## Tag: Numerical Analysis

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Pi and products

This is my Pi Day post for 2021. Every year on March 14th (written 3/14 in the US), geeky mathematicians and their friends celebrate "all things pi-related" because 3.14 is the three-decimal approximation to pi. Most years I write about lower-case pi (π), which is the ratio of a circle's

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Trap and map: Trapping invalid values

Finite-precision computations can be tricky. You might know, mathematically, that a certain result must be non-negative or must be within a certain interval. However, when you actually compute that result on a computer that uses finite-precision, you might observe that the value is slightly negative or slightly outside of the

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The Kolmogorov D distribution and exact critical values

If you have ever run a Kolmogorov-Smirnov test for normality, you have encountered the Kolmogorov D statistic. The Kolmogorov D statistic is used to assess whether a random sample was drawn from a specified distribution. Although it is frequently used to test for normality, the statistic is "distribution free" in

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The Kullback–Leibler divergence between continuous probability distributions

In a previous article, I discussed the definition of the Kullback-Leibler (K-L) divergence between two discrete probability distributions. For completeness, this article shows how to compute the Kullback-Leibler divergence between two continuous distributions. When f and g are discrete distributions, the K-L divergence is the sum of f(x)*log(f(x)/g(x)) over all

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Bilinear interpolation in SAS

This article shows how to perform two-dimensional bilinear interpolation in SAS by using a SAS/IML function. It is assumed that you have observed the values of a response variable on a regular grid of locations. A previous article showed how to interpolate inside one rectangular cell. When you have a

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What is bilinear interpolation?

I've previously written about linear interpolation in one dimension. Bilinear interpolation is a method for two-dimensional interpolation on a rectangle. If the value of a function is known at the four corners of a rectangle, an interpolation scheme gives you a way to estimate the function at any point in

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Find points where a regression curve has zero slope

This article shows how to find local maxima and maxima on a regression curve, which means finding points where the slope of the curve is zero. An example appears at the right, which shows locations where the loess smoother in a scatter plot has local minima and maxima. Except for

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Cubic spline interpolation in SAS

I recently showed how to use linear interpolation in SAS. Linear interpolation is a common way to interpolate between a set of planar points, but the interpolating function (the interpolant) is not smooth. If you want a smoother interpolant, you can use cubic spline interpolation. This article describes how to

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Linear interpolation in SAS

SAS programmers sometimes ask about ways to perform one-dimensional linear interpolation in SAS. This article shows three ways to perform linear interpolation in SAS: PROC IML (in SAS/IML software), PROC EXPAND (in SAS/ETS software), and PROC TRANSREG (in SAS/STAT software). Of these, PROC IML Is the simplest to use and

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The circular law for eigenvalues

I've previously written about how to generate points that are uniformly distributed in the unit disk. A seemingly unrelated topic is the distribution of eigenvalues (in the complex plane) of various kinds of random matrices. However, I recently learned that these topics are somewhat related! A mathematical result called the

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Round to even

In grade school, students learn how to round numbers to the nearest integer. In later years, students learn variations, such as rounding up and rounding down by using the greatest integer function and least integer function, respectively. My sister, who is an engineer, learned a rounding method that rounds half-integers

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Linearly spaced vectors in SAS

The SAS/IML language and the MATLAB language are similar. Both provide a natural syntax for performing high-level computations on vectors and matrices, including basic linear algebra subroutines. Sometimes a SAS programmer will convert an algorithm from MATLAB into SAS/IML. Because the languages are not identical, I am sometimes asked, "what

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Compute derivatives for nonparametric regression models

SAS enables you to evaluate a regression model at any location within the range of the data. However, sometimes you might be interested in how the predicted response is increasing or decreasing at specified locations. You can use finite differences to compute the slope (first derivative) of a regression model.

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Fit a distribution from quantiles

Data analysts often fit a probability distribution to data. When you have access to the data, a common technique is to use maximum likelihood estimation (MLE) to compute the parameters of a distribution that are "most likely" to have produced the observed data. However, how can you fit a distribution

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Solve a system of nonlinear equations with SAS

This article shows how to use SAS to solve a system of nonlinear equations. When there are n unknowns and n equations, this problem is equivalent to finding a multivariate root of a vector-valued function F(x) = 0 because you can always write the system as f1(x1, x2, ..., xn)

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Compute the quantiles of any distribution

Your statistical software probably provides a function that computes quantiles of common probability distributions such as the normal, exponential, and beta distributions. Because there are infinitely many probability distributions, you might encounter a distribution for which a built-in quantile function is not implemented. No problem! This article shows how to

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Simulate data from the beta-binomial distribution in SAS

This article shows how to simulate beta-binomial data in SAS and how to compute the density function (PDF). The beta-binomial distribution is a discrete compound distribution. The "binomial" part of the name means that the discrete random variable X follows a binomial distribution with parameters N (number of trials) and

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Symbolic derivatives in SAS

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

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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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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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Monte Carlo estimates of joint probabilities

Monte Carlo techniques have many applications, but a primary application is to approximate the probability that some event occurs. The idea is to simulate data from the population and count the proportion of times that the event occurs in the simulated data. For continuous univariate distributions, the probability of an

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Need to log-transform a distribution? There's a SAS function for that!

At a conference last week, a presenter showed SAS statements that compute the logarithm of a probability density function (PDF). The log-PDF is a a common computation because it occurs when maximizing the log-likelihood function. The presenter computed the expression in SAS by using an expression that looked like y

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The Lambert W function in SAS

This article describes how you can evaluate the Lambert W function in SAS/IML software. The Lambert W function is defined implicitly: given a real value x, the function's value w = W(x) is the value of w that satisfies the equation w exp(w) = x. Thus W is the inverse

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Halley's method for finding roots

Edmond Halley (1656-1742) is best known for computing the orbit and predicting the return of the short-period comet that bears his name. However, like many scientists of his era, he was involved in a variety of mathematical and scientific activities. One of his mathematical contributions is a numerical method for

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The sensitivity of Newton's method to an initial guess

In my article about finding an initial guess for root-finding algorithms, I stated that Newton's root-finding method "might not converge or might converge to a root that is far away from the root that you wanted to find." A reader wanted more information about that statement. I have previously shown

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Finding roots: Automating the search for an initial guess

A SAS programmer asked an interesting question on a SAS Support Community. The programmer had a nonlinear function with 12 parameters. He also had file that contained 4,000 lines, where each line contained values for the 12 parameters. In other words, the file specified 4,000 different functions. The programmer wanted