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Rick WicklinAugust 24, 2026 0

A direct method to generate correlation matrices with specified eigenvalues

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

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Rick WicklinAugust 17, 2026 0

A visual introduction to the Genz method for computing multivariate normal probabilities

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

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Rick WicklinAugust 10, 2026 0

Constructing orthogonal vectors in SAS

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,

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Rick WicklinAugust 3, 2026 0

Nonnegative matrix factorization with sparseness constraints

A previous article describes the nonnegative matrix factorization (NMF). You can use the NMF to reveal important features in data that can be used to reduce the dimensionality of the problem. NMF is useful when the data are nonnegative, such as counts or pixel values in an image. The goal

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Rick WicklinJuly 27, 2026 5

Hoyer's sparseness measure

In his 2004 paper, "Non-negative Matrix Factorization with Sparseness Constraints," Patrick Hoyer introduced a function that measures the sparseness of a nonzero vector. The paper does not explain or motivate the formula, so this article describes the geometry and intuition behind Hoyer's formula, along with a visualization and examples. Hoyer's

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Rick WicklinJuly 20, 2026 2

The log-KDE method: Density estimates for positive data

A previous article discusses the problem of fitting a kernel density estimate (KDE) to data that are strictly positive. If you use a standard KDE, the resulting density curve might estimate non-zero probability for negative values. This is unsatisfactory because quantities like lengths and mass cannot be negative. The previous

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Rick WicklinJuly 13, 2026 1

Kernel density estimates for positive data

The kernel density estimate (KDE) is a powerful tool for estimating the density of univariate data. The KDE is a flexible model. For example, it can fit data distributions that are multimodal or have long tails. It is nonparametric, which means that you do not need to assume any form

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Rick WicklinJuly 6, 2026 0

An alternative Pareto chart in SAS

Pareto charts are used by quality engineers to visually display a set of causes that produce defects in a manufacturing process. The chart is based on the famous Pareto Principle (the 80/20 rule), which states that 20% of the causes often produce 80% of the defects. If you focus on

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Rick WicklinJune 29, 2026 0

Confidence limits for counts in a multinomial distribution

A recent article discussed Pareto charts and how to create them in SAS. A Pareto chart is used in quality control to identify the most likely reasons that a manufacturing process produces defective items. Suppose there are k different reasons why a process can fail, and you intend to randomly

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Rick WicklinJune 24, 2026 3

Sort the rows or columns of a matrix independently

When you have a data matrix, the rows represent observations and the columns represent variables. If you sort the matrix by one or more columns, the sorting occurs in a way that preserves the elements within rows. Although the rows of the sorted matrix are permuted by the sort, the

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