In this class we're going to talk about
the multivariate Gaussian. So, in all our
past lectures we've looked at the
one-dimensional case, so one-dimensional
distributions both discrete and
continuous. When we talk about the
multivariate Gaussian distribution we're
talking about two or more dimensions, and
of course MATLAB is perfect for this
because it works with all matrices and
vectors. So now what happens when you
extend the Gaussian like this? So the
first thing is that the mean becomes a
vector, and again this doesn't really
change the definition of the mean. You
still add up all the values of X and
divide by n to get your sample mean. So
as an example let's say we have a random
matrix with 100 values and with 10
dimensions. Alright, so, if we do mean(X)
we get a 1 by 10 matrix, and each
component is the 100 values in that
column summed up together and divided by 100. The
thing that is kind of more different
than the one dimensional case is the
variance. So we see this Sigma symbol
here and what this stands for is the
covariance matrix. So, with the variance
it told us how far X was spread out from the
mean, the covariance does that too but only
on the diagonals. So, on the off diagonals
it tells us how correlated one dimension
is with the other, and so you can see
here
the definition of the covariance matrix
is the covariance of the if
dimension with the jf dimension, and that
would be the ij of element of the
covariance matrix. We don't really need to
worry about how to calculate it because
we are in the end going to end up using
the matlab function cov. So let's check
that covariance is 10 by 10, so our
dimensionality is 10. What's interesting
about the PDF of the Gaussian in the
multi-dimensional case is that we never
use the covariance directly. We use the
inverse right here in the exponent and
the determinant, and again since we
already wrote a function to calculate
Gaussians before I'm not going to ask
you to do it again. There's already a
function called mvnpdf in matlab that
will do this for you. So I've got an
example here that I want to show you. We
set our mean and our covariance...
Don't worry about this code too much
since it is not the focus of this
lecture, we're simply setting the x-axis,
so the X1 and X2 values. Note that the
dimensionality of X is 2, so mu is
two-dimensional and Sigma is two by two.
So here we calculate PDF value using
mvnpdf, and we're going to do a surface plot
of F versus X1 and X2. So now some
points worth talking about is this
distribution is more spread out on the X2
axis than the X1 axis, and that's
because you see here this distribution
has a variance of 1 on the X2 axis,
but a variance of point .25 on the X 1
axis, and since the off diagonals are not
zero what we get is a Gaussian that is
not perpendicular to the X1 and X2
axes.