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 is the last article in a series about computing probabilities for the multivariate normal distribution. A previous article shows how to compute the probability of a standardized multivariate normal (MVN) distribution in an orthant for dimensions k=1, 2, and 3 by using the Childs-Sun formulas. The formulas are exact
In general, it is very difficult to compute a probability for a multivariate continuous distribution. For all continuous distributions, the probability requires solving a complicated multiple integral. For example, the probability for a bivariate normal distribution requires integrating the bivariate normal density over a two-dimensional (2-D) area. The probability for
A course in elementary statistics always introduces the "Z-score." A Z-score is the result of standardizing a normally distributed random variable. By subtracting the distribution's mean and dividing by its standard deviation, you transform a general normal random variable into a standardized variable that has zero mean and unit standard
A previous article discusses regression splines and how to use the EFFECT statement in SAS regression procedures to specify the location of knots for a regressor variable, X. Knots are breakpoints that partition the range of X into subintervals. The splines are defined on a set of adjacent subintervals. The
Splines are useful tools for fitting regression models to data. A spline replaces a single variable (call it X) with several other variables, which are a spline basis for X. When using a spline basis, the shape and location of the basis functions depend on the placement of knots. Knots
Did you know that, for skewed distributions, the sample median is a biased estimator of the population median? You can show this in two ways. For any distribution, you can run a Monte Carlo simulation that reveals the bias. And, for some distributions (such as the exponential distribution), you can
A previous article discusses Welford's one-pass method for computing the sample mean and variance. The article mentions that a useful application of Welford's formulas is to monitor the convergence of a Monte Carlo simulation. This article shows how to implement that strategy in SAS. The Goldilocks Principle for Monte Carlo
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
