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CompactGeneralizedLinearModel

Compact generalized linear regression model class

Description

CompactGeneralizedLinearModel is a compact version of a full generalized linear regression model object GeneralizedLinearModel. Because a compact model does not store the input data used to fit the model or information related to the fitting process, a CompactGeneralizedLinearModel object consumes less memory than a GeneralizedLinearModel object. You can still use a compact model to predict responses using new input data, but some GeneralizedLinearModel object functions do not work with a compact model.

Creation

Create a CompactGeneralizedLinearModel model from a full, trained GeneralizedLinearModel model by using compact.

fitglm returns CompactGeneralizedLinearModel when you work with tall arrays, and returns GeneralizedLinearModel when you work with in-memory tables and arrays.

Properties

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Coefficient Estimates

This property is read-only.

Covariance matrix of coefficient estimates, specified as a p-by-p matrix of numeric values. p is the number of coefficients in the fitted model, as given by NumCoefficients.

For details, see Coefficient Standard Errors and Confidence Intervals.

Data Types: single | double

This property is read-only.

Coefficient names, specified as a cell array of character vectors, each containing the name of the corresponding term.

Data Types: cell

This property is read-only.

Coefficient values, specified as a table. Coefficients contains one row for each coefficient and these columns:

  • Estimate — Estimated coefficient value

  • SE — Standard error of the estimate

  • tStatt-statistic for a two-sided test with the null hypothesis that the coefficient is zero

  • pValuep-value for the t-statistic

Use anova (only for a linear regression model) or coefTest to perform other tests on the coefficients. Use coefCI to find the confidence intervals of the coefficient estimates.

To obtain any of these columns as a vector, index into the property using dot notation. For example, obtain the estimated coefficient vector in the model mdl:

beta = mdl.Coefficients.Estimate

Data Types: table

This property is read-only.

Number of model coefficients, specified as a positive integer. NumCoefficients includes coefficients that are set to zero when the model terms are rank deficient.

Data Types: double

This property is read-only.

Number of estimated coefficients in the model, specified as a positive integer. NumEstimatedCoefficients does not include coefficients that are set to zero when the model terms are rank deficient. NumEstimatedCoefficients is the degrees of freedom for regression.

Data Types: double

Summary Statistics

This property is read-only.

Deviance of the fit, specified as a numeric value. The deviance is useful for comparing two models when one model is a special case of the other model. The difference between the deviance of the two models has a chi-square distribution with degrees of freedom equal to the difference in the number of estimated parameters between the two models. For more information, see Deviance.

Data Types: single | double

This property is read-only.

Degrees of freedom for the error (residuals), equal to the number of observations minus the number of estimated coefficients, specified as a positive integer.

Data Types: double

This property is read-only.

Scale factor of the variance of the response, specified as a numeric scalar.

If the 'DispersionFlag' name-value pair argument of fitglm or stepwiseglm is true, then the function estimates the Dispersion scale factor in computing the variance of the response. The variance of the response equals the theoretical variance multiplied by the scale factor.

For example, the variance function for the binomial distribution is p(1–p)/n, where p is the probability parameter and n is the sample size parameter. If Dispersion is near 1, the variance of the data appears to agree with the theoretical variance of the binomial distribution. If Dispersion is larger than 1, the data set is “overdispersed” relative to the binomial distribution.

Data Types: double

This property is read-only.

Flag to indicate whether fitglm used the Dispersion scale factor to compute standard errors for the coefficients in Coefficients.SE, specified as a logical value. If DispersionEstimated is false, fitglm used the theoretical value of the variance.

  • DispersionEstimated can be false only for the binomial and Poisson distributions.

  • Set DispersionEstimated by setting the 'DispersionFlag' name-value pair argument of fitglm or stepwiseglm.

Data Types: logical

Penalty for the likelihood estimate, specified as "none" or "jeffreys-prior".

  • "none" — The likelihood estimate is not penalized during model fitting.

  • "jeffreys-prior" — The likelihood estimate is penalized using the Jeffreys prior.

For logistic models, setting LikelihoodPenalty to "jeffreys-prior" is called Firth's regression. To reduce the coefficient estimate bias when you have a small number of samples, or when you are performing binomial (logistic) regression on a separable data set, set LikelihoodPenalty to "jeffreys-prior" during training.

Example: LikelihoodPenalty="jeffreys-prior"

Data Types: char | string

This property is read-only.

Loglikelihood of the model distribution at the response values, specified as a numeric value. The mean is fitted from the model, and other parameters are estimated as part of the model fit.

Data Types: single | double

This property is read-only.

Criterion for model comparison, specified as a structure with these fields:

  • AIC — Akaike information criterion. AIC = –2*logL + 2*m, where logL is the loglikelihood and m is the number of estimated parameters.

  • AICc — Akaike information criterion corrected for the sample size. AICc = AIC + (2*m*(m + 1))/(n – m – 1), where n is the number of observations.

  • BIC — Bayesian information criterion. BIC = –2*logL + m*log(n).

  • CAIC — Consistent Akaike information criterion. CAIC = –2*logL + m*(log(n) + 1).

Information criteria are model selection tools that you can use to compare multiple models fit to the same data. These criteria are likelihood-based measures of model fit that include a penalty for complexity (specifically, the number of parameters). Different information criteria are distinguished by the form of the penalty.

When you compare multiple models, the model with the lowest information criterion value is the best-fitting model. The best-fitting model can vary depending on the criterion used for model comparison.

To obtain any of the criterion values as a scalar, index into the property using dot notation. For example, obtain the AIC value aic in the model mdl:

aic = mdl.ModelCriterion.AIC

Data Types: struct

This property is read-only.

R-squared value for the model, specified as a structure with five fields.

FieldDescriptionEquation
OrdinaryOrdinary (unadjusted) R-squared

ROrdinary2=1SSESST

SSE is the sum of squared errors, and SST is the total sum of squared deviations of the response vector from the mean of the response vector.

AdjustedR-squared adjusted for the number of coefficients

RAdjusted2=1SSESSTN1DFE

N is the number of observations (NumObservations), and DFE is the degrees of freedom for the error (residuals).

LLRLoglikelihood ratio

RLLR2=1LL0

L is the loglikelihood of the fitted model (LogLikelihood), and L0 is the loglikelihood of a model that includes only a constant term. R2LLR is the McFadden pseudo R-squared value [1] for logistic regression models.

DevianceDeviance R-squared

RDeviance2=1DD0

D is the deviance of the fitted model (Deviance), and D0 is the deviance of a model that includes only a constant term.

AdjGeneralizedAdjusted generalized R-squared

RAdjGeneralized2=1exp(2(L0L)N)1exp(2L0N)

R2AdjGeneralized is the Nagelkerke adjustment [2] to a formula proposed by Maddala [3], Cox and Snell [4], and Magee [5] for logistic regression models.

To obtain any of these values as a scalar, index into the property using dot notation. For example, to obtain the adjusted R-squared value in the model mdl, enter:

r2 = mdl.Rsquared.Adjusted

Data Types: struct

This property is read-only.

Sum of squared errors (residuals), specified as a numeric value. If the model was trained with observation weights, the sum of squares in the SSE calculation is the weighted sum of squares.

Data Types: single | double

This property is read-only.

Regression sum of squares, specified as a numeric value. SSR is equal to the sum of the squared deviations between the fitted values and the mean of the response. If the model was trained with observation weights, the sum of squares in the SSR calculation is the weighted sum of squares.

Data Types: single | double

This property is read-only.

Total sum of squares, specified as a numeric value. SST is equal to the sum of squared deviations of the response vector y from the mean(y). If the model was trained with observation weights, the sum of squares in the SST calculation is the weighted sum of squares.

Data Types: single | double

Input Data

This property is read-only.

Generalized distribution information, specified as a structure with the fields described in this table.

FieldDescription
NameName of the distribution: 'normal', 'binomial', 'poisson', 'gamma', or 'inverse gaussian'
DevianceFunctionFunction that computes the components of the deviance as a function of the fitted parameter values and the response values
VarianceFunctionFunction that computes the theoretical variance for the distribution as a function of the fitted parameter values. When DispersionEstimated is true, the software multiplies the variance function by Dispersion in the computation of the coefficient standard errors.

Data Types: struct

This property is read-only.

Model information, specified as a LinearFormula object.

Display the formula of the fitted model mdl using dot notation:

mdl.Formula

This property is read-only.

Number of observations the fitting function used in fitting, specified as a positive integer. NumObservations is the number of observations supplied in the original table, dataset, or matrix, minus any excluded rows (set with the 'Exclude' name-value pair argument) or rows with missing values.

Data Types: double

This property is read-only.

Number of predictor variables used to fit the model, specified as a positive integer.

Data Types: double

This property is read-only.

Number of variables in the input data, specified as a positive integer. NumVariables is the number of variables in the original table or dataset, or the total number of columns in the predictor matrix and response vector.

NumVariables also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: double

This property is read-only.

Names of predictors used to fit the model, specified as a cell array of character vectors.

Data Types: cell

This property is read-only.

Response variable name, specified as a character vector.

Data Types: char

This property is read-only.

Information about variables contained in Variables, specified as a table with one row for each variable and the columns described in this table.

ColumnDescription
ClassVariable class, specified as a cell array of character vectors, such as 'double' and 'categorical'
Range

Variable range, specified as a cell array of vectors

  • Continuous variable — Two-element vector [min,max], the minimum and maximum values

  • Categorical variable — Vector of distinct variable values

InModelIndicator of which variables are in the fitted model, specified as a logical vector. The value is true if the model includes the variable.
IsCategoricalIndicator of categorical variables, specified as a logical vector. The value is true if the variable is categorical.

VariableInfo also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: table

This property is read-only.

Names of variables, specified as a cell array of character vectors.

  • If the fit is based on a table or dataset, this property provides the names of the variables in the table or dataset.

  • If the fit is based on a predictor matrix and response vector, VariableNames contains the values specified by the 'VarNames' name-value pair argument of the fitting method. The default value of 'VarNames' is {'x1','x2',...,'xn','y'}.

VariableNames also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: cell

Object Functions

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fevalPredict responses of generalized linear regression model using one input for each predictor
predictPredict responses of generalized linear regression model
randomSimulate responses with random noise for generalized linear regression model
coefCIConfidence intervals of coefficient estimates of generalized linear regression model
coefTestLinear hypothesis test on generalized linear regression model coefficients
devianceTestAnalysis of deviance for generalized linear regression model
partialDependenceCompute partial dependence
plotPartialDependenceCreate partial dependence plot (PDP) and individual conditional expectation (ICE) plots
plotSlicePlot of slices through fitted generalized linear regression surface
gatherGather properties of Statistics and Machine Learning Toolbox object from GPU

Examples

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Fit a generalized linear regression model to data and reduce the size of a full, fitted model by discarding the sample data and some information related to the fitting process.

Load the largedata4reg data set, which contains 15,000 observations and 45 predictor variables.

load largedata4reg

Fit a generalized linear regression model to the data using the first 15 predictor variables.

mdl = fitglm(X(:,1:15),Y);

Compact the model.

compactMdl = compact(mdl);

The compact model discards the original sample data and some information related to the fitting process, so it uses less memory than the full model.

Compare the size of the full model mdl and the compact model compactMdl.

vars = whos('compactMdl','mdl');
[vars(1).bytes,vars(2).bytes]
ans = 1×2

       17060     4384077

The compact model consumes less memory than the full model.

More About

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References

[1] McFadden, Daniel. "Conditional logit analysis of qualitative choice behavior." in Frontiers in Econometrics, edited by P. Zarembka,105–42. New York: Academic Press, 1974.

[2] Nagelkerke, N. J. D. "A Note on a General Definition of the Coefficient of Determination." Biometrika 78, no. 3 (1991): 691–92.

[3] Maddala, Gangadharrao S. Limited-Dependent and Qualitative Variables in Econometrics. Econometric Society Monographs. New York, NY: Cambridge University Press, 1983.

[4] Cox, D. R., and E. J. Snell. Analysis of Binary Data. 2nd ed. Monographs on Statistics and Applied Probability 32. London; New York: Chapman and Hall, 1989.

[5] Magee, Lonnie. "R 2 Measures Based on Wald and Likelihood Ratio Joint Significance Tests." The American Statistician 44, no. 3 (August 1990): 250–53.

Extended Capabilities

Version History

Introduced in R2016b