data. Can. J . F i s h . A q u a t . S c i . 42: 147-149. Walters, C.J. 1986. A d a p t i v e management of resources. M a c m i l l a n . New Y o r k . 374 renewable pages. Watt, K.E.F. 1968. E c o l o g y and R e s o u r c e Management. Hill. New Y o r k . 450 pages. Zubin, T. 1938. A technique f o r measuring J . Abnorm. Soc. P s y c h o l . 33:508-516. McGraw- 1ike-mindedness.
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Opportunities for management created by spatial structures...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Opportunities for management created by spatial structures : a case study of Finnish reindeer Berkson, James Meyer 1988
pdf
Page Metadata
Item Metadata
Title | Opportunities for management created by spatial structures : a case study of Finnish reindeer |
Creator |
Berkson, James Meyer |
Publisher | University of British Columbia |
Date Issued | 1988 |
Description | This study examines opportunities for renewable resource management when population data are collected by spatial subdivisions. In particular I look at potential applications for the design of management experiments, the distribution of monitoring resources, and the improvement of parameter estimation. Methods are developed to rank possible groupings of subdivisions for use as experimental units. Factors external to the experiment can cause differences between experimental units. Selecting subdivisions that have reacted similarly in the past to external factors could minimize the risk of external factors creating differences in experimental units. Methods are developed to identify subdivisions that could provide information about similar subdivisions when monitoring resources are low or when stratified sampling is being used. The use of these subdivisions as "index units" could notify managers of extremely good or bad years in a large number of subdivisions. Two methods developed by Walters (1986) provide innovative estimation techniques that can be used with subdivided populations. A Bayesian approach allows parameter estimates to be adjusted using a known distribution. Another approach allows similar subdivisions to be estimated jointly more accurately than would be possible individually. Not all renewable resource data sets provide reliable information for use with these applications. Data sets where there is little common variation, high levels of autocorrelation in the noise, or even modest amounts of measurement error are inappropriate for most methods. A series of steps is introduced for managers to test the reliability of the methods on their particular data sets. Data on Finnish reindeer (Rangifer tarandus tarandus) are used throughout the thesis to illustrate the methods. The reindeer data appear to be appropriate for these methods when tested using the steps developed. Possible experimental units and index units for monitoring are identified. Walters' (1986) methods of parameter estimation are used on the data set as well. The reindeer data show that subdivisions with similar external effects were located close to one another. This pattern was at least partially caused by the existence of extremely bad years occurring within geographic regions. The reindeer subdivisions are very highly managed and provide little evidence of any kind of density dependence. Managers could potentially benefit by conducting experiments to test the biological limits of the population growth rates and carrying capacities within subdivisions. |
Subject |
Renewable natural resources Reindeer Animal populations |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0097656 |
URI | http://hdl.handle.net/2429/27799 |
Degree |
Master of Science - MSc |
Program |
Zoology |
Affiliation |
Science, Faculty of Zoology, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1988_A6_7 B47.pdf [ 4.44MB ]
- Metadata
- JSON: 831-1.0097656.json
- JSON-LD: 831-1.0097656-ld.json
- RDF/XML (Pretty): 831-1.0097656-rdf.xml
- RDF/JSON: 831-1.0097656-rdf.json
- Turtle: 831-1.0097656-turtle.txt
- N-Triples: 831-1.0097656-rdf-ntriples.txt
- Original Record: 831-1.0097656-source.json
- Full Text
- 831-1.0097656-fulltext.txt
- Citation
- 831-1.0097656.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0097656/manifest