Spatial Point Patterns. Potential questions. Application areas. Examples in pictures:

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1 Spatial Point Pattens Obseved locations of events: datum is the location Majo shift in inteest fom pevious mateial! Up to now, location has been fixed point o fixed aea, Location abitay o happenstance, often contolled by the investigato (whee to take point samples) Random quantity has been the value at each location Random quantity is now the location of an event May ecod additional infomation at each location maked point pocess Sometimes small # of classes Examples: species of tee, live / dead plant, successful / unsuccessful bid nest, disease case / not diseased peson O, may be continuous quantity Examples: diamete of tee, angle of a cystal c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Potential questions 1 is intensity (# events / unit aea) const. o vay. ove the study aea 2 how does intensity vay as function of potential covaiates EX: does intensity of duck nests decline with distance to wetland? 3 ae events andomly scatteed, clusteed, o egula EX: ae duck nests independently located in space, o do they cluste nea othe duck nests, o do they avoid being nea othe nests? 4 how can we descibe patten at multiple scales? 5 how can we descibe el. between two (o moe) types of points? EX: do cypess tees tend to occu nea othe cypess tees? 6 how can we descibe the co. between maks as a function of distance? EX: Do small tees tend to occu nea othe small tees? Histoically, 3) was most impotant Q Now, moving beyond to all the othe Q. We ll begin with 3 and 4, then 5 and 6, end w/intensity c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Application aeas Many, including: Ecology: histoically impotant field of application, many diffeent applications, including: spatial patten (andom / clusteed / avoidance) of a single species pattens of motality (clusteed o not?) tanspotation: locations of accidents neuology: locations of neuons geology: locations of eathquakes (space, o space/time) geogaphy: do simila types of stoes tend to cluste nea each othe? epidemiology: do cases of a paticula disease cluste? If so, suggests contagious disease o single spatial cause c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Examples in pictues: c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

2 c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Cypess tees in Good plot 1 td$x td$y c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Amacine cells, on c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

3 Homogeneous Poisson Pocess = Complete Spatial Randomness imagine a vey small aea, da, with P[event occus in da] = λda da small enough that: at most 1 event in da most aeas have 0 events λ = expected # events / unit aea λ is the intensity of the spatial pocess Two assumptions that give HPP = CSR λ constant ove study aea the outcome (0/1) in da 1 is independent of the outcome in non-ovelapping aea da 2 c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Homogeneous Poisson Pocesses Some mathematical esults: Define N A = # points in aea A (no longe small) N A Poiss(λ A) mean #: λ A va #: λ A pmf P[X λa] = e λa (λa) X X! examples: CSR, obseve 196 obs on (0,10), (0,10) Look at individual 1x1 quadats mean count pe 1x1 quadats = 1.96 Va count = 1.90 Histogam close to theoetical pmf c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Numbe in 1x1 cell Density c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

4 Clusteing Clusteed pocesses/patten: points moe likely to occu nea othe points. Fo quadats, means that: some quadats contain a cluste, have moe points than expected othe quadats have no points same mean, lage vaiance fo clusteed pocess with 196 points on next two slides: mean = 1.96, vaiance = 4.70 c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 A clusteed pocess c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Density Numbe in 1x1 cell c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Regula pocesses P[event in da] lowe if da close to anothe point Tends to space out points c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

5 Amacine cells, on c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 A bette appoach Histoically: Quadats used extensively But, vey limited. Resticted to one specific scale (size of quadat) Bette appoach ecod locations of events, not just count in a box usually all events in a pedefined aea can be andom sample of events But, had to take a simple andom sample Can convet to quadat counts, but can do a LOT moe with (x,y) data c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Summay functions Concept: measue something as a function of distance Vaious choices of summay Distance to neaest neighbo (event - event distance) Distance to neaest point (point - event distance) Combination of these two Ripley s K function pai coelation function Each have uses c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Distance to neaest neighbo How close is each obs. to its neaest neighbo? clusteing: NN distances tend to be small andom (CSR): intemediate egula: NN distances tend to be lage Histoical: calculate mean NN distance, compae to theoetical value (Clak-Evans test) Cuent: estimate cdf of NN distance: G(x) = P[NN distance x] fo each event: find NN, calculate distance to NN had pat is finding NN. Some fancy and fast algoithms (see NN aticle) compae estimated Ĝ(x) to theoetical G(x) fo CSR c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

6 Theoetical CDF on NN distance P[D < x] = 1 e λπx2 x is distance of concen, λ is intensity (events pe unit aea) πx 2 is aea of cicle, adius x (Fo the statisticians). Nice ex. of CDF method fo deiving tansfoming a andom vaiable Define D = distance to NN dist. G(x) = P[D x] = 1 P[D > x] = P[no obs in cicle of adius x] N A Poiss(λA), so: P[0] = e λa (λa) 0 0! P[D x] = 1 e λπx2 Simila ideas, diffeent fomula fo 1D o 3D. c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Edge effects Above assumes infinite plane Real study aeas have edges When a point is close to edge of mapped aea, what is the distance to the NN? oveestimate D. tue NN may be just ove the bounday (close to event) obseved NN (inside study aea) is lage than it should be So undeestimate G(x) especially fo lage x c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Edge effects Thee appoaches to edge effects 1) Ignoe poblem. Study aea edge is a eal edge (e.g. lake shoeline) eally cae about distance to neaest valid event 2) Taditional: adjust estimato edge-coected estimato of Ĝ(x) Usual: use the Kaplan-Meie estimato fo censoed data othes have been poposed, avoid educed sample method but, bias coection inceases Va Ĝ(x) 3) Radical: adjust expectation Use uncoected estimato Change theoetical G(x) to account fo edge effects If goal is to est. G(x), 2) much bette If goal is to test CSR (o some othe pocess), 3) has highe powe Estimate theoetical G(x) by simulation c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 CSR G() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

7 clusteing Moe points close: shote NN distance G() G^ km() G^ bod() G^ han() G^ aw() G pois() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 inhibitation / egulaity Fewe points close: longe NN distance Had coe pocess: no points within a minimum distance G() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Sampling vaiance in Ĝ(x) How to calculate Va Ĝ(x)? Quite a had poblem: 1) edge effects 2) Reflexive NN s : pai of points B is A s NN, A is B s NN same NN distance supisingly common: P[eflexive] = 0.63 fo CSR inceases Va Ĝ(x) Va Ĝ(x) has been deived unde CSR, ignoing edge effects Now, computed by simulation Simulate a ealization of null hypothesis pocess (e.g. CSR) Estimate Ĝ(x) Repeat simulate/estimate 99 o 999 times Calculate Va G(x) at vaious x O go staight to a confidence inteval Calculate0.025 and quantiles of Ĝ(x) at a specified x value epeat fo vaious x s c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Cypess tees G() G^ km() G pois () c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

8 G() G obs () G theo () G hi () G lo () c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Point-event distance = empty-space function cdf of distance fom andomly chosen point (not an event) to neaest event Usually denoted F (x) Unde CSR, ignoing edge effects: F (x) = 1 e λπx2 same deivation as fo G(x) But now: lage distances clusteing, because big aeas of empty space small distances egulaity Evaluated in same way as G(x) F (x) moe poweful than G(x) to detect clusteing G(x) moe poweful to detect egulaity c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 F() F^km() F pois () 14 c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 F() F obs () F theo () F hi () F lo () 14 c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

9 Baddeley s J function Can combine F (x) and G(x) Intepetation: clusteing: J(x) < 1 CSR: J(x) = 1 egulaity: J(x) > 1 J(x) = 1 G(x) 1 F (x) Much newe than F (x) o G(x): 1996 pape Few have much expeience with it c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 J() J^km() J pois () 14 c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 J() J obs () J theo () J hi () J lo () 14 c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Looking at multiple scales simulataneously I ve emphasized neaest neighbo (of an event, of a point). closest event (to the event, to the point) Staightfowad extension to 2nd NN (next closest), 3d NN,... Gets hade to intepet and you have a sepaate plot fo each NN ank Rethink how to compute the summay Instead of how fa to closest point think of how many points within a specified distance? leads to Ripley s K statistic (Ripley 1976, J. Appl. Pob) c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

10 Ripley s K function Like F (x), G(x), and J(x), looks at 2nd ode chaacteistics of a point patten Now, the most commonly used point patten analysis function Povides infomation at multiple scales simultaneously K(x) = 1 E (# events w/i x of an event) λ Intepetation: Clusteing: K(x) lage. Many events close to othe events Regulaity: K(x) small o 0 at shot distances. Notes: K(x) can detect clustes of egulaly spaced points i.e., diffeent pattens at diffeent scales but it is cumulative (numbe of points within distance x we ll see a efinement, the pai coelation function, that looks at points at distance x which simplifies (geatly) infeing the scale of a patten c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 K(x) unde CSR Expected value, E ˆK(x): Unde CSR, events ae independent, E # in aea A = λa E ˆK(x) = E # in aea πx 2 /λ = λπx 2 /λ = πx 2 Vaiance, Va ˆK(x): Va # in aea A = λa so, Va ˆK(x) = 1 λ 2 Va # in πx 2 = πx 2 /λ smalle with moe expected points (lage λ) inceases with distance, x c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 K() K^ iso() K pois () c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 K() K obs () K theo () K hi () K lo () c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

11 Besag s L function L(x) is a vaiance equalizing tansfomation if Y Poiss(X β), then Y has constant vaiance Besag s oiginal vesion L(x) = K(x)/π Wiegand and Moloney (2014) call this L 1 (x) Unde CSR: L(x) = x, Va ˆL(x) appox constant. I pefe L (x) = L(x) x Wiegand and Moloney (2014) call this L 2 (x) Nice featue of L = L 2 (x): unde CSR, L (x) = 0 I believe plots of L ae much cleane (but you decide which you pefe) c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 CSR L() L() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Clusteed L() 0.30 L() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 inhibition L() L() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

12 L() L^iso() L pois () c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 L() L^iso() L pois () c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 L() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Pai coelation functions K(x) and L(x) ae cumulative measues Coe is numbe of events within x of anothe event What if you want to descibe association at distance x? Close to intuition about spatial scale Can untangle multiple pocesses inhibition at shot distances clusteing at lage distances pai-coelation function, g(x) o ρ(x) g(x) = 1 dk(x) 2πx dx unde CSR (K(x) = πx 2 ), dk(x) dx = 2πx, and g(x) = 1 g(x) > 1 events moe likely AT distance x than unde CSR clusteing at a scale of x g(x) < 1 events less likely AT distance x than unde CSR epulsion at a scale of x ange is (0, ) with 1 as the neutal point so often log tansfom: evaluate log g(x) c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

13 Estimatos of g(x) g(x) is much hade to estimate than K(x) ˆK(x) is a sum (# events within distance < x) g(x) depends on 0/1 vaiable: is thee an event at distance = x o not paallel to the issue that a cdf: P[X < x] is easie to estimate than a pdf: f [X = x] Two poposed estimatos: Wiegand and Moloney O-ing estimato: # events within (x, x + dx) equivalent to binning obs. to make a histogam kenel smoothing: much bette (both fo density estimation and ĝ(x)) What is the histogam of 5,10,11,11, 12, 16? choice of bin width eally mattes see next two slides c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Histogam with wide categoies Fequency c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Histogam with naow categoies Fequency c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Kenel smoothing Histogams estimate pobability density fo a ange of X using only the values in that ange the vaiance in the estimated pobability depends on the numbe of obs in the bin wide bin: many points, low vaiance, but biased estimate (one numbe fo many X values) naow bin: low bias (small ange of X values), but lage vaiance (few obs in bin) Density estimation patially avoids this tadeoff and is less dependent on the beaks between categoies Concept: supeimpose little bumps of pobability aound each obs. Add up the pobability to estimate f (x) esult depends on sd of each bump sd called bandwidth esult also depends kenel, i.e. the shape of the bump notice that ange of density estimate is wide than data ange. c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

14 Density Density N = 6 Bandwidth = N = 6 Bandwidth = 0.70 Density Density N = 6 Bandwidth = 2.11 N = 6 Bandwidth = 2.11 c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 CSR g() log(g()) c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Clusteed L() log(g()) c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 inhibition L() log(g()) c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

15 log(g()) log(g obs()) log(g theo()) log(g hi()) log(g lo()) c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Intepeting / using measues 1) to test CSR: pointwise tests estimate ˆL(x) at a ange of distances, x use simulation to calculate point-wise quantiles of ˆL(x) plot ˆL(x) and simulation envelope intepet deviations above and below expected conside distance x 1, then distance x 2 called pointwise-tests. Type I eo ate, α level, coect fo one test One issue (seious): multiple testing doing many tests, one at each distance P[eject CSR at any distance] is much lage than P[eject CSR] especially fo cumulative summay functions, K(x) and L(x) Quite had to do a tue level α test usual appoaches don t wok well because ˆL(x 1) and ˆL(x 2) ae coelated c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Intepeting / using measues 2) summay tests of CSR Calculate a summay statistic acoss elevant ange of distances Two common choices, using L(x) as example: s = max x[ ˆL(x) L(x) ] (maximum statistic, Maximum Absolute Deviation) s = x [ˆL(x) L(x)] 2 (integal statistic, Loosmoe and Fod test) Both computed by evaluating inteesting set of x, finding max o sum L(x) can be theoetical expectation (K(x) = πx 2, L 2 (x) = 0) O, L(x) computed as aveage of n simulations (see below) accounts fo bias due to edge coections Integal bette when consistent but small deviations above expected cuve moe commonly used Max bette when lage excusion fom theoetical value fo a small ange of distances c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Summay tests Tuning into a hypothesis test Have s obs fom the obseved patten Simulate many (39, 99, 999) andom pattens unde H0 (e.g., CSR) Calculate summay statistic fo obseved data and each simulated data set Calculate P[as o moe exteme summay statistic] = p-value Usually one-sided definition of moe exteme (only cae about lage s) This avoids multiple testing issues and gives valid p-value c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

16 Summay tests: Cypess pointwise tests L() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Summay tests: Cypess summay tests Fequency Integal though 12 m Fequency Integal though 4.2 m c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Summay tests Need to choose uppe and lowe distances best when chosen to be biologically elevant. DON T look fo the most significant egion most commonly used to test CSR But you specify the null hypothesis same appoach can be used fo any point pocess model (examples coming soon) The integal and especially the maximum statistic assume Va ˆL(x) appoximately constant Don t use K(x): because Va K(x) is definitely not constant Use L(x) instead, appoximately constant vaiance But not pefect (see next two plots) Can tansfom G(x) o F (x) (both popotions), e.g. sin 1 G(x) ˆ Thee ae studentized summay statistics, if unequal Va is bad c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 L() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

17 L() Lobs() Ltheo() Lhi() Llo() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 A poo use of a test of CSR Until a few yeas ago, it was fashionable to map locations of all things in an aea. Usually tees o othe plants, could be animal nests usually many species fo each species, test CSR (usually using K/L functions) tabulate # species that ae clusteed, # andom, # egula then make ecological conclusions about the community do you see the issue hee? c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 A poo use of a test of CSR If you don t eject H0: CSR fo a species, do you know that species is andomly distibuted? What if you only had 10 individuals fo that species? Statistical powe to detect not-csr is eally small failue to eject H0 does not H0 is tue In my expeience (mostly with tees) lage # events: detect clusteing, sometimes egulaity small # events: accept H0 If you expect intensity to vay ove a study aea, that intoduces clusteing. If you believe that non-andom spatial pattens ae the nom, the hypothesis test is eally telling you only whethe you have a sufficiently lage sample size to detect that non-andom patten. c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Intepeting / using measues 3) Estimating # excess events if patten is CSR, expect λπx 2 events within adius x of anothe event data says an aveage of λ ˆK(x) events within adius x of anothe event λ ˆK(x) λπx 2 is aveage excess events descibes magnitude of clusteing in subject-matte tems less fequently used is ˆK(x) πx 1 2 popotion of excess events at distance x Cypess tee illustation ˆλ = 98/(50 200) = at distance of 10m, ˆK(10) = ave. of 4.7 cypess tees within 10m of anothe cypess tee π10 2 = excess cypess tees within 10m of anothe. 1 = % moe cypess tees within 10m of anothe. O, c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

18 Intepeting / using measues 4) Descibing spatial scale scale is a ticky concept. Vaious definitions Hee, scale = distance(s) at which events epulse each othe o attact each othe A distance-specific concept Many studies have used ˆK(x) o ˆL(x) to estimate scale, e.g. find x whee L(x) is most diffeent fom theoetical value Inceasingly undestood to be wong Both ˆK(x) and ˆL(x) ae cumulative functions: # points within cicle of adius x Small # at distance x may be because epulsion (fewe pts.) at distances < x, even if stong clusteing at x Really want to know what is going on AT distance x, not x Use pai-coelation function c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Intepeting / using measues 5) How pecise is ˆL(x) o ĝ(x)? Not the width of the Null hypothesis envelopes Pecision of ˆL(x) o ĝ(x) Cetainly depends on N = # points But also on the spatial patten ˆK(x) moe vaiable fo clusteed pattens If you know the tue spatial patten, simulate fom that patten and calculate envelope If you don t know the tue patten, use a bootstap Point patten bootstap poposed by Loh, 2008 c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 The bootstap Geneal method fo estimating pecision of a statistic Uses esampling the data to appoximate the unknown sampling distibution of a statistic Gives you the se of a statistic o a confidence inteval fo a statistic CI much moe common Not the same as a andomization test o a null hypothesis test Hypothesis test: simulate / esample assuming H0 (CSR, no diff. in means) Bootstap: simulate / esample assuming Ha (abitay patten, non-zeo diff) Extemely useful tool fo difficult poblems Usual foms of bootstap don t wok fo point patten data Poblem is that one point contibutes to many L(x) Loh devised something that (so fa) is acceptable sometimes esample contibutions to ĝ(x) o ˆL(x) Issues when bootstap aveage cuve not same as data cuve (see below) c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Cypess L(x) bootstap L() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75

19 Cypess L(x) CSR (null) envelope L() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Cypess g(x) bootstap g() c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75 Cypess g(x) CSR (null) envelope log(g()) log(g obs()) log(g theo()) log(g hi()) log(g lo()) c Philip M. Dixon (Iowa State Univ.) Spatial Data Analysis - Pat 6 Sping / 75