Generating Random Matrices

1
Generating Random Matrices

• BIOS 524 Project
• Brett Kliner
• Abigail Robinson

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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.

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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.

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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.

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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.

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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

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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).

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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 .

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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).

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The SAS Code

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