You can use the Cholesky decomposition of a covariance matrix to simulate data from a correlated multivariate normal distribution. This method is encapsulated in the RANDNORMAL function in SAS/IML software, but you can also perform the computations manually by calling the ROOT function to get the Cholesky root and then
In a previous article, I discussed various ways to solve a least-square linear regression model. I discussed the SWEEP operator (used by many SAS regression routines), the LU-based methods (SOLVE and INV in SAS/IML), and the QR decomposition (CALL QR in SAS/IML). Each method computes the estimates for the regression
In computational statistics, there are often several ways to solve the same problem. For example, there are many ways to solve for the least-squares solution of a linear regression model. A SAS programmer recently mentioned that some open-source software uses the QR algorithm to solve least-squares regression problems and asked
Look at the following matrices. Do you notice anything that these matrices have in common? If you noticed that the rows of each matrix are arithmetic progressions, good for you. For each row, there is a constant difference (also called the "increment") between adjacent elements. For these examples: In the
A data analyst recently asked a question about restricted least square regression in SAS. Recall that a restricted regression puts linear constraints on the coefficients in the model. Examples include forcing a coefficient to be 1 or forcing two coefficients to equal each other. Each of these problems can be
Matrix balancing is an interesting problem that has a long history. Matrix balancing refers to adjusting the cells of a frequency table to match known values of the row and column sums. One of the early algorithms for matrix balancing is known as the RAS algorithm, but it is also
The Kronecker product (also called the direct product) is a binary operation that combines two matrices to form a new matrix. The Kronecker product appears in textbooks about the design of experiments and multivariate statistics. The Kronecker product seems intimidating at first, but often one of the matrices in the
If you have ever run a Kolmogorov-Smirnov test for normality, you have encountered the Kolmogorov D statistic. The Kolmogorov D statistic is used to assess whether a random sample was drawn from a specified distribution. Although it is frequently used to test for normality, the statistic is "distribution free" in
Sometimes in matrix computations, it is important to display the nonzero elements of a matrix. This can be useful for visualizing the structure of a sparse matrix (one that has many zeros) and it is also useful when describing a matrix algorithm (such as Gaussian elimination) that introduces zeros at
Rockin' around the Christmas tree At the Christmas party hop. – Brenda Lee Last Christmas, I saw a fun blog post that used optimization methods to de-noise an image of a Christmas tree. Although there are specialized algorithms that remove random noise from an image, I am not going to
