LinearModelFit

LinearModelFit[{y1,y2,},{f1,f2,},x]
constructs a linear model of the form that fits the yi for successive x values 1, 2, .

LinearModelFit[{{x11,x12,,y1},{x21,x22,,y2},},{f1,f2,},{x1,x2,}]
constructs a linear model of the form where the fi depend on the variables xk.

LinearModelFit[{m,v}]
constructs a linear model from the design matrix m and response vector v.

Details and OptionsDetails and Options

  • LinearModelFit returns a symbolic FittedModel object to represent the linear model it constructs. The properties and diagnostics of the model can be obtained from model["property"].
  • The value of the best-fit function from LinearModelFit at a particular point x1, can be found from model[x1,].
  • With data in the form {{x_(11),x_(12),... ,y_(1)},{x_(21),x_(22),... ,y_(2)},...}, the number of coordinates xi1, xi2, should equal the number of variables xi.
  • Data in the form {y1,y2,} is equivalent to data in the form {{1,y1},{2,y2},}.
  • LinearModelFit produces a linear model of the form under the assumption that the original are independent normally distributed with mean and common standard deviation.
  • LinearModelFit takes the following options:
  • ConfidenceLevel95/100confidence level to use for parameters and predictions
    IncludeConstantBasisTruewhether to include a constant basis function
    LinearOffsetFunctionNoneknown offset in the linear predictor
    NominalVariablesNonevariables considered as nominal or categorical
    VarianceEstimatorFunctionAutomaticfunction for estimating the error variance
    WeightsAutomaticweights for data elements
    WorkingPrecisionAutomaticprecision used in internal computations
  • With the setting IncludeConstantBasis->False, a model of the form is fitted. The option IncludeConstantBasis is ignored if the design matrix is specified in the input.
  • With the setting LinearOffsetFunction->h, a model of the form is fitted.
  • With ConfidenceLevel->p, probability-p confidence intervals are computed for parameter and prediction intervals.
  • With the setting Weights->{w1,w2,}, the error variance for yi is assumed to be . By default, unit weights are used.
  • With the setting VarianceEstimatorFunction->f, the variance is estimated by f[res,w], where res={y1-1,y2-2,} is the list of residuals and w={w1,w2,} is the list of weights for the measurements yi.
  • Using VarianceEstimatorFunction->(1&) and Weights->{1/Δy12,1/Δy22,}, Δyi is treated as the known uncertainty of measurement yi and parameter standard errors are effectively computed only from the weights.
  • Properties related to data and the fitted function obtained using model["property"] include:
  • "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
  • Types of residuals include:
  • "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
  • Properties related to the sum of squared errors include:
  • "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"estimated standard deviation divided by the response mean
    "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 and diagnostics for parameter estimates include:
  • "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
    "ParameterTable"table of fitted parameter information
    "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
    "ParameterTableEntries"unformatted array of values from the table
    "ParameterTStatistics"statistics for parameter estimates
    "VarianceInflationFactors"list of inflation factors for the estimated parameters
  • Properties related to influence measures include:
  • "BetaDifferences"DFBETAS measures of influence on parameter values
    "CatcherMatrix"catcher matrix
    "CookDistances"list of Cook distances
    "CovarianceRatios"COVRATIO measures of observation influence
    "DurbinWatsonD"DurbinWatson 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 of predicted values include:
  • "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 that measure goodness of fit include:
  • "AdjustedRSquared" adjusted for the number of model parameters
    "AIC"Akaike Information Criterion
    "AICc"finite sample corrected AIC
    "BIC"Bayesian Information Criterion
    "RSquared"coefficient of determination
  • In LinearModelFit[{m,v }], the design matrix m is formed from the values of basis functions fi at data points in the form {{f1,f2,},{f1,f2,},}. The response vector v is the list of responses {y1,y2,}.
  • For a design matrix m and response vector v, the model is , where is the vector of parameters to be estimated.
  • When a design matrix is used, the basis functions fi can be specified using the form LinearModelFit[{m,v},{f1,f2,}].

ExamplesExamplesopen allclose all

Basic Examples  (1)Basic Examples  (1)

Fit a linear model to some data:

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Obtain the functional form:

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Evaluate the model at a point:

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Visualize the fitted function with the data:

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Extract information about the fitting:

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Plot the residuals:

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Introduced in 2008
(7.0)
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