# Multivariate Gaussian Distributions in Julia

February 17th 2016Tagged: julia

I recently worked my way through the latest incarnation of Andrew Ng's Machine Learning Course on Coursera. One particular method stood out to me as having some potential real-life applications for me: anomaly detection with multivariate Gaussian distributions.

The class is conducted in Octave, which I find a little annoying to
deal with as a language, so to play around with some data, I wanted to
convert the procedure to Julia. Not having
found anything prebuilt from quick Googling, I translated it myself,
and leave it here for anyone who'd like to refer to it later. The
mathematical formulas I'm working from are in the above-referenced
Wikipedia article, at "Non-degenerate
case".
The set of data I'll represent as `X`

.

First we calculate our μ, just the mean of each column in `X`

:

```
mu = mean(X, 1)
```

Having found our mean, we can now calculate the covariance matrix:

```
function covariance(x, mu)
n = length(x[:,1])
(x .- mu)' * (x .- mu) ./ n
end
sigma = covariance(X, mu)
```

And having both μ and Σ, we can calculate the probability for any
vector `x`

against our distribution:

```
function probability(mu, sigma, x)
n = length(mu)
exp(-0.5 * sum((x .- mu)' .* inv(sigma) .* (x .- mu))) *
(2 * pi) ^ (-0.5 * n) *
det(sigma) ^ -0.5
end
```