Estatística espacial - GET/IME/UFF
Patrícia Lusié Velozo da Costa
2023-2
Processos Pontuais
1 Exemplo
###################################################
#### Aula Computacional de Processos Pontuais ####
###################################################
library(spatstat)
library(splancs)
###################################################
# Método dos quadrantes
###################################################
# Nas bibliotecas spatstat e splancs há alguns conjuntos de dados.
# Usaremos o conjunto cardiff, que é um padrão de pontos que fornece
# as localizações das casas de 168 jovens infratores
# em um conjunto habitacional de Cardiff.
data(cardiff)
dados = as.ppp(cardiff)
# Plotando os pontos e o polígono da regiao D
#plot(cardiff$poly, asp=1, type="n")
plot(cardiff$poly, type="n", bty="n")
points(as.points(cardiff),pch="*")
polymap(cardiff$poly,add=TRUE,border="grey",lwd=4,lty=3)
# Definicao dos quadrantes
nx = 5
ny = 5
quadrantes1 = quadratcount(dados,nx=nx,ny=ny)
quadrantes1 ## x
## y [2.35,20.7) [20.7,39) [39,57.3) [57.3,75.7) [75.7,94]
## [77.1,94.8] 1 20 16 11 8
## [59.5,77.1) 0 11 9 3 0
## [41.8,59.5) 2 7 8 10 7
## [24.2,41.8) 7 8 14 8 3
## [6.49,24.2) 0 7 5 3 0# Plotando os dados e os quadrantes
par(mar=c(0,0,0,0))
plot(cardiff,pch="+",main="")
plot(quadrantes1,add=TRUE, col="red", cex=1.5, lty=2,border="red")
plot(cardiff$poly, bty="n", type="n")
points(as.points(cardiff),pch="*")
polymap(cardiff$poly,add=TRUE,border="gray80",lwd=3,lty=3)
plot(quadrantes1,add=TRUE, col="red", cex=1.5, lty=2,border="red")
# Teste de quadrantes para CSR
t_quadrantes <- quadrat.test(dados,nx=nx,ny=ny)
t_quadrantes##
## Chi-squared test of CSR using quadrat counts
##
## data: dados
## X2 = 97.476, df = 24, p-value = 1.613e-10
## alternative hypothesis: two.sided
##
## Quadrats: 5 by 5 grid of tilest_quadrantes$p.value## [1] 1.613223e-10c(qchisq(0.95,nx*ny-1), t_quadrantes$statistic)## X2
## 36.41503 97.47619#########################################################
# ICS (Indice de tamanho de cluster):
#########################################################
#obtendo o número médio de árvores encontradas em uma região com área
#igual a dos quadrantes
media1 = mean(quadrantes1)
media1## [1] 6.72#obtendo a variância
variancia1 = sum((quadrantes1 - media1)^2/(nx*ny-1))
variancia1## [1] 27.29333#outra forma de calcular a variância
var(as.vector(quadrantes1))## [1] 27.29333#calculando o ICS
ICS = (variancia1/media1) - 1
ICS## [1] 3.061508#como ICS maior que zero, há indícios de haver aglomeração
#########################################################
# Intensidade estimada em cada quadrante: lambda_i
#########################################################
#obtendo a área de cada quadrante
(max(dados$x)-min(dados$x))/nx## [1] 18.328(max(dados$y)-min(dados$y))/ny## [1] 17.664quadrantes1## x
## y [2.35,20.7) [20.7,39) [39,57.3) [57.3,75.7) [75.7,94]
## [77.1,94.8] 1 20 16 11 8
## [59.5,77.1) 0 11 9 3 0
## [41.8,59.5) 2 7 8 10 7
## [24.2,41.8) 7 8 14 8 3
## [6.49,24.2) 0 7 5 3 0i = 1
j = 1
names(quadrantes1[1,])[i]## [1] "[2.35,20.7)"names(quadrantes1[,1])[j]## [1] "[77.1,94.8]"area = (20.7-2.35)*(94.8-77.1)
area## [1] 324.795quadrantes1[i,j]/area## [1] 0.003078865intensity(quadrantes1)[i,j]## [1] 0.003088843i = 3
j = 1
names(quadrantes1[1,])[i]## [1] "[39,57.3)"names(quadrantes1[,1])[j]## [1] "[77.1,94.8]"area = (57.3-39)*(94.8-77.1)
area## [1] 323.91quadrantes1[i,j]/area## [1] 0.006174555intensity(quadrantes1)[i,j]## [1] 0.006177686#obtendo a intensidade em cada quadrante
intensity(quadrantes1)## x
## y [2.35,20.7) [20.7,39) [39,57.3) [57.3,75.7) [75.7,94]
## [77.1,94.8] 0.003088843 0.061776865 0.049421492 0.033977276 0.024710746
## [59.5,77.1) 0.000000000 0.033977276 0.027799589 0.009266530 0.000000000
## [41.8,59.5) 0.006177686 0.021621903 0.024710746 0.030888432 0.021621903
## [24.2,41.8) 0.021621903 0.024710746 0.043243805 0.024710746 0.009266530
## [6.49,24.2) 0.000000000 0.021621903 0.015444216 0.009266530 0.000000000#visualizando essas quantidades
par(mar=c(0,0,0,2))
plot(intensity(quadrantes1, image=TRUE), main="",
col=terrain.colors(256))
polymap(cardiff$poly,add=TRUE,border="black",lwd=3,lty=3)
##############################
# Intensidade via Kernel #
##############################
tau = 5
lambda <- kernel2d(as.points(cardiff), cardiff$poly, tau,
nx=100,ny=100, kernel="quartic")## Xrange is 0.8 95.6
## Yrange is 4.8 97.4
## Doing quartic kernelpar(mar=c(2,2,0.5,0.5))
image(lambda,col=terrain.colors(256))
pointmap(as.points(cardiff),pch="*",add=TRUE,cex=0.75)
tau = 10
lambda <- kernel2d(as.points(cardiff), cardiff$poly, tau,
nx=100,ny=100, kernel="quartic")## Xrange is 0.8 95.6
## Yrange is 4.8 97.4
## Doing quartic kernelpar(mar=c(2,2,0.5,0.5))
image(lambda,col=terrain.colors(256))
pointmap(as.points(cardiff),pch="*",add=TRUE,cex=0.75)
tau = 20
lambda <- kernel2d(as.points(cardiff), cardiff$poly, tau,
nx=100,ny=100, kernel="quartic")## Xrange is 0.8 95.6
## Yrange is 4.8 97.4
## Doing quartic kernelpar(mar=c(2,2,0.5,0.5))
image(lambda,col=terrain.colors(256))
pointmap(as.points(cardiff),pch="*",add=TRUE,cex=0.75)
n = length(cardiff$x)
n## [1] 168tau = 0.68*(n^(-0.2))
tau## [1] 0.2440322lambda <- kernel2d(as.points(cardiff), cardiff$poly, tau,
nx=100,ny=100, kernel="quartic")## Xrange is 0.8 95.6
## Yrange is 4.8 97.4
## Doing quartic kernelpar(mar=c(2,2,0.5,0.5))
image(lambda,col=terrain.colors(256))
pointmap(as.points(cardiff),pch="*",add=TRUE,cex=0.75)
############################
# Vizinho mais próximo #
############################
# Função G
dists <- nndistG(as.points(cardiff))
dists ## $dists
## [1] 13.9992071 4.0067069 1.6570154 1.6570154 1.8300000 3.1918803
## [7] 3.1918803 1.1292475 1.6181780 3.9772352 3.1465855 1.0135581
## [13] 1.0135581 2.2217111 2.2217111 1.3559130 1.1684605 1.1684605
## [19] 1.6633100 1.6633100 0.8700000 0.8700000 0.8984431 1.4038875
## [25] 0.8955445 0.6811755 0.6811755 1.7861971 1.8064606 1.8064606
## [31] 0.8994443 0.8994443 2.1600000 1.7861971 1.6648123 1.1346365
## [37] 1.1346365 3.1287857 5.3049505 2.1239821 0.7900000 0.9080198
## [43] 0.7900000 0.8984431 0.8984431 2.2447494 1.1684605 1.1684605
## [49] 3.3942599 1.6181780 4.3340166 1.7901117 2.5347584 2.0306649
## [55] 2.0306649 1.8977091 1.8977091 2.6100000 2.6100000 7.6735520
## [61] 6.8442458 8.4212410 4.4617485 3.8086481 4.1046559 2.3973319
## [67] 2.3973319 1.1610340 1.1610340 1.1384639 1.1384639 1.1610340
## [73] 2.6702996 3.7522527 3.4080053 3.4080053 3.9086571 3.9086571
## [79] 6.7430705 1.1292475 2.2389730 4.1268026 4.1268026 4.0508024
## [85] 1.6648123 1.6648123 1.8384776 1.8384776 1.6648123 1.6648123
## [91] 2.2600221 6.9348180 2.2600221 1.3559130 1.3559130 3.1598101
## [97] 1.6710775 1.1346365 0.7800000 0.7800000 0.7800000 0.7800000
## [103] 0.4300000 0.4300000 0.6747592 0.6747592 1.0187247 1.4001428
## [109] 1.4001428 6.3105071 3.1465855 3.7055229 1.6648123 1.6648123
## [115] 2.3093289 3.4125943 3.4125943 4.2847754 2.0237095 1.1610340
## [121] 1.1610340 2.8309892 4.1734638 1.5160475 1.5160475 1.6710775
## [127] 1.6710775 2.8002857 5.6619785 1.1253888 1.1253888 1.1684605
## [133] 1.1684605 2.9181158 2.9181158 2.7511816 2.7511816 3.6400549
## [139] 3.6400549 4.1734638 5.0082831 2.0185391 2.0185391 7.5106990
## [145] 1.8701069 1.8701069 1.5160475 0.2600000 0.2600000 1.3559130
## [151] 1.3559130 3.1312937 3.0056114 1.5670673 1.5670673 4.1734638
## [157] 3.8148263 1.4001428 1.4001428 3.5468296 5.1864535 3.1312937
## [163] 3.0128060 3.0056114 1.5700000 0.9080198 0.9080198 0.9080198
##
## $neighs
## [1] 2 4 4 3 4 7 6 80 50 9 12 13 12 15 14 17 18 17
## [19] 20 19 22 21 22 23 27 27 26 34 28 29 32 31 34 28 36 37
## [37] 36 37 40 41 43 43 41 45 44 47 48 47 48 9 49 9 54 55
## [55] 54 57 56 59 58 57 63 61 64 67 64 67 66 69 68 71 70 71
## [73] 72 76 76 75 78 77 77 8 80 83 82 85 86 85 88 87 90 89
## [91] 90 91 94 95 94 95 99 99 100 99 102 101 104 103 106 105 106 109
## [109] 108 111 114 111 114 113 104 117 116 160 120 121 120 121 156 125 124 127
## [127] 126 127 147 131 130 133 132 135 134 137 136 139 138 139 140 143 142 143
## [145] 146 145 148 149 148 151 150 164 164 155 154 123 158 159 158 158 118 163
## [163] 164 153 168 168 166 166maximo <- max(dists$dists)
minimo <- min(dists$dists)
w <- seq(minimo,maximo,by=((maximo - minimo)/100))
## o comando Ghat recebe 2 parâmetros, o primeiro o objeto pontos,
## o segundo um vetor com a lista de distâncias onde se estima a função G
funcao_G <- Ghat(as.points(cardiff),w)
par(mar=c(4,4,0.5,0.5))
plot(w,funcao_G,type="l",xlab="distâncias", ylab="G estimada",col=4,lwd=2)
# Função F
dists2 <- nndistF(as.points(csr(cardiff$poly, length(cardiff$x))), as.points(cardiff))
maximo <- max(dists2)
minimo <- min(dists2)
x <- seq(minimo,maximo,by=((maximo - minimo)/100))
funcao_F <- Fhat(as.points(csr(cardiff$poly, length(cardiff$x))), as.points(cardiff),x)
plot(x,funcao_F,type="l", xlab="distâncias", ylab="F estimada",col=3,lwd=2)
#Função G x F
plot(funcao_G, funcao_F, type="l", xlab="G estimada", ylab="F estimada",lwd=2)
abline(0,1,lty=2,col=2,lwd=2)
############################
# Função K #
############################
h <- seq(0,30,1)
funcaoK<-khat(as.points(cardiff), cardiff$poly, h)
funcaoL<- sqrt(funcaoK/pi) - h
m <- 99
funcaoK.i<-matrix(NA,length(h),m)
funcaoL.i<-matrix(NA,length(h),m)
for (i in 1:m) {
y<-as.points(csr(cardiff$poly, length(cardiff$x)))
funcaoK.i[,i]<- khat(y, cardiff$poly, h)
funcaoL.i[,i]<- sqrt(funcaoK.i[,i]/pi) - h
}
env.sup<-0
for (i in 1:length(h)) {
env.sup[i]<-max(funcaoL.i[i,])
}
env.inf<-0
for (i in 1:length(h)) {
env.inf[i]<-min(funcaoL.i[i,])
}
par(mar=c(4.5,4.5,0.5,0.5))
plot(h,funcaoL, type="l", xlab="Distâncias", lwd=2, ylab=expression(hat(L)(h)),ylim=c(-1,1.4))
lines(h,rep(0,length(h)),lty=2,col=2,lwd=2)
lines(h,env.sup,lty=3,col=4,lwd=2)
lines(h,env.inf,lty=3,col=4,lwd=2)
2 Exercício
###################################################
#### Aula Computacional de Processos Pontuais ####
###################################################
#Carregando os pacotes
library(spatstat)
library(splancs)
#Simulando um conjunto de dados
n = 1000
x = runif(n, 0, 5)
y = runif(n, 0, 5)
dados = list(x = x,
y = y,
area = c(xl = 0, xu = 5, yl = 0, yu = 5),
poly = as.matrix(data.frame(x = c(5, 5, 0, 0),
y = c(5, 0, 0, 5))))
dados2 = as.ppp(dados)
#dados2 = ppp(x, y, window = owin(c(0, 5), c(0, 5)))
###################################################
# Método dos quadrantes
###################################################
# Plotando os pontos e o polígono da regiao D
plot(dados$poly, type="n", bty="n")
points(as.points(dados),pch="*")
polymap(dados$poly,add=TRUE,border="grey",lwd=4,lty=3)
# Definicao dos quadrantes
nx = 5
ny = 5
quadrantes1 = quadratcount(dados2,nx=nx,ny=ny)
quadrantes1 ## x
## y [0,1) [1,2) [2,3) [3,4) [4,5]
## [4,5] 42 44 42 42 45
## [3,4) 31 42 34 32 35
## [2,3) 42 43 39 47 45
## [1,2) 37 39 29 53 38
## [0,1) 37 37 46 40 39# Plotando os dados e os quadrantes
par(mar=c(0,0,0,0))
plot(dados,pch="+",main="")
plot(quadrantes1,add=TRUE, col="red", cex=1.5, lty=2,border="red")
plot(dados$poly, bty="n", type="n")
points(as.points(dados),pch="*")
polymap(dados$poly,add=TRUE,border="gray80",lwd=3,lty=3)
plot(quadrantes1,add=TRUE, col="red", cex=1.5, lty=2,border="red")
# Teste de quadrantes para CSR
t_quadrantes <- quadrat.test(dados2,nx=nx,ny=ny)
t_quadrantes##
## Chi-squared test of CSR using quadrat counts
##
## data: dados2
## X2 = 17.75, df = 24, p-value = 0.3701
## alternative hypothesis: two.sided
##
## Quadrats: 5 by 5 grid of tilest_quadrantes$p.value## [1] 0.3700705#c(qchisq(0.95,nx*ny-1), t_quadrantes$statistic)
#########################################################
# ICS (Indice de tamanho de cluster):
#########################################################
#obtendo o número médio de árvores encontradas em uma região com área
#igual a dos quadrantes
media1 = mean(quadrantes1)
media1## [1] 40#obtendo a variância
variancia1 = sum((quadrantes1 - media1)^2/(nx*ny-1))
variancia1## [1] 29.58333#outra forma de calcular a variância
var(as.vector(quadrantes1))## [1] 29.58333#calculando o ICS
ICS = (variancia1/media1) - 1
ICS## [1] -0.2604167#como ICS maior que zero, há indícios de haver aglomeração
#########################################################
# Intensidade estimada em cada quadrante: lambda_i
#########################################################
#obtendo a intensidade em cada quadrante
intensity(quadrantes1)## x
## y [0,1) [1,2) [2,3) [3,4) [4,5]
## [4,5] 42 44 42 42 45
## [3,4) 31 42 34 32 35
## [2,3) 42 43 39 47 45
## [1,2) 37 39 29 53 38
## [0,1) 37 37 46 40 39#visualizando essas quantidades
par(mar=c(0,0,0,2))
plot(dados$poly, bty="n", type="n")
plot(intensity(quadrantes1, image=TRUE), main="",
col=terrain.colors(256), add=T)
polymap(dados$poly,add=TRUE,border="gray80",lwd=3,lty=3)
##############################
# Intensidade via Kernel #
##############################
tau = 5
lambda <- kernel2d(as.points(dados), dados$poly, tau,
nx=100,ny=100, kernel="quartic")## Xrange is 0 5
## Yrange is 0 5
## Doing quartic kernelpar(mar=c(2,2,0.5,0.5))
image(lambda,col=terrain.colors(256))
pointmap(as.points(dados),pch="*",add=TRUE,cex=0.75)
tau = 10
lambda <- kernel2d(as.points(dados), dados$poly, tau,
nx=100,ny=100, kernel="quartic")## Xrange is 0 5
## Yrange is 0 5
## Doing quartic kernelpar(mar=c(2,2,0.5,0.5))
image(lambda,col=terrain.colors(256))
pointmap(as.points(dados),pch="*",add=TRUE,cex=0.75)
tau = 20
lambda <- kernel2d(as.points(dados), dados$poly, tau,
nx=100,ny=100, kernel="quartic")## Xrange is 0 5
## Yrange is 0 5
## Doing quartic kernelpar(mar=c(2,2,0.5,0.5))
image(lambda,col=terrain.colors(256))
pointmap(as.points(dados),pch="*",add=TRUE,cex=0.75)
############################
# Vizinho mais próximo #
############################
# Função G
dists <- nndistG(as.points(dados))
dists ## $dists
## [1] 0.046322662 0.032598563 0.086075290 0.057975375 0.076134054 0.120605358
## [7] 0.031217103 0.083849710 0.072575744 0.100227476 0.062018488 0.142586236
## [13] 0.082562518 0.007673871 0.088034597 0.070900422 0.059926678 0.034742209
## [19] 0.004371273 0.025600109 0.014177921 0.070201857 0.150170649 0.065052001
## [25] 0.070569349 0.089439084 0.086081002 0.027628092 0.151850576 0.047689125
## [31] 0.097257277 0.130181735 0.130019778 0.053587019 0.057747733 0.023988615
## [37] 0.105781372 0.092419447 0.100386816 0.132619187 0.078432454 0.059788386
## [43] 0.091763790 0.043283231 0.091619511 0.088294929 0.066920816 0.108992229
## [49] 0.113009531 0.021571013 0.067522652 0.153227649 0.125812779 0.191195687
## [55] 0.021304500 0.073991207 0.107212943 0.094375257 0.036109258 0.053356859
## [61] 0.057649673 0.060029880 0.043372079 0.081386704 0.069610243 0.043684928
## [67] 0.072562877 0.066602604 0.079792001 0.083849621 0.112064648 0.093495607
## [73] 0.077422139 0.008851637 0.048065493 0.139607690 0.063517469 0.076502239
## [79] 0.132407470 0.054036056 0.042088807 0.077352224 0.054860793 0.032315154
## [85] 0.048065493 0.101413603 0.043283231 0.130019778 0.024977103 0.135877016
## [91] 0.057649673 0.090432851 0.114972085 0.093005409 0.165132419 0.067681915
## [97] 0.079623175 0.075265323 0.080564642 0.067638412 0.059311775 0.006086593
## [103] 0.070201857 0.128574032 0.045654274 0.132407877 0.038289037 0.122309603
## [109] 0.098767036 0.045941827 0.086478500 0.042447908 0.075265323 0.026023692
## [115] 0.016634581 0.154197429 0.038505715 0.029005499 0.050156814 0.059354082
## [121] 0.079730301 0.074274895 0.052941677 0.171344455 0.094063178 0.037712382
## [127] 0.115201910 0.092043283 0.099722471 0.021304500 0.105983887 0.145803061
## [133] 0.016154187 0.057039718 0.021568736 0.085800421 0.012624570 0.141303045
## [139] 0.126826130 0.032117277 0.093449914 0.074123213 0.115201910 0.015347961
## [145] 0.086081002 0.052941677 0.025600109 0.042447908 0.098741265 0.130835256
## [151] 0.036971228 0.076384360 0.116168044 0.079466749 0.050689027 0.029540763
## [157] 0.093603839 0.094572164 0.041518435 0.036059131 0.075213179 0.018385232
## [163] 0.143262500 0.027628092 0.118196752 0.097257277 0.065951953 0.047252660
## [169] 0.035865345 0.198365961 0.030196734 0.076481552 0.147987681 0.020632638
## [175] 0.077352224 0.046885650 0.063990055 0.092043283 0.029761552 0.034209479
## [181] 0.091453748 0.024278877 0.032924474 0.084447708 0.075358509 0.105588440
## [187] 0.020913541 0.055945230 0.119014963 0.018234220 0.016154187 0.066950247
## [193] 0.125632815 0.142647938 0.050661633 0.049317745 0.090288347 0.107758767
## [199] 0.141203341 0.092658528 0.078734598 0.091133910 0.049869941 0.100461422
## [205] 0.085497517 0.137610681 0.094528074 0.106967228 0.031217103 0.073451576
## [211] 0.069878106 0.085221266 0.059654565 0.148465610 0.028507637 0.040160935
## [217] 0.078275137 0.156149353 0.013049530 0.055207254 0.017995508 0.126895533
## [223] 0.053812853 0.158636363 0.087929680 0.135802349 0.133402850 0.090428495
## [229] 0.113555683 0.147174478 0.060422229 0.151309399 0.102384600 0.093772926
## [235] 0.123500251 0.043171781 0.094429131 0.075372395 0.113243137 0.062436487
## [241] 0.043117506 0.098560004 0.073553650 0.025020909 0.075858089 0.053978840
## [247] 0.056952414 0.152721020 0.104828656 0.070896992 0.163199968 0.134324833
## [253] 0.042737062 0.039445453 0.102728533 0.081652119 0.070896992 0.061711109
## [259] 0.024706851 0.111307374 0.043171781 0.095660171 0.116894000 0.218055593
## [265] 0.039833360 0.006086593 0.094365127 0.061026449 0.094528074 0.079466850
## [271] 0.076402851 0.087767383 0.039951144 0.067378780 0.087981408 0.001051062
## [277] 0.029666742 0.093772926 0.024706851 0.052238690 0.123612473 0.113275826
## [283] 0.090357441 0.070266793 0.131789554 0.202123343 0.113704819 0.073024618
## [289] 0.050954504 0.085141331 0.049156180 0.031310190 0.057975375 0.123439294
## [295] 0.165881917 0.051869791 0.088818958 0.032107413 0.024336119 0.028507637
## [301] 0.131961410 0.019815194 0.063753155 0.140064806 0.062958707 0.154197429
## [307] 0.099397823 0.051991120 0.135877016 0.116032201 0.068780127 0.030184318
## [313] 0.090316578 0.041518435 0.142503790 0.022953096 0.047593828 0.059776152
## [319] 0.062911997 0.063753155 0.058574406 0.074123213 0.102422272 0.038046656
## [325] 0.061711109 0.134106618 0.109612678 0.041594073 0.079466850 0.040870760
## [331] 0.105996658 0.040870760 0.116032201 0.136809372 0.163261925 0.076580295
## [337] 0.209357428 0.101192490 0.073087701 0.047689125 0.068135164 0.073451576
## [343] 0.077667260 0.053356859 0.051869791 0.145247221 0.097431256 0.046542793
## [349] 0.101826122 0.088441094 0.047252660 0.121520290 0.098670878 0.100227476
## [355] 0.070650508 0.059654565 0.065820259 0.079990346 0.100844240 0.029761552
## [361] 0.083849621 0.056551693 0.039833360 0.054954856 0.014007055 0.088517244
## [367] 0.066937118 0.149164393 0.051124066 0.048169593 0.113760681 0.111307374
## [373] 0.171853196 0.075350117 0.031310190 0.075681646 0.063409105 0.056041368
## [379] 0.083571195 0.115093860 0.084514904 0.080743620 0.014177921 0.107195205
## [385] 0.032107413 0.128542367 0.059428139 0.073615240 0.046713490 0.071628562
## [391] 0.052238690 0.066937118 0.097390387 0.101646978 0.110091747 0.086218196
## [397] 0.030184318 0.040161631 0.113976221 0.071839369 0.044416606 0.103981646
## [403] 0.016634581 0.116054476 0.094004472 0.138488269 0.079623175 0.138939078
## [409] 0.113704819 0.078899701 0.088517244 0.149022507 0.036702106 0.071130360
## [415] 0.066930248 0.058974716 0.023389693 0.075225232 0.024439506 0.084011191
## [421] 0.057747733 0.088294929 0.099933611 0.024439506 0.029590010 0.036073136
## [427] 0.095763537 0.071740303 0.130357299 0.036702106 0.106967228 0.036971228
## [433] 0.029538744 0.029903879 0.079198495 0.019731265 0.044379452 0.112499769
## [439] 0.046988222 0.019439121 0.248191525 0.142769408 0.112413712 0.082074380
## [445] 0.076295728 0.063425950 0.076384360 0.095123826 0.036059131 0.013007619
## [451] 0.099397823 0.087205067 0.124538749 0.138534812 0.074679356 0.104661381
## [457] 0.159595465 0.035671797 0.127722080 0.076421499 0.082562518 0.036332196
## [463] 0.108957541 0.049869941 0.067676757 0.083479114 0.030520294 0.067358425
## [469] 0.060093761 0.068834694 0.070698751 0.075201020 0.095123826 0.044991989
## [475] 0.181049070 0.085830307 0.075358509 0.054036056 0.080045846 0.122516103
## [481] 0.063425950 0.113669555 0.014864901 0.044827449 0.125294403 0.064792422
## [487] 0.018863562 0.018394218 0.085446891 0.136809372 0.094093277 0.067971315
## [493] 0.056388728 0.032117277 0.095701244 0.034209479 0.014007055 0.037767083
## [499] 0.095763537 0.045721440 0.098767036 0.138488269 0.106822266 0.058792572
## [505] 0.133411967 0.084866022 0.106955527 0.144520797 0.102097710 0.052163677
## [511] 0.072575744 0.085063855 0.072130197 0.092057109 0.109568646 0.110506102
## [517] 0.141543827 0.083721660 0.093005409 0.083721660 0.105484937 0.100386816
## [523] 0.082556589 0.095876077 0.134324833 0.078275137 0.090772570 0.137920319
## [529] 0.105295348 0.074077603 0.070347893 0.048661939 0.070698751 0.091292171
## [535] 0.100518488 0.084514904 0.033337832 0.037712382 0.079466749 0.028543865
## [541] 0.079702984 0.038046656 0.084891813 0.009221303 0.081713022 0.066993779
## [547] 0.024977103 0.073276646 0.085446891 0.075884398 0.044460269 0.011661355
## [553] 0.046542793 0.056075652 0.026980325 0.129798573 0.064158624 0.128542367
## [559] 0.117847629 0.081394083 0.118163419 0.090108871 0.047942253 0.084454223
## [565] 0.019971453 0.062082939 0.050954504 0.084483893 0.107765183 0.018234220
## [571] 0.115093860 0.112466493 0.067971315 0.014864901 0.067638412 0.117320652
## [577] 0.259132677 0.085042908 0.102922985 0.073024618 0.056075652 0.109141735
## [583] 0.030302688 0.016816416 0.047213587 0.165955570 0.075677293 0.207754591
## [589] 0.139599191 0.042707672 0.063928621 0.040505784 0.152287438 0.053812853
## [595] 0.073553650 0.173816744 0.049317745 0.057039718 0.087929680 0.044548646
## [601] 0.140187599 0.060422229 0.051789925 0.046708905 0.060093761 0.097431256
## [607] 0.095347886 0.071740303 0.059776152 0.033337832 0.059354082 0.029666742
## [613] 0.047776895 0.092057109 0.124723918 0.067284912 0.101826122 0.070900422
## [619] 0.076402851 0.029540763 0.127798762 0.073615240 0.071960922 0.094050846
## [625] 0.020913541 0.029207289 0.113555683 0.118799568 0.106172650 0.155120806
## [631] 0.021568736 0.028916262 0.240045135 0.083479114 0.078734598 0.022838905
## [637] 0.023167173 0.013049530 0.054746666 0.075884398 0.012624570 0.057154015
## [643] 0.116054476 0.121523028 0.072232688 0.070786967 0.013007619 0.159595465
## [649] 0.061026449 0.152721020 0.064630411 0.100518488 0.099587532 0.109013247
## [655] 0.066920816 0.165127275 0.075634207 0.102200152 0.094375257 0.126826130
## [661] 0.038305924 0.028543865 0.109974522 0.076481552 0.086478500 0.028946786
## [667] 0.069669968 0.032598563 0.075201020 0.015347961 0.086774981 0.087160999
## [673] 0.093664669 0.097509303 0.035865345 0.202123343 0.044548646 0.094189922
## [679] 0.070786967 0.128574032 0.029207289 0.089691571 0.066930248 0.158684212
## [685] 0.074423610 0.046885650 0.069013430 0.016427117 0.029538744 0.050144243
## [691] 0.163066815 0.022293594 0.113587454 0.102778053 0.025020909 0.114234076
## [697] 0.107591208 0.149164393 0.145334210 0.037767083 0.023167173 0.089439084
## [703] 0.134355137 0.090316578 0.063326735 0.130835256 0.127608211 0.122657261
## [709] 0.038305924 0.106728634 0.067676757 0.119014963 0.105285340 0.008851637
## [715] 0.054746666 0.062608042 0.076134054 0.028619495 0.089564644 0.083849710
## [721] 0.050026991 0.054023191 0.025844775 0.099819746 0.090432851 0.133838562
## [727] 0.045654274 0.036332196 0.085830307 0.108165148 0.024675236 0.060116198
## [733] 0.064276565 0.023848155 0.023988615 0.028619495 0.094572164 0.101126345
## [739] 0.099506235 0.087995033 0.091619511 0.120824087 0.101172778 0.091580426
## [745] 0.062460411 0.040106338 0.044416606 0.081726430 0.138776867 0.059788386
## [751] 0.023389693 0.199078271 0.100589664 0.107498650 0.219378408 0.068780127
## [757] 0.090428495 0.030520294 0.133411967 0.044827449 0.057154015 0.075372395
## [763] 0.056980564 0.017995508 0.050292078 0.007994400 0.099506235 0.124884150
## [769] 0.209397965 0.090129738 0.051124066 0.066602604 0.043684928 0.016816416
## [775] 0.085042908 0.071130360 0.034742209 0.094569253 0.029903879 0.118163419
## [781] 0.090072691 0.081352829 0.120397276 0.105983887 0.063409105 0.078900419
## [787] 0.133077102 0.113149513 0.113344587 0.114972085 0.062958707 0.120642782
## [793] 0.134577492 0.040106338 0.016427117 0.074323768 0.018394218 0.044460269
## [799] 0.209638887 0.079508152 0.118911010 0.056388728 0.094429131 0.020632638
## [805] 0.018385232 0.080564642 0.040161631 0.130636876 0.111116685 0.085497517
## [811] 0.070650508 0.061081608 0.159422317 0.092035181 0.062436487 0.084483893
## [817] 0.144366640 0.076502239 0.096543105 0.086346167 0.122516103 0.047130008
## [823] 0.060029880 0.055945230 0.032903638 0.113149513 0.018863562 0.035893034
## [829] 0.125398746 0.036073136 0.101646978 0.056995291 0.026023692 0.237164992
## [835] 0.007673871 0.133077102 0.084011191 0.033902423 0.135466036 0.050144243
## [841] 0.092708265 0.039951144 0.125632815 0.140187599 0.025844775 0.145334210
## [847] 0.050156814 0.103199426 0.050661633 0.127999634 0.108142838 0.067681915
## [853] 0.030206866 0.058574406 0.034625302 0.029590010 0.081652119 0.112267221
## [859] 0.158998180 0.082556589 0.009221303 0.089490085 0.056980564 0.022838905
## [865] 0.024675236 0.075961774 0.040160935 0.018598284 0.116450760 0.046708905
## [871] 0.062018488 0.086075290 0.036109258 0.105683222 0.024278877 0.170322719
## [877] 0.136098439 0.101658304 0.145572056 0.077256012 0.090129738 0.108278068
## [883] 0.118834331 0.062082939 0.047776895 0.029925916 0.085162678 0.118468479
## [889] 0.073276646 0.034625302 0.076421499 0.096196520 0.096699452 0.126755131
## [895] 0.128317684 0.109013247 0.099236638 0.028946786 0.095715583 0.093759615
## [901] 0.059598370 0.107758767 0.092035181 0.059645689 0.046322662 0.051991120
## [907] 0.152440999 0.019815194 0.022953096 0.065052001 0.096302909 0.054210160
## [913] 0.054954856 0.172258291 0.068827085 0.089521199 0.153893294 0.021866851
## [919] 0.072260698 0.131813601 0.047213587 0.094823441 0.086252915 0.134269263
## [925] 0.050233203 0.032903638 0.001051062 0.056551693 0.063326735 0.029925916
## [931] 0.044152068 0.151564616 0.087261664 0.102384600 0.097022164 0.072168571
## [937] 0.114539616 0.011661355 0.112595209 0.068135164 0.112985754 0.071960922
## [943] 0.097951414 0.192386341 0.029803585 0.099819746 0.100195270 0.170965904
## [949] 0.047593828 0.070347893 0.028916262 0.057080740 0.141351323 0.120605358
## [955] 0.109855502 0.071655147 0.100998420 0.150170649 0.073772791 0.126018646
## [961] 0.054210160 0.029803585 0.085152622 0.144962114 0.088818958 0.084891813
## [967] 0.019731265 0.004371273 0.052163677 0.043372079 0.019971453 0.018598284
## [973] 0.030302688 0.062911997 0.177916956 0.021866851 0.090072691 0.047466577
## [979] 0.059428139 0.086460730 0.073087701 0.068834694 0.113009531 0.101413603
## [985] 0.124615787 0.085344848 0.060995354 0.051789925 0.019439121 0.072562877
## [991] 0.120763432 0.040505784 0.047942253 0.145294346 0.077705169 0.064630411
## [997] 0.124884150 0.067712116 0.089521199 0.060619767
##
## $neighs
## [1] 905 668 872 293 717 954 209 720 511 354 871 672 461 835 479 618 945 777
## [19] 968 147 383 103 958 910 547 702 145 164 233 340 166 409 88 773 421 735
## [37] 451 240 522 382 936 750 640 87 741 422 655 810 983 766 598 681 606 973
## [55] 130 852 509 659 873 344 91 823 970 159 344 773 990 772 25 361 1 87
## [73] 796 714 85 757 291 818 621 478 50 175 195 971 75 984 44 33 547 309
## [91] 61 725 790 519 297 852 407 113 806 575 147 266 22 680 727 926 647 58
## [109] 501 875 665 148 98 833 403 306 700 219 847 611 172 679 146 265 760 538
## [127] 143 178 563 55 784 144 191 598 631 468 641 570 660 494 669 322 127 670
## [145] 27 123 20 112 911 706 432 447 864 539 555 620 217 737 314 449 246 805
## [163] 695 28 655 31 921 351 675 683 433 664 63 804 82 686 134 128 360 496
## [181] 625 875 764 118 477 744 625 824 712 570 133 425 843 824 849 597 435 902
## [199] 78 488 635 275 464 126 810 509 269 431 7 342 855 180 356 141 300 867
## [217] 526 430 638 488 764 53 594 923 599 57 579 757 627 871 602 173 934 278
## [235] 841 261 803 762 838 815 292 781 595 695 319 603 637 650 392 257 521 525
## [253] 709 419 228 857 250 325 279 372 236 642 272 787 363 102 27 649 207 329
## [271] 619 474 842 546 277 927 612 234 259 391 654 988 952 823 411 676 409 580
## [289] 567 171 727 375 4 242 906 345 965 385 21 215 595 908 320 158 791 116
## [307] 451 906 90 333 756 397 704 159 550 909 949 609 974 303 854 142 481 542
## [325] 258 843 217 735 270 332 694 330 310 490 851 362 45 522 981 30 940 210
## [343] 377 60 296 43 606 553 617 560 168 372 169 10 811 213 87 66 181 179
## [361] 70 928 265 913 497 411 392 698 771 84 602 260 156 925 292 179 785 60
## [379] 811 571 536 101 21 929 298 558 979 622 930 231 280 367 719 831 256 683
## [397] 312 807 847 1 747 803 115 643 43 502 97 36 287 566 366 425 430 776
## [415] 683 779 751 311 424 837 35 46 775 419 856 830 499 608 71 413 208 151
## [433] 689 779 566 967 254 34 183 989 415 282 854 325 722 481 152 473 160 647
## [451] 307 513 423 455 594 922 648 637 893 891 13 728 413 203 711 634 758 35
## [469] 605 982 533 669 448 59 448 729 185 80 389 821 446 602 574 760 935 777
## [487] 827 797 549 334 830 573 802 140 688 180 365 700 427 989 109 406 901 731
## [505] 759 364 317 935 862 969 9 126 791 614 359 903 35 520 94 518 947 39
## [523] 860 664 252 217 461 987 349 360 950 190 471 464 652 381 610 126 154 662
## [541] 316 324 966 861 80 794 89 889 489 640 798 938 348 581 130 960 715 386
## [559] 509 797 780 544 993 216 971 884 289 816 323 190 380 501 492 483 100 790
## [577] 175 775 552 288 554 928 973 774 921 368 824 976 860 183 60 992 280 223
## [595] 243 333 196 134 225 677 844 231 988 870 469 347 811 428 318 537 120 277
## [613] 885 514 967 651 349 16 271 156 576 388 942 685 187 681 229 891 455 790
## [631] 135 951 76 466 201 864 701 219 715 550 137 761 404 498 100 679 450 457
## [649] 268 248 996 535 69 896 47 777 156 204 58 139 709 540 467 172 111 898
## [667] 873 2 472 144 420 880 292 511 169 286 600 494 646 104 626 140 415 817
## [685] 951 176 973 795 433 840 913 670 462 134 244 853 908 368 846 498 637 26
## [703] 819 313 929 150 465 434 661 526 465 189 290 74 639 827 5 736 11 8
## [721] 835 626 845 946 92 776 105 462 476 521 865 498 462 565 36 718 158 276
## [739] 767 657 45 229 612 771 636 794 401 919 489 42 417 295 358 273 120 311
## [757] 228 467 505 484 642 238 863 221 993 14 739 997 868 881 369 68 66 584
## [775] 578 414 18 818 434 561 977 432 121 131 377 201 836 826 609 93 305 837
## [793] 806 746 688 904 488 551 93 68 527 493 237 174 162 99 398 702 881 205
## [811] 355 584 469 903 240 568 150 78 775 833 480 169 62 188 926 788 487 312
## [829] 903 426 394 690 114 51 14 787 420 968 765 690 174 273 193 601 723 699
## [847] 119 520 195 872 354 96 164 321 890 425 256 570 536 523 544 16 763 636
## [865] 731 118 216 972 472 604 11 3 59 269 182 82 240 739 7 339 770 225
## [883] 30 566 613 930 13 977 548 855 460 472 135 988 791 654 716 666 356 89
## [901] 387 198 814 7 1 308 649 302 316 24 73 961 364 772 510 999 320 976
## [919] 42 351 585 238 329 259 840 825 276 362 705 886 63 191 201 233 191 176
## [937] 671 552 646 341 950 623 207 768 962 724 300 75 317 531 632 266 561 6
## [955] 321 303 210 23 622 479 912 945 407 55 297 543 436 19 510 63 565 868
## [973] 583 319 667 918 781 221 387 344 339 470 49 86 160 758 731 603 440 67
## [991] 940 592 563 500 78 651 768 436 916 688maximo <- max(dists$dists)
minimo <- min(dists$dists)
w <- seq(minimo,maximo,by=((maximo - minimo)/100))
## o comando Ghat recebe 2 parâmetros, o primeiro o objeto pontos,
## o segundo um vetor com a lista de distâncias onde se estima a função G
funcao_G <- Ghat(as.points(dados),w)
par(mar=c(4,4,0.5,0.5))
plot(w,funcao_G,type="l",xlab="distâncias", ylab="G estimada",col=4,lwd=2)
# Função F
dists2 <- nndistF(as.points(csr(dados$poly, length(dados$x))), as.points(dados))
maximo <- max(dists2)
minimo <- min(dists2)
x <- seq(minimo,maximo,by=((maximo - minimo)/100))
funcao_F <- Fhat(as.points(csr(dados$poly, length(dados$x))), as.points(dados),x)
plot(x,funcao_F,type="l", xlab="distâncias", ylab="F estimada",col=3,lwd=2)
#Função G x F
plot(funcao_G, funcao_F, type="l", xlab="G estimada", ylab="F estimada",lwd=2)
abline(0,1,lty=2,col=2,lwd=2)
############################
# Função K #
############################
h <- seq(0,30,1)
funcaoK<-khat(as.points(dados), dados$poly, h)
funcaoL<- sqrt(funcaoK/pi) - h
m <- 99
funcaoK.i<-matrix(NA,length(h),m)
funcaoL.i<-matrix(NA,length(h),m)
for (i in 1:m) {
y<-as.points(csr(dados$poly, length(dados$x)))
funcaoK.i[,i]<- khat(y, dados$poly, h)
funcaoL.i[,i]<- sqrt(funcaoK.i[,i]/pi) - h
}
env.sup<-0
for (i in 1:length(h)) {
env.sup[i]<-max(funcaoL.i[i,])
}
env.inf<-0
for (i in 1:length(h)) {
env.inf[i]<-min(funcaoL.i[i,])
}
par(mar=c(4.5,4.5,0.5,0.5))
plot(h,funcaoL, type="l", xlab="Distâncias", lwd=2, ylab=expression(hat(L)(h)),ylim=c(-1,1.4))
lines(h,rep(0,length(h)),lty=2,col=2,lwd=2)
lines(h,env.sup,lty=3,col=4,lwd=2)
lines(h,env.inf,lty=3,col=4,lwd=2)