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

Coefficient Estimates

CoefficientCovariance — Covariance matrix of coefficient estimates

numeric matrix

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

CoefficientNames — Coefficient names

cell array of character vectors

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

Coefficients — Coefficient values

table

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

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

• pValue — p-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

NumCoefficients — Number of model coefficients

positive integer

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

NumEstimatedCoefficients — Number of estimated coefficients

positive integer

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

Deviance — Deviance of fit

numeric value

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

DFE — Degrees of freedom for error

positive integer

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

Dispersion — Scale factor of variance of response

numeric scalar

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

DispersionEstimated — Flag to indicate use of dispersion scale factor

logical value

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

LikelihoodPenalty — Penalty for likelihood estimate

"none" (default) | "jeffreys-prior"

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

LogLikelihood — Loglikelihood

numeric value

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

ModelCriterion — Criterion for model comparison

structure

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

Rsquared — R-squared value for model

structure

This property is read-only.

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

Field	Description	Equation
Ordinary	Ordinary (unadjusted) R-squared	$$R_{\text{Ordinary}}^2=1-\frac{\text{SSE}}{\text{SST}}$$ 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.
Adjusted	R-squared adjusted for the number of coefficients	$$R_{\text{Adjusted}}^2=1-\frac{\text{SSE}}{\text{SST}}\cdot \frac{N-1}{\text{DFE}}$$ N is the number of observations (NumObservations), and DFE is the degrees of freedom for the error (residuals).
LLR	Loglikelihood ratio	$$R_{\text{LLR}}^2=1-\frac{L}{L_0}$$ 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.
Deviance	Deviance R-squared	$$R_{\text{Deviance}}^2=1-\frac{D}{D_0}$$ D is the deviance of the fitted model (Deviance), and D0 is the deviance of a model that includes only a constant term.
AdjGeneralized	Adjusted generalized R-squared	$$R_{\text{AdjGeneralized}}^2=\frac{1-\exp \left(\frac{2\left(L_0-L\right)}{N}\right)}{1-\exp \left(\frac{2L_0}{N}\right)}$$ 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

SSE — Sum of squared errors

numeric value

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

SSR — Regression sum of squares

numeric value

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

SST — Total sum of squares

numeric value

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

Distribution — Generalized distribution information

structure

This property is read-only.

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

Field	Description
Name	Name of the distribution: 'normal', 'binomial', 'poisson', 'gamma', or 'inverse gaussian'
DevianceFunction	Function that computes the components of the deviance as a function of the fitted parameter values and the response values
VarianceFunction	Function 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

Formula — Model information

LinearFormula object

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

Link — Link function

structure

This property is read-only.

Link function, specified as a structure with the fields described in this table.

Field	Description
Name	Name of the link function, specified as a character vector. If you specify the link function using a function handle, then Name is ''.
Link	Function f that defines the link function, specified as a function handle
Derivative	Derivative of f, specified as a function handle
Inverse	Inverse of f, specified as a function handle

The link function is a function f that links the distribution parameter μ to the fitted linear combination Xb of the predictors:

f(μ) = Xb.

Data Types: struct

NumObservations — Number of observations

positive integer

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

NumPredictors — Number of predictor variables

positive integer

This property is read-only.

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

Data Types: double

NumVariables — Number of variables

positive integer

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

PredictorNames — Names of predictors used to fit model

cell array of character vectors

This property is read-only.

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

Data Types: cell

ResponseName — Response variable name

character vector

This property is read-only.

Response variable name, specified as a character vector.

Data Types: char

VariableInfo — Information about variables

table

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.

Column	Description
Class	Variable 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
InModel	Indicator of which variables are in the fitted model, specified as a logical vector. The value is true if the model includes the variable.
IsCategorical	Indicator 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

VariableNames — Names of variables

cell array of character vectors

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

Predict Responses

feval	Predict responses of generalized linear regression model using one input for each predictor
predict	Predict responses of generalized linear regression model
random	Simulate responses with random noise for generalized linear regression model

Evaluate Generalized Linear Model

coefCI	Confidence intervals of coefficient estimates of generalized linear regression model
coefTest	Linear hypothesis test on generalized linear regression model coefficients
devianceTest	Analysis of deviance for generalized linear regression model
partialDependence	Compute partial dependence

Visualize Generalized Linear Model and Summary Statistics

plotPartialDependence	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
plotSlice	Plot of slices through fitted generalized linear regression surface

Gather Properties of Generalized Linear Model

gather

Gather properties of Statistics and Machine Learning Toolbox object from GPU

Examples

Compact Generalized Linear Regression Model

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

Deviance

Deviance is a generalization of the residual sum of squares. It measures the goodness of fit compared to a saturated model.

The deviance of a model M₁ is twice the difference between the loglikelihood of the model M₁ and the saturated model M_s. A saturated model is a model with the maximum number of parameters that you can estimate.

For example, if you have n observations (y_i, i = 1, 2, ..., n) with potentially different values for X_i^Tβ, then you can define a saturated model with n parameters. Let L(b,y) denote the maximum value of the likelihood function for a model with the parameters b. Then the deviance of the model M₁ is

$$-2\left(\log L\left(b_1,y\right)-\log L\left(b_S,y\right)\right),$$

where b₁ and b_s contain the estimated parameters for the model M₁ and the saturated model, respectively. The deviance has a chi-squared distribution with n – p degrees of freedom, where n is the number of parameters in the saturated model and p is the number of parameters in the model M₁.

Assume you have two different generalized linear regression models M₁ and M₂, and M₁ has a subset of the terms in M₂. You can assess the fit of the models by comparing their deviances D₁ and D₂. The difference of the deviances is

$$\begin{array}{l}D=D_2-D_1=-2\left(\log L\left(b_2,y\right)-\log L\left(b_S,y\right)\right)+2\left(\log L\left(b_1,y\right)-\log L\left(b_S,y\right)\right)\\ =-2\left(\log L\left(b_2,y\right)-\log L\left(b_1,y\right)\right).\end{array}$$

Asymptotically, the difference D has a chi-squared distribution with degrees of freedom v equal to the difference in the number of parameters estimated in M₁ and M₂. You can obtain the p-value for this test by using 1  —  chi2cdf(D,v).

Typically, you examine D using a model M₂ with a constant term and no predictors. Therefore, D has a chi-squared distribution with p – 1 degrees of freedom. If the dispersion is estimated, the difference divided by the estimated dispersion has an F distribution with p – 1 numerator degrees of freedom and n – p denominator degrees of freedom.

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

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The predict and random functions support code generation.

• When you fit a model by using fitglm or stepwiseglm, you cannot specify Link, Derivative, and Inverse fields of the 'Link' name-value pair argument as anonymous functions. That is, you cannot generate code using a generalized linear model that was created using anonymous functions for links. Instead, define functions for link components.

For more information, see Introduction to Code Generation.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• The object functions of the CompactGeneralizedLinearModel model fully support GPU arrays.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2016b