Generating Random Matrices BIOS 524 Project Brett Kliner Abigail Robinson

Goals of Project To use simulation to create a random vector X, where X~N(μ, Σ). To simulate the probability that W >= w, where W is a scalar generated from the X matrix. W is generated from the mean vector, μ. W is generated fro the a k x 1 zero vector.

Applications This exercise is mostly academic with uses in matrix algorithms and general linear models. Hypothesis testing that the mean vector is equal to the zero vector. This will be useful in Dr. Johnson’s General Linear Models class next semester.

The Random X Vector The X vector (k x 1) will be replicated n times. X will have a mean vector μ, k x 1. X will be formed using the covariance matrix Σ, k x k. The user may specify n, μ and Σ. The mean vector μ (k x 1) replicated n times gives us an n x k matrix.

The Random X Vector μ and Σ must match on dimension so matrix multiplication can occur. The covariance matrix must be symmetric, that is Σ = Σ’. Σ must also be positive definite which means that all of the eigenvalues must be positive.

The Random X Vector Each column of the new n x k matrix will be averaged using PROC MEANS. Mean of each column Standard Deviation 95% Confidence interval on the mean The n x k matrix will be compared to the Vnormal matrix.

The Random X Vector The call Vnormal function will be used to generate an n x k Vnormal matrix. PROC Means will be used to analyze each column. Mean of each column Standard Deviation 95% Confidence Interval

Computing W A quadratic form occurs when q = x’Ax. W is a quadratic form where: W = (x - v)’ -1 (x – v) v is a k x 1 vector of constants. We will consider two cases of v: v = μ v = 0

When v = μ When v = μ, the distribution of W is considered to be Chi-Square with k degrees of freedom. The value of w is specified by the user. The probability that (W>=w) is compared to the call function 1 - ProbChi (w,k).

When v = 0 ncp is calculated by: When v = 0, the distribution of W is considered to be a non-central chi-squared distribution with k degrees of freedom and non-centrality parameter ncp. Notice that when v = 0, W = x’ -1 x. ncp is calculated by: ncp = v’ -1 v where v = μ .

When v = 0 The probability that (W >= w), where W = x’ -1 x, can be compared to the call function 1 – ProbChi (w, k, ncp).

The SAS Code Let’s take a look at the SAS code that accomplishes these tasks. Please ask questions when they arise.