Rick Wicklin, PhD, is a distinguished researcher in computational statistics at SAS and is a principal developer of SAS/IML software. His areas of expertise include computational statistics, simulation, statistical graphics, and modern methods in statistical data analysis. Rick is author of the books Statistical Programming with SAS/IML Software and Simulating Data with SAS.
Let X be any rectangular matrix. What is the trace of the crossproducts matrix, X'*X? Interestingly, you do not need to form the crossproducts matrix to compute the answer! It turns out that tr(X'*X) equals the sum of the squared elements of X. Theorem: For any matrix, X, the trace
In a previous article, I discussed the Wilcoxon signed rank test, which is a nonparametric test for the location of the median. The Wikipedia article about the signed rank test mentions a variation of the test due to Pratt (1959). Whereas the standard Wilcoxon test excludes values that equal μ0
Wilcoxon's signed rank test is a popular nonparametric alternative to a paired t test. In a paired t test, you analyze measurements for subjects before and after some treatment or intervention. You analyze the difference in the measurements for each subject, and test whether the mean difference is significantly different
A previous article discusses standardized coefficients in linear regression models and shows how to compute standardized regression coefficients in SAS by using the STB option on the MODEL statement in PROC REG. It also discusses how to interpret a standardized regression coefficient. Recently, a SAS user wanted to know how
A previous article shows an example of a Markov chain model and computes the probability that the system ends up in a terminal state (called an absorbing state). As explained previously, you can often compute exact probabilities for questions about Markov chains. Nevertheless, it can be useful to know how
A previous article shows how to model the probabilities in a discrete-time Markov chain by using a Markov transition matrix. A Markov chain is a discrete-time stochastic process for which the current state of the system determines the probability of the next state. In this process, the probabilities for transitioning
Given a set of N points in k-dimensional space, can you find the location that minimizes the sum of the distances to the points? The location that minimizes the distances is called the geometric median of the points. For univariate data, the "points" are merely a set of numbers \(\{p_1,
While writing an article about labeling a polygon by using the centroid, I almost made a false claim about the centroid. I almost claimed that that the centroid is the point in a polygon that minimizes the sum of the distances to the vertices. It is not. The point that
A colleague asked how to compute the barycentric coordinates of a point inside a triangle. Given a triangle in the plane with vertices p1, p2, and p3, every point in the triangle can be represented as a convex combination of the vertices: c1*p1 + c2*p2 + c3*p3, where c1,c2,c3 ≥
Part of the power of the SAS ODS system is the ability to visualize data by using ODS templates. An ODS template describes how to render data as a table or as a graph. A lot of papers and documentation have been written about how to define a custom template
