When you implement a numerical algorithm, it is helpful to write tests for which the answer is known analytically. Because I work in computational statistics, I am always looking for test matrices that are symmetric and positive definite because those matrices can be used as covariance matrices. I have previously
The first-order autoregressive (AR(1)) correlation structure is important for applications in time series modeling and for repeated measures analysis. The AR(1) model provides a simple situations where measurements (on the same subject) that are closer in time are correlated more strongly than measurements recorded far apart. The AR(1) model uses
A colleague asked me an interesting question: Suppose you have a structured correlation matrix, such as a matrix that has a compound symmetric, banded, or an AR1(ρ) structure. If you generate a random correlation matrix that has the same eigenvalues as the structured matrix, does the random matrix have the
A previous article discusses covariance matrices that have the same set of eigenvalues. The set of eigenvalues is called the spectrum of the matrix. For symmetric matrices, the spectrum contains real numbers. For covariance matrices, which are positive semidefinite, the eigenvalues are nonnegative. It turns out that two symmetric matrices
A previous article discusses how to generate a random covariance matrix with a specified set of (positive) eigenvalues. A SAS programmer asked whether it is possible to produce a correlation matrix that has a specified set of eigenvalues. After discussing the problem with a friend, I am happy to report
A colleague remarked that my recent article about using Jacobi's iterative method for solving a linear system of equations "seems like magic." Specifically, it seems like magic that you can solve a certain class of linear systems by using only matrix multiplication. For any initial guess, the iteration converges to
In a first course in numerical analysis, students often encounter a simple iterative method for solving a linear system of equations, known as Jacobi's method (or Jacobi's iterative method). Although Jacobi's method is not used much in practice, it is introduced because it is easy to explain, easy to implement,
In several previous articles, I've shown how to use SAS to fit models to data by using maximum likelihood estimation (MLE). However, I have not previously shown how to obtain standard errors for the estimates. This article combines two previous articles to show how to obtain MLE estimates and the
Many useful matrices in applied math and statistics have a banded structure. Examples include diagonal matrices, tridiagonal matrices, banded matrices, and Toeplitz matrices. An example of an unsymmetric Toeplitz matrix is shown to the right. Notice that the matrix is constant along each diagonal, including sub- and superdiagonals. Recently, I
I have previously written about how to efficiently generate points uniformly at random inside a sphere (often called a ball by mathematicians). The method uses a mathematical fact from multivariate statistics: If X is drawn from the uncorrelated multivariate normal distribution in dimensiond, then S = r*X / ||X|| has
