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Model Building and Assessment

Feature selection, hyperparameter optimization, cross-validation, residual diagnostics, plots

When building a high-quality regression model, it is important to select the right features (or predictors), tune hyperparameters (model parameters not fit to the data), and assess model assumptions through residual diagnostics.

You can tune hyperparameters by iterating between choosing values for them, and cross-validating a model using your choices. This process yields multiple models, and the best model among them can be the one that minimizes the estimated generalization error. For example, to tune an SVM model, choose a set of box constraints and kernel scales, cross-validate a model for each pair of values, and then compare their 10-fold cross-validated mean-squared error estimates.

Certain nonparametric regression functions in Statistics and Machine Learning Toolbox™ additionally offer automatic hyperparameter tuning through Bayesian optimization, grid search, or random search. However, bayesopt, which is the main function to implement Bayesian optimization, is flexible enough for many other applications. For more details, see Bayesian Optimization Workflow.


Regression LearnerTrain regression models to predict data using supervised machine learning


sequentialfsSequential feature selection
relieffImportance of attributes (predictors) using ReliefF algorithm
plotPartialDependenceCreate partial dependence plot (PDP) and individual conditional expectation (ICE) plots
stepwiselm Create linear regression model using stepwise regression
stepwiseglmCreate generalized linear regression model by stepwise regression
bayesoptSelect optimal machine learning hyperparameters using Bayesian optimization
hyperparametersVariable descriptions for optimizing a fit function
crossvalLoss estimate using cross validation
cvpartitionCreate cross validation partition for data
repartitionRepartition data for cross-validation
testTest indices for cross-validation
trainingTraining indices for cross-validation
coefCIConfidence intervals of coefficient estimates of linear model
coefTestLinear hypothesis test on linear regression model coefficients
dwtestDurbin-Watson test of linear model
plotScatter plot or added variable plot of linear model
plotAddedAdded variable plot or leverage plot for linear model
plotAdjustedResponseAdjusted response plot for linear regression model
plotDiagnosticsPlot diagnostics of linear regression model
plotEffectsPlot main effects of each predictor in linear regression model
plotInteractionPlot interaction effects of two predictors in linear regression model
plotResidualsPlot residuals of linear regression model
plotSlicePlot of slices through fitted linear regression surface
coefCIConfidence intervals of coefficient estimates of generalized linear model
coefTestLinear hypothesis test on generalized linear regression model coefficients
devianceTestAnalysis of deviance
plotDiagnosticsPlot diagnostics of generalized linear regression model
plotResidualsPlot residuals of generalized linear regression model
plotSlicePlot of slices through fitted generalized linear regression surface
coefCIConfidence intervals of coefficient estimates of nonlinear regression model
coefTestLinear hypothesis test on nonlinear regression model coefficients
plotDiagnosticsPlot diagnostics of nonlinear regression model
plotResidualsPlot residuals of nonlinear regression model
plotSlicePlot of slices through fitted nonlinear regression surface
linhyptestLinear hypothesis test

Using Objects

BayesianOptimizationBayesian optimization results
optimizableVariableVariable description for bayesopt or other optimizers
cvpartitionData partitions for cross validation


Regression Learner App Workflow

Train Regression Models in Regression Learner App

Workflow for training, comparing and improving regression models, including automated, manual, and parallel training.

Choose Regression Model Options

In Regression Learner, automatically train a selection of models, or compare and tune options of linear regression models, regression trees, support vector machines, Gaussian process regression models, and ensembles of regression trees.

Feature Selection and Feature Transformation Using Regression Learner App

Identify useful predictors using plots, manually select features to include, and transform features using PCA in Regression Learner.

Assess Model Performance in Regression Learner App

Compare model statistics and visualize results.

Feature Selection

Feature Selection

Learn about feature selection algorithms, such as sequential feature selection.

Hyperparameter Optimization

Bayesian Optimization Workflow

Perform Bayesian optimization using a fit function or by calling bayesopt directly.

Variables for a Bayesian Optimization

Create variables for Bayesian optimization.

Bayesian Optimization Objective Functions

Create the objective function for Bayesian optimization.

Constraints in Bayesian Optimization

Set different types of constraints for Bayesian optimization.

Optimize a Boosted Regression Ensemble

Minimize cross-validation loss of a regression ensemble.

Bayesian Optimization Plot Functions

Visually monitor a Bayesian optimization.

Bayesian Optimization Output Functions

Monitor a Bayesian optimization.

Bayesian Optimization Algorithm

Understand the underlying algorithms for Bayesian optimization.

Parallel Bayesian Optimization

How Bayesian optimization works in parallel.


Implement Cross-Validation Using Parallel Computing

Speed up cross-validation using parallel computing.

Linear Model Diagnostics

Interpret Linear Regression Results

Display and interpret linear regression output statistics.

Examine Quality and Adjust the Fitted Model

Linear Regression with Interaction Effects

Construct and analyze a linear regression model with interaction effects and interpret the results.

Summary of Output and Diagnostic Statistics

F-statistic and t-statistic

In linear regression, the F-statistic is the test statistic for the analysis of variance (ANOVA) approach to test the significance of the model or the components in the model. The t-statistic is useful for making inferences about the regression coefficients

Coefficient of Determination (R-Squared)

Coefficient of determination (R-squared) indicates the proportionate amount of variation in the response variable y explained by the independent variables X in the linear regression model.

Coefficient Standard Errors and Confidence Intervals

Estimated coefficient variances and covariances capture the precision of regression coefficient estimates.


Residuals are useful for detecting outlying y values and checking the linear regression assumptions with respect to the error term in the regression model.

Durbin-Watson Test

The Durbin-Watson test assesses whether there is autocorrelation among the residuals or not.

Cook’s Distance

Cook's distance is useful for identifying outliers in the X values (observations for predictor variables).

Hat Matrix and Leverage

The hat matrix provides a measure of leverage.

Delete-1 Statistics

Delete-1 change in covariance (covratio) identifies the observations that are influential in the regression fit.

Generalized Linear Model Diagnostics

Examine Quality and Adjust the Fitted Model

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