3
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I am learning ML. I have encountered a lot probability density functions. I want to use MATLAB to illustrate them. For example,the density function of the Normal distribution function in two dimensions:

$$p(\vec{x};\mu,\Sigma) = \frac{1}{2\pi \sqrt{\det(\Sigma)}}\exp\left(-\frac{1}{2}(\vec{x}-\mu)^T\Sigma^{-1}(\vec{x}-\mu)\right)$$

When we use MATLAB to plot three-dimensional graphs, it plots a lot of points in space, and then uses straight lines to connect adjacent points. The steps are:

  1. generating points of the domain in the XY plane: [X,Y]=meshgrid(-3;.5;3)

  2. using vectors $X$ and $Y$ to generate $Z$. If the function is $z= \sin(xy)$, it is very simple. Z=sin(X.*Y) is OK. But here we need to do $(x,y)\cdot\Sigma^{-1}\cdot(x,y)^T$ for every element in $[X,Y]$. If I do it directly: X.*$\Sigma^{-1}$`.*Y`. I will encounter the error that the dimensions do not match, because the matrix $\Sigma$ is $2 \times 2$. and the vector $X$ is about $12 \times 1$.

How can I solve it in MATLAB? Thanks.

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2
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It is not terribly enlightening, I am afraid, but you can permute the parallel dimensions out to the third and fourth, use bsxfun to do the broadcasting, then bring the results back. It involves a few shiftdim operations:

% set the random seed:
randn('state',[2979743726;1541610269]);

% create a random sigma, mu:
Sigma = cov(randn(100,2));
mu = randn(2,1);

% compute, plot the cdf
[X,Y] = meshgrid(-3:.2:3);
XY = cat(3,X,Y);
% subtract mu
XYmmu = bsxfun(@minus,XY,shiftdim(mu(:),-2));

isigXY = squeeze(sum(bsxfun(@times,shiftdim(inv(Sigma),-2),XYmmu),3));
XYisXY = sum(isigXY .* XYmmu,3);

normcdf = (1/(2*pi*sqrt(det(Sigma)))) * exp(-0.5 * XYisXY);
surf(X,Y,normcdf);

enter image description here

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0
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To do that operation I would try something along the lines of

product = [X(:), Y(:)] * ( SigmaInv * [X(:)'; Y(:)'] )';

where SigmaInv is your 2x2 matrix. The variable product should be a matrix having the values for all the points of the grid.

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