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 use 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

`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.

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 test 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

`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:

Ordinary — Ordinary (unadjusted) R-squared
Adjusted — R-squared adjusted for the number of coefficients
LLR — Loglikelihood ratio
Deviance — Deviance
AdjGeneralized — Adjusted generalized R-squared

The R-squared value is the proportion of the total sum of squares explained by the model. The ordinary R-squared value relates to the SSR and SST properties:

Rsquared = SSR/SST

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

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.

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. The regression sum of squares is equal to the sum of squared deviations of the fitted values from their mean.

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. The total sum of squares is equal to the sum of squared deviations of the response vector y from the mean(y).

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`, `Dispersion` multiplies the variance function 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 `''`.
`LinkFunction`	Function f that defines the link function, specified as a function handle
`DevianceFunction`	Derivative of f, specified as a function handle
`VarianceFunction`	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

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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 linear or generalized linear regression model

Examples

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Compact Generalized Linear Regression Model

Open Live Script

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

       15518     4382502

The compact model consumes less memory than the full model.

More About

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Deviance

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 the 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

$- 2 (\log L (b_{1}, y) - \log L (b_{S}, y)),$

where b₁ and b_s contain the estimated parameters for the model M₁ and the saturated model, respectively. The deviance has a chi-square 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 the deviances D₁ and D₂ of the two models. The difference of the deviances is

$\begin{array}{l} D = D_{2} - D_{1} = - 2 (\log L (b_{2}, y) - \log L (b_{S}, y)) + 2 (\log L (b_{1}, y) - \log L (b_{S}, y)) \\ = - 2 (\log L (b_{2}, y) - \log L (b_{1}, y)) . \end{array}$

Asymptotically, the difference D has a chi-square 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-square 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.

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, the following restrictions apply.
- Code generation does not support categorical predictors. You cannot supply training data in a table that contains a logical vector, character array, categorical array, string array, or cell array of character vectors. Also, you cannot use the 'CategoricalVars' name-value pair argument. To include categorical predictors in a model, preprocess the categorical predictors by using dummyvar before fitting the model.
- The Link, Derivative, and Inverse fields of the 'Link' name-value pair argument cannot be 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 following object functions fully support GPU arrays:
The following object functions support model objects fitted with GPU array input arguments:

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

Documentation

CompactGeneralizedLinearModel

Description

Creation

Properties

Coefficient Estimates

CoefficientCovariance — Covariance matrix of coefficient estimates numeric matrix

CoefficientNames — Coefficient names cell array of character vectors

Coefficients — Coefficient values table

NumCoefficients — Number of model coefficients positive integer

NumEstimatedCoefficients — Number of estimated coefficients positive integer

Summary Statistics

Deviance — Deviance of fit numeric value

DFE — Degrees of freedom for error positive integer

Dispersion — Scale factor of variance of response numeric scalar

DispersionEstimated — Flag to indicate use of dispersion scale factor logical value

LogLikelihood — Loglikelihood numeric value

ModelCriterion — Criterion for model comparison structure

Rsquared — R-squared value for model structure

SSE — Sum of squared errors numeric value

SSR — Regression sum of squares numeric value

SST — Total sum of squares numeric value

Input Data

Distribution — Generalized distribution information structure

Formula — Model information LinearFormula object

Link — Link function structure

NumObservations — Number of observations positive integer

NumPredictors — Number of predictor variables positive integer

NumVariables — Number of variables positive integer

PredictorNames — Names of predictors used to fit model cell array of character vectors

ResponseName — Response variable name character vector

VariableInfo — Information about variables table

VariableNames — Names of variables cell array of character vectors

Object Functions

Predict Responses

Evaluate Generalized Linear Model

Visualize Generalized Linear Model and Summary Statistics

Gather Properties of Generalized Linear Model

Examples

Compact Generalized Linear Regression Model

More About

Deviance

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

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

See Also

Topics

Statistics and Machine Learning Toolbox Documentation

Support

`CoefficientCovariance` — Covariance matrix of coefficient estimates
numeric matrix

`CoefficientNames` — Coefficient names
cell array of character vectors

`Coefficients` — Coefficient values
table

`NumCoefficients` — Number of model coefficients
positive integer

`NumEstimatedCoefficients` — Number of estimated coefficients
positive integer

`Deviance` — Deviance of fit
numeric value

`DFE` — Degrees of freedom for error
positive integer

`Dispersion` — Scale factor of variance of response
numeric scalar

`DispersionEstimated` — Flag to indicate use of dispersion scale factor
logical value

`LogLikelihood` — Loglikelihood
numeric value

`ModelCriterion` — Criterion for model comparison
structure

`Rsquared` — R-squared value for model
structure

`SSE` — Sum of squared errors
numeric value

`SSR` — Regression sum of squares
numeric value

`SST` — Total sum of squares
numeric value

`Distribution` — Generalized distribution information
structure

`Formula` — Model information
`LinearFormula` object

`Link` — Link function
structure

`NumObservations` — Number of observations
positive integer

`NumPredictors` — Number of predictor variables
positive integer

`NumVariables` — Number of variables
positive integer

`PredictorNames` — Names of predictors used to fit model
cell array of character vectors

`ResponseName` — Response variable name
character vector

`VariableInfo` — Information about variables
table

`VariableNames` — Names of variables
cell array of character vectors

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

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