1. Construct a 9×2 matrix X with some missing values, such that SX computed using all available data

1. Construct
a 9×2 matrix X with some missing values, such that SX computed using all
available data for the covariance or correlation matrix is not nonnegative
definite.

2. Consider an m×m,
symmetric nonsingular matrix, R, with 1s on the diagonal and with all
off-diagonal elements less than 1 in absolute value. If this matrix is positive
definite, it is a correlation matrix. Suppose, however, that some of the
eigenvalues are negative. Iman and Davenport (1982) describe a method of
adjusting the matrix to a “near-by” matrix that is positive definite. (See
Ronald L. Iman and James
»

 

1. Construct
a 9×2 matrix X with some missing values, such that SX computed using all
available data for the covariance or correlation matrix is not nonnegative
definite.

2. Consider an m×m,
symmetric nonsingular matrix, R, with 1s on the diagonal and with all
off-diagonal elements less than 1 in absolute value. If this matrix is positive
definite, it is a correlation matrix. Suppose, however, that some of the
eigenvalues are negative. Iman and Davenport (1982) describe a method of
adjusting the matrix to a “near-by” matrix that is positive definite. (See
Ronald L. Iman and James M. Davenport, 1982, An Iterative Algorithm to Produce
a Positive Definite Correlation Matrix from an “Approximate Correlation
Matrix”, Sandia Report SAND81-1376, Sandia National Laboratories, Albuquerque,
New Mexico.) For their method, they assumed the eigenvalues are unique, but
this is not necessary in the algorithm. Before beginning the algorithm, choose
a small positive quantity,, to use in the adjustments, set k = 0, and set R(k)
= R.

3. Form
R(k) from R_ by setting all diagonal elements to 1.

4. Set
k = k + 1, and go to step 1. (The algorithm iterates on k until     

p = 0.)

Write a
program to implement this adjustment algorithm. Write your program to accept
any size matrix and a user-chosen value for. Test your program on the
correlation matrix from Exercise 9.14

 

»

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