# Numerical Operations on Data

Basic Statistics
 Mean[list] mean (average) Median[list] median (central value) Max[list] maximum value Variance[list] variance StandardDeviation[list] standard deviation Quantile[list,q] q th quantile Total[list] total
Basic descriptive statistics operations.
Given a list with elements , the mean Mean[list] is defined to be .
The variance Variance[list] is defined to be , for real data. (For complex data .)
The standard deviation StandardDeviation[list] is defined to be .
If the elements in list are thought of as being selected at random according to some probability distribution, then the mean gives an estimate of where the center of the distribution is located, while the standard deviation gives an estimate of how wide the dispersion in the distribution is.
The median Median[list] effectively gives the value at the halfway point in the sorted version of list. It is often considered a more robust measure of the center of a distribution than the mean, since it depends less on outlying values.
The th quantile Quantile[list,q] effectively gives the value that is of the way through the sorted version of list.
For a list of length , the Wolfram Language defines Quantile[list,q] to be s[[Ceiling[n q]]], where is Sort[list,Less].
There are, however, about 10 other definitions of quantile in use, all potentially giving slightly different results. The Wolfram Language covers the common cases by introducing four quantile parameters in the form Quantile[list,q,{{a,b},{c,d}}]. The parameters and in effect define where in the list should be considered a fraction of the way through. If this corresponds to an integer position, then the element at that position is taken to be the th quantile. If it is not an integer position, then a linear combination of the elements on either side is used, as specified by and .
The position in a sorted list for the th quantile is taken to be . If is an integer, then the quantile is . Otherwise, it is , with the indices taken to be or if they are out of range.
 {{0,0},{1,0}} inverse empirical CDF (default) {{0,0},{0,1}} linear interpolation (California method) {{1/2,0},{0,0}} element numbered closest to {{1/2,0},{0,1}} linear interpolation (hydrologist method) {{0,1},{0,1}} mean‐based estimate (Weibull method) {{1,-1},{0,1}} mode‐based estimate {{1/3,1/3},{0,1}} median‐based estimate {{3/8,1/4},{0,1}} normal distribution estimate
Common choices for quantile parameters.
Whenever , the value of the th quantile is always equal to some actual element in list, so that the result changes discontinuously as varies. For , the th quantile interpolates linearly between successive elements in list. Median is defined to use such an interpolation.
Note that Quantile[list,q] yields quartiles when and percentiles when .
 Mean[{x1,x2,…}] the mean of the xi Mean[{{x1,y1,…},{x2,y2,…},…}] a list of the means of the xi,yi,…
Handling multidimensional data.
Sometimes each item in your data may involve a list of values. The basic statistics functions in the Wolfram Language automatically apply to all corresponding elements in these lists.
This separately finds the mean of each "column" of data:
Note that you can extract the elements in the th "column" of a multidimensional list using list[[All,i]].
Descriptive Statistics
Descriptive statistics refers to properties of distributions, such as location, dispersion, and shape. The functions described here compute descriptive statistics of lists of data. You can calculate some of the standard descriptive statistics for various known distributions by using the functions described in "Continuous Distributions" and "Discrete Distributions".
The statistics are calculated assuming that each value of data has probability equal to , where is the number of elements in the data.
 Mean[data] average value Median[data] median (central value) Commonest[data] list of the elements with highest frequency GeometricMean[data] geometric mean HarmonicMean[data] harmonic mean RootMeanSquare[data] root mean square TrimmedMean[data,f] mean of remaining entries, when a fraction is removed from each end of the sorted list of data TrimmedMean[data,{f1,f2}] mean of remaining entries, when fractions and are dropped from each end of the sorted data Quantile[data,q] th quantile Quartiles[data] list of the th, th, th quantiles of the elements in list
Location statistics.
Location statistics describe where the data is located. The most common functions include measures of central tendency like the mean, median, and mode. Quantile[data,q] gives the location before which percent of the data lie. In other words, Quantile gives a value such that the probability that is less than or equal to and the probability that is greater than or equal to .
Here is a dataset:
This finds the mean and median of the data:
This is the mean when the smallest entry in the list is excluded. TrimmedMean allows you to describe the data with removed outliers:
 Variance[data] unbiased estimate of variance, StandardDeviation[data] unbiased estimate of standard deviation MeanDeviation[data] mean absolute deviation, MedianDeviation[data] median absolute deviation, median of values InterquartileRange[data] difference between the first and third quartiles QuartileDeviation[data] half the interquartile range
Dispersion statistics.
Dispersion statistics summarize the scatter or spread of the data. Most of these functions describe deviation from a particular location. For instance, variance is a measure of deviation from the mean, and standard deviation is just the square root of the variance.
This gives an unbiased estimate for the variance of the data with as the divisor:
This compares three types of deviation:
 Covariance[v1,v2] covariance coefficient between lists v1 and v2 Covariance[m] covariance matrix for the matrix m Covariance[m1,m2] covariance matrix for the matrices m1 and m2 Correlation[v1,v2] correlation coefficient between lists v1 and v2 Correlation[m] correlation matrix for the matrix m Correlation[m1,m2] correlation matrix for the matrices m1 and m2
Covariance and correlation statistics.
Covariance is the multivariate extension of variance. For two vectors of equal length, the covariance is a number. For a single matrix m, the i,j th element of the covariance matrix is the covariance between the i th and j th columns of m. For two matrices m1 and m2, the i,j th element of the covariance matrix is the covariance between the i th column of m1 and the j th column of m2.
While covariance measures dispersion, correlation measures association. The correlation between two vectors is equivalent to the covariance between the vectors divided by the standard deviations of the vectors. Likewise, the elements of a correlation matrix are equivalent to the elements of the corresponding covariance matrix scaled by the appropriate column standard deviations.
This gives the covariance between data and a random vector:
Here is a random matrix:
This is the correlation matrix for the matrix m:
This is the covariance matrix:
Scaling the covariance matrix terms by the appropriate standard deviations gives the correlation matrix:
 CentralMoment[data,r] r th central moment Skewness[data] coefficient of skewness Kurtosis[data] kurtosis coefficient QuartileSkewness[data] quartile skewness coefficient
Shape statistics.
You can get some information about the shape of a distribution using shape statistics. Skewness describes the amount of asymmetry. Kurtosis measures the concentration of data around the peak and in the tails versus the concentration in the flanks.
Skewness is calculated by dividing the third central moment by the cube of the population standard deviation. Kurtosis is calculated by dividing the fourth central moment by the square of the population variance of the data, equivalent to CentralMoment[data,2]. (The population variance is the second central moment, and the population standard deviation is its square root.)
QuartileSkewness is calculated from the quartiles of data. It is equivalent to , where , , and are the first, second, and third quartiles respectively.
Here is the second central moment of the data:
A negative value for skewness indicates that the distribution underlying the data has a long leftsided tail:
 Expectation[f[x],xlist] expected value of the function f of x with respect to the values of list
Expected values.
The expectation or expected value of a function is for the list of values , , , . Many descriptive statistics are expectations. For instance, the mean is the expected value of , and the th central moment is the expected value of where is the mean of the .
Here is the expected value of the Log of the data:
Discrete Distributions
The functions described here are among the most commonly used discrete univariate statistical distributions. You can compute their densities, means, variances, and other related properties. The distributions themselves are represented in the symbolic form name[param1,param2,]. Functions such as Mean, which give properties of statistical distributions, take the symbolic representation of the distribution as an argument. "Continuous Distributions" describes many continuous statistical distributions.
 BernoulliDistribution[p] Bernoulli distribution with mean p BetaBinomialDistribution[α,β,n] binomial distribution where the success probability is a BetaDistribution[α,β] random variable BetaNegativeBinomialDistribution[α,β,n] negative binomial distribution where the success probability is a BetaDistribution[α,β] random variable BinomialDistribution[n,p] binomial distribution for the number of successes that occur in n trials, where the probability of success in a trial is p DiscreteUniformDistribution[{imin,imax}] discrete uniform distribution over the integers from imin to imax GeometricDistribution[p] geometric distribution for the number of trials before the first success, where the probability of success in a trial is p HypergeometricDistribution[n,nsucc,ntot] hypergeometric distribution for the number of successes out of a sample of size n, from a population of size ntot containing nsucc successes LogSeriesDistribution[θ] logarithmic series distribution with parameter θ NegativeBinomialDistribution[n,p] negative binomial distribution with parameters n and p PoissonDistribution[μ] Poisson distribution with mean μ ZipfDistribution[ρ] Zipf distribution with parameter ρ
Discrete statistical distributions.
Most of the common discrete statistical distributions can be understood by considering a sequence of trials, each with two possible outcomes, for example, success and failure.
The Bernoulli distribution is the probability distribution for a single trial in which success, corresponding to value 1, occurs with probability p, and failure, corresponding to value 0, occurs with probability 1-p.
The binomial distribution BinomialDistribution[n,p] is the distribution of the number of successes that occur in n independent trials, where the probability of success in each trial is p.
The negative binomial distribution for positive integer n is the distribution of the number of failures that occur in a sequence of trials before n successes have occurred, where the probability of success in each trial is p. The distribution is defined for any positive n, though the interpretation of n as the number of successes and p as the success probability no longer holds if n is not an integer.
The beta binomial distribution BetaBinomialDistribution[α,β,n] is a mixture of binomial and beta distributions. A BetaBinomialDistribution[α,β,n] random variable follows a BinomialDistribution[n,p] distribution, where the success probability p is itself a random variable following the beta distribution BetaDistribution[α,β]. The beta negative binomial distribution is a similar mixture of the beta and negative binomial distributions.
The geometric distribution is the distribution of the total number of trials before the first success occurs, where the probability of success in each trial is p.
The hypergeometric distribution HypergeometricDistribution[n,nsucc,ntot] is used in place of the binomial distribution for experiments in which the n trials correspond to sampling without replacement from a population of size ntot with nsucc potential successes.
The discrete uniform distribution DiscreteUniformDistribution[{imin,imax}] represents an experiment with multiple equally probable outcomes represented by integers imin through imax.
The Poisson distribution describes the number of events that occur in a given time period where μ is the average number of events per period.
The terms in the series expansion of about are proportional to the probabilities of a discrete random variable following the logarithmic series distribution . The distribution of the number of items of a product purchased by a buyer in a specified interval is sometimes modeled by this distribution.
The Zipf distribution , sometimes referred to as the zeta distribution, was first used in linguistics and its use has been extended to model rare events.
 PDF[dist,x] probability mass function at x CDF[dist,x] cumulative distribution function at x InverseCDF[dist,q] the largest integer x such that CDF[dist,x] is at most q Quantile[dist,q] q th quantile Mean[dist] mean Variance[dist] variance StandardDeviation[dist] standard deviation Skewness[dist] coefficient of skewness Kurtosis[dist] coefficient of kurtosis CharacteristicFunction[dist,t] characteristic function Expectation[f[x],xdist] expectation of f[x] for x distributed according to dist Median[dist] median Quartiles[dist] list of the th, th, th quantiles for dist InterquartileRange[dist] difference between the first and third quartiles QuartileDeviation[dist] half the interquartile range QuartileSkewness[dist] quartile‐based skewness measure RandomVariate[dist] pseudorandom number with specified distribution RandomVariate[dist,dims] pseudorandom array with dimensionality dims, and elements from the specified distribution
Some functions of statistical distributions.
Distributions are represented in symbolic form. PDF[dist,x] evaluates the mass function at x if x is a numerical value, and otherwise leaves the function in symbolic form whenever possible. Similarly, CDF[dist,x] gives the cumulative distribution and Mean[dist] gives the mean of the specified distribution. The table above gives a sampling of some of the more common functions available for distributions. For a more complete description of these functions, see the description of their continuous analogues in "Continuous Distributions".
Here is a symbolic representation of the binomial distribution for 34 trials, each having probability 0.3 of success:
This is the mean of the distribution:
You can get the expression for the mean by using symbolic variables as arguments:
Here is the 50% quantile, which is equal to the median:
This gives the expected value of with respect to the binomial distribution:
The elements of this matrix are pseudorandom numbers from the binomial distribution:
Continuous Distributions
The functions described here are among the most commonly used continuous univariate statistical distributions. You can compute their densities, means, variances, and other related properties. The distributions themselves are represented in the symbolic form name[param1,param2,]. Functions such as Mean, which give properties of statistical distributions, take the symbolic representation of the distribution as an argument. "Discrete Distributions" describes many common discrete univariate statistical distributions.
 NormalDistribution[μ,σ] normal (Gaussian) distribution with mean μ and standard deviation σ HalfNormalDistribution[θ] half‐normal distribution with scale inversely proportional to parameter θ LogNormalDistribution[μ,σ] lognormal distribution based on a normal distribution with mean μ and standard deviation σ InverseGaussianDistribution[μ,λ] inverse Gaussian distribution with mean μ and scale λ
Distributions related to the normal distribution.
The lognormal distribution is the distribution followed by the exponential of a normally distributed random variable. This distribution arises when many independent random variables are combined in a multiplicative fashion. The half-normal distribution is proportional to the distribution NormalDistribution[0,1/(θ Sqrt[2/π])] limited to the domain .
The inverse Gaussian distribution , sometimes called the Wald distribution, is the distribution of first passage times in Brownian motion with positive drift.
 ChiSquareDistribution[ν] distribution with ν degrees of freedom InverseChiSquareDistribution[ν] inverse distribution with ν degrees of freedom FRatioDistribution[n,m] -ratio distribution with n numerator and m denominator degrees of freedom StudentTDistribution[ν] Student t distribution with ν degrees of freedom NoncentralChiSquareDistribution[ν,λ] noncentral distribution with ν degrees of freedom and noncentrality parameter λ NoncentralStudentTDistribution[ν,δ] noncentral Student t distribution with ν degrees of freedom and noncentrality parameter δ NoncentralFRatioDistribution[n,m,λ] noncentral -ratio distribution with n numerator degrees of freedom and m denominator degrees of freedom and numerator noncentrality parameter λ
Distributions related to normally distributed samples.
If , , are independent normal random variables with unit variance and mean zero, then has a distribution with degrees of freedom. If a normal variable is standardized by subtracting its mean and dividing by its standard deviation, then the sum of squares of such quantities follows this distribution. The distribution is most typically used when describing the variance of normal samples.
If follows a distribution with degrees of freedom, follows the inverse distribution . A scaled inverse distribution with degrees of freedom and scale can be given as . Inverse distributions are commonly used as prior distributions for the variance in Bayesian analysis of normally distributed samples.
A variable that has a Student distribution can also be written as a function of normal random variables. Let and be independent random variables, where is a standard normal distribution and is a variable with degrees of freedom. In this case, has a distribution with degrees of freedom. The Student distribution is symmetric about the vertical axis, and characterizes the ratio of a normal variable to its standard deviation. Location and scale parameters can be included as μ and σ in StudentTDistribution[μ,σ,ν]. When , the distribution is the same as the Cauchy distribution.
The ratio distribution is the distribution of the ratio of two independent variables divided by their respective degrees of freedom. It is commonly used when comparing the variances of two populations in hypothesis testing.
Distributions that are derived from normal distributions with nonzero means are called noncentral distributions.
The sum of the squares of normally distributed random variables with variance and nonzero means follows a noncentral distribution . The noncentrality parameter is the sum of the squares of the means of the random variables in the sum. Note that in various places in the literature, or is used as the noncentrality parameter.
The noncentral Student distribution describes the ratio where is a central random variable with degrees of freedom, and is an independent normally distributed random variable with variance and mean .
The noncentral ratio distribution NoncentralFRatioDistribution[n,m,λ] is the distribution of the ratio of to , where is a noncentral random variable with noncentrality parameter and degrees of freedom and is a central random variable with degrees of freedom.
 TriangularDistribution[{a,b}] symmetric triangular distribution on the interval {a,b} TriangularDistribution[{a,b},c] triangular distribution on the interval {a,b} with maximum at c UniformDistribution[{min,max}] uniform distribution on the interval {min,max}
Piecewise linear distributions.
The triangular distribution TriangularDistribution[{a,b},c] is a triangular distribution for with maximum probability at and . If is , TriangularDistribution[{a,b},c] is the symmetric triangular distribution TriangularDistribution[{a,b}].
The uniform distribution UniformDistribution[{min,max}], commonly referred to as the rectangular distribution, characterizes a random variable whose value is everywhere equally likely. An example of a uniformly distributed random variable is the location of a point chosen randomly on a line from min to max.
 BetaDistribution[α,β] continuous beta distribution with shape parameters α and β CauchyDistribution[a,b] Cauchy distribution with location parameter a and scale parameter b ChiDistribution[ν] distribution with ν degrees of freedom ExponentialDistribution[λ] exponential distribution with scale inversely proportional to parameter λ ExtremeValueDistribution[α,β] extreme maximum value (Fisher–Tippett) distribution with location parameter α and scale parameter β GammaDistribution[α,β] gamma distribution with shape parameter α and scale parameter β GumbelDistribution[α,β] Gumbel minimum extreme value distribution with location parameter α and scale parameter β InverseGammaDistribution[α,β] inverse gamma distribution with shape parameter α and scale parameter β LaplaceDistribution[μ,β] Laplace (double exponential) distribution with mean μ and scale parameter β LevyDistribution[μ,σ] Lévy distribution with location parameter μ and dispersion parameter σ LogisticDistribution[μ,β] logistic distribution with mean μ and scale parameter β MaxwellDistribution[σ] Maxwell (Maxwell – Boltzmann) distribution with scale parameter σ ParetoDistribution[k,α] Pareto distribution with minimum value parameter k and shape parameter α RayleighDistribution[σ] Rayleigh distribution with scale parameter σ WeibullDistribution[α,β] Weibull distribution with shape parameter α and scale parameter β
Other continuous statistical distributions.
If is uniformly distributed on [-π,π], then the random variable follows a Cauchy distribution CauchyDistribution[a,b], with and .
When and , the gamma distribution GammaDistribution[α,λ] describes the distribution of a sum of squares of -unit normal random variables. This form of the gamma distribution is called a distribution with degrees of freedom. When , the gamma distribution takes on the form of the exponential distribution , often used in describing the waiting time between events.
If a random variable follows the gamma distribution GammaDistribution[α,β], follows the inverse gamma distribution InverseGammaDistribution[α,1/β]. If a random variable follows InverseGammaDistribution[1/2,σ/2], follows a Lévy distribution LevyDistribution[μ,σ].
When and have independent gamma distributions with equal scale parameters, the random variable follows the beta distribution BetaDistribution[α,β], where and are the shape parameters of the gamma variables.
The distribution is followed by the square root of a random variable. For , the distribution is identical to with . For , the distribution is identical to the Rayleigh distribution with . For , the distribution is identical to the MaxwellBoltzmann distribution with .
The Laplace distribution LaplaceDistribution[μ,β] is the distribution of the difference of two independent random variables with identical exponential distributions. The logistic distribution LogisticDistribution[μ,β] is frequently used in place of the normal distribution when a distribution with longer tails is desired.
The Pareto distribution ParetoDistribution[k,α] may be used to describe income, with representing the minimum income possible.
The Weibull distribution WeibullDistribution[α,β] is commonly used in engineering to describe the lifetime of an object. The extreme value distribution is the limiting distribution for the largest values in large samples drawn from a variety of distributions, including the normal distribution. The limiting distribution for the smallest values in such samples is the Gumbel distribution, GumbelDistribution[α,β]. The names "extreme value" and "Gumbel distribution" are sometimes used interchangeably because the distributions of the largest and smallest extreme values are related by a linear change of variable. The extreme value distribution is also sometimes referred to as the logWeibull distribution because of logarithmic relationships between an extreme value-distributed random variable and a properly shifted and scaled Weibull-distributed random variable.
 PDF[dist,x] probability density function at x CDF[dist,x] cumulative distribution function at x InverseCDF[dist,q] the value of x such that CDF[dist,x] equals q Quantile[dist,q] q th quantile Mean[dist] mean Variance[dist] variance StandardDeviation[dist] standard deviation Skewness[dist] coefficient of skewness Kurtosis[dist] coefficient of kurtosis CharacteristicFunction[dist,t] characteristic function Expectation[f[x],xdist] expectation of f[x] for x distributed according to dist Median[dist] median Quartiles[dist] list of the th, th, th quantiles for dist InterquartileRange[dist] difference between the first and third quartiles QuartileDeviation[dist] half the interquartile range QuartileSkewness[dist] quartile‐based skewness measure RandomVariate[dist] pseudorandom number with specified distribution RandomVariate[dist,dims] pseudorandom array with dimensionality dims, and elements from the specified distribution
Some functions of statistical distributions.
The preceding table gives a list of some of the more common functions available for distributions in the Wolfram Language.
The cumulative distribution function (CDF) at is given by the integral of the probability density function (PDF) up to . The PDF can therefore be obtained by differentiating the CDF (perhaps in a generalized sense). In this package the distributions are represented in symbolic form. PDF[dist,x] evaluates the density at if is a numerical value, and otherwise leaves the function in symbolic form. Similarly, CDF[dist,x] gives the cumulative distribution.
The inverse CDF InverseCDF[dist,q] gives the value of at which CDF[dist,x] reaches . The median is given by InverseCDF[dist,1/2]. Quartiles, deciles, and percentiles are particular values of the inverse CDF. Quartile skewness is equivalent to , where , , and are the first, second, and third quartiles, respectively. Inverse CDFs are used in constructing confidence intervals for statistical parameters. InverseCDF[dist,q] and Quantile[dist,q] are equivalent for continuous distributions.
The mean Mean[dist] is the expectation of the random variable distributed according to dist and is usually denoted by . The mean is given by , where is the PDF of the distribution. The variance Variance[dist] is given by . The square root of the variance is called the standard deviation, and is usually denoted by .
The Skewness[dist] and Kurtosis[dist] functions give shape statistics summarizing the asymmetry and the peakedness of a distribution, respectively. Skewness is given by and kurtosis is given by .
The characteristic function CharacteristicFunction[dist,t] is given by . In the discrete case, . Each distribution has a unique characteristic function, which is sometimes used instead of the PDF to define a distribution.
The expected value Expectation[g[x],xdist] of a function g is given by . In the discrete case, the expected value of g is given by .
RandomVariate[dist] gives pseudorandom numbers from the specified distribution.
This gives a symbolic representation of the gamma distribution with and :
Here is the cumulative distribution function evaluated at 10:
This is the cumulative distribution function. It is given in terms of the builtin function GammaRegularized:
Here is a plot of the cumulative distribution function:
This is a pseudorandom array with elements distributed according to the gamma distribution:
Partitioning Data into Clusters
Cluster analysis is an unsupervised learning technique used for classification of data. Data elements are partitioned into groups called clusters that represent proximate collections of data elements based on a distance or dissimilarity function. Identical element pairs have zero distance or dissimilarity, and all others have positive distance or dissimilarity.
 FindClusters[data] partition data into lists of similar elements FindClusters[data,n] partition data into at most n lists of similar elements
General clustering function.
The data argument of FindClusters can be a list of data elements, associations, or rules indexing elements and labels.
 {e1,e2,…} data specified as a list of data elements ei {e1v1,e2v2,…} data specified as a list of rules between data elements ei and labels vi {e1,e2,…}{v1,v2,…} data specified as a rule mapping data elements ei to labels vi key1e1,key2e2…|> data specified as an association mapping elements ei to labels keyi
Ways of specifying data in FindClusters.
FindClusters works for a variety of data types, including numerical, textual, and image, as well as Boolean vectors, dates and times. All data elements ei must have the same dimensions.
Here is a list of numbers:
FindClusters clusters the numbers based on their proximity:
The rule-based data syntax allows for clustering data elements and returning labels for those elements.
Here two-dimensional points are clustered and labeled with their positions in the data list:
The rule-based data syntax can also be used to cluster data based on parts of each data entry. For instance, you might want to cluster data in a data table while ignoring particular columns in the table.
Here is a list of data entries:
This clusters the data while ignoring the first two elements in each data entry:
In principle, it is possible to cluster points given in an arbitrary number of dimensions. However, it is difficult at best to visualize the clusters above two or three dimensions. To compare optional methods in this documentation, an easily visualizable set of two-dimensional data will be used.
The following commands define a set of 300 two-dimensional data points chosen to group into four somewhat nebulous clusters:
This clusters the data based on the proximity of points:
Here is a plot of the clusters:
With the default settings, FindClusters has found the four clusters of points.
You can also direct FindClusters to find a specific number of clusters.
This shows the effect of choosing 3 clusters:
This shows the effect of choosing 5 clusters:
 option name default value CriterionFunction Automatic criterion for selecting a method DistanceFunction Automatic the distance function to use Method Automatic the clustering method to use PerformanceGoal Automatic aspect of performance to optimize Weights Automatic what weight to give to each example
Options for FindClusters.
In principle, clustering techniques can be applied to any set of data. All that is needed is a measure of how far apart each element in the set is from other elements, that is, a function giving the distance between elements.
FindClusters[{e1,e2,},DistanceFunction->f] treats pairs of elements as being less similar when their distances f[ei,ej] are larger. The function f can be any appropriate distance or dissimilarity function. A dissimilarity function f satisfies the following:
If the ei are vectors of numbers, FindClusters by default uses a squared Euclidean distance. If the ei are lists of Boolean True and False (or 0 and 1) elements, FindClusters by default uses a dissimilarity based on the normalized fraction of elements that disagree. If the ei are strings, FindClusters by default uses a distance function based on the number of point changes needed to get from one string to another.
 EuclideanDistance[u,v] the Euclidean norm SquaredEuclideanDistance[u,v] squared Euclidean norm ManhattanDistance[u,v] the Manhattan distance ChessboardDistance[u,v] the chessboard or Chebyshev distance CanberraDistance[u,v] the Canberra distance CosineDistance[u,v] the cosine distance CorrelationDistance[u,v] the correlation distance 1-(u-Mean[u]).(v-Mean[v])/(Abs[u-Mean[u]]Abs[v-Mean[v]]) BrayCurtisDistance[u,v] the Bray–Curtis distance
Distance functions for numerical data.
This shows the clusters in datapairs found using a Manhattan distance:
Dissimilarities for Boolean vectors are typically calculated by comparing the elements of two Boolean vectors and pairwise. It is convenient to summarize each dissimilarity function in terms of , where is the number of corresponding pairs of elements in and , respectively, equal to and . The number counts the pairs in , with and being either 0 or 1. If the Boolean values are True and False, True is equivalent to 1 and False is equivalent to 0.
 MatchingDissimilarity[u,v] simple matching (n10+n01)/Length[u] JaccardDissimilarity[u,v] the Jaccard dissimilarity RussellRaoDissimilarity[u,v] the Russell–Rao dissimilarity (n10+n01+n00)/Length[u] SokalSneathDissimilarity[u,v] the Sokal–Sneath dissimilarity RogersTanimotoDissimilarity[u,v] the Rogers–Tanimoto dissimilarity DiceDissimilarity[u,v] the Dice dissimilarity YuleDissimilarity[u,v] the Yule dissimilarity
Dissimilarity functions for Boolean data.
Here is some Boolean data:
These are the clusters found using the default dissimilarity for Boolean data:
 EditDistance[u,v] the number of edits to transform u into string v DamerauLevenshteinDistance[u,v] Damerau–Levenshtein distance between u and v HammingDistance[u,v] the number of elements whose values disagree in u and v
Dissimilarity functions for string data.
The edit distance is determined by counting the number of deletions, insertions, and substitutions required to transform one string into another while preserving the ordering of characters. In contrast, the DamerauLevenshtein distance counts the number of deletions, insertions, substitutions, and transpositions, while the Hamming distance counts only the number of substitutions.
Here is some string data:
This clusters the string data using the edit distance:
The Method option can be used to specify different methods of clustering.
 "Agglomerate" find clustering hierarchically "DBSCAN" density-based spatial clustering of applications with noise "GaussianMixture" variational Gaussian mixture algorithm "JarvisPatrick" Jarvis–Patrick clustering algorithm "KMeans" k-means clustering algorithm "KMedoids" partitioning around medoids "MeanShift" mean-shift clustering algorithm "NeighborhoodContraction" shift data points toward high-density regions "SpanningTree" minimum spanning tree-based clustering algorithm "Spectral" spectral clustering algorithm
Explicit settings for the Method option.
By default, FindClusters tries different methods and selects the best clustering.
The methods "KMeans" and "KMedoids" determine how to cluster the data for a particular number of clusters k.
The methods "DBSCAN", "JarvisPatrick", "MeanShift", "SpanningTree", "NeighborhoodContraction", and "GaussianMixture" determine how to cluster the data without assuming any particular number of clusters.
The methods "Agglomerate", "Spectral" and "SpanningTree" can be used in both cases.
This shows the clusters in datapairs found using the "KMeans" method:
This shows the clusters in datapairs found using the "GaussianMixture" method:
Additional Method suboptions are available to allow for more control over the clustering. Available suboptions depend on the Method chosen.
 "NeighborhoodRadius" specifies the average radius of a neighborhood of a point "NeighborsNumber" specifies the average number of points in a neighborhood "InitialCentroids" specifies the initial centroids/medoids "SharedNeighborsNumber" specifies the minimum number of shared neighbors "MaxEdgeLength" specifies the pruning length threshold ClusterDissimilarityFunction specifies the intercluster dissimilarity
Suboption for all methods.
The suboption "NeighborhoodRadius" can be used in methods "DBSCAN", "MeanShift", "JarvisPatrick", "NeighborhoodContraction", and "Spectral".
The suboptions "NeighborsNumber" and "SharedNeighborsNumber" can be used in methods "DBSCAN" and "JarvisPatrick", respectively.
The suboption "MaxEdgeLength" can be used in the method "SpanningTree".
The suboption "InitialCentroids" can be used in methods "KMeans" and "KMedoids".
The suboption ClusterDissimilarityFunction can be used in the method "Agglomerate".
The "NeighborhoodRadius" suboption can be used to control the average radius of the neighborhood of a generic point.
This shows different clusterings of datapairs found using the "NeighborhoodContraction" method by varying the "NeighborhoodRadius":
The "NeighborsNumber" suboption can be used to control the number of neighbors in the neighborhood of a generic point.
This shows different clusterings of datapairs found using the "DBSCAN" method by varying the "NeighborsNumber":
The "InitialCentroids" suboption can be used to change the initial configuration in the "KMeans" and "KMedoids" methods. Bad initial configurations may result in bad clusterings.
This shows different clusterings of datapairs found using the "KMeans" method by varying the "InitialCentroids":
With Method->{"Agglomerate",ClusterDissimilarityFunction->f}, the specified linkage function f is used for agglomerative clustering.
 "Single" smallest intercluster dissimilarity "Average" average intercluster dissimilarity "Complete" largest intercluster dissimilarity "WeightedAverage" weighted average intercluster dissimilarity "Centroid" distance from cluster centroids "Median" distance from cluster medians "Ward" Ward's minimum variance dissimilarity f a pure function
Possible values for the ClusterDissimilarityFunction suboption.
Linkage methods determine this intercluster dissimilarity, or fusion level, given the dissimilarities between member elements.
With , f is a pure function that defines the linkage algorithm. Distances or dissimilarities between clusters are determined recursively using information about the distances or dissimilarities between unmerged clusters to determine the distances or dissimilarities for the newly merged cluster. The function f defines a distance from a cluster k to the new cluster formed by fusing clusters i and j. The arguments supplied to f are dik, djk, dij, ni, nj, and nk, where d is the distance between clusters and n is the number of elements in a cluster.
This shows different clusterings of datapairs found using the "Agglomerate" method by varying the ClusterDissimilarityFunction:
The CriterionFunction option can be used to select both the method to use and the best number of clusters.
 "StandardDeviation" root-mean-square standard deviation "RSquared" R-squared "Dunn" Dunn index "CalinskiHarabasz" Calinski–Harabasz index "DaviesBouldin" Davies–Bouldin index Automatic internal index
This shows the result of clustering using different settings for CriterionFunction:
These are the clusters found using the default CriterionFunction with automatically selected number of clusters:
These are the clusters found using the "CalinskiHarabasz" index:
Using Nearest
Nearest is used to find elements in a list that are closest to a given data point.
 Nearest[{elem1,elem2,…},x] give the list of elemi to which x is nearest Nearest[{elem1->v1,elem2->v2,…},x] give the vi corresponding to the elemi to which x is nearest Nearest[{elem1,elem2,…}->{v1,v2,…},x] give the same result Nearest[{elem1,elem2,…}->Automatic,x] take the vi to be the integers 1, 2, 3, … Nearest[data,x,n] give the n nearest elements to x Nearest[data,x,{n,r}] give up to the n nearest elements to x within a radius r Nearest[data] generate a which can be applied repeatedly to different x
Nearest function.
Nearest works with numeric lists, tensors, or a list of strings.
This finds the elements nearest to 4.5:
This finds 3 elements nearest to 4.5:
This finds all elements nearest to 4.5 within a radius of 2:
This finds the points nearest to {1,2} in 2D:
This finds the nearest string to "cat":
The rule-based data syntax lets you use nearest elements to return their labels.
Here two-dimensional points are labeled:
This labels the elements using successive integers:
If Nearest is to be applied repeatedly to the same numerical data, you can get significant performance gains by first generating a NearestFunction.
This generates a set of 10,000 points in 2D and a NearestFunction:
This finds points in the set that are closest to the 10 target points:
It takes much longer if NearestFunction is not used:
 option name default value DistanceFunction Automatic the distance metric to use
Option for Nearest.
For numerical data, by default Nearest uses the EuclideanDistance. For strings, EditDistance is used.
Manipulating Numerical Data
When you have numerical data, it is often convenient to find a simple formula that approximates it. For example, you can try to "fit" a line or curve through the points in your data.
 Fit[{y1,y2,…},{f1 , f2,…},x] fit the values yn to a linear combination of functions fi Fit[{{x1,y1},{x2,y2},…},{f1 , f2,…},x] fit the points (xn,yn) to a linear combination of the fi
Fitting curves to linear combinations of functions.
This generates a table of the numerical values of the exponential function. Table is discussed in "Making Tables of Values":
This finds a leastsquares fit to data of the form . The elements of data are assumed to correspond to values , , of :
This finds a fit of the form :
This gives a table of , pairs:
This finds a fit to the new data, of the form :
 FindFit[data,form,{p1,p2,…},x] find a fit to form with parameters pi
Fitting data to general forms.
This finds the best parameters for a linear fit:
This does a nonlinear fit:
One common way of picking out "signals" in numerical data is to find the Fourier transform, or frequency spectrum, of the data.
 Fourier[data] numerical Fourier transform InverseFourier[data] inverse Fourier transform
Fourier transforms.
Here is a simple square pulse:
This takes the Fourier transform of the pulse:
Note that the Fourier function in the Wolfram Language is defined with the sign convention typically used in the physical sciencesopposite to the one often used in electrical engineering. "Discrete Fourier Transforms" gives more details.
Curve Fitting
There are many situations where one wants to find a formula that best fits a given set of data. One way to do this in the Wolfram Language is to use Fit.
 Fit[{f1,f2,…},{fun1,fun2,…},x] find a linear combination of the funi that best fits the values fi
Basic linear fitting.
Here is a table of the first 20 primes:
Here is a plot of this "data":
This gives a linear fit to the list of primes. The result is the best linear combination of the functions 1 and x:
Here is a plot of the fit:
Here is the fit superimposed on the original data:
This gives a quadratic fit to the data:
Here is a plot of the quadratic fit:
This shows the fit superimposed on the original data. The quadratic fit is better than the linear one:
 {f1,f2,…} data points obtained when a single coordinate takes on values {{x1,f1},{x2,f2},…} data points obtained when a single coordinate takes on values {{x1,y1,…,f1},{x2,y2,…,f2},…} data points obtained with values of a sequence of coordinates
Ways of specifying data.
If you give data in the form then Fit will assume that the successive correspond to values of a function at successive integer points . But you can also give Fit data that corresponds to the values of a function at arbitrary points, in one or more dimensions.
 Fit[data,{fun1,fun2,…},{x,y,…}] fit to a function of several variables
Multivariate fitting.
This gives a table of the values of , , and . You need to use Flatten to get it in the right form for Fit:
This produces a fit to a function of two variables:
Fit takes a list of functions, and uses a definite and efficient procedure to find what linear combination of these functions gives the best leastsquares fit to your data. Sometimes, however, you may want to find a nonlinear fit that does not just consist of a linear combination of specified functions. You can do this using FindFit, which takes a function of any form, and then searches for values of parameters that yield the best fit to your data.
 FindFit[data,form,{par1,par2,…},x] search for values of the pari that make form best fit data FindFit[data,form,pars,{x,y,…}] fit multivariate data
Searching for general fits to data.
This fits the list of primes to a simple linear combination of terms:
The result is the same as from Fit:
This fits to a nonlinear form, which cannot be handled by Fit:
By default, both Fit and FindFit produce leastsquares fits, which are defined to minimize the quantity , where the are residuals giving the difference between each original data point and its fitted value. One can, however, also consider fits based on other norms. If you set the option NormFunction->u, then FindFit will attempt to find the fit that minimizes the quantity u[r], where r is the list of residuals. The default is , corresponding to a leastsquares fit.
This uses the norm, which minimizes the maximum distance between the fit and the data. The result is slightly different from leastsquares:
FindFit works by searching for values of parameters that yield the best fit. Sometimes you may have to tell it where to start in doing this search. You can do this by giving parameters in the form . FindFit also has various options that you can set to control how it does its search.
 FindFit[data,{form,cons},pars,vars] finds a best fit subject to the parameter constraints cons
Searching for general fits to data.
This gives a best fit subject to constraints on the parameters:
 option name default value NormFunction Norm the norm to use AccuracyGoal Automatic number of digits of accuracy to try to get PrecisionGoal Automatic number of digits of precision to try to get WorkingPrecision Automatic precision to use in internal computations MaxIterations Automatic maximum number of iterations to use StepMonitor None expression to evaluate whenever a step is taken EvaluationMonitor None expression to evaluate whenever form is evaluated Method Automatic method to use
Options for FindFit.
Statistical Model Analysis
When fitting models to data, it is often useful to analyze how well the model fits the data and how well the fitting meets the assumptions of the model. For a number of common statistical models, this is accomplished in the Wolfram System by way of fitting functions that construct FittedModel objects.
 FittedModel represent a symbolic fitted model
Object for fitted model information.
FittedModel objects can be evaluated at a point or queried for results and diagnostic information. Diagnostics vary somewhat across model types. Available model fitting functions fit linear, generalized linear, and nonlinear models.
 LinearModelFit construct a linear model GeneralizedLinearModelFit construct a generalized linear model LogitModelFit construct a binomial logistic regression model ProbitModelFit construct a binomial probit regression model NonlinearModelFit construct a nonlinear least-squares model
Functions that generate FittedModel objects.
This fits a linear model assuming values are 1, 2, :
Here is the functional form of the fitted model:
This evaluates the model at :
Here is a shortened list of available results for the linear fitted model:
The major difference between model fitting functions such as LinearModelFit and functions such as Fit and FindFit is the ability to easily obtain diagnostic information from the FittedModel objects. The results are accessible without refitting the model.
This gives the residuals for the fitting:
Here multiple results are obtained at once in a list:
Fitting options relevant to property computations can be passed to FittedModel objects to override defaults.
This gives default 95% confidence intervals:
Here 90% intervals are obtained:
Typical data for these model-fitting functions takes the same form as data in other fitting functions such as Fit and FindFit.
 {y1,y2,…} data points with a single predictor variable taking values 1, 2, … {{x11,x12,…,y1},{x21,x22,…,y2},…} data points with explicit coordinates
Data specifications.

### Linear Models

Linear models with assumed independent normally distributed errors are among the most common models for data. Models of this type can be fitted using the LinearModelFit function.
 LinearModelFit[{y1,y2,…},{f1,f2,…},x] obtain a linear model with basis functions fi and a single predictor variable x LinearModelFit[{{x11,x12,…,y1},{x21,x22,…,y2}},{f1,f2,…},{x1,x2,…}] obtain a linear model of multiple predictor variables xi LinearModelFit[{m,v}] obtain a linear model based on a design matrix m and a response vector v
Linear model fitting.
Linear models have the form , where is the fitted or predicted value, the are parameters to be fitted, and the are functions of the predictor variables . The models are linear in the parameters . The can be any functions of the predictor variables. Quite often the are simply the predictor variables .
This fits a linear model to the first 20 primes:
Options for model specification and for model analysis are available.
 option name default value ConfidenceLevel 95/100 confidence level to use for parameters and predictions IncludeConstantBasis True whether to include a constant basis function LinearOffsetFunction None known offset in the linear predictor NominalVariables None variables considered as nominal or categorical VarianceEstimatorFunction Automatic function for estimating the error variance Weights Automatic weights for data elements WorkingPrecision Automatic precision used in internal computations
Options for LinearModelFit.
The Weights option specifies weight values for weighted linear regression. The NominalVariables option specifies which predictor variables should be treated as nominal or categorical. With , the model is an analysis of variance (ANOVA) model. With NominalVariables->{x1,,xi-1,xi+1,,xn} the model is an analysis of covariance (ANCOVA) model with all but the th predictor treated as nominal. Nominal variables are represented by a collection of binary variables indicating equality and inequality to the observed nominal categorical values for the variable.
ConfidenceLevel, VarianceEstimatorFunction, and WorkingPrecision are relevant to the computation of results after the initial fitting. These options can be set within LinearModelFit to specify the default settings for results obtained from the FittedModel object. These options can also be set within an already constructed FittedModel object to override the option values originally given to LinearModelFit.
Here are the default and mean-squared error variance estimates:
IncludeConstantBasis, LinearOffsetFunction, NominalVariables, and Weights are relevant only to the fitting. Setting these options within an already constructed FittedModel object will have no further impact on the result.
A major feature of the model-fitting framework is the ability to obtain results after the fitting. The full list of available results can be obtained using "Properties".
This is the number of properties available for linear models:
The properties include basic information about the data, fitted model, and numerous results and diagnostics.
 "BasisFunctions" list of basis functions "BestFit" fitted function "BestFitParameters" parameter estimates "Data" the input data or design matrix and response vector "DesignMatrix" design matrix for the model "Function" best-fit pure function "Response" response values in the input data
Properties related to data and the fitted function.
The "BestFitParameters" property gives the fitted parameter values {β0,β1,}. "BestFit" is the fitted function and "Function" gives the fitted function as a pure function. "BasisFunctions" gives the list of functions , with being the constant 1 when a constant term is present in the model. The "DesignMatrix" is the design or model matrix for the data. "Response" gives the list of the response or values from the original data.
 "FitResiduals" difference between actual and predicted responses "StandardizedResiduals" fit residuals divided by the standard error for each residual "StudentizedResiduals" fit residuals divided by single deletion error estimates
Types of residuals.
Residuals give a measure of the pointwise difference between the fitted values and the original responses. "FitResiduals" gives the differences between the observed and fitted values {y1-,y2-,}. "StandardizedResiduals" and "StudentizedResiduals" are scaled forms of the residuals. The th standardized residual is , where is the estimated error variance, is the th diagonal element of the hat matrix, and is the weight for the th data point. The th studentized residual uses the same formula with replaced by , the variance estimate omitting the th data point.
 "ANOVATable" analysis of variance table "ANOVATableDegreesOfFreedom" degrees of freedom from the ANOVA table "ANOVATableEntries" unformatted array of values from the table "ANOVATableFStatistics" F‐statistics from the table "ANOVATableMeanSquares" mean square errors from the table "ANOVATablePValues" ‐values from the table "ANOVATableSumsOfSquares" sums of squares from the table "CoefficientOfVariation" response mean divided by the estimated standard deviation "EstimatedVariance" estimate of the error variance "PartialSumOfSquares" changes in model sum of squares as nonconstant basis functions are removed "SequentialSumOfSquares" the model sum of squares partitioned componentwise
Properties related to the sum of squared errors.
"ANOVATable" gives a formatted analysis of variance table for the model. "ANOVATableEntries" gives the numeric entries in the table and the remaining ANOVATable properties give the elements of columns in the table so individual parts of the table can easily be used in further computations.
This gives a formatted ANOVA table for the fitted model:
Here are the elements of the MS column of the table:
 "CorrelationMatrix" parameter correlation matrix "CovarianceMatrix" parameter covariance matrix "EigenstructureTable" eigenstructure of the parameter correlation matrix "EigenstructureTableEigenvalues" eigenvalues from the table "EigenstructureTableEntries" unformatted array of values from the table "EigenstructureTableIndexes" index values from the table "EigenstructureTablePartitions" partitioning from the table "ParameterConfidenceIntervals" parameter confidence intervals "ParameterConfidenceIntervalTable" table of confidence interval information for the fitted parameters "ParameterConfidenceIntervalTableEntries" unformatted array of values from the table "ParameterConfidenceRegion" ellipsoidal parameter confidence region "ParameterErrors" standard errors for parameter estimates "ParameterPValues" ‐values for parameter ‐statistics "ParameterTable" table of fitted parameter information "ParameterTableEntries" unformatted array of values from the table "ParameterTStatistics" ‐statistics for parameter estimates "VarianceInflationFactors" list of inflation factors for the estimated parameters
Properties and diagnostics for parameter estimates.
"CovarianceMatrix" gives the covariance between fitted parameters. The matrix is , where is the variance estimate, is the design matrix, and is the diagonal matrix of weights. "CorrelationMatrix" is the associated correlation matrix for the parameter estimates. "ParameterErrors" is equivalent to the square root of the diagonal elements of the covariance matrix.
"ParameterTable" and "ParameterConfidenceIntervalTable" contain information about the individual parameter estimates, tests of parameter significance, and confidence intervals.
Here is some data:
This fits a model using both predictor variables:
These are the formatted parameter and parameter confidence interval tables:
Here 99% confidence intervals are used in the table:
The Estimate column of these tables is equivalent to "BestFitParameters". The -statistics are the estimates divided by the standard errors. Each value is the twosided value for the -statistic and can be used to assess whether the parameter estimate is statistically significantly different from 0. Each confidence interval gives the upper and lower bounds for the parameter confidence interval at the level prescribed by the ConfidenceLevel option. The various ParameterTable and ParameterConfidenceIntervalTable properties can be used to get the columns or the unformatted array of values from the table.
"VarianceInflationFactors" is used to measure the multicollinearity between basis functions. The th inflation factor is equal to , where is the coefficient of variation from fitting the th basis function to a linear function of the other basis functions. With , the first inflation factor is for the constant term.
"EigenstructureTable" gives the eigenvalues, condition indices, and variance partitions for the nonconstant basis functions. The Index column gives the square root of the ratios of the eigenvalues to the largest eigenvalue. The column for each basis function gives the proportion of variation in that basis function explained by the associated eigenvector. "EigenstructureTablePartitions" gives the values in the variance partitioning for all basis functions in the table.
 "BetaDifferences" DFBETAS measures of influence on parameter values "CatcherMatrix" catcher matrix "CookDistances" list of Cook distances "CovarianceRatios" COVRATIO measures of observation influence "DurbinWatsonD" Durbin–Watson ‐statistic for autocorrelation "FitDifferences" DFFITS measures of influence on predicted values "FVarianceRatios" FVARATIO measures of observation influence "HatDiagonal" diagonal elements of the hat matrix "SingleDeletionVariances" list of variance estimates with the th data point omitted
Properties related to influence measures.
Pointwise measures of influence are often employed to assess whether individual data points have a large impact on the fitting. The hat matrix and catcher matrix play important roles in such diagnostics. The hat matrix is the matrix such that , where is the observed response vector and is the predicted response vector. "HatDiagonal" gives the diagonal elements of the hat matrix. "CatcherMatrix" is the matrix such that , where is the fitted parameter vector.
"FitDifferences" gives the DFFITS values that provide a measure of influence of each data point on the fitted or predicted values. The th DFFITS value is given by , where is the th hat diagonal and is the th studentized residual.
"BetaDifferences" gives the DFBETAS values that provide measures of influence of each data point on the parameters in the model. For a model with parameters, the th element of "BetaDifferences" is a list of length with the th value giving the measure of the influence of data point on the th parameter in the model. The th "BetaDifferences" vector can be written as , where is the , th element of the catcher matrix.
"CookDistances" gives the Cook distance measures of leverage. The th Cook distance is given by , where is the th standardized residual.
The th element of "CovarianceRatios" is given by and the th "FVarianceRatios" value is equal to , where is the th single deletion variance.
The DurbinWatson statistic "DurbinWatsonD" is used for testing the existence of a first-order autoregressive process. The statistic is equivalent to , where is the th residual.
This plots the Cook distances for the bivariate model:
 "MeanPredictionBands" confidence bands for mean predictions "MeanPredictionConfidenceIntervals" confidence intervals for the mean predictions "MeanPredictionConfidenceIntervalTable" table of confidence intervals for the mean predictions "MeanPredictionConfidenceIntervalTableEntries" unformatted array of values from the table "MeanPredictionErrors" standard errors for mean predictions "PredictedResponse" fitted values for the data "SinglePredictionBands" confidence bands based on single observations "SinglePredictionConfidenceIntervals" confidence intervals for the predicted response of single observations "SinglePredictionConfidenceIntervalTable" table of confidence intervals for the predicted response of single observations "SinglePredictionConfidenceIntervalTableEntries" unformatted array of values from the table "SinglePredictionErrors" standard errors for the predicted response of single observations
Properties of predicted values.
Tabular results for confidence intervals are given by "MeanPredictionConfidenceIntervalTable" and "SinglePredictionConfidenceIntervalTable". These include the observed and predicted responses, standard error estimates, and confidence intervals for each point. Mean prediction confidence intervals are often referred to simply as confidence intervals and single prediction confidence intervals are often referred to as prediction intervals.
Mean prediction intervals give the confidence interval for the mean of the response at fixed values of the predictors and are given by , where is the quantile of the Student distribution with degrees of freedom, is the vector of basis functions evaluated at fixed predictors, and is the estimated covariance matrix for the parameters. Single prediction intervals provide the confidence interval for predicting at fixed values of the predictors, and are given by , where is the estimated error variance.
"MeanPredictionBands" and "SinglePredictionBands" give formulas for mean and single prediction confidence intervals as functions of the predictor variables.
Here is the mean prediction table:
This gives the 90% mean prediction intervals:
 "AdjustedRSquared" adjusted for the number of model parameters "AIC" Akaike Information Criterion "BIC" Bayesian Information Criterion "RSquared" coefficient of determination
Goodness-of-fit measures.
Goodness-of-fit measures are used to assess how well a model fits or to compare models. The coefficient of determination "RSquared" is the ratio of the model sum of squares to the total sum of squares. "AdjustedRSquared" penalizes for the number of parameters in the model and is given by .
"AIC" and "BIC" are likelihoodbased goodness-of-fit measures. Both are equal to times the log-likelihood for the model plus , where is the number of parameters to be estimated including the estimated variance. For "AIC" is , and for "BIC" is .

### Generalized Linear Models

The linear model can be seen as a model with each response value being an observation from a normal distribution with mean value . The generalized linear model extends to models of the form , with each assumed to be an observation from a distribution of known exponential family form with mean , and being an invertible function over the support of the exponential family. Models of this sort can be obtained via GeneralizedLinearModelFit.
 GeneralizedLinearModelFit[{y1,y2,…},{f1,f2,…},x] obtain a generalized linear model with basis functions fi and a single predictor variable x GeneralizedLinearModelFit[{{x11,x12,…,y1},{x21,x22,…,y2}},{f1,f2,…},{x1,x2,…}] obtain a generalized linear model of multiple predictor variables xi GeneralizedLinearModelFit[{m,v}] obtain a generalized linear model based on a design matrix m and response vector v
Generalized linear model fitting.
The invertible function is called the link function and the linear combination is referred to as the linear predictor. Common special cases include the linear regression model with the identity link function and Gaussian or normal exponential family distribution, logit and probit models for probabilities, Poisson models for count data, and gamma and inverse Gaussian models.
The error variance is a function of the prediction and is defined by the distribution up to a constant , which is referred to as the dispersion parameter. The error variance for a fitted value can be written as , where is an estimate of the dispersion parameter obtained from the observed and predicted response values, and is the variance function associated with the exponential family evaluated at the value .
This fits a linear regression model:
This fits a canonical gamma regression model to the same data:
Here are the functional forms of the models:
Logit and probit models are common binomial models for probabilities. The link function for the logit model is and the link for the probit model is the inverse CDF for a standard normal distribution . Models of this type can be fitted via GeneralizedLinearModelFit with ExponentialFamily->"Binomial" and the appropriate LinkFunction or via LogitModelFit and ProbitModelFit.
 LogitModelFit[data,funs,vars] obtain a logit model with basis functions funs and predictor variables vars LogitModelFit[{m,v}] obtain a logit model based on a design matrix m and response vector v ProbitModelFit[data,funs,vars] obtain a probit model fit to data ProbitModelFit[{m,v}] obtain a probit model fit to a design matrix m and response vector v
Logit and probit model fitting.
Parameter estimates are obtained via iteratively reweighted least squares with weights obtained from the variance function of the assumed distribution. Options for GeneralizedLinearModelFit include options for iteration fitting such as PrecisionGoal, options for model specification such as LinkFunction, and options for further analysis such as ConfidenceLevel.
 option name default value AccuracyGoal Automatic the accuracy sought ConfidenceLevel 95/100 confidence level to use for parameters and predictions CovarianceEstimatorFunction "ExpectedInformation" estimation method for the parameter covariance matrix DispersionEstimatorFunction Automatic function for estimating the dispersion parameter ExponentialFamily Automatic exponential family distribution for y IncludeConstantBasis True whether to include a constant basis function LinearOffsetFunction None known offset in the linear predictor LinkFunction Automatic link function for the model MaxIterations Automatic maximum number of iterations to use NominalVariables None variables considered as nominal or categorical PrecisionGoal Automatic the precision sought Weights Automatic weights for data elements WorkingPrecision Automatic precision used in internal computations
The options for LogitModelFit and ProbitModelFit are the same as for GeneralizedLinearModelFit except that ExponentialFamily and LinkFunction are defined by the logit or probit model and so are not options to LogitModelFit and ProbitModelFit.
ExponentialFamily can be "Binomial", "Gamma", "Gaussian", "InverseGaussian", "Poisson", or "QuasiLikelihood". Binomial models are valid for responses from 0 to 1. Poisson models are valid for non-negative integer responses. Gaussian or normal models are valid for real responses. Gamma and inverse Gaussian models are valid for positive responses. Quasi-likelihood models define the distributional structure in terms of a variance function such that the log of the quasilikelihood function for the th data point is given by . The variance function for a "QuasiLikelihood" model can be optionally set via ExponentialFamily->{"QuasiLikelihood", "VarianceFunction"->fun}, where fun is a pure function to be applied to fitted values.
DispersionEstimatorFunction defines a function for estimating the dispersion parameter . The estimate is analogous to in linear and nonlinear regression models.
ExponentialFamily, IncludeConstantBasis, LinearOffsetFunction, LinkFunction, NominalVariables, and Weights all define some aspect of the model structure and optimization criterion and can only be set within GeneralizedLinearModelFit. All other options can be set either within GeneralizedLinearModelFit or passed to the FittedModel object when obtaining results and diagnostics. Options set in evaluations of FittedModel objects take precedence over settings given to GeneralizedLinearModelFit at the time of the fitting.
This gives 95% and 99% confidence intervals for the parameters in the gamma model:
 "BasisFunctions" list of basis functions "BestFit" fitted function "BestFitParameters" parameter estimates "Data" the input data or design matrix and response vector "DesignMatrix" design matrix for the model "Function" best fit pure function "LinearPredictor" fitted linear combination "Response" response values in the input data
Properties related to data and the fitted function.
"BestFitParameters" gives the parameter estimates for the basis functions. "BestFit" gives the fitted function , and "LinearPredictor" gives the linear combination . "BasisFunctions" gives the list of functions , with being the constant 1 when a constant term is present in the model. "DesignMatrix" is the design or model matrix for the basis functions.
 "Deviances" deviances "DevianceTable" deviance table "DevianceTableDegreesOfFreedom" degrees of freedom differences from the table "DevianceTableDeviances" deviance differences from the table "DevianceTableEntries" unformatted array of values from the table "DevianceTableResidualDegreesOfFreedom" residual degrees of freedom from the table "DevianceTableResidualDeviances" residual deviances from the table "EstimatedDispersion" estimated dispersion parameter "NullDeviance" deviance for the null model "NullDegreesOfFreedom" degrees of freedom for the null model "ResidualDeviance" difference between the deviance for the fitted model and the deviance for the full model "ResidualDegreesOfFreedom" difference between the model degrees of freedom and null degrees of freedom
Properties related to dispersion and model deviances.
Deviances and deviance tables generalize the model decomposition given by analysis of variance in linear models. The deviance for a single data point is , where is the log-likelihood function for the fitted model. "Deviances" gives a list of the deviance values for all data points. The sum of all deviances gives the model deviance. The model deviance can be decomposed as sums of squares, which are in an ANOVA table for linear models. The full model is the model whose predicted values are the same as the data.
Here is some data with two predictor variables:
This fits the data to an inverse Gaussian model:
Here is the deviance table for the model:
As with sums of squares, deviances are additive. The Deviance column of the table gives the increase in the model deviance when the given basis function is added. The Residual Deviance column gives the difference between the model deviance and the deviance for the submodel containing all previous terms in the table. For large samples, the increase in deviance is approximately distributed with degrees of freedom equal to that for the basis function in the table.
"NullDeviance" is the deviance for the null model, the constant model equal to the mean of all observed responses for models including a constant, or if a constant term is not included.
As with "ANOVATable", a number of properties are included to extract the columns or unformatted array of entries from "DevianceTable".
 "AnscombeResiduals" Anscombe residuals "DevianceResiduals" deviance residuals "FitResiduals" difference between actual and predicted responses "LikelihoodResiduals" likelihood residuals "PearsonResiduals" Pearson residuals "StandardizedDevianceResiduals" standardized deviance residuals "StandardizedPearsonResiduals" standardized Pearson residuals "WorkingResiduals" working residuals
Types of residuals.
"FitResiduals" is the list of residuals, differences between the observed and predicted responses. Given the distributional assumptions, the magnitude of the residuals is expected to change as a function of the predicted response value. Various types of scaled residuals are employed in the analysis of generalized linear models.
If and are the deviance and residual for the th data point, the th deviance residual is given by . The th Pearson residual is defined as , where is the variance function for the exponential family distribution. Standardized deviance residuals and standardized Pearson residuals include division by , where is the th diagonal of the hat matrix. "LikelihoodResiduals" values combine deviance and Pearson residuals. The th likelihood residual is given by .
"AnscombeResiduals" provide a transformation of the residuals toward normality, so a plot of these residuals should be expected to look roughly like white noise. The th Anscombe residual can be written as .
"WorkingResiduals" gives the residuals from the last step of the iterative fitting. The th working residual can be obtained as evaluated at .
This plots the residuals and Anscombe residuals for the inverse Gaussian model:
 "CorrelationMatrix" asymptotic parameter correlation matrix "CovarianceMatrix" asymptotic parameter covariance matrix "ParameterConfidenceIntervals" parameter confidence intervals "ParameterConfidenceIntervalTable" table of confidence interval information for the fitted parameters "ParameterConfidenceIntervalTableEntries" unformatted array of values from the table "ParameterConfidenceRegion" ellipsoidal parameter confidence region "ParameterTableEntries" unformatted array of values from the table "ParameterErrors" standard errors for parameter estimates "ParameterPValues" ‐values for parameter ‐statistics "ParameterTable" table of fitted parameter information "ParameterZStatistics" ‐statistics for parameter estimates
Properties and diagnostics for parameter estimates.
"CovarianceMatrix" gives the covariance between fitted parameters and is very similar to the definition for linear models. With CovarianceEstimatorFunction->"ExpectedInformation", the expected information matrix obtained from the iterative fitting is used. The matrix is , where is the design matrix and is the diagonal matrix of weights from the final stage of the fitting. The weights include both weights specified via the Weights option and the weights associated with the distribution's variance function. With CovarianceEstimatorFunction->"ObservedInformation", the matrix is given by , where is the observed Fisher information matrix, which is the Hessian of the loglikelihood function with respect to parameters of the model.
"CorrelationMatrix" is the associated correlation matrix for the parameter estimates. "ParameterErrors" is equivalent to the square root of the diagonal elements of the covariance matrix. "ParameterTable" and "ParameterConfidenceIntervalTable" contain information about the individual parameter estimates, tests of parameter significance, and confidence intervals. The test statistics for generalized linear models asymptotically follow normal distributions.
 "CookDistances" list of Cook distances "HatDiagonal" diagonal elements of the hat matrix
Properties related to influence measures.
"CookDistances" and "HatDiagonal" extend the leverage measures from linear regression to generalized linear models. The hat matrix from which the diagonal elements are extracted is defined using the final weights of the iterative fitting.
The Cook distance measures of leverage are defined as in linear regression with standardized residuals replaced by standardized Pearson residuals. The th Cook distance is given by , where is the th standardized Pearson residual.
 "PredictedResponse" fitted values for the data
Properties of predicted values.
 "AdjustedLikelihoodRatioIndex" Ben‐Akiva and Lerman's adjusted likelihood ratio index "AIC" Akaike Information Criterion "BIC" Bayesian Information Criterion "CoxSnellPseudoRSquared" Cox and Snell's pseudo "CraggUhlerPseudoRSquared" Cragg and Uhler's pseudo "EfronPseudoRSquared" Efron's pseudo "LikelihoodRatioIndex" McFadden's likelihood ratio index "LikelihoodRatioStatistic" likelihood ratio "LogLikelihood" log likelihood for the fitted model "PearsonChiSquare" Pearson's statistic
Goodness-of-fit measures.
"LogLikelihood" is the loglikelihood for the fitted model. "AIC" and "BIC" are penalized loglikelihood measures , where is the loglikelihood for the fitted model, is the number of parameters estimated including the dispersion parameter, and is for "AIC" and for "BIC" for a model of data points. "LikelihoodRatioStatistic" is given by , where is the loglikelihood for the null model.
A number of the goodness-of-fit measures generalize from linear regression as either a measure of explained variation or as a likelihoodbased measure. "CoxSnellPseudoRSquared" is given by . "CraggUhlerPseudoRSquared" is a scaled version of Cox and Snell's measure . "LikelihoodRatioIndex" involves the ratio of loglikelihoods , and "AdjustedLikelihoodRatioIndex" adjusts by penalizing for the number of parameters . "EfronPseudoRSquared" uses the sum of squares interpretation of and is given as , where is the th residual and is the mean of the responses .
"PearsonChiSquare" is equal to , where the are Pearson residuals.

### Nonlinear Models

A nonlinear least-squares model is an extension of the linear model where the model need not be a linear combination of basis function. The errors are still assumed to be independent and normally distributed. Models of this type can be fitted using the NonlinearModelFit function.
 NonlinearModelFit[{y1,y2,…},form,{β1,…},x] obtain a nonlinear model of the function form with parameters βi a single parameter predictor variable x NonlinearModelFit[{{x11,…,y1},{x21,…,y2}},form,{β1,…},{x1,…}] obtain a nonlinear model as a function of multiple predictor variables xi NonlinearModelFit[data,{form,cons},{β1,…},{x1,…}] obtain a nonlinear model subject to the constraints cons
Nonlinear model fitting.
Nonlinear models have the form , where is the fitted or predicted value, the are parameters to be fitted, and the are predictor variables. As with any nonlinear optimization problem, a good choice of starting values for the parameters may be necessary. Starting values can be given using the same parameter specifications as for FindFit.
This fits a nonlinear model to a sequence of square roots:
Options for model fitting and for model analysis are available.
 option name default value AccuracyGoal Automatic the accuracy sought ConfidenceLevel 95/100 confidence level to use for parameters and predictions EvaluationMonitor None expression to evaluate whenever expr is evaluated MaxIterations Automatic maximum number of iterations to use Method Automatic method to use PrecisionGoal Automatic the precision sought StepMonitor None the expression to evaluate whenever a step is taken VarianceEstimatorFunction Automatic function for estimating the error variance Weights Automatic weights for data elements WorkingPrecision Automatic precision used in internal computations
Options for NonlinearModelFit.
General numeric options such as AccuracyGoal, Method, and WorkingPrecision are the same as for FindFit.
The Weights option specifies weight values for weighted nonlinear regression. The optimal fit is for a weighted sum of squared errors.
All other options can be relevant to computation of results after the initial fitting. They can be set within NonlinearModelFit for use in the fitting and to specify the default settings for results obtained from the FittedModel object. These options can also be set within an already constructed FittedModel object to override the option values originally given to NonlinearModelFit.
 "BestFit" fitted function "BestFitParameters" parameter estimates "Data" the input data "Function" best fit pure function "Response" response values in the input data
Properties related to data and the fitted function.
Basic properties of the data and fitted function for nonlinear models behave like the same properties for linear and generalized linear models with the exception that "BestFitParameters" returns a rule as is done for the result of FindFit.
This gives the fitted function and rules for the parameter estimates:
Many diagnostics for nonlinear models extend or generalize concepts from linear regression. These extensions often rely on linear approximations or large sample approximations.
 "FitResiduals" difference between actual and predicted responses "StandardizedResiduals" fit residuals divided by the standard error for each residual "StudentizedResiduals" fit residuals divided by single deletion error estimates
Types of residuals.
As in linear regression, "FitResiduals" gives the differences between the observed and fitted values , and "StandardizedResiduals" and "StudentizedResiduals" are scaled forms of these differences.
The th standardized residual is , where is the estimated error variance, is the th diagonal element of the hat matrix, and is the weight for the th data point, and the th studentized residual is obtained by replacing with the th single deletion variance . For nonlinear models a first-order approximation is used for the design matrix, which is needed to compute the hat matrix.
 "ANOVATable" analysis of variance table "ANOVATableDegreesOfFreedom" degrees of freedom from the ANOVA table "ANOVATableEntries" unformatted array of values from the table "ANOVATableMeanSquares" mean square errors from the table "ANOVATableSumsOfSquares" sums of squares from the table "EstimatedVariance" estimate of the error variance
Properties related to the sum of squared errors.
"ANOVATable" provides a decomposition of the variation in the data attributable to the fitted function and to the errors or residuals.
This gives the ANOVA table for the nonlinear model:
The uncorrected total sums of squares gives the sum of squared responses, while the corrected total gives the sum of squared differences between the responses and their mean value.
 "CorrelationMatrix" asymptotic parameter correlation matrix "CovarianceMatrix" asymptotic parameter covariance matrix "ParameterBias" estimated bias in the parameter estimates "ParameterConfidenceIntervals" parameter confidence intervals "ParameterConfidenceIntervalTable" table of confidence interval information for the fitted parameters "ParameterConfidenceIntervalTableEntries" unformatted array of values from the table "ParameterConfidenceRegion" ellipsoidal parameter confidence region "ParameterErrors" standard errors for parameter estimates "ParameterPValues" ‐values for parameter ‐statistics "ParameterTable" table of fitted parameter information "ParameterTableEntries" unformatted array of values from the table "ParameterTStatistics" ‐statistics for parameter estimates
Properties and diagnostics for parameter estimates.
"CovarianceMatrix" gives the approximate covariance between fitted parameters. The matrix is , where is the variance estimate, is the design matrix for the linear approximation to the model, and is the diagonal matrix of weights. "CorrelationMatrix" is the associated correlation matrix for the parameter estimates. "ParameterErrors" is equivalent to the square root of the diagonal elements of the covariance matrix.
"ParameterTable" and "ParameterConfidenceIntervalTable" contain information about the individual parameter estimates, tests of parameter significance, and confidence intervals obtained using the error estimates.
 "CurvatureConfidenceRegion" confidence region for curvature diagnostics "FitCurvatureTable" table of curvature diagnostics "FitCurvatureTableEntries" unformatted array of values from the table "MaxIntrinsicCurvature" measure of maximum intrinsic curvature "MaxParameterEffectsCurvature" measure of maximum parameter effects curvature
Curvature diagnostics.
The first-order approximation used for many diagnostics is equivalent to the model being linear in the parameters. If the parameter space near the parameter estimates is sufficiently flat, the linear approximations and any results that rely on first-order approximations can be deemed reasonable. Curvature diagnostics are used to assess whether the approximate linearity is reasonable. "FitCurvatureTable" is a table of curvature diagnostics.
"MaxIntrinsicCurvature" and "MaxParameterEffectsCurvature" are scaled measures of the normal and tangential curvatures of the parameter spaces at the best-fit parameter values. "CurvatureConfidenceRegion" is a scaled measure of the radius of curvature of the parameter space at the best-fit parameter values. If the normal and tangential curvatures are small relative to the value of the "CurvatureConfidenceRegion", the linear approximation is considered reasonable. Some rules of thumb suggest comparing the values directly, while others suggest comparing with half the "CurvatureConfidenceRegion".
Here is the curvature table for the nonlinear model:
 "HatDiagonal" diagonal elements of the hat matrix "SingleDeletionVariances" list of variance estimates with the th data point omitted
Properties related to influence measures.
The hat matrix is the matrix such that , where is the observed response vector and is the predicted response vector. "HatDiagonal" gives the diagonal elements of the hat matrix. As with other properties, uses the design matrix for the linear approximation to the model.
The th element of "SingleDeletionVariances" is equivalent to , where is the number of data points, is the number of parameters, is the th hat diagonal, is the variance estimate for the full dataset, and is the th residual.
 "MeanPredictionBands" confidence bands for mean predictions "MeanPredictionConfidenceIntervals" confidence intervals for the mean predictions "MeanPredictionConfidenceIntervalTable" table of confidence intervals for the mean predictions "MeanPredictionConfidenceIntervalTableEntries" unformatted array of values from the table "MeanPredictionErrors" standard errors for mean predictions "PredictedResponse" fitted values for the data "SinglePredictionBands" confidence bands based on single observations "SinglePredictionConfidenceIntervals" confidence intervals for the predicted response of single observations "SinglePredictionConfidenceIntervalTable" table of confidence intervals for the predicted response of single observations "SinglePredictionConfidenceIntervalTableEntries" unformatted array of values from the table "SinglePredictionErrors" standard errors for the predicted response of single observations
Properties of predicted values.
Tabular results for confidence intervals are given by "MeanPredictionConfidenceIntervalTable" and "SinglePredictionConfidenceIntervalTable". These results are analogous to those for linear models obtained via LinearModelFit, again with first-order approximations used for the design matrix.
"MeanPredictionBands" and "SinglePredictionBands" give functions of the predictor variables.
Here the fitted function and mean prediction bands are obtained:
This plots the fitted curve and confidence bands:
 "AdjustedRSquared" adjusted for the number of model parameters "AIC" Akaike Information Criterion "BIC" Bayesian Information Criterion "RSquared" coefficient of determination
Goodness-of-fit measures.
"AdjustedRSquared", "AIC", "BIC", and "RSquared" are all direct extensions of the measures as defined for linear models. The coefficient of determination "RSquared" is , where is the residual sum of squares and is the uncorrected total sum of squares. The coefficient of determination does not have the same interpretation as the percentage of explained variation in nonlinear models as it does in linear models because the sum of squares for the model and for the residuals do not necessarily sum to the total sum of squares. "AdjustedRSquared" penalizes for the number of parameters in the model and is given by .
"AIC" and "BIC" are equal to times the log-likelihood for the model plus , where is the number of parameters to be estimated including the estimated variance. For "AIC" is , and for "BIC" is .
Approximate Functions and Interpolation
In many kinds of numerical computations, it is convenient to introduce approximate functions. Approximate functions can be thought of as generalizations of ordinary approximate real numbers. While an approximate real number gives the value to a certain precision of a single numerical quantity, an approximate function gives the value to a certain precision of a quantity which depends on one or more parameters. The Wolfram Language uses approximate functions, for example, to represent numerical solutions to differential equations obtained with NDSolve, as discussed in "Numerical Differential Equations".
Approximate functions in the Wolfram Language are represented by InterpolatingFunction objects. These objects work like the pure functions discussed in "Pure Functions". The basic idea is that when given a particular argument, an InterpolatingFunction object finds the approximate function value that corresponds to that argument.
The InterpolatingFunction object contains a representation of the approximate function based on interpolation. Typically it contains values and possibly derivatives at a sequence of points. It effectively assumes that the function varies smoothly between these points. As a result, when you ask for the value of the function with a particular argument, the InterpolatingFunction object can interpolate to find an approximation to the value you want.
 Interpolation[{f1,f2,…}] construct an approximate function with values fi at successive integers Interpolation[{{x1,f1},{x2,f2},…}] construct an approximate function with values fi at points xi
Constructing approximate functions.
Here is a table of the values of the sine function:
This constructs an approximate function which represents these values:
The approximate function reproduces each of the values in the original table:
It also allows you to get approximate values at other points:
In this case the interpolation is a fairly good approximation to the true sine function:
You can work with approximate functions much as you would with any other Wolfram Language functions. You can plot approximate functions, or perform numerical operations such as integration or root finding.
If you give a nonnumerical argument, the approximate function is left in symbolic form:
Here is a numerical integral of the approximate function:
Here is the same numerical integral for the true sine function:
A plot of the approximate function is essentially indistinguishable from the true sine function:
If you differentiate an approximate function, the Wolfram Language will return another approximate function that represents the derivative.
This finds the derivative of the approximate sine function, and evaluates it at :
The result is close to the exact one:
InterpolatingFunction objects contain all the information the Wolfram Language needs about approximate functions. In standard Wolfram Language output format, however, only the part that gives the domain of the InterpolatingFunction object is printed explicitly. The lists of actual parameters used in the InterpolatingFunction object are shown only in iconic form.
In standard output format, the only parts of an InterpolatingFunction object printed explicitly are its domain and output type:
If you ask for a value outside of the domain, the Wolfram Language prints a warning, then uses extrapolation to find a result:
The more information you give about the function you are trying to approximate, the better the approximation the Wolfram Language constructs can be. You can, for example, specify not only values of the function at a sequence of points, but also derivatives.
 Interpolation[{{{x1},f1,df1,ddf1,…},…}] construct an approximate function with specified derivatives at points xi
Constructing approximate functions with specified derivatives.
This interpolates through the values of the sine function and its first derivative:
This finds a better approximation to the derivative than the previous interpolation:
Interpolation works by fitting polynomial curves between the points you specify. You can use the option InterpolationOrder to specify the degree of these polynomial curves. The default setting is , yielding cubic curves.
This makes a table of values of the cosine function:
This creates an approximate function using linear interpolation between the values in the table:
The approximate function consists of a collection of straightline segments:
With the default setting , cubic curves are used, and the function looks smooth:
Increasing the setting for InterpolationOrder typically leads to smoother approximate functions. However, if you increase the setting too much, spurious wiggles may develop.
 ListInterpolation[{{f11,f12,…},{f21,…},…}] construct an approximate function from a two‐dimensional grid of values at integer points ListInterpolation[list,{{xmin,xmax},{ymin,ymax}}] assume the values are from an evenly spaced grid with the specified domain ListInterpolation[list,{{x1,x2,…},{y1,y2,…}}] assume the values are from a grid with the specified grid lines
Interpolating multidimensional arrays of data.
This interpolates an array of values from integer grid points:
Here is the value at a particular position:
Here is another array of values:
To interpolate this array you explicitly have to tell the Wolfram Language the domain it covers:
ListInterpolation works for arrays of any dimension, and in each case it produces an InterpolatingFunction object which takes the appropriate number of arguments.
This interpolates a threedimensional array:
The resulting InterpolatingFunction object takes three arguments:
The Wolfram Language can handle not only purely numerical approximate functions, but also ones which involve symbolic parameters.
This generates an InterpolatingFunction that depends on the parameters a and b:
This shows how the interpolated value at 2.2 depends on the parameters:
With the default setting for InterpolationOrder used, the value at this point no longer depends on a:
In working with approximate functions, you can quite often end up with complicated combinations of InterpolatingFunction objects. You can always tell the Wolfram Language to produce a single InterpolatingFunction object valid over a particular domain by using FunctionInterpolation.
This generates a new InterpolatingFunction object valid in the domain 0 to 1:
This generates a nested InterpolatingFunction object:
This produces a pure twodimensional InterpolatingFunction object:
 FunctionInterpolation[expr,{x,xmin,xmax}] construct an approximate function by evaluating expr with x ranging from xmin to xmax FunctionInterpolation[expr,{x,xmin,xmax},{y,ymin,ymax},…] construct a higher‐dimensional approximate function
Constructing approximate functions by evaluating expressions.
Discrete Fourier Transforms
A common operation in analyzing various kinds of data is to find the discrete Fourier transform (or spectrum) of a list of values. The idea is typically to pick out components of the data with particular frequencies or ranges of frequencies.
 Fourier[{u1,u2,…,un}] discrete Fourier transform InverseFourier[{v1,v2,…,vn}] inverse discrete Fourier transform
Discrete Fourier transforms.
Here is some data, corresponding to a square pulse:
Here is the discrete Fourier transform of the data. It involves complex numbers:
Here is the inverse discrete Fourier transform:
Fourier works whether or not your list of data has a length which is a power of two:
This generates a list of 200 elements containing a periodic signal with random noise added:
The data looks fairly random if you plot it directly:
The discrete Fourier transform, however, shows a strong peak at , and a symmetric peak at , reflecting the frequency component of the original signal near :
In the Wolfram Language, the discrete Fourier transform of a list of length is by default defined to be . Notice that the zero frequency term appears at position 1 in the resulting list.
The inverse discrete Fourier transform of a list of length is by default defined to be .
In different scientific and technical fields different conventions are often used for defining discrete Fourier transforms. The option FourierParameters allows you to choose any of these conventions you want.
 common convention setting discrete Fourier transform inverse discrete Fourier transform Wolfram Language default {0,1} data analysis {-1,1} signal processing {1,-1} general case {a,b}
Typical settings for FourierParameters with various conventions.
 Fourier[{{u11,u12,…},{u21,u22,…},…}] two‐dimensional discrete Fourier transform
Twodimensional discrete Fourier transform.
The Wolfram Language can find discrete Fourier transforms for data in any number of dimensions. In dimensions, the data is specified by a list nested levels deep. Twodimensional discrete Fourier transforms are often used in image processing.
One issue with the usual discrete Fourier transform for real data is that the result is complex-valued. There are variants of real discrete Fourier transforms that have real results. The Wolfram Language has commands for computing the discrete cosine transform and the discrete sine transform.
 FourierDCT[list] Fourier discrete cosine transform of a list of real numbers FourierDST[list] Fourier discrete sine transform of a list of real numbers
Discrete real Fourier transforms.
Here is some data corresponding to a square pulse:
Here is the Fourier discrete cosine transform of the data:
Here is the Fourier discrete sine transform of the data:
There are four types each of Fourier discrete sine and cosine transforms typically in use, denoted by number or sometimes roman numeral as in "DCTII" for the discrete cosine transform of type 2.
 FourierDCT[list,m] Fourier discrete cosine transform of type m FourierDST[list,m] Fourier discrete sine transform of type m
Discrete real Fourier transforms of different types.
The default is type 2 for both FourierDCT and FourierDST.
The Wolfram Language does not need InverseFourierDCT or InverseFourierDST functions because FourierDCT and FourierDST are their own inverses when used with the appropriate type. The inverse transforms for types 1, 2, 3, 4 are types 1, 3, 2, 4, respectively.
Check that the type 3 transform is the inverse of the type 2 transform:
The discrete real transforms are convenient to use for data or image compression.
Here is data that might be like a front or an edge:
The discrete cosine transform has most of the information in the first few modes:
Reconstruct the front from only the first 20 modes (1/10 of the original data size). The oscillations are a consequence of the truncation and are known to show up in image processing applications as well:
Convolutions and Correlations
Convolution and correlation are central to many kinds of operations on lists of data. They are used in such areas as signal and image processing, statistical data analysis, and approximations to partial differential equations, as well as operations on digit sequences and power series.
In both convolution and correlation the basic idea is to combine a kernel list with successive sublists of a list of data. The convolution of a kernel with a list has the general form , while the correlation has the general form .
 ListConvolve[kernel,list] form the convolution of kernel with list ListCorrelate[kernel,list] form the correlation of kernel with list
Convolution and correlation of lists.
This forms the convolution of the kernel {x,y} with a list of data:
This forms the correlation:
In this case reversing the kernel gives exactly the same result as ListConvolve:
This forms successive differences of the data:
In forming sublists to combine with a kernel, there is always an issue of what to do at the ends of the list of data. By default, ListConvolve and ListCorrelate never form sublists which would "overhang" the ends of the list of data. This means that the output you get is normally shorter than the original list of data.
With an input list of length 6, the output is in this case of length 4:
In practice one often wants to get output that is as long as the original list of data. To do this requires including sublists that overhang one or both ends of the list of data. The additional elements needed to form these sublists must be filled in with some kind of "padding". By default, the Wolfram Language takes copies of the original list to provide the padding, thus effectively treating the list as being cyclic.
 ListCorrelate[kernel,list] do not allow overhangs on either side (result shorter than list ) ListCorrelate[kernel,list,1] allow an overhang on the right (result same length as list ) ListCorrelate[kernel,list,-1] allow an overhang on the left (result same length as list ) ListCorrelate[kernel,list,{-1,1}] allow overhangs on both sides (result longer than list ) ListCorrelate[kernel,list,{kL,kR}] allow particular overhangs on left and right
Controlling how the ends of the list of data are treated.
The default involves no overhangs:
The last term in the last element now comes from the beginning of the list:
Now the first term of the first element and the last term of the last element both involve wraparound:
In the general case ListCorrelate[kernel,list,{kL,kR}] is set up so that in the first element of the result, the first element of list appears multiplied by the element at position kL in kernel, and in the last element of the result, the last element of list appears multiplied by the element at position kR in kernel. The default case in which no overhang is allowed on either side thus corresponds to ListCorrelate[kernel,list,{1,-1}].
With a kernel of length 3, alignments {-1,2} always make the first and last elements of the result the same:
For many kinds of data, it is convenient to assume not that the data is cyclic, but rather that it is padded at either end by some fixed element, often 0, or by some sequence of elements.
 ListCorrelate[kernel,list,klist,p] pad with element p ListCorrelate[kernel,list,klist,{p1,p2,…}] pad with cyclic repetitions of the pi ListCorrelate[kernel,list,klist,list] pad with cyclic repetitions of the original data
Controlling the padding for a list of data.
A common case is to pad with zero:
When the padding is indicated by {p,q}, the list {a,b,c} overlays {,p,q,p,q,} with a p aligned under the a:
Different choices of kernel allow ListConvolve and ListCorrelate to be used for different kinds of computations.
This finds a moving average of data:
Here is a Gaussian kernel:
This generates some "data":
Here is a plot of the data:
This convolves the kernel with the data:
The result is a smoothed version of the data:
You can use ListConvolve and ListCorrelate to handle symbolic as well as numerical data.
This forms the convolution of two symbolic lists:
The result corresponds exactly with the coefficients in the expanded form of this product of polynomials:
ListConvolve and ListCorrelate work on data in any number of dimensions.
This imports image data from a file:
Here is the image:
This convolves the data with a twodimensional kernel:
This shows the image corresponding to the data:
Cellular Automata
Cellular automata provide a convenient way to represent many kinds of systems in which the values of cells in an array are updated in discrete steps according to a local rule.
 CellularAutomaton[rnum,init,t] evolve rule rnum from init for t steps
Generating a cellular automaton evolution.
This starts with the list given, then evolves rule 30 for four steps:
This shows 100 steps of rule 30 evolution from random initial conditions:
 {a1,a2,…} explicit list of values ai {{a1,a2,…},b} values ai superimposed on a b background {{a1,a2,…},blist} values ai superimposed on a background of repetitions of blist {{{{a11,a12,…},{d1}},…},blist} values aij at offsets di
Ways of specifying initial conditions for onedimensional cellular automata.
If you give an explicit list of initial values, CellularAutomaton will take the elements in this list to correspond to all the cells in the system, arranged cyclically.
The right neighbor of the cell at the end is the cell at the beginning:
It is often convenient to set up initial conditions in which there is a small "seed" region, superimposed on a constant "background". By default, CellularAutomaton automatically fills in enough background to cover the size of the pattern that can be produced in the number of steps of evolution you specify.
This shows rule 30 evolving from an initial condition containing a single black cell:
This shows rule 30 evolving from an initial condition consisting of a {1,1} seed on a background of repeated {1,0,1,1} blocks:
Particularly in studying interactions between structures, you may sometimes want to specify initial conditions for cellular automata in which certain blocks are placed at particular offsets.
This sets up an initial condition with black cells at offsets :
 n , , elementary rule {n,k} general nearest‐neighbor rule with k colors {n,k,r} general rule with k colors and range r {n,{k,1}} k‐color nearest‐neighbor totalistic rule {n,{k,1},r} k‐color range-r totalistic rule {n,{k,{wt1,wt2,…}},r} rule in which neighbor i is assigned weight wti {n,kspec,{{off1},{off2},…,{offs}}} rule with neighbors at specified offsets {lhs1->rhs1,lhs2->rhs2,…} explicit replacements for lists of neighbors {fun,{},rspec} rule obtained by applying function fun to each neighbor list
Specifying rules for onedimensional cellular automata.
In the simplest cases, a cellular automaton allows k possible values or "colors" for each cell, and has rules that involve up to r neighbors on each side. The digits of the "rule number" n then specify what the color of a new cell should be for each possible configuration of the neighborhood.
This evolves a single neighborhood for 1 step:
Here are the 8 possible neighborhoods for a , cellular automaton:
This shows the new color of the center cell for each of the 8 neighborhoods:
For rule 30, this sequence corresponds to the base2 digits of the number 30:
This runs the general , rule with rule number 921408:
For a general cellular automaton rule, each digit of the rule number specifies what color a different possible neighborhood of cells should yield. To find out which digit corresponds to which neighborhood, one effectively treats the cells in a neighborhood as digits in a number. For an cellular automaton, the number is obtained from the list of elements neig in the neighborhood by neig.{k^2,k,1}.
It is sometimes convenient to consider totalistic cellular automata, in which the new value of a cell depends only on the total of the values in its neighborhood. One can specify totalistic cellular automata by rule numbers or "codes" in which each digit refers to neighborhoods with a given total value, obtained for example from neig.{1,1,1}.
In general, CellularAutomaton allows one to specify rules using any sequence of weights. Another choice sometimes convenient is {k,1,k}, which yields outer totalistic rules.
This runs the , totalistic rule with code number 867:
Rules with range involve all cells with offsets through . Sometimes it is convenient to think about rules that involve only cells with specific offsets. You can do this by replacing a single with a list of offsets.
Any cellular automaton rule can be thought of as corresponding to a Boolean function. In the simplest case, basic Boolean functions like And or Nor take two arguments. These are conveniently specified in a cellular automaton rule as being at offsets {{0},{1}}. Note that for compatibility with handling higherdimensional cellular automata, offsets must always be given in lists, even for onedimensional cellular automata.
This generates the truth table for 2cellneighborhood rule number 7, which turns out to be the Boolean function Nand:
Rule numbers provide a highly compact way to specify cellular automaton rules. But sometimes it is more convenient to specify rules by giving an explicit function that should be applied to each possible neighborhood.
This runs an additive cellular automaton whose rule adds all values in each neighborhood modulo 4:
The function is given the step number as a second argument:
When you specify rules by functions, the values of cells need not be integers:
They can even be symbolic:
 CellularAutomaton[rnum,init,t] evolve for t steps, keeping all steps CellularAutomaton[rnum,init,{{t}}] evolve for t steps, keeping only the last step CellularAutomaton[rnum,init,{spect}] keep only steps specified by spect CellularAutomaton[rnum,init] evolve rule for one step, giving only the last step
Selecting which steps to keep.
This runs rule 30 for 5 steps, keeping only the last step:
This keeps the last 2 steps:
This gives one step:
The step specification spect works very much like taking elements from a list with Take. One difference, though, is that the initial condition for the cellular automaton is considered to be step 0. Note that any step specification of the form {} must be enclosed in an additional list.
 u steps 0 through u {u} step u {u1,u2} steps u1 through u2 {u1,u2,du} steps u1, u1+du, …
Cellular automaton step specifications.
This evolves for 100 steps, but keeps only every other step:
 CellularAutomaton[rnum,init,t] keep all steps, and all relevant cells CellularAutomaton[rnum,init,{spect,specx}] keep only specified steps and cells
Selecting steps and cells to keep.
Much as you can specify which steps to keep in a cellular automaton evolution, so also you can specify which cells to keep. If you give an initial condition such as {{a1,a2,},blist}, then rd is taken to have offset 0 for the purpose of specifying which cells to keep.
 All all cells that can be affected by the specified initial condition Automatic all cells in the region that differs from the background (default) 0 cell aligned with beginning of aspec x cells at offsets up to x on the right -x cells at offsets up to x on the left {x} cell at offset x to the right {-x} cell at offset x to the left {x1,x2} cells at offsets x1 through x2 {x1,x2,dx} cells x1, x1+dx, …
Cellular automaton cell specifications.
This keeps all steps, but drops cells at offsets more than 20 on the left:
This keeps just the center column of cells:
If you give an initial condition such as {{a1,a2,},blist}, then CellularAutomaton will always effectively do the cellular automaton as if there were an infinite number of cells. By using a specx such as {x1,x2} you can tell CellularAutomaton to include only cells at specific offsets x1 through x2 in its output. CellularAutomaton by default includes cells out just far enough that their values never simply stay the same as in the background blist.
In general, given a cellular automaton rule with range , cells out to distance on each side could in principle be affected in the evolution of the system. With specx being All, all these cells are included; with the default setting of Automatic, cells whose values effectively stay the same as in blist are trimmed off.
By default, only the parts that are not constant black are kept:
Using All for specx includes all cells that could be affected by a cellular automaton with this range:
CellularAutomaton generalizes quite directly to any number of dimensions. Above two dimensions, however, totalistic and other special types of rules tend to be more useful, since the number of entries in the rule table for a general rule rapidly becomes astronomical.
 {n,k,{r1,r2,…,rd}} ‐dimensional rule with neighborhood {n,{k,1},{1,1}} two‐dimensional 9‐neighbor totalistic rule {n,{k,{{0,1,0},{1,1,1},{0,1,0}}},{1,1}} two‐dimensional 5‐neighbor totalistic rule {n,{k,{{0,k,0},{k,1,k},{0,k,0}}},{1,1}} two‐dimensional 5‐neighbor outer totalistic rule
Higherdimensional rule specifications.
This is the rule specification for the twodimensional 9neighbor totalistic cellular automaton with code 797:
This gives steps 0 and 1 in its evolution:
This shows step 70 in the evolution:
This shows all steps in a slice along the axis: