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.

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 .

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.

Note that you can extract the elements in the ^th "column" of a multidimensional list using list[[All,i]].

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.

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 .

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.

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.

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.

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 .

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.

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 BernoulliDistribution[p] 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 NegativeBinomialDistribution[n,p] 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 BetaNegativeBinomialDistribution[α,β,n] is a similar mixture of the beta and negative binomial distributions.

The geometric distribution GeometricDistribution[p] 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,n_succ,n_tot] 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 n_tot with n_succ potential successes.

The discrete uniform distribution DiscreteUniformDistribution[{i_min,i_max}] represents an experiment with multiple equally probable outcomes represented by integers i_min through i_max.

The Poisson distribution PoissonDistribution[μ] 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 LogSeriesDistribution[θ]. 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 ZipfDistribution[ρ], sometimes referred to as the zeta distribution, was first used in linguistics and its use has been extended to model rare events.

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".

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.

The lognormal distribution LogNormalDistribution[μ,σ] 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 HalfNormalDistribution[θ] is proportional to the distribution NormalDistribution[0,1/(θ Sqrt[2/π])] limited to the domain .

The inverse Gaussian distribution InverseGaussianDistribution[μ,λ], sometimes called the Wald distribution, is the distribution of first passage times in Brownian motion with positive drift.

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 InverseChiSquareDistribution[ν]. A scaled inverse distribution with degrees of freedom and scale can be given as InverseChiSquareDistribution[ν,ξ]. 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 NoncentralChiSquareDistribution[ν,λ]. 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 NoncentralStudentTDistribution[ν,δ] 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.

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.

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 ExponentialDistribution[λ], 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 ChiDistribution[ν] is followed by the square root of a random variable. For , the distribution is identical to HalfNormalDistribution[θ] with . For , the distribution is identical to the Rayleigh distribution RayleighDistribution[σ] with . For , the distribution is identical to the Maxwell–Boltzmann distribution MaxwellDistribution[σ] 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 ExtremeValueDistribution[α,β] 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 log‐Weibull distribution because of logarithmic relationships between an extreme value-distributed random variable and a properly shifted and scaled Weibull-distributed random variable.

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],xdist] 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.

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.

The data argument of FindClusters can be a list of data elements, associations, or rules indexing elements and labels.

FindClusters works for a variety of data types, including numerical, textual, and image, as well as Boolean vectors, dates and times. All data elements e_i must have the same dimensions.

The rule-based data syntax allows for clustering data elements and returning labels for those elements.

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.

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.

With the default settings, FindClusters has found the four clusters of points.

You can also direct FindClusters to find a specific number of clusters.

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[e_i,e_j] are larger. The function f can be any appropriate distance or dissimilarity function. A dissimilarity function f satisfies the following:

If the e_i are vectors of numbers, FindClusters by default uses a squared Euclidean distance. If the e_i 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 e_i are strings, FindClusters by default uses a distance function based on the number of point changes needed to get from one string to another.

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.

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 Damerau–Levenshtein distance counts the number of deletions, insertions, substitutions, and transpositions, while the Hamming distance counts only the number of substitutions.

The Method option can be used to specify different methods of clustering.

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.

Additional Method suboptions are available to allow for more control over the clustering. Available suboptions depend on the Method chosen.

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.

The "NeighborsNumber" suboption can be used to control the number of neighbors in the neighborhood of a generic point.

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.

With Method->{"Agglomerate",ClusterDissimilarityFunction->f}, the specified linkage function f is used for agglomerative clustering.

Linkage methods determine this intercluster dissimilarity, or fusion level, given the dissimilarities between member elements.

With ClusterDissimilarityFunction->f, 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 d_ik, d_jk, d_ij, n_i, n_j, and n_k, where d is the distance between clusters and n is the number of elements in a cluster.

The CriterionFunction option can be used to select both the method to use and the best number of clusters.

Nearest is used to find elements in a list that are closest to a given data point.

Nearest works with numeric lists, tensors, or a list of strings.

The rule-based data syntax lets you use nearest elements to return their labels.

If Nearest is to be applied repeatedly to the same numerical data, you can get significant performance gains by first generating a NearestFunction.

For numerical data, by default Nearest uses the EuclideanDistance. For strings, EditDistance is used.

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.

One common way of picking out "signals" in numerical data is to find the Fourier transform, or frequency spectrum, of the data.

Note that the Fourier function in the Wolfram Language is defined with the sign convention typically used in the physical sciences—opposite to the one often used in electrical engineering. "Discrete Fourier Transforms" gives more details.

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.

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 takes a list of functions, and uses a definite and efficient procedure to find what linear combination of these functions gives the best least‐squares 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.

By default, both Fit and FindFit produce least‐squares 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 NormFunction->Norm, corresponding to a least‐squares fit.

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.

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 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.

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.

Fitting options relevant to property computations can be passed to FittedModel objects to override defaults.

Typical data for these model-fitting functions takes the same form as data in other fitting functions such as Fit and FindFit.

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.

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 .

Options for model specification and for model analysis are available.

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 NominalVariables->All, the model is an analysis of variance (ANOVA) model. With NominalVariables->{x₁,…,x_i-1,x_i+1,…,x_n} 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.

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".

The properties include basic information about the data, fitted model, and numerous results and diagnostics.

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.

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" 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.

"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.

The Estimate column of these tables is equivalent to "BestFitParameters". The -statistics are the estimates divided by the standard errors. Each ‐value is the two‐sided ‐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 IncludeConstantBasis->True, 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.

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 Durbin–Watson ‐statistic "DurbinWatsonD" is used for testing the existence of a first-order autoregressive process. The ‐statistic is equivalent to , where is the ^th residual.

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.

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 likelihood‐based 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 .

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.

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 .

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.

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.

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 quasi‐likelihood 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.

"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 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.

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".

"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 .

"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 log‐likelihood 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" 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.

"LogLikelihood" is the log‐likelihood for the fitted model. "AIC" and "BIC" are penalized log‐likelihood measures , where is the log‐likelihood 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 log‐likelihood 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 likelihood‐based measure. "CoxSnellPseudoRSquared" is given by . "CraggUhlerPseudoRSquared" is a scaled version of Cox and Snell's measure . "LikelihoodRatioIndex" involves the ratio of log‐likelihoods , 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.

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.