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Geometric interpretation of the singular value decomposition (SVD) as the product of a rotation/reflection, followed by a scaling, followed by another rotation/reflection.

Rick WicklinAugust 28, 2017 8

The singular value decomposition: A fundamental technique in multivariate data analysis

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

Programming Tips

Rick WicklinAugust 23, 2017 1

The arithmetic-geometric mean

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

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Programming Tips

Rick WicklinAugust 21, 2017 1

6 tips for timing the performance of algorithms

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

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Analytics | Data Visualization

Bar chart of pairwise correlations between variables

Rick WicklinAugust 16, 2017 2

Use a bar chart to visualize pairwise correlations

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

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Analytics | Learn SAS

Rick WicklinAugust 14, 2017 7

What is rank correlation?

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

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Advanced Analytics

Classical and robust principal component scores for crime data, computed in SAS

Rick WicklinAugust 9, 2017 10

Robust principal component analysis in SAS

Recently, I was asked whether SAS can perform a principal component analysis (PCA) that is robust to the presence of outliers in the data. A PCA requires a data matrix, an estimate for the center of the data, and an estimate for the variance/covariance of the variables. Classically, these estimates

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Programming Tips

Rick WicklinAugust 7, 2017 6

The curse of non-unique eigenvectors

A SAS customer asked, "I computed the eigenvectors of a matrix in SAS and in another software package. I got different answers? How do I know which answer is correct?" I've been asked variations of this question dozens of times. The answer is usually "both answers are correct." The mathematical

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Programming Tips

Broken-stick method for retaining principal components

Rick WicklinAugust 2, 2017 7

Dimension reduction: Guidelines for retaining principal components

Last week I blogged about the broken-stick problem in probability, which reminded me that the broken-stick model is one of the many techniques that have been proposed for choosing the number of principal components to retain during a principal component analysis. Recall that for a principal component analysis (PCA) of

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Learn SAS | Programming Tips

Rick WicklinJuly 31, 2017 4

Flip it. Flip it good.

A SAS user needed to convert a program from MATLAB into the SAS/IML matrix language and asked whether there is a SAS/IML equivalent to the fliplr and flipud functions in MATLAB. These functions flip the columns or rows (respectively) of a matrix; "LR" stands for "left-right" and "UD" stands for

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Programming Tips

Broken stick problem: What is the probability that three randomly chosen points will break a segment into a triangle?

Rick WicklinJuly 26, 2017 1

Random segments and broken sticks

A classical problem in elementary probability asks for the expected lengths of line segments that result from randomly selecting k points along a segment of unit length. It is both fun and instructive to simulate such problems. This article uses simulation in the SAS/IML language to estimate solutions to the

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