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.
This article shows how to simulate data from a mixture of multivariate normal distributions, which is also called a Gaussian mixture. You can use this simulation to generate clustered data. The adjacent graph shows three clusters, each simulated from a four-dimensional normal distribution. Each cluster has its own within-cluster covariance,
Did you know that you can get SAS to compute symbolic (analytical) derivatives of simple functions, including applying the product rule, quotient rule, and chain rule? SAS can form the symbolic derivatives of single-variable functions and partial derivatives of multivariable functions. Furthermore, the derivatives are output in a form that
If you use SAS regression procedures, you are probably familiar with the "stars and bars" notation, which enables you to construct interaction effects in regression models. Although you can construct many regression models by using that classical notation, a friend recently reminded me that the EFFECT statement in SAS provides
Correlation is a fundamental statistical concept that measures the linear association between two variables. There are multiple ways to think about correlation: geometrically, algebraically, with matrices, with vectors, with regression, and more. To paraphrase the great songwriter Paul Simon, there must be 50 ways to view your correlation! But don't
A previous article discussed the mathematical properties of the singular value decomposition (SVD) and showed how to use the SVD subroutine in SAS/IML software. This article uses the SVD to construct a low-rank approximation to an image. Applications include image compression and denoising an image. Construct a grayscale image The
The singular value decomposition (SVD) could be called the "billion-dollar algorithm" since it provides the mathematical basis for many modern algorithms in data science, including text mining, recommender systems (think Netflix and Amazon), image processing, and classification problems. Although the SVD was mathematically discovered in the late 1800s, computers have
All statisticians are familiar with the classical arithmetic mean. Some statisticians are also familiar with the geometric mean. Whereas the arithmetic mean of n numbers is the sum divided by n, the geometric mean of n nonnegative numbers is the n_th root of the product of the numbers. The geometric
When you implement a statistical algorithm in a vector-matrix language such as SAS/IML, R, or MATLAB, you should measure the performance of your implementation, which means that you should time how long a program takes to analyze data of varying sizes and characteristics. There are some general tips that can
Visualizing the correlations between variables often provides insight into the relationships between variables. I've previously written about how to use a heat map to visualize a correlation matrix in SAS/IML, and Chris Hemedinger showed how to use Base SAS to visualize correlations between variables. Recently a SAS programmer asked how
When someone refers to the correlation between two variables, they are probably referring to the Pearson correlation, which is the standard statistic that is taught in elementary statistics courses. Elementary courses do not usually mention that there are other measures of correlation. Why would anyone want a different estimate of
