Transcript Generating Random Matrices

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
• 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.