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.
Bessel functions appear in surprising places. Although originally studied in physics as the solution to differential equations that describe waves and heat diffusion, Bessel functions also have applications in probability, statistics, and signal processing. SAS software provides the JBESSEL function, which is a Bessel function of the first kind, and
A convolution is a mathematical operation on two discrete sequences (or continuous functions) that produces a new sequence (or function). In probability and statistics, a convolution is closely associated with the concept of a "moving window," which time series analysts often call a "filter." If you have ever smoothed a
While researching a topic in probability, I encountered a special mathematical function that I had not previously known about. The function is called Owen's T function (Owen, 1956, Ann. Math. Stat.). It is useful for computing multivariate probabilities, for defining the skew-normal distribution, and for computing probabilities associated with certain
I write a lot of Monte Carlo simulations. This article discusses how to assess whether two independent Monte Carlo estimates are close to each other. This result can be used to test the correctness of a Monte Carlo simulation. It can also be used to compare different Monte Carlo algorithms
SAS has more than 25 common probability distributions that are supported in the PDF, CDF, QUANTILE, and RAND functions. If you want to work with a less common distribution, you can implement these functions yourself. For example, I previously showed how to use PROC FCMP in Base SAS to implement
The classical multivariate normal (MVN) distribution is a standard model for correlated data. It is a simple model, it is easy to fit the MVN model to data, and the parameters in the model (locations and correlations) are intuitive. Of course, normality is a strong assumption that is not always
A previous article discusses how to generate a random correlation matrix. On average, in a random correlation matrix, half of the off-diagonal entries are negative and half are positive. For any realization, the proportion of negative correlations might be greater than (or less than) half. This is in contrast to
In a previous article, I implemented an algorithm due to Niels Waller (TAS, 2020) that uses the method of alternating projections (MAP) to generate random correlation matrices that have a specified set of eigenvalues. The algorithm is iterative, and the MAP method is not guaranteed to converge, although Waller claims
I've been working on a project that uses quasi-Monte Carlo (QMC) techniques to estimate probabilities for multivariate normal (MVN) distributions on finite or infinite rectangular regions. The goal is to enable SAS users to compute these probabilities accurately and efficiently. My implementation is based on a numerical technique called the
One of the great algorithms of linear algebra is the Gram-Schmidt orthogonalization process, which enables you to construct an orthogonal basis for a linear subspace from any set of linearly independent vectors that span the subspace. The Gram-Schmidt process is the basis for the QR decomposition in numerical linear algebra,
