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→Multivariate Statistics: Added basics of multivariate stats.
== Multivariate Statistics ==
The 1-variable Gaussian distribution works well for one measurement. What if we're working with more than one random variable? Suppose we'TODOre measuring two voltages that have some amount of noise on them. We'll call them <math>\mathbf{X}</math> and <math>\mathbf{Y}</math>. As a first attempt, we could write down a model for <math>\mathbf{X}</math> using a normal distribution and a separate model for <math>\mathbf{Y}</math> using a different distribution. However, this might not always make sense. If we write two separate distributions, what we're saying is that the two variables are independent: basics when <math>\mathbf{X}</math> goes up, there's no guarantee that <math>\mathbf{Y}</math> will follow it. Multivariate distributions let us model multiple random variables that may or may not be correlated. In a multivariate distribution, instead of multivariable statswriting down a single variance <math>\sigma</math>, we keep track of a whole matrix of covariances. For example, to model three random variables (<math>\mathbf{X}, \mathbf{Y}, \mathbf{Z}</math>), this matrix would be <math>\mathbf{\Sigma} = \begin{bmatrix}Var(\mathbf{X}) & Cov(\mathbf{X}, \mathbf{Y}) & Cov(\mathbf{X}, \mathbf{Z}) \\Cov(\mathbf{Y}, \mathbf{X}) & Var(\mathbf{Y}) & Cov(\mathbf{Y}, \mathbf{Z}) \\Cov(\mathbf{Z}, \mathbf{X}) & Cov(\mathbf{Z}, \mathbf{Y}) & Var(\mathbf{Z}) \end{bmatrix}</math> Also, note that this distribution needs to have a mean for each random variable: <math>\mathbf{\mu} = \begin{bmatrix}\mu_X \\\mu_Y \\\mu_Z\end{bmatrix}</math> The PDF of this distribution is more complicated: instead of using a single number as an argument, it uses a vector with all of the variables in it (<math>\mathbf{x} = [x, y, z, \dots]^T</math>). The equation for <math>k</math> random variables is <math>f(\mathbf{x})= \frac{1}{\sqrt{(2\pi)^k |\mathbf{\Sigma}|}} e^{-(\mathbf{(x - \mu)}^T \mathbf{\Sigma}^{-1} \mathbf{(x - \mu)} / 2}</math> Don't worry if this looks crazy - the SciPy package in Python will do all the heavy lifting for us. As with the single-variable distributions, we're going to use this to find how likely a certain observation is. In other words, if we put <math>k</math> points of our power trace into <math>\mathbf{x}</math> and we find that <math>f(\mathbf{x})</math> is very high, then we've probably found a good guess.
= Profiling a Device =