Hamise.mixt, Hmise.mixt, amise.mixt, ise.mixt, mise.mixt {ks}R Documentation

MISE- and AMISE-optimal bandwidth matrix selectors for normal mixture densities

Description

Normal mixture densities have closed form expressions for the MISE and AMISE. So in these cases, we can numerically minimise these criteria to find MISE- and AMISE-optimal matrices.

The global errors ISE (Integrated Squared Error), MISE (Mean Integrated Squared Error) of kernel density estimates for normal densities, for 2- to 6-dimensional data, and AMISE (Asymptotic Mean Integrated Squared Error) for 2-dimensional data.

Usage

Hmise.mixt(mus, Sigmas, props, samp, Hstart)
Hamise.mixt(mus, Sigmas, props, samp, Hstart)

ise.mixt(x, H, mus, Sigmas, props)  
mise.mixt(H, mus, Sigmas, props, samp)
amise.mixt(H, mus, Sigmas, props, samp)

Arguments

mus (stacked) matrix of mean vectors
Sigmas (stacked) matrix of variance matrices
props vector of mixing proportions
samp sample size
Hstart initial bandwidth matrix, used in numerical optimisation
x matrix of data values
H bandwidth matrix

Details

For normal mixture densities, ISE and MISE have exact formulas for all dimensions, and AMISE has an exact form for 2 dimensions. See Wand & Jones (1995).

If Hstart is not given then it defaults to k*var(x) where k = 4/(n*(d + 2))^(2/(d+ 4)), n = sample size, d = dimension of data.

Value

– Full MISE- or AMISE-optimal bandwidth matrix. Diagonal forms of these matrices are not available.
– ISE, MISE or AMISE value.

Note

ISE is a random variable that depends on the data x. MISE and AMISE are non-random and don't depend on the data.

References

Wand, M.P. & Jones, M.C. (1995) Kernel Smoothing. Chapman & Hall. London.

Examples

mus <- rbind(c(0,0,0), c(2,2,2))
Sigma <- matrix(c(1, 0.7, 0.7, 0.7, 1, 0.7, 0.7, 0.7, 1), nr=3, nc=3) 
Sigmas <- rbind(Sigma, Sigma)
props <- c(1/2, 1/2)
samp <- 1000
x <- rmvnorm.mixt(n=samp, mus=mus, Sigmas=Sigmas, props=props)

H1 <- Hmise.mixt(mus, Sigmas, props, samp)
H2 <- Hamise.mixt(mus, Sigmas, props, samp)

ise.mixt(x, H2, mus, Sigmas, props)
mise.mixt(H2, mus, Sigmas, props, samp)

[Package ks version 1.5.6 Index]