3.2.5 Multivariate Gaussian Distribution

$$\fbox{$ p(x_{1}, \dots, x_{n}) = \frac{1}{\sqrt{(2 \pi)^{n}\, S}} \, e^{\textstyle -\frac{1}{2} \sum \sum g_{ij}\, x_{i}\, x_{j}} $}$$

or

$$p(\mbox{$\bf x$}) = \frac{1}{\sqrt{(2 \pi)^{n}\, S}} \, e^{-\frac{1}{2} \mbox{$\bf x$... ...f x$}} \equiv \frac{1}{\sqrt{(2 \pi)^{n}\,S}} \, e^{\textstyle -\frac{1}{2}\,Q}$$

with the covariance matrix of the $x_{i}$

$$\mbox{${\bf S}$} \equiv \mbox{${\bf G}$}^{\,-1} \equiv \left( \begin{array}{ccc} \langle ... ...vdots & \langle x_{2}^{2} \rangle & \dots \\ & & \ddots \\ \end{array}\right)$$

$S \equiv \vert \mbox{${\bf S}$}\vert$ is the determinant of this matrix. $\mbox{${\bf S}$}$ and $\mbox{${\bf G}$}$ are symmetric, their eigenvalues are called $\sigma_{i}^{2}$ and $\gamma_{i}$ (sorry!).



EXAMPLE: Assume that two Gaussian variates have the variances $s_{11} \equiv \langle x_{1}^{2} \rangle =3$, $s_{22} \equiv \langle x_{2}^{2} \rangle =4$, and the covariance $s_{12} \equiv \langle x_{1}\,x_{2} \rangle =2$:

$$\mbox{${\bf S}$}=\mbox{$\left( \begin{array}{cc}3&2\\ \vspac... ... \vspace{-9pt}\\ -\frac{1}{4}&\frac{3}{8}\end{array} \right)$}$$

The quadratic form $Q$ in the exponent is then $Q=(1/2) \, x_{1}^{2} - (1/2)\,x_{1}\,x_{2} + (3/8)\,x_{2}^{2}\, .$, and the lines of equal density (that is, of equal $Q$) are ellipses which are inclined with respect to the $x_{1,2}\,$ coordinate axes:

Rotate the axes of the ellipsoids $Q=const$ to coincide with the coordinate axes: $\Longrightarrow$ cross correlations vanish!







Having found $\mbox{${\bf T}$}$, we arrive at the following prescription for the production of correlated Gaussian variables: $\Longrightarrow$





Let's try it out: $\Longrightarrow$



EXAMPLE: Once more, let

$$\mbox{${\bf S}$}=\mbox{$\left( \begin{array}{cc}3&2\\ \vspac... ... \vspace{-9pt}\\ -\frac{1}{4}&\frac{3}{8}\end{array} \right)$}$$

Principal axis transformation: The eigenvalues of ${\bf S}$ are $\sigma_{1,2}^{2}= (7\pm\sqrt{17})/2 = 5.562\vert 1.438$, and the corresponding eigenvectors are

$$\mbox{$\bf s$}_{1}=\mbox{$\left( \begin{array}{r} 0.615 \\ \... ....615&0.788\\ \vspace{-9pt}\\ 0.788&-0.615\end{array} \right)$}$$

Generator: To produce pairs $(x_{1},x_{2})$ of Gaussian random numbers with the given covariance matrix:

• Draw $y_{1}$ and $y_{2}$ Gaussian, uncorrelated, with variances $5.562$ and $1.438$.
  
• Compute $x_{1}$ and $x_{2}$ according to
  
  $$\mbox{$\left( \begin{array}{r} x_{1} \\ \vspace{-9 pt}\\ x_{... ...rray}{r} y_{1} \\ \vspace{-9 pt}\\ y_{2} \end{array} \right)$}$$
  
  

EXERCISE: Write a program that generates a sequence of bivariate Gaussian random numbers with the statistical properties as assumed in the foregoing example. Determine $\langle x_{1}^{2}\rangle$, $\langle x_{2}^{2}\rangle$, and $\langle x_{1}x_{2}\rangle$ to see if they indeed approach the given values of $3$, $4$, and $2$.