Random generation from multivariate Gaussian kernel density
Random generation from multivariate Gaussian kernel density
rmvg(n, y, bw = bw.silv(y), weights = NULL, adjust = 1)
n |
number of observations. If |
y |
numeric matrix or data.frame. |
bw |
numeric matrix with number of rows and columns equal to
|
weights |
numeric vector of length equal to |
adjust |
scalar; the bandwidth used is actually |
Multivariate kernel density estimator with multivariate Gaussian (normal) kernels KH is defined as
f(x) = sum[i](w[i] * KH(x-y[i]))
where w is a vector of weights such that all w[i] ≥ 0 and sum(w) = 1 (by default uniform 1/n weights are used), KH is kernel K parametrized by bandwidth matrix H and y is a matrix of data points used for estimating the kernel density.
Random generation from multivariate normal distribution is possible by taking
x = A' z + μ
where z is a vector of m i.i.d. standard normal deviates, μ is a vector of means and A is a m*m matrix such that A'A=Σ (A is a Cholesky factor of Σ). In the case of multivariate Gaussian kernel density, μ, is the i-th row of y, where i is drawn randomly with replacement with probability proportional to w[i], and Σ is the bandwidth matrix H.
For functions estimating kernel densities please check KernSmooth, ks, or other packages reviewed by Deng and Wickham (2011).
Deng, H. and Wickham, H. (2011). Density estimation in R. http://vita.had.co.nz/papers/density-estimation.pdf
set.seed(1)
dat <- mtcars[, c(1,3)]
bw <- bw.silv(dat)
X <- rmvg(5000, dat, bw = bw)
if (requireNamespace("ks", quietly = TRUE)) {
pal <- colorRampPalette(c("chartreuse4", "yellow", "orange", "brown"))
col <- pal(10)[cut(ks::kde(dat, H = bw, eval.points = X)$estimate, breaks = 10)]
plot(X, col = col, pch = 19, axes = FALSE,
main = "Multivariate Gaussian Kernel")
points(dat, pch = 2, col = "blue")
axis(1); axis(2)
} else {
plot(X, pch = 16, axes = FALSE, col = "#458B004D",
main = "Multivariate Gaussian Kernel")
points(dat, pch = 2, col = "red", lwd = 2)
axis(1); axis(2)
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