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# ECO MOD&SPAT ANLY QERM 550

UW

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This 116 page Class Notes was uploaded by Ms. Bart Lind on Wednesday September 9, 2015. The Class Notes belongs to QERM 550 at University of Washington taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/192199/qerm-550-university-of-washington in Quantitative Ecology And Resource Management at University of Washington.

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Date Created: 09/09/15

j Inference for Point Pattern Spatial Statistics N Bert Loosmore nhluwashington edu QERM 550 University of Washington May 11 amp 13 2005 J Inference for Point Pattern Spatial Statistics p 149 Outline r 0 Use of Point Pattern Statistics in Ecology L J Inference for Point Pattern Spatial Statistics p249 Outline r 0 Use of Point Pattern Statistics in Ecology J The Failure of the Simulation Envelope L J Inference for Point Pattern Spatial Statistics p249 Outline r Use of Point Pattern Statistics in Ecology J The Failure of the Simulation Envelope Diggle s 1983 2003 Goodness of Fit Test L J Inference for Point Pattern Spatial Statistics p249 Outline r Use of Point Pattern Statistics in Ecology The Failure of the Simulation Envelope Diggle s 1983 2003 Goodness of Fit Test 1 Unresolved Implementation Issues L J Inference for Point Pattern Spatial Statistics p249 Outline Use of Point Pattern Statistics in Ecology The Failure of the Simulation Envelope Diggle s 1983 2003 Goodness of Fit Test Unresolved Implementation Issues Parameterization Based on the Ecological Research Question A Inference for Point Pattern Spatial Statistics p249 Outline Use of Point Pattern Statistics in Ecology The Failure of the Simulation Envelope DiggIe s 1983 2003 Goodness of Fit Test Unresolved Implementation Issues Parameterization Based on the Ecological Research Question Characterizing Type I II Error Rate Performance A Inference for Point Pattern Spatial Statistics p249 Point Pattern Statistics in Ecology V 7 Spatial processes gt Ecological processes MB 688 mocc o 8 g 09 33 ng 0 90 386 98 e t was a 0 oegoa 09 gt 05b 900 63 150 100 I Northingm L Eastingm J Inference for Point Pattern Spatial Statistics p349 Northingm Point Pattern Statistics in Ecology 1 Spatial processes gt Ecological processes 150 100 I What pattern for the green points 50 J Inference for Point Pattern Spatial Statistics p349 Eastingm r Northingm Point Pattern Statistics in Ecology 1 Spatial processes gt Ecological processes g a 0 0 0 0 gig 55 amp 33 28 W g Q o 3 lt9g o What pattern for the red points 50 J Inference for Point Pattern Spatial Statistics p349 Eastingm Northingm Point Pattern Statistics in Ecology 1 Spatial processes gt Ecological processes 150 100 I Do we see or expect stationarity 50 J Inference for Point Pattern Spatial Statistics p349 Eastingm Point Pattern Spatial Stats HOW V 7 Evaluate observed pattern against ideas of L J Inference for Point Pattern Spatial Statistics p449 Point Pattern Spatial Stats HOW V 7 Evaluate observed pattern against ideas of aggregation rMatClustO with 105 points radius 01 Oct 0 o 00 0 o o o O 000 o 0 Q o o o 0 lt50 9 80 o o 00 o oo o O o 0000 I o 3oo Inference for Point Pattern Spatial Statistics p449 Point Pattern Spatial Stats HOW V 7 Evaluate observed pattern against ideas of aggregation o CSR CSR pattern with 100 points C O o O o O O Inference for Point Pattern Spatial Statistics p449 Point Pattern Spatial Stats HOW r Evaluate observed pattern against ideas of aggregation o CSR inhibition rSS with 100 points radius005 L o o o n J Inference for Point Pattern Spatial Statistics p449 Point Pattern Spatial Stats HOW V 7 Evaluate observed pattern against ideas of aggregation o CSR o inhibition 9 Analyze distances between events L J Inference for Point Pattern Spatial Statistics p449 Point Pattern Spatial Stats How V 7 Evaluate observed pattern against ideas of aggregation o CSR o inhibition Analyze distances between events G nearest neighbor L J Inference for Point Pattern Spatial Statistics p449 Point Pattern Spatial Stats How 7 7 Evaluate observed pattern against ideas of aggregation o CSR o inhibition Analyze distances between events G nearest neighbor Fgrid to nearest point L J Inference for Point Pattern Spatial Statistics p449 Point Pattern Spatial Stats How 7 7 Evaluate observed pattern against ideas of aggregation o CSR o inhibition Analyze distances between events G nearest neighbor Fgrid to nearest point o KL all neighbors L J Inference for Point Pattern Spatial Statistics p449 Point Pattern Spatial Stats How 7 7 Evaluate observed pattern against ideas of aggregation o CSR inhibition Analyze distances between events G nearest neighbor Fgrid to nearest point o KL all neighbors 9 Typically perform analysis using Simulation Envelope L J Inference for Point Pattern Spatial Statistics p449 De nition of the G and F Statistics 7 U G statistic uses the nearest neighbor distances yi for eac of n sample points as F statistic uses the distances 5 from each of m sample points typically located on a grid to their nearest event as A 1 F i lt x m o 90gt Under CSR both the G and F statistic is approximated as X 7T2 L 1 pgtt J Inference for Point Pattern Spatial Statistics p549 De nition of the K and L Statistics F i K statistic uses the distances between all neighbors uZj as 1 TL MW EZZIWU S 15 2 1 375239 Under CSR K statistic can be approximated by Kt 71752 L statistic used to set mean 0 and supposedly stabilize variance as Lt xKm t L J Inference for Point Pattern Spatial Statistics p649 Building the Simulation Envelope VA 7 CSR pattern with A 100 08 04 02 000 005 010 015 020 L Distance J Inference for Point Pattern Spatial Statistics p749 Building the Simulation Envelope V 7 99 CSR patterns with A 100 08 04 02 000 005 010 015 020 L Distance J Inference for Point Pattern Spatial Statistics p749 Using the Simulation Envelope 9 o 01 00 01 02 02 Plot after subtracting CW 1 1 rSSIr003 n100 000 005 010 Distance 015 020 J n erence for Point Pattern Spatial Statistics p849 Perceived a Level Performance V i 12m 81 3 9 Using all results from 19 simulations yields 32 005 or Throwing out upper and lower 2 simulations at each distance m 2 from 99 simulations also yields 32 005 L l Inference for Point Pattern Spatial Statistics p949 Kenkel 1988 Methods V 7 Evaluated spatial locations of all live trees all live standing dead trees in a jack pine Pinus Bansana forest L J Inference for Point Pattern Spatial Statistics p1049 Kenkel 1988 Methods 7 7 Evaluated spatial locations of all live trees all live standing dead trees in a jack pine Pinus Bansana forest 0 Map of live standing dead represents distribution following early sapling mortality but prior to the onset of densitydepending mortality L J Inference for Point Pattern Spatial Statistics p1049 Kenkel 1988 Methods 7 7 Evaluated spatial locations of all live trees all live standing dead trees in a jack pine Pinus Bansana forest 0 Map of live standing dead represents distribution following early sapling mortality but prior to the onset of densitydepending mortality 1 Methods Used MC techniques for the G and L statistics to evaluate observed results against H0 of i random locations CSR and ii random mortality L J Inference for Point Pattern Spatial Statistics p1049 Kenkel 1988 Conclusions V 7 o G live dead shows no departure from randomness whereas live trees only shows significant regularity L J Inference for Point Pattern Spatial Statistics p 1 149 Kenkel 1988 Conclusions 7 7 o G live dead shows no departure from randomness whereas live trees only shows significant regularity o L live dead shows no departure from CSR at small scales live trees show regularity at smaller scales L J Inference for Point Pattern Spatial Statistics p 1 149 Kenkel 1988 Conclusions 7 7 o G live dead shows no departure from randomness whereas live trees only shows significant regularity o L live dead shows no departure from CSR at small scales live trees show regularity at smaller scales o But is this interpretation correct L J Inference for Point Pattern Spatial Statistics p 1 149 Examples in Ecological Research r Author Year Statistics Patterns in CI Marginal Used Sim Env s Results yn Batista and Maguire 1998 G K 19 95 n Dolezal etal 2004 K 99 95 y Freeman and Ford 2002 G K 99 99 n Grassi etal 2004 K 99 95 n Hirayama and Sakimoto 2003 K 1999 95 99 n Martens etal 1997 L 99 95 n Moeur 1997 G K 200 90 n Parish etal 1999 G K 19 95 n Salvador Van Eysenrode etal 2000 G K 1000 95 y Srutek et al 2002 L 99 95 Tirado and Pugnaire 2003 K 1000 99 L J Inference for Point Pattern Spatial Statistics p1249 Outline The Failure of the Simulation Envelope J Inference for Point Pattern Spatial Statistics p1349 Sim EnV 1 Level Performance V i 1 Simulation study with independent trials of a CSR pattern against a CSR envelope 9 Designate failure if pattern exceeds envelope at any distance Type I error 9 Expected type I error rate 005 L l Inference for Point Pattern Spatial Statistics p1449 Sim EnV d Level Performance V i 1 Simulation study with independent trials of a CSR pattern against a CSR envelope 9 Designate failure if pattern exceeds envelope at any distance Type I error 9 Expected type I error rate 005 actual type I error rate 0507 L l Inference for Point Pattern Spatial Statistics p1449 Monte Carlo Simulation Theory V 7 For a univariate continuous distribution Cultd Prsgti z391s 1 L J Inference for Point Pattern Spatial Statistics p1549 Monte Carlo Simulation Theory V 7 For a univariate continuous distribution Cultd Prsgti z391s 1 But does the simulation envelope comprise a univariate distribution L J Inference for Point Pattern Spatial Statistics p1549 How the Envelope is Really Made Simulation envelope built from 100 patterns 02 01 00 01 02 03 03 1 55 patterns comprising the simulation envelope I I I I I I 010 015 020 025 Distance J Inference for Point Pattern Spatial Statistics p1649 Failure of the Simulation Envelope rA 1 G F andor K statistics and 2 spatial patterns j Ithough built from 8 patterns complexity of both yields a multivariate result Since evaluation of the observed pattern occurs at many distances we are performing simultaneous inference and thus Oz is increased Further if the simulation envelope is invalid then how can we use it to determine scale L J Inference for Point Pattern Spatial Statistics p1749 b Outline Diggle s 1983 2003 Goodness of Fit Test A Inference for Point Pattern Spatial Statistics p1849 Proper Statistical Methods 7 7 From Diggle 1983 2003 for a given H0 1 At a single a priori distance use upper and lower simulated values 2 Across a range of distances use Goodness of Fit test L J Inference for Point Pattern Spatial Statistics p1949 The Goodness of Fit Test 1 V 7 1 Represent the empirical results as GM observed pattern and A Mt forz39 2 s simulated patterns L J Inference for Point Pattern Spatial Statistics p2049 The Goodness of Fit Test 2 f 7 2 Calculate dzt am at 1 tori1s Summary statistic indicative of the total deviation of the given pattern from the theoretical result L J Inference for Point Pattern Spatial Statistics p2149 The Goodness of Fit Test 2 V i 2 Calculate W C Zt 124152 dt 1 tori1s butuse to reduce bias Inference for Point Pattern Spatial Statistics p2149 The Goodness of Fit Test 3 F i 3 Reject fail to H0 based on the rank of m using the pvalue calculated as ranku1 1 pobs 8 2310 gt uj 8 1 forj 1 s 1 So if ranku1 100 the largest then 19055 001 Now we have quantitative results to evaluate a pattern s significance based on an exact a level test because of Lproper MC methods I Inference for Point Pattern Spatial Statistics p2249 Outline Unresolved Implementation Issues J Inference for Point Pattern Spatial Statistics p2349 Unresolved Implementation Issues V i What is the optimal method to calculate W W 2 ow an dt 75 How to replace integration with summation 9 incorporate edge correction methods 9 choose limits 15mm distance list 5 9 simulate patterns from null process L l Inference for Point Pattern Spatial Statistics p2449 Replacing Integration With Summation V i We can rewrite Eqn 1 as W an an dt 75 Z z t CM 515 2 22 But how accurate is this approximation Inference for Point Patte 39stics p2549 Edge Correction j rUsed to eliminate bias from edge interfering with detecting a points neighbor Reduced Sample edge correction approach Let dig be the distance for point z39 to the closest boundary 0 Remove point z39 from calculation at distance twhere t gt dig Other approaches toroidal isotropic etc L J Inference for Point Pattern Spatial Statistics p2649 Choice of Limits tmm Distance List t V 7 Recommended default for tmm 025 but application dependen L J Inference for Point Pattern Spatial Statistics p2749 Choice of Limits tmm Distance List t V 7 Recommended default for tmm 025 but application dependen CW fat are discrete change where L J Inference for Point Pattern Spatial Statistics p2749 Choice of Limits tmm Distance List t r 0 Recommended default for tmm 025 but application I dependen CW fat are discrete change where 0 new neighbor detected or L J Inference for Point Pattern Spatial Statistics p2749 Choice of Limits tmm Distance List t V 7 0 Recommended default for tmm 025 but application dependen CW fat are discrete change where 0 new neighbor detected or point removed from sample L J Inference for Point Pattern Spatial Statistics p2749 Choice of Limits tmm Distance List t V 7 0 Recommended default for imam 025 but application dependen CW fat are discrete change where 0 new neighbor detected or point removed from sample Use empirical distance list for exact results from a single pattern L J Inference for Point Pattern Spatial Statistics p2749 Choice of Limits tmm Distance List t r Recommended default for imam 025 but application j dependen CW fat are discrete change where 0 new neighbor detected or 4 point removed from sample Use empirical distance list for exact results from a single pattern Because of ui calculation especially Mt for exact solution need to use complete empirical distance list ie from all patterns for evaluation of each pattern A Inference for Point Pattern Spatial Statistics p2749 Resolution of Simulated Patterns V 7 Complexity Number of distances grows with A s L J Inference for Point Pattern Spatial Statistics p2849 Resolution of Simulated Patterns 7 7 Complexity Number of distances grows with A s Resolution ie 23956 2 001 vs 23956 2 001000 of simulated patterns should be equivalent to that of observed pattern L J Inference for Point Pattern Spatial Statistics p2849 Resolution of Simulated Patterns 7 7 Complexity Number of distances grows with A s Resolution ie 23956 2 001 vs 23956 2 001000 of simulated patterns should be equivalent to that of observed pattern 0 Limiting resolution helps constrain complexity L J Inference for Point Pattern Spatial Statistics p2849 Resolution of Simulated Patterns j Complexity Number of distances grows with A s Resolution ie 23956 2 001 vs 23956 2 001000 of simulated patterns should be equivalent to that of observed pattern 0 Limiting resolution helps constrain complexity 0 01cm100m 2 le 5 is highly accurate for ecological data Freeman and Ford 2002 L J Inference for Point Pattern Spatial Statistics p2849 Resolution of Simulated Patterns j Complexity Number of distances grows with A s Resolution ie 23956 2 001 vs 23956 2 001000 of simulated patterns should be equivalent to that of observed pattern 0 Limiting resolution helps constrain complexity 0 01cm100m 2 le 5 is highly accurate for ecological data Freeman and Ford 2002 gt Combining resolution and default tmm leads to at most 25000 distances in t39 regardless of A s or test statistic and provides an exact solution L J Inference for Point Pattern Spatial Statistics p2849 Outline 0 Parameterization Based on the Ecological Research Question L J Inference for Point Pattern Spatial Statistics p2949 Parameterization 1 V 7 How to run any given test based on the ecological research question 1 Number of simulations 8 L J Inference for Point Pattern Spatial Statistics p3049 Parameterization 1 r How to run any given test based on the ecological research question 9 Number of simulations 3 Choice of H0 including choice of tmax L J Inference for Point Pattern Spatial Statistics p3049 P0198 versus P0198 F i Uncertainly in realized pvalue palm results from the use of MC simulations Ramifications of s Affects precision of pobs through 9 actual simulated patterns against which observed pattern tested and 9 number of those patterns Note about exact or level performance across many tests vs variation of pvalue for single test L l Inference for Point Pattern Spatial Statistics p3149 Distribution of P0198 V i Let Y m and Xj uj forj 1 s 1 The pvalue for the test is then ZjIYgtXj 8 P1 L J Inference for Point Pattern Spatial Statistics p3249 Distribution of P0198 F i Let Y m and Xj uj forj 1 s 1 The pvalue for the test is then any gt X3 8 P1 The expected value of P is EP E1 ELMgt393 8 11 p mp 8 Assuming Y comes from H0 then p PrX gt YY So each Of the Y gt Xj N Bernoulli p l Inference for Point Pattern Spatial Statistics p3249 Variance of P 0 V i Looking at the variance of 3015 we have 0 Var 1 Zj H gt Xj Sigarr 210 gt X3 88 1p1 p g 291 p L J Inference for Point Pattern Spatial Statistics p3349 Variance of P 0 V i Looking at the variance of 3015 we have 0 Var 1 Zj H gt Xj Sigarr 210 gt X3 88 1p1 p g 291 p Hence we can model the theoretical distribution of 3015 as lfrom a binomialps distribution l Inference for Point Pattern Spatial Statistics p3349 Managing Uncertainty in pobs V 7 Q Rem that binomial quickly converges to Normal L J Inference for Point Pattern Spatial Statistics p3449 Managing Uncertainty in pobs i 1 Rem that binomial quickly converges to Normal j 9 Create 95 CI on p055 true pvalue nearp 005 as 330195 3 196 X 0 J Inference for Point Pattern Spatial Statistics p3449 Managing Uncertainty in pobs f 7 9 Rem that binomial quickly converges to Normal 9 Create 95 Cl on p055 true pvalue nearp 005 as 13055 196 X 0 9 95 of Cl created this way should contain the true value of p055 and so set decision rule eg reject H0 if Cl contains or fully below 005 L J Inference for Point Pattern Spatial Statistics p3449 Managing Uncertainty in pobs j Rem that binomial quickly converges to Normal Create 95 Cl on p055 true pvalue nearp 005 as 13055 196 X 0 95 of Cl created this way should contain the true value of p055 and so set decision rule eg reject H0 if Cl contains or fully below 005 Choose acceptable range of uncertainty for p055 J Inference for Point Pattern Spatial Statistics p3449 Managing Uncertainty in pobs j Rem that binomial quickly converges to Normal Create 95 Cl on p055 true pvalue nearp 005 as 13055 196 X 0 95 of Cl created this way should contain the true value of p055 and so set decision rule eg reject H0 if Cl contains or fully below 005 Choose acceptable range of uncertainty for p055 For example it 003 lt 10055 lt 007 is ok use a 001 J Inference for Point Pattern Spatial Statistics p3449 Managing Uncertainty in pobs j Rem that binomial quickly converges to Normal Create 95 Cl on p055 true pvalue nearp 005 as 13055 196 X 0 95 of Cl created this way should contain the true value of p055 and so set decision rule eg reject H0 if Cl contains or fully below 005 Choose acceptable range of uncertainty for p055 For example it 003 lt 10055 lt 007 is ok use a 001 Use relationship between 0 and s to find value of s l Inference for Point Pattern Spatial Statistics p3449 B 002 003 004 005 006 007 001 2 P as a function of s 1000 of Simulations 1500 Inference for Point Pattern Spatial Statistics p3549 Choice of H0 V 7 9 Use all available ecological knowledge for a more informative test L J Inference for Point Pattern Spatial Statistics p3649 Choice of H0 f 7 3 Use all available ecological knowledge for a more informative test Null point process just needs to be able to be simulated many models available eg spatstat or write your own L J Inference for Point Pattern Spatial Statistics p3649 Choice of H0 W 7 1 Use all available ecological knowledge for a more informative test 3 Null point process just needs to be able to be simulated many models available eg spatstat or write your own 9 At the very least choose simple inhibition model based on physical separation L J Inference for Point Pattern Spatial Statistics p3649 Choice of H0 j Use all available ecological knowledge for a more informative test Null point process just needs to be able to be simulated many models available eg spatstat or write your own At the very least choose simple inhibition model based on physical separation EDA vs confirmatory analysis results in iterative nature of research with hopefully tests on independent data sets A Inference for Point Pattern Spatial Statistics p3649 Choice of H0 j Use all available ecological knowledge for a more informative test Null point process just needs to be able to be simulated many models available eg spatstat or write your own At the very least choose simple inhibition model based on physical separation EDA vs confirmatory analysis results in iterative nature of research with hopefully tests on independent data sets Use the model to determine information on scale I Inference for Point Pattern Spatial Statistics p3649 Example of model tting 7 7 Attempt to fit a Clustered model representing establishment processes to the lower SW quadrant of the WRCCRF data for all trees 3 6m in height L J Inference for Point Pattern Spatial Statistics p3749 Example of model tting 7 7 Attempt to fit a clustered model representing establishment processes to the lower SW quadrant of the WRCCRF data for all trees 3 6m in height 0 Used Poisson Clustered model with p represents the number of parents and it represents the expected number of children per parent and where clustering of children around each parent are described as 1 2 exp 932 312202 M93731 2m L J Inference for Point Pattern Spatial Statistics p3749 Example of model tting If Attempt to fit a clustered model representing 7 establishment processes to the lower SW quadrant of the WRCCRF data for all trees 3 6m in height 0 Used Poisson Clustered model with p represents the number of parents and it represents the expected number of children per parent and where clustering of children around each parent are described as 1 2 exp 932 312202 M93731 2m How to choose values for p u and a A 2 up L J Inference for Point Pattern Spatial Statistics p3749 Example of model tting j Attempt to fit a clustered model representing establishment processes to the lower SW quadrant of the WRCCRF data for all trees 3 6m in height Used Poisson Clustered model with p represents the number of parents and u represents the expected number of children per parent and where clustering of children around each parent are described as 1 2 exp 932 312202 M93731 2m How to choose values for p u and a A 2 pp Note that my null model here describes not only the process but also the parameter values A Inference for Point Pattern Spatial Statistics p3749 Example of model tting 2 V 7 0 This is Exploratory Data Analysis L J Inference for Point Pattern Spatial Statistics p3849 Example of model tting 2 V 7 9 This is Exploratory Data Analysis 0 If we knew the theoretical value of G K for this model use Diggle s Least Squares Estimation method L J Inference for Point Pattern Spatial Statistics p3849 Example of model tting 2 If 9 This is Exploratory Data Analysis j 0 If we knew the theoretical value of G K for this model use Diggle s Least Squares Estimation method 0 Otherwise use GoF test to estimate parameter space L J Inference for Point Pattern Spatial Statistics p3849 Example of model tting 2 This is Exploratory Data Analysis j If we knew the theoretical value of G K for this model use Diggle s Least Squares Estimation method Otherwise use GoF test to estimate parameter space Find 13 for different combinations of p a and accept model where 13 gt 0106 A Inference for Point Pattern Spatial Statistics p3849 Example of model tting 2 V 7 o This is Exploratory Data Analysis 0 If we knew the theoretical value of G K for this model use Diggle s Least Squares Estimation method 0 Otherwise use GoF test to estimate parameter space Inference for Point Pattern Spatial Statistics p3849 Example of model tting 3 V 7 Inference For the observed data if this model fits then larger 0 suggests lower p ie few parents and so more childrenparent L J Inference for Point Pattern Spatial Statistics p3949 Example of model tting 3 j If lnference For the observed data if this model fits then larger 0 suggests lower p ie few parents and so more childrenparent 0 Conversely a smaller clustering radius requires higher p and so fewer children per parent L J Inference for Point Pattern Spatial Statistics p3949 Example of model tting 3 j If lnference For the observed data if this model fits then larger 0 suggests lower p ie few parents and so more childrenparent Conversely a smaller clustering radius requires higher p and so fewer children per parent o Is this model a good fit What might the physiological andor ecological implications be L J Inference for Point Pattern Spatial Statistics p3949 Example of model tting 3 j Inference For the observed data if this model fits then larger 0 suggests lower p ie few parents and so more childrenparent Conversely a smaller clustering radius requires higher p and so fewer children per parent Is this model a good fit What might the physiological andor ecological implications be 0 gives us hints about scale A Inference for Point Pattern Spatial Statistics p3949 imam Variance stabilization j tmax should be chosen before the test and based on research question ie what is the interaction distance of interest A Inference for Point Pattern Spatial Statistics p4049 imam Variance stabilization j tmax should be chosen before the test and based on research question ie what is the interaction distance of interest l 000 005 010 015 020 Distance LVariance stabilization to make variance independent of t J Inference for Point Pattern Spatial Statistics p4049 Outline 0 Characterizing Type I II Error Rate Performance A Inference for Point Pattern Spatial Statistics p4149 Type I Error Rate a 1 V i 1 Simulation study of Type I error rate performance 9 Evaluated different a levels for different point pattern intensities A 50 100 200 9 Results within LRT boundaries L l Inference for Point Pattern Spatial Statistics p4249 Type I Error Rate 04 2 V i Simulations of 1000 independent trials using 5 1999 a Type I error rates for G b Type I error rates for K L2 J O O O O O 0 d d 0 A A OI OI 8 8 O9 9 O O O O O O O O I I I I I I I I I 0 50 100 150 200 250 0 50 100 150 200 250 l points A points A l Inference for Point Pattern Spatial Statistics p4349 Type 11 Error Rate lPower V 7 Type II error rate is the prob of accepting H0 given that HA is really true L J Inference for Point Pattern Spatial Statistics p4449 Type 11 Error Rate lPower V 7 Type II error rate is the prob of accepting H0 given that HA is really true p Requires definition of HA L J Inference for Point Pattern Spatial Statistics p4449 Type 11 Error Rate lPower V 7 Type II error rate is the prob of accepting H0 given that HA is really true o Requires definition of HA 9 Power will be a function of how far H0 is from HA Easy to think of this distance when using Normal distribution but more difficult to conceptualize here L J Inference for Point Pattern Spatial Statistics p4449 Type 11 Error Rate lPower j Type II error rate is the prob of accepting H0 given that HA is really true Requires definition of HA Power will be a function of how far H0 is from HA Easy to think of this distance when using Normal distribution but more difficult to conceptualize here Often overlooked for spatial point process analysis but can be simulated J Inference for Point Pattern Spatial Statistics p4449 Analysis of Type 11 Error Rate If 9 Analysis of power against HA of CSR for WRCCRF j example for different parameterizations of H0 Type II error rate tells us the ability to distinguish the pattern from CSR As 0 increases larger clusters are more like CSR ap20 bp40 Power Power ex 2 xx 0 o Om Oo ovox d 7 o 7 l l l l l l l l l l l l l l 005 015 025 035 005 015 025 035 C5 C5 Inference for Point Pattern Spatial Statistics p4549 02 01 00 01 02 03 03 Power of the G Statistic Large deviation at small distances may be swamped out 1 rSSIr002 rSSIr003 l 000 005 010 015 020 Distance J Inference for Point Pattern Spatial Statistics p4649 Parameters that may improve Power V i Rewriting Equation 2 in its full form Diggle 2003 tmazc W Z wandtr 040 615 75 0 Parameters that may improve Power V i Rewriting Equation 2 in its full form Diggle 2003 tmazc W Z wandtr 040 615 75 0 wt c as parameters to improve Power against certain HA Parameters that may improve Power V i Rewriting Equation 2 in its full form Diggle 2003 m Eutr 6400 615 75 0 39 Use of wt not well explored but could be used to emphasize certain distances For my calculations wt 1 L l Inference for Point Pattern Spatial Statistics p4749 Parameters that may improve Power V i Rewriting Equation 2 in its full form Diggle 2003 tmazc W Z wtmt mug 32 615 75 0 39 For Kt 9 use c 05 for L statistic 9 use c 025 for power against clustered patterns Diggle 2003 9 other L l Inference for Point Pattern Spatial Statistics p4749 Conclusions 7 7 0 Simulation envelope does not result in expected Type I error rates Limits are not confidence intervals L J Inference for Point Pattern Spatial Statistics p4849 Conclusions 7 7 Simulation envelope does not result in expected Type I error rates Limits are not confidence intervals o For more precise reliable results implement Diggle s goodness of fit test L J Inference for Point Pattern Spatial Statistics p4849 Conclusions 7 7 Simulation envelope does not result in expected Type I error rates Limits are not confidence intervals o For more precise reliable results implement Diggle s goodness of fit test 0 Previous marginal results should be reexamined L J Inference for Point Pattern Spatial Statistics p4849 Conclusions j Simulation envelope does not result in expected Type I error rates Limits are not confidence intervals For more precise reliable results implement Diggle s goodness of fit test Previous marginal results should be reexamined Choice of H0 tmm based on research question and previous knowledge A Inference for Point Pattern Spatial Statistics p4849 Conclusions j Simulation envelope does not result in expected Type I error rates Limits are not confidence intervals For more precise reliable results implement Diggle s goodness of fit test Previous marginal results should be reexamined Choice of H0 tmm based on research question and previous knowledge Evaluate the Power of your test A Inference for Point Pattern Spatial Statistics p4849 Conclusions j Simulation envelope does not result in expected Type I error rates Limits are not confidence intervals For more precise reliable results implement Diggle s goodness of fit test Previous marginal results should be reexamined Choice of H0 tmm based on research question and previous knowledge Evaluate the Power of your test R software availability http students washington edunhlmasters html Inference for Point Pattern Spatial Statistics p4849 R software I CSOUI CCS rCRAN W httpcranr project org Comprehensive R Archive Network site A Baddeley s spatstat package http wwwmaths uwa edu auNadrianspatstat html P Diggle s splancs package http wwwmaths lance ac ukNrowlingsSplancs UW R and S plus user support group 3 http mailmanl uwashington edumailmanlistinfosplus L J Inference for Point Pattern Spatial Statistics p4949

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