我想在3d内核密度估计中绘制特定百分比轮廓的等值面。然后,我想知道该3d形状内的点。
我将展示我采用第2种情况来说明我的问题(从R - How to find points within specific Contour和How to plot a contour line showing where 95% of values fall within, in R and in ggplot2模仿的代码)。
library(MASS)
library(misc3d)
library(rgl)
library(sp)
# Create dataset
set.seed(42)
Sigma <- matrix(c(15, 8, 5, 8, 15, .2, 5, .2, 15), 3, 3)
mv <- data.frame(mvrnorm(400, c(100, 100, 100),Sigma))
### 2d ###
# Create kernel density
dens2d <- kde2d(mv[, 1], mv[, 2], n = 40)
# Find the contour level defined in prob
dx <- diff(dens2d$x[1:2])
dy <- diff(dens2d$y[1:2])
sd <- sort(dens2d$z)
c1 <- cumsum(sd) * dx * dy
prob <- .5
levels <- sapply(prob, function(x) {
approx(c1, sd, xout = 1 - x)$y
})
# Find which values are inside the defined polygon
ls <- contourLines(dens2d, level = levels)
pinp <- point.in.polygon(mv[, 1], mv[, 2], ls[[1]]$x, ls[[1]]$y)
# Plot it
plot(mv[, 1], mv[, 2], pch = 21, bg = "gray")
contour(dens2d, levels = levels, labels = prob,
add = T, col = "red")
points(mv[pinp == 1, 1], mv[pinp == 1, 2], pch = 21, bg = "orange")
因此,使用近似值定义50%的轮廓,使用轮廓线创建轮廓,然后point.in.polygon查找该轮廓内的点。
我想做同样的事情,但是在3d情况下。这是我所管理的:
### 3d ###
# Create kernel density
dens3d <- kde3d(mv[,1], mv[,2], mv[,3], n = 40)
# Find the contour level defined in prob
dx <- diff(dens3d$x[1:2])
dy <- diff(dens3d$y[1:2])
dz <- diff(dens3d$z[1:2])
sd3d <- sort(dens3d$d)
c3d <- cumsum(sd3d) * dx * dy * dz
levels <- sapply(prob, function(x) {
approx(c3d, sd3d, xout = 1 - x)$y
})
# Find which values are inside the defined polygon
# # No idea
# Plot it
points3d(mv[,1], mv[,2], mv[,3], size = 2)
box3d(col = "gray")
contour3d(dens3d$d, level = levels, x = dens3d$x, y = dens3d$y, z = dens3d$z, #exp(-12)
alpha = .3, color = "red", color2 = "gray", add = TRUE)
title3d(xlab = "x", ylab = "y", zlab = "z")
所以,我还没走。
[我意识到在3d情况下定义级别的方法不正确,我猜问题出在c3d <- cumsum(sd3d) * dx * dy * dz
之内,但老实说我不知道如何进行。
而且,一旦正确定义了3d轮廓,我将感谢您提供任何有关如何逼近该轮廓内的点的技巧。
非常感谢!
Edit:基于user2554330的建议,我将编辑我的问题,以添加测试代码,将他或她的建议与我在此处发布的建议进行比较。 (我确实意识到,将轮廓用作新数据点的推论的目的不在最初的问题中,对此致歉。)
[此外,我在下面的评论中有些仓促。两种方法在2D情况下的效果取决于样本的大小。在样本n = 48左右时,user2554330的方法捕获了大约69%的人口(而我发布的方法捕获了大约79%的人口),但是在样本n = 400左右时,user2554330的方法捕获了大约79%的人口(相比83% )。
# Load libraries
library(MASS)
library(misc3d)
library(rgl)
library(sp)
library(oce)
library(akima)
# Create dataset
set.seed(42)
tn <- 1000 # number in pop
Sigma <- matrix(c(15, 8, 5, 8, 15, .2, 5, .2, 15), 3, 3)
mv <- data.frame(mvrnorm(tn, c(100, 100, 100),Sigma)) # population
prob <- .8 # rather than .5
simn <- 100 # number of simulations
pinp <- rep(NA, simn)
cuts <- pinp
sn <- 48 # sample size, at n = 400 user2554330 performs better
### 2d scenario
for (isim in 1:simn) {
# Sample
smv <- mv[sample(1:tn, sn), ]
# Create kernel density
dens2d <- kde2d(smv[, 1], smv[, 2], n = 40,
lims = c(min(smv[, 1]) - abs(max(smv[, 1]) - min(smv[, 1])) / 2,
max(smv[, 1]) + abs(max(smv[, 1]) - min(smv[, 1])) / 2,
min(smv[, 2]) - abs(max(smv[, 2]) - min(smv[, 2])) / 2,
max(smv[, 2]) + abs(max(smv[, 2]) - min(smv[, 2])) / 2))
# Approach based on https://stackoverflow.com/questions/30517160/r-how-to-find-points-within-specific-contour
# Find the contour level defined in prob
dx <- diff(dens2d$x[1:2])
dy <- diff(dens2d$y[1:2])
sd <- sort(dens2d$z)
c1 <- cumsum(sd) * dx * dy
levels <- sapply(prob, function(x) {
approx(c1, sd, xout = 1 - x)$y
})
# Find which values are inside the defined polygon
ls <- contourLines(dens2d, level = levels)
# Note below that I check points from "population"
pinp[isim] <- sum(point.in.polygon(mv[, 1], mv[, 2], ls[[1]]$x, ls[[1]]$y)) / tn
# Approach based on user2554330
# Find the estimated density at each observed point
sdatadensity<- bilinear(dens2d$x, dens2d$y, dens2d$z,
smv[,1], smv[,2])$z
# Find the contours
levels2 <- quantile(sdatadensity, probs = 1- prob, na.rm = TRUE)
# Find within
# Note below that I check points from "population"
datadensity <- bilinear(dens2d$x, dens2d$y, dens2d$z,
mv[,1], mv[,2])$z
cuts[isim] <- sum(as.numeric(cut(datadensity, c(0, levels2, Inf))) == 2, na.rm = T) / tn
}
summary(pinp)
summary(cuts)
> summary(pinp)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0030 0.7800 0.8205 0.7950 0.8565 0.9140
> summary(cuts)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.5350 0.6560 0.6940 0.6914 0.7365 0.8120
我还尝试通过以下代码查看user2554330在3D情况下的性能如何:
# 3d scenario
for (isim in 1:simn) {
# Sample
smv <- mv[sample(1:tn, sn), ]
# Create kernel density
dens3d <- kde3d(smv[,1], smv[,2], smv[,3], n = 40,
lims = c(min(smv[, 1]) - abs(max(smv[, 1]) - min(smv[, 1])) / 2,
max(smv[, 1]) + abs(max(smv[, 1]) - min(smv[, 1])) / 2,
min(smv[, 2]) - abs(max(smv[, 2]) - min(smv[, 2])) / 2,
max(smv[, 2]) + abs(max(smv[, 2]) - min(smv[, 2])) / 2,
min(smv[, 3]) - abs(max(smv[, 3]) - min(smv[, 3])) / 2,
max(smv[, 3]) + abs(max(smv[, 3]) - min(smv[, 3])) / 2))
# Approach based on user2554330
# Find the estimated density at each observed point
sdatadensity <- approx3d(dens3d$x, dens3d$y, dens3d$z, dens3d$d,
smv[,1], smv[,2], smv[,3])
# Find the contours
levels <- quantile(sdatadensity, probs = 1 - prob, na.rm = TRUE)
# Find within
# Note below that I check points from "population"
datadensity <- approx3d(dens3d$x, dens3d$y, dens3d$z, dens3d$d,
mv[,1], mv[,2], mv[,3])
cuts[isim] <- sum(as.numeric(cut(datadensity, c(0, levels, Inf))) == 2, na.rm = T) / tn
}
summary(cuts)
> summary(cuts)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.1220 0.1935 0.2285 0.2304 0.2620 0.3410
我更喜欢定义轮廓,使得即使样本n相对较小(即<50),指定的概率也(接近)捕获从相同总体中提取的未来数据点的概率。
而不是尝试查找轮廓中的哪些点,而是尝试评估每个点的密度,并根据该值与轮廓水平的比较方式为这些点着色。在边界附近的几个点可能会做出不同的决定,但应该非常接近。
要进行该评估,可以在密度估计上使用oce::approx3d
函数。
我要做的另一件事是根据观察到的密度的分位数来选择轮廓,而不是尝试模拟估计密度的3维积分。
这里是完成所有操作的代码:
library(MASS)
library(misc3d)
library(rgl)
library(oce)
#> Loading required package: testthat
#> Loading required package: gsw
# Create dataset
set.seed(42)
Sigma <- matrix(c(15, 8, 5, 8, 15, .2, 5, .2, 15), 3, 3)
mv <- data.frame(mvrnorm(400, c(100, 100, 100),Sigma))
### 3d ###
# Create kernel density
dens3d <- kde3d(mv[,1], mv[,2], mv[,3], n = 40)
# Find the estimated density at each observed point
datadensity <- approx3d(dens3d$x, dens3d$y, dens3d$z, dens3d$d,
mv[,1], mv[,2], mv[,3])
# Find the contours
prob <- .5
levels <- quantile(datadensity, probs = prob, na.rm = TRUE)
# Plot it
colours <- c("gray", "orange")
cuts <- cut(datadensity, c(0, levels, Inf))
for (i in seq_along(levels(cuts))) {
gp <- as.numeric(cuts) == i
spheres3d(mv[gp,1], mv[gp,2], mv[gp,3], col = colours[i], radius = 0.2)
}
box3d(col = "gray")
contour3d(dens3d$d, level = levels, x = dens3d$x, y = dens3d$y, z = dens3d$z, #exp(-12)
alpha = .1, color = "red", color2 = "gray", add = TRUE)
title3d(xlab = "x", ylab = "y", zlab = "z")
这是产生的剧情: