Fast scaling of Gaussian Kernel by the Covariance of the Inputs - matlab

Question

I am currently fiddling with multivariate kernel density estimations for estimating the probability density functions (PDF) of hydrological data sets using Matlab. I am most familiar with kernel density estimation using Gaussian kernels as outlined in Sharma (2000 and 2014) (where the kernel bandwidths are set using the Gaussian Reference Rule (GRR)). The GRR is written as follows (Sharma, 2000):
where lambda_ref = GRR bandwidth of kernel, n is the sample size, and d is the dimension of the data set we are using for density estimation. To estimate the multivariate density of our data set X we use the following formula (Sharma, 2000):
where lamda is the same as lamda_ref above, S is the sample covariance of X and det() stands for determinant.
My question is: I understand that there are many "fast" methods for calculating the Gaussian kernel function represented by the term exp() such as the method proposed here (using Matlab): http://mrmartin.net/?p=218. Since I will be working with data sets that are quite large in sample size (1000-10,000) I am looking for a fast code. Is anyone aware how I can write a fast code for the second equation that takes into account the inverse of the sample covariance matrix (S^-1)?
I greatly appreciate any help that can be provided on this issue. Thank you!
Note(s):
I understand that there is a Matlab code for calculating the second equation, found as a sub-function in: http://www.mathworks.com/matlabcentral/fileexchange/29039-mutual-information-2-variablle/content/MutualInfo.m. However this code has a bottleneck in how it calculates the kernel matrix.
References:
1 A. Sharma, Seasonal to interannual rainfall probabilistic forecasts for improved water supply management: Part 3 — A nonparametric probabilistic forecast model, Journal of Hydrology, Volume 239, Issues 1–4, 20 December 2000, Pages 249-258, ISSN 0022-1694, http://dx.doi.org/10.1016/S0022-1694(00)00348-6.
2 Sharma, A., and R. Mehrotra (2014), An information theoretic alternative to model a natural system using observational information alone, Water Resour. Res., 50, 650–660, doi:10.1002/2013WR013845.

Answer 1

I have found a code that I am able to modify for my purposes. The original code is listed at the following link: http://www.kernel-methods.net/matlab/kernels/rbf.m.
Code
function K = rbf(coord,sig)
%function K = rbf(coord,sig)
%
% Computes an rbf kernel matrix from the input coordinates
%
%INPUTS
% coord = a matrix containing all samples as rows
% sig = sigma, the kernel width; squared distances are divided by
% squared sig in the exponent
%
%OUTPUTS
% K = the rbf kernel matrix ( = exp(-1/(2*sigma^2)*(coord*coord')^2) )
%
%
% For more info, see www.kernel-methods.net
%
%Author: Tijl De Bie, february 2003. Adapted: october 2004 (for speedup).
n=size(coord,1);
K=coord*coord'/sig^2;
d=diag(K);
K=K-ones(n,1)*d'/2;
K=K-d*ones(1,n)/2;
K=exp(K);
Modified Code incorporating sample covariance scaling:
xcov = cov(x.'); % sample covariance of the data
invxc = pinv(xcov); % inversion of data sample covariance
coord = x.';
sig = sigma; % kernel bandwidth
n = size(coord,1);
K = coord*invxc*coord'/sig^2;
d = diag(K);
K = K-ones(n,1)*d'/2;
K = K-d*ones(1,n)/2;
K = exp(K); % kernel matrix
I hope this helps someone else looking into the same problem.