I am using the package ks for kernel density estimation. Here's an easy example:
n <- 70
x <- rnorm(n)
library(ks)
f_kde <- kde(x)
I am actually interested in the respective exceeding probabilities of my input data, which can be easily returned by ks having f_kde:
p_kde <- pkde(x, f_kde)
This is done in ks with a numerical integration using Simpson's rule. Unfortunately, they only implemented this for a 1d case. In a bivariate case, there's no implementation in ks of any method for returning the probabilities :
y <- rnorm(n)
f_kde <- kde(data.frame(x,y))
# does not work, but it's what I am looking for:
p_kde <- pkde(data.frane(x,y), f_kde)
I couldnt find any package or help searching in stackoverflow to solve this issue in R (some suggestions for Python exist, but I would like to keep it in R). Any line of code or package recommendation is appreciated. Even though I am mostly interested in the bivariate case, any ideas for a multivariate case are appreciated as well.
kde allows multidimensional kernel estimate, so we could use kde to calculate pkde.
For this, we calculate kde on small enough dx and dy steps using eval.points parameter : this gives us the local density estimate on a dx*dy
square.
We verify that the sum of estimates mutiplied by the surface of the squares almost equals 1:
library(ks)
set.seed(1)
n <- 10000
x <- rnorm(n)
y <- rnorm(n)
xy <- cbind(x,y)
xmin <- -10
xmax <- 10
dx <- .1
ymin <- -10
ymax <- 10
dy <- .1
pts.x <- seq(xmin, xmax, dx)
pts.y <- seq(ymin, ymax, dy)
pts <- as.data.frame(expand.grid(x = pts.x, y = pts.y))
f_kde <- kde(xy,eval.points=pts)
pts$est <- f_kde$estimate
sum(pts$est)*dx*dy
[1] 0.9998778
You can now query the pts dataframe for the cumulative probability on the area of your choice :
library(data.table)
setDT(pts)
# cumulative density
pts[x < 1 & y < 2 , .(pkde=sum(est)*dx*dy)]
pkde
1: 0.7951228
# average density around a point
tolerance <-.1
pts[pmin(abs(x-1))<tolerance & pmin(abs(y-2))<tolerance, .(kde = mean(est))]
kde
1: 0.01465478