Documentation

# NonLinearModel class

Nonlinear regression model class

## Description

An object comprising training data, model description, diagnostic information, and fitted coefficients for a nonlinear regression. Predict model responses with the predict or feval methods.

## Construction

Create a NonLinearModel object using fitnlm.

## Properties

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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

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

Data Types: cell

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 test 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

Diagnostic information for the model, specified as a table. Diagnostics can help identify outliers and influential observations. Diagnostics contains the following fields.

FieldMeaningUtility
LeverageDiagonal elements of HatMatrixLeverage indicates to what extent the predicted value for an observation is determined by the observed value for that observation. A value close to 1 indicates that the prediction is largely determined by that observation, with little contribution from the other observations. A value close to 0 indicates the fit is largely determined by the other observations. For a model with P coefficients and N observations, the average value of Leverage is P/N. An observation with Leverage larger than 2*P/N can be regarded as having high leverage.
CooksDistanceCook's measure of scaled change in fitted valuesCooksDistance is a measure of scaled change in fitted values. An observation with CooksDistance larger than three times the mean Cook's distance can be an outlier.
HatMatrixProjection matrix to compute fitted from observed responsesHatMatrix is an N-by-N matrix such that Fitted = HatMatrix*Y, where Y is the response vector and Fitted is the vector of fitted response values.

Data Types: table

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

Fitted (predicted) values based on the input data, specified as a numeric vector. fitnlm attempts to make Fitted as close as possible to the response data.

Data Types: single | double

Model information, specified as a NonLinearFormula object.

Display the formula of the fitted model mdl by using dot notation.

mdl.Formula

Information about the fitting process, specified as a structure with the following fields:

• InitialCoefs — Initial coefficient values (the beta0 vector)

• IterOpts — Options included in the Options name-value pair argument for fitnlm.

Data Types: struct

Log likelihood 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

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

Mean squared error, specified as a numeric value. The mean squared error is an estimate of the variance of the error term in the model.

Data Types: single | double

Number of coefficients in the fitted model, specified as a positive integer. NumCoefficients is the same as NumEstimatedCoefficients for NonLinearModel objects. NumEstimatedCoefficients is equal to the degrees of freedom for regression.

Data Types: double

Number of estimated coefficients in the fitted model, specified as a positive integer. NumEstimatedCoefficients is the same as NumCoefficients for NonLinearModel objects. NumEstimatedCoefficients is equal to the degrees of freedom for regression.

Data Types: double

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

Data Types: double

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

Observation information, specified as an n-by-4 table, where n is equal to the number of rows of input data. The ObservationInfo contains the columns described in this table.

ColumnDescription
WeightsObservation weight, specified as a numeric value. The default value is 1.
ExcludedIndicator of excluded observation, specified as a logical value. The value is true if you exclude the observation from the fit by using the 'Exclude' name-value pair argument.
MissingIndicator of missing observation, specified as a logical value. The value is true if the observation is missing.
SubsetIndicator of whether or not a fitting function uses the observation, specified as a logical value. The value is true if the observation is not excluded or missing, meaning that fitting function uses the observation.

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

w = mdl.ObservationInfo.Weights

Data Types: table

Observation names, specified as a cell array of character vectors containing the names of the observations used in the fit.

• If the fit is based on a table or dataset containing observation names, ObservationNames uses those names.

• Otherwise, ObservationNames is an empty cell array.

Data Types: cell

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

Data Types: cell

Residuals for the fitted model, specified as a table that contains one row for each observation and the columns described in this table.

ColumnDescription
RawObserved minus fitted values
PearsonRaw residuals divided by the root mean squared error (RMSE)
StandardizedRaw residuals divided by their estimated standard deviation
StudentizedRaw residual divided by an independent estimate of the residual standard deviation. The residual for observation i is divided by an estimate of the error standard deviation based on all observations except observation i.

Use plotResiduals to create a plot of the residuals. For details, see Residuals.

Rows not used in the fit because of missing values (in ObservationInfo.Missing) or excluded values (in ObservationInfo.Excluded) contain NaN values.

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

r = mdl.Residuals.Raw

Data Types: table

Response variable name, specified as a character vector.

Data Types: char

Root mean squared error, specified as a numeric value. The root mean squared error is an estimate of the standard deviation of the error term in the model.

Data Types: single | double

Robust fit information, specified as a structure with the following fields:

FieldDescription
WgtFunRobust weighting function, such as 'bisquare' (see robustfit)
TuneValue specified for tuning parameter (can be [])
WeightsVector of weights used in final iteration of robust fit

This structure is empty unless fitnlm constructed the model using robust regression.

Data Types: struct

R-squared value for the model, specified as a structure with two fields:

• Ordinary — Ordinary (unadjusted) 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 = 1 – SSE/SST,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

For details, see Coefficient of Determination (R-Squared).

To obtain either 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:

Data Types: struct

Sum of squared errors (residuals), specified as a numeric value.

The Pythagorean theorem implies

SST = SSE + SSR,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

Data Types: single | double

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.

The Pythagorean theorem implies

SST = SSE + SSR,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

Data Types: single | double

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).

The Pythagorean theorem implies

SST = SSE + SSR,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

Data Types: single | double

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

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

Input data, specified as a table. Variables contains both predictor and response values. If the fit is based on a table or dataset array, Variables contains all the data from the table or dataset array. Otherwise, Variables is a table created from the input data matrix X and response the vector y.

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

Data Types: table

## Methods

 coefCI Confidence intervals of coefficient estimates of nonlinear regression model coefTest Linear hypothesis test on nonlinear regression model coefficients disp Display nonlinear regression model feval Evaluate nonlinear regression model prediction fit (Not Recommended) Fit nonlinear regression model plotDiagnostics Plot diagnostics of nonlinear regression model plotResiduals Plot residuals of nonlinear regression model plotSlice Plot of slices through fitted nonlinear regression surface predict Predict response of nonlinear regression model random Simulate responses for nonlinear regression model

## Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).

## Examples

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Fit a nonlinear regression model for auto mileage based on the carbig data. Predict the mileage of an average car.

Load the sample data. Create a matrix X containing the measurements for the horsepower (Horsepower) and weight (Weight) of each car. Create a vector y containing the response values in miles per gallon (MPG).

X = [Horsepower,Weight];
y = MPG;

Fit a nonlinear regression model.

modelfun = @(b,x)b(1) + b(2)*x(:,1).^b(3) + ...
b(4)*x(:,2).^b(5);
beta0 = [-50 500 -1 500 -1];
mdl = fitnlm(X,y,modelfun,beta0)
mdl =
Nonlinear regression model:
y ~ b1 + b2*x1^b3 + b4*x2^b5

Estimated Coefficients:
Estimate      SE        tStat       pValue
________    _______    ________    ________

b1     -49.383     119.97    -0.41164     0.68083
b2      376.43     567.05     0.66384     0.50719
b3    -0.78193    0.47168     -1.6578    0.098177
b4      422.37     776.02     0.54428     0.58656
b5    -0.24127    0.48325    -0.49926     0.61788

Number of observations: 392, Error degrees of freedom: 387
Root Mean Squared Error: 3.96
F-statistic vs. constant model: 283, p-value = 1.79e-113

Find the predicted mileage of an average auto. Since the sample data contains some missing (NaN) observations, compute the mean using nanmean.

Xnew = nanmean(X)
Xnew = 1×2
103 ×

0.1051    2.9794

MPGnew = predict(mdl,Xnew)
MPGnew = 21.8073