Multivariate Normal Distribution - Affine Transformation

Affine Transformation

If y = c + Bx is an affine transformation of where c is an vector of constants and B is a constant matrix, then y has a multivariate normal distribution with expected value c + Bμ and variance BΣBT i.e., . In particular, any subset of the x_i has a marginal distribution that is also multivariate normal. To see this, consider the following example: to extract the subset (x₁, x₂, x₄)T, use

$\mathbf{B} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & \ldots & 0 \\ 0 & 1 & 0 & 0 & 0 & \ldots & 0 \\ 0 & 0 & 0 & 1 & 0 & \ldots & 0 \end{bmatrix}$

which extracts the desired elements directly.

Another corollary is that the distribution of Z = b · x, where b is a constant vector of the same length as x and the dot indicates a vector product, is univariate Gaussian with . This result follows by using

$\mathbf{B}=\begin{bmatrix} b_1 & b_2 & \ldots & b_n \\ 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & 0 \end{bmatrix}$

and considering only the first component of the product (the first row of B is the vector b). Observe how the positive-definiteness of Σ implies that the variance of the dot product must be positive.

An affine transformation of x such as 2x is not the same as the sum of two independent realisations of x.

Read more about this topic: Multivariate Normal Distribution