Question

Is there any built in function calculating the value of a gradient of multivariate normal probability density function for a given point?

Edit: found this how to evaluate derivative of function in matlab? but that is not what I am looking for

Edit2: owkay, that’s what I’m using http://www.mathworks.co.uk/help/stats/mvnpdf.html case 3, looking for value of derivative with respect to X

Answer 1

Might I suggest you check The Matrix Cookbook by Peterson and Pedersen (available for free online – just google it). The analytical solution to your problem is on p39, equation 325 (2008 edition).

We didn’t even need Matlab for this one!

EDIT: As YBE implies, perhaps I should include the solution in my answer. So, let p(x) denote the multivariate Gaussian pdf, characterized by mean vector m and covariance matrix S. Then:

dp(x) / dx = -p(x) * S^(-1) * (x – m)

and

d^2p / dx dx’ = p(x) * (S^(-1) (x – m)(x – m)’ S^(-1) – S^(-1))

If you want a Matlab function, then:

function Gradient = MultNormD1(x, Mu, Sigma)
Gradient = -1 * mvnpdf(x, Mu, Sigma) * (Sigma \ (x - Mu));