In statistical programming, we often assume that the data are stored in a matrix or a data set and can be read at will. For example, if you want to compute a sample mean or standard deviation, you simply pass a vector to a built-in function, and the software spits
This is the last article in a series about the nonnegative matrix factorization (NMF). In this article, I run and visualize an NMF analysis of the Scotch whisky data and compare it to a principal component analysis (PCA). Previous articles in the series provide information about the whisky data, the
The classical singular value decomposition (SVD) has a long and venerable history. I have described it as a fundamental theorem of linear algebra. Nevertheless, the mathematical property that makes it so useful in general (namely, the orthogonality of the matrix factors) also makes it less useful for certain applications. In
A common task in statistics is to approximate a large data matrix by using a low-rank approximation. Low-rank approximations are used for reducing the dimension of a problem (for example, in principal component analysis), for image analysis via the singular value decomposition (SVD), and for decomposing large matrices that are
Welcome to Pi Day 2026! Every year on March 14th (written 3/14 in the US), people in the mathematical sciences celebrate "all things pi-related" because 3.14 is the three-decimal approximation to π ≈ 3.14159265358979.... The purpose of this day is to have fun and celebrate the importance and ubiquity of
SAS procedures automatically generate many graphs when you turn on ODS graphics. For example, I have written about how to interpret the graphs that are produced automatically when you use PROC PRINCOMP to perform a principal component analysis (PCA). The built-in graphs are well-designed and informative, but this article discusses
Good old Scotch drink! Inspire me, until I lisp and wink, To sing your name! -- Robert Burns (1785) Scotch whisky (spelled without an 'e') is a popular drink that makes up a multi-billion dollar industry. Scotch whisky accounts for almost 75% of Scotland's food and drink exports! Poets
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
A previous article shows how to compute various robust estimates of scale in SAS. In that article, I show how to scale these robust estimators so that they become consistent estimators of the standard deviation (σ) when the data are normally distributed. The scaling factor is related to the expected
Years ago, I wrote an article about the "trap and cap" programming technique. The idea is that programmers should "trap" inputs to functions (like SQRT, LOG, and QUANTILE functions) to avoid domain errors. In addition, when visualizing a function's range, you should "cap" the output to improve graphs of functions
