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Generating Random Matrices

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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( , ). – PowerPoint PPT presentation

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Title: Generating Random Matrices


1
Generating Random Matrices
  • BIOS 524 Project
  • Brett Kliner
  • Abigail Robinson

2
Goals of Project
  • To use simulation to create a random vector X,
    where XN(µ, S).
  • To simulate the probability that W gt 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.

3
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. Johnsons General
    Linear Models class next semester.

4
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 S, k
    x k.
  • The user may specify n, µ and S.
  • The mean vector µ (k x 1) replicated n times
    gives us an n x k matrix.

5
The Random X Vector
  • µ and S must match on dimension so matrix
    multiplication can occur.
  • The covariance matrix must be symmetric, that is
    S S.
  • S must also be positive definite which means that
    all of the eigenvalues must be positive.

6
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.

7
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

8
Computing W
  • A quadratic form occurs when q xAx.
  • 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

9
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 (Wgtw) is compared to the
    call function 1 - ProbChi (w,k).

10
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 .

11
When v 0
  • The probability that (W gt w), where
  • W x ?-1 x, can be compared to the call
    function 1 ProbChi (w, k, ncp).

12
The SAS Code
  • Lets take a look at the SAS code that
    accomplishes these tasks.
  • Please ask questions when they arise.
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