r - R：将 alpha 包添加到 2d 或 3d 散点图

Question

我知道ggplot2可以将凸包按组添加到散点图中，如

library(ggplot2)
library(plyr)
data(iris)
df<-iris
find_hull <- function(df) df[chull(df$Sepal.Length, df$Sepal.Width), ]
hulls <- ddply(df, "Species", find_hull)
plot <- ggplot(data = df, aes(x = Sepal.Length, y = Sepal.Width, colour=Species, fill = Species)) +
    geom_point() + 
    geom_polygon(data = hulls, alpha = 0.5) +
    labs(x = "Sepal.Length", y = "Sepal.Width")
plot

我想知道如何计算和添加 alpha 包，即最大的凸包，其中至少包含所有点的 1-alpha 比例？在 2d（用 ggplot2 显示）或 3d（用 rgl 显示）中。

编辑：我最初的想法是继续“剥离”凸包，只要满足包含至少给定百分比点的标准，尽管在这里的论文中似乎他们使用了不同的算法（等深，这似乎是在 R 包深度中实现的，在函数isodepth和aplpack::plothulls 中似乎也接近我想要的（尽管它产生了一个完整的图而不仅仅是轮廓），所以我认为这些我可能会被排序。虽然这些功能仅适用于 2D，但我也对 3D 扩展感兴趣（以 rgl 绘制）。如果有人有任何指示，请告诉我！

EDIT2：通过功能，depth::isodepth我找到了一个 2d 解决方案（见下面的帖子），虽然我仍在寻找 3D 解决方案 - 如果有人碰巧知道如何做到这一点，请告诉我！

Answer 1

哈在函数的帮助下，depth::isodepth我想出了以下解决方案 - 在这里我找到了包含所有点的至少 1-alpha 比例的 alpha 包：

library(mgcv)
library(depth)
library(plyr)
library(ggplot2)
data(iris)
df=iris[,c(1,2,5)]
alph=0.05
find_bag = function(x,alpha=alph) {
    n=nrow(x)
    target=1-alpha
    propinside=1
    d=1
    while (propinside>target) {
        p=isodepth(x[,1:2],dpth=d,output=T, mustdith=T)[[1]]
        ninside=sum(in.out(p,as.matrix(x[,1:2],ncol=2))*1)
        nonedge=sum(sapply(1:nrow(p),function (row)
            nrow(merge(round(setNames(as.data.frame(p[row,,drop=F]),names(x)[1:2]),5),as.data.frame(x[,1:2])))>0)*1)-3
        propinside=(ninside+nonedge)/n
        d=d+1
    }
    p=isodepth(x[,1:2],dpth=d-1,output=T, mustdith=T)[[1]]
    p }
bags <- ddply(df, "Species", find_bag,alpha=alph)
names(bags) <- c("Species",names(df)[1:2])
plot <- ggplot(data = df, aes(x = Sepal.Length, y = Sepal.Width, colour=Species, fill = Species)) +
    geom_point() + 
    geom_polygon(data = bags, alpha = 0.5) +
    labs(x = "Sepal.Length", y = "Sepal.Width")
plot

EDIT2：使用我最初的凸包剥离想法，我还提出了以下解决方案，现在可以在 2d 和 3d 中使用；结果与 isodepth 算法不太一样，但非常接近：

# in 2d
library(plyr)
library(ggplot2)
data(iris)
df=iris[,c(1,2,5)]
alph=0.05
find_bag = function(x,alpha=alph) {
  n=nrow(x)
  propinside=1
  target=1-alpha
  x2=x
  while (propinside>target) {
    propinside=nrow(x2)/n
    hull=chull(x2)
    x2old=x2
    x2=x2[-hull,]
  }
  x2old[chull(x2old),] }
bags <- ddply(df, "Species", find_bag, alpha=alph)
plot <- ggplot(data = df, aes(x = Sepal.Length, y = Sepal.Width, colour=Species, fill = Species)) +
  geom_point() +
  geom_polygon(data = bags, alpha = 0.5) +
  labs(x = "Sepal.Length", y = "Sepal.Width")
plot

# in 3d
library(plyr)
library(ggplot2)
data(iris)
df=iris[,c(1,2,3,5)]
levels=unique(df[,"Species"])
nlevels=length(levels)
zoom=0.8
cex=1
aspectr=c(1,1,0.7)
pointsalpha=1
userMatrix=matrix(c(0.80,-0.60,0.022,0,0.23,0.34,0.91,0,-0.55,-0.72,0.41,0,0,0,0,1),ncol=4,byrow=T)
windowRect=c(0,29,1920,1032)
cols=c("red","forestgreen","blue")
alph=0.05
plotbag = function(x,alpha=alph,grp=1,cols=c("red","forestgreen","blue"),transp=0.2) {
  propinside=1
  target=1-alpha
  x2=x
  levels=unique(x2[,ncol(x2)])
  x2=x2[x2[,ncol(x2)]==levels[[grp]],]
  n=nrow(x2)
  while (propinside>target) {
    propinside=nrow(x2)/n
    hull=unique(as.vector(convhulln(as.matrix(x2[,1:3]), options = "Tv")))
    x2old=x2
    x2=x2[-hull,]
  }
  ids=t(convhulln(as.matrix(x2old[,1:3]), options = "Tv"))
  rgl.triangles(x2old[ids,1],x2old[ids,2],x2old[ids,3],col=cols[[grp]],alpha=transp,shininess=50)
}

open3d(zoom=zoom,userMatrix=userMatrix,windowRect=windowRect,antialias=8)
for (i in 1:nlevels) { 
  plot3d(x=df[df[,ncol(df)]==levels[[i]],][,1],
         y=df[df[,ncol(df)]==levels[[i]],][,2],
         z=df[df[,ncol(df)]==levels[[i]],][,3],
         type="s", 
         col=cols[[i]],
         size=cex,
         lit=TRUE,
         alpha=pointsalpha,point_antialias=TRUE,
         line_antialias=TRUE,shininess=50, add=TRUE)
  plotbag(df,alpha=alph, grp=i, cols=c("red","forestgreen","blue"), transp=0.3) }
axes3d(color="black",drawfront=T,box=T,alpha=1)
title3d(color="black",xlab=names(df)[[1]],ylab=names(df)[[2]],zlab=names(df)[[3]],alpha=1)
aspect3d(aspectr)

Answer 2

我们可以修改aplpack::plothulls函数以接受一个参数来表示要包围的点的比例（在 aplpack 中它设置为 50%）。然后我们可以使用这个修改后的函数为 ggplot 定制一个 geom。

这是自定义几何图形：

library(ggplot2)
StatBag <- ggproto("Statbag", Stat,
                   compute_group = function(data, scales, prop = 0.5) {

                     #################################
                     #################################
                     # originally from aplpack package, plotting functions removed
                     plothulls_ <- function(x, y, fraction, n.hull = 1,
                                            col.hull, lty.hull, lwd.hull, density=0, ...){
                       # function for data peeling:
                       # x,y : data
                       # fraction.in.inner.hull : max percentage of points within the hull to be drawn
                       # n.hull : number of hulls to be plotted (if there is no fractiion argument)
                       # col.hull, lty.hull, lwd.hull : style of hull line
                       # plotting bits have been removed, BM 160321
                       # pw 130524
                       if(ncol(x) == 2){ y <- x[,2]; x <- x[,1] }
                       n <- length(x)
                       if(!missing(fraction)) { # find special hull
                         n.hull <- 1
                         if(missing(col.hull)) col.hull <- 1
                         if(missing(lty.hull)) lty.hull <- 1
                         if(missing(lwd.hull)) lwd.hull <- 1
                         x.old <- x; y.old <- y
                         idx <- chull(x,y); x.hull <- x[idx]; y.hull <- y[idx]
                         for( i in 1:(length(x)/3)){
                           x <- x[-idx]; y <- y[-idx]
                           if( (length(x)/n) < fraction ){
                             return(cbind(x.hull,y.hull))
                           }
                           idx <- chull(x,y); x.hull <- x[idx]; y.hull <- y[idx];
                         }
                       }
                       if(missing(col.hull)) col.hull <- 1:n.hull
                       if(length(col.hull)) col.hull <- rep(col.hull,n.hull)
                       if(missing(lty.hull)) lty.hull <- 1:n.hull
                       if(length(lty.hull)) lty.hull <- rep(lty.hull,n.hull)
                       if(missing(lwd.hull)) lwd.hull <- 1
                       if(length(lwd.hull)) lwd.hull <- rep(lwd.hull,n.hull)
                       result <- NULL
                       for( i in 1:n.hull){
                         idx <- chull(x,y); x.hull <- x[idx]; y.hull <- y[idx]
                         result <- c(result, list( cbind(x.hull,y.hull) ))
                         x <- x[-idx]; y <- y[-idx]
                         if(0 == length(x)) return(result)
                       }
                       result
                     } # end of definition of plothulls
                     #################################


                     # prepare data to go into function below
                     the_matrix <- matrix(data = c(data$x, data$y), ncol = 2)

                     # get data out of function as df with names
                     setNames(data.frame(plothulls_(the_matrix, fraction = prop)), nm = c("x", "y"))
                     # how can we get the hull and loop vertices passed on also?
                   },

                   required_aes = c("x", "y")
)

#' @inheritParams ggplot2::stat_identity
#' @param prop Proportion of all the points to be included in the bag (default is 0.5)
stat_bag <- function(mapping = NULL, data = NULL, geom = "polygon",
                     position = "identity", na.rm = FALSE, show.legend = NA, 
                     inherit.aes = TRUE, prop = 0.5, alpha = 0.3, ...) {
  layer(
    stat = StatBag, data = data, mapping = mapping, geom = geom, 
    position = position, show.legend = show.legend, inherit.aes = inherit.aes,
    params = list(na.rm = na.rm, prop = prop, alpha = alpha, ...)
  )
}


geom_bag <- function(mapping = NULL, data = NULL,
                     stat = "identity", position = "identity",
                     prop = 0.5, 
                     alpha = 0.3,
                     ...,
                     na.rm = FALSE,
                     show.legend = NA,
                     inherit.aes = TRUE) {
  layer(
    data = data,
    mapping = mapping,
    stat = StatBag,
    geom = GeomBag,
    position = position,
    show.legend = show.legend,
    inherit.aes = inherit.aes,
    params = list(
      na.rm = na.rm,
      alpha = alpha,
      prop = prop,
      ...
    )
  )
}

#' @rdname ggplot2-ggproto
#' @format NULL
#' @usage NULL
#' @export
GeomBag <- ggproto("GeomBag", Geom,
                   draw_group = function(data, panel_scales, coord) {
                     n <- nrow(data)
                     if (n == 1) return(zeroGrob())

                     munched <- coord_munch(coord, data, panel_scales)
                     # Sort by group to make sure that colors, fill, etc. come in same order
                     munched <- munched[order(munched$group), ]

                     # For gpar(), there is one entry per polygon (not one entry per point).
                     # We'll pull the first value from each group, and assume all these values
                     # are the same within each group.
                     first_idx <- !duplicated(munched$group)
                     first_rows <- munched[first_idx, ]

                     ggplot2:::ggname("geom_bag",
                                      grid:::polygonGrob(munched$x, munched$y, default.units = "native",
                                                         id = munched$group,
                                                         gp = grid::gpar(
                                                           col = first_rows$colour,
                                                           fill = alpha(first_rows$fill, first_rows$alpha),
                                                           lwd = first_rows$size * .pt,
                                                           lty = first_rows$linetype
                                                         )
                                      )
                     )


                   },

                   default_aes = aes(colour = "NA", fill = "grey20", size = 0.5, linetype = 1,
                                     alpha = NA, prop = 0.5),

                   handle_na = function(data, params) {
                     data
                   },

                   required_aes = c("x", "y"),

                   draw_key = draw_key_polygon
)

这是一个如何使用它的示例：

ggplot(iris, aes(Sepal.Length,  Petal.Length, colour = Species, fill = Species)) + 
  geom_point() + 
  stat_bag(prop = 0.95) +  # enclose 95% of points
  stat_bag(prop = 0.5, alpha = 0.5) + # enclose 50% of points
  stat_bag(prop = 0.05, alpha = 0.9) # enclose 5% of points