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.
While writing an article about Toeplitz matrices, I saw an interesting fact about the eigenvalues of tridiagonal Toeplitz matrices on Nick Higham's blog. Recall that a Toeplitz matrix is a banded matrix that is constant along each diagonal. A tridiagonal Toeplitz matrix is zero except for the main diagonal, the
A Toeplitz matrix is a banded matrix. You can construct it by specifying the parameters that are constant along each diagonal, including sub- and super-diagonals. For a square N x N matrix, there is one main diagonal, N-1 sub-diagonals, and N-1 super-diagonals, for a total of 2N-1 parameters. In statistics and applied
A previous article explains the Spearman rank correlation, which is a robust cousin to the more familiar Pearson correlation. I've also discussed why you might want to use rank correlation, and how to interpret the strength of a rank correlation. This article gives a short example that helps you to
Since the COVID-19 pandemic began, video presentations and webcasts have become a regular routine for many of us. On days that I will be using my webcam, I wear a solid-color shirt. If I don't plan to be on camera, I can wear a pinstripe Oxford shirt. Why the difference?
Real-world data often exhibits extreme skewness. It is not unusual to have data span many orders of magnitude. Classic examples are the distributions of incomes (impoverished and billionaires) and population sizes (small countries and populous nations). The readership of books and blog posts show a similar distribution, which is sometimes
Labeling objects in graphs can be difficult. SAS has a long history of providing support for labeling markers in scatter plots and for labeling regions on a map. This article discusses how the SGPLOT procedure decides where to put a label for a polygon. It discusses the advantages and disadvantages
SAS supports many ways to compute the rank of a numeric variable and to handle tied values. However, sometimes I need to rank the values in a character categorical variable. For example, the values {"Male", "Female", "Male"} have ranks {2, 1, 2} because, in alphabetical order, "Female" is the first-ranked
A previous article defines the silhouette statistic (Rousseeuw, 1987) and shows how to use it to identify observations in a cluster analysis that are potentially misclassified. The article provides many graphs, including the silhouette plot, which is a bar chart or histogram that displays the distribution of the silhouette statistic
Assigning observations into clusters can be challenging. One challenge is deciding how many clusters are in the data. Another is identifying which observations are potentially misclassified because they are on the boundary between two different clusters. Ralph Abbey's 2019 paper ("How to Evaluate Different Clustering Results") is a good way
A lot of programmers have been impressed by the ability of ChatGPT, GPT-4, and Bing Chat to write computer programs. Recently, I wrote an article that discusses an elementary programming assignment, called FizzBuzz, which is sometimes used as part of a hiring process to assess a candidate's basic knowledge of
