numerical partial derivative in MatLab

Question

How can I compute the numerical partial derivative of a probability density function (PDF) in Matlab? I'm not looking for a solution using automatic differences or a symbolic solution.

Given the following example:

arg = (-1:.01:1)';
mu = 0;
sigma = 0.5;

f = normpdf(arg,mu,sigma);

Is it possible to compute the numerical partial derivative of df/dsigma? Or am I stuck to having to use the automatic differences or the Symbolic Math toolbox?

Answer 1

I assume that the actual function is not the PDF of the normal distribution. You might try using complex step differentiation if you only need the first derivative:

mu = 0;
sigma = 0.5;
f = @(x)normpdf(x,mu,sigma);
x = -1:0.01:1;
h = 2^-28;
dx = imag(f(x+1i*h))/h;

Or if you hold x and mu constant and vary sigma:

mu = 0;
x = 0;
g = @(sigma)normpdf(x,mu,sigma);
sigma = 0.25:0.01:0.75;
h = 2^-28;
dsigma = imag(g(sigma+1i*h))/h;

This technique is fast, simple, and very accurate. You can download this as a convenient function, cdiff, from my GitHub.

Answer 2

My understanding is that arg and mu are constant and sigma varies. This is an obvious approximation:

arg = 0;
mu = 0;
incr = 0.0001;
sigma = incr:incr:1.5;

f = normpdf(arg,mu,sigma);
d = diff(f) / incr;
plot(sigma(2:end), d, '.')
set(gca, 'XLim', [0.05 1.5])