Gaussian Process

Normal Distribution

Examples on classification research,change point test. Using non-parametric method first, then with bigger sample size to apply CLT.

Example 14.3

library(RNifti),library(fabisearch)

Multivariate Normal Distribution

\[ X \sim N(\mu,\sum) \]

inverse of covariance matrix is called precision matrix

Example 15.1

Compute the covariance matrix, given: vector \(Z(X_1+X_2,X_3-X_2,X_2) ^T\)

\[ Cov(Z) = \begin{matrix}Cov(Z_1,Z_1)&Cov(Z_1,Z2) & Cov(Z_1,Z_3)\\Cov(Z_2,Z_1) & Cov(Z_2,Z_2) & Cov(Z_2,Z_3) \\ Cov(Z_3,Z_1) & Cov(Z_3,Z_2) & Cov(Z_3,Z_3)\end{matrix} \]

Note: Taking the inverse of covariance matrix is the precision matrix and its non diagonal zero elements means there is no node between the two points, and its edges are (1,3),(2,3) if (1,2),(2,1) of the Q matrix are zero, denoted by

\[ Z_1 \perp Z_2 |Z_3 \]

Backwards

Let \(Z_2,Y_1,Y_2 \sim N(0,1)\)

They have

\[ \begin{align*} Z_1 = aY_1+bZ_2 \\ Z_3 = cY_2+dZ_2 \end{align*} \]

Use the covariance matrix