NonLinearModel

Nonlinear regression model

Description

NonLinearModel is a fitted nonlinear regression model object. A regression model describes the relationship between a response and predictors. The nonlinearity in a nonlinear regression model refers to a nonlinear combination of predictors and predictor coefficients.

Use the properties of a NonLinearModel object to investigate a fitted nonlinear regression model. The object properties include information about coefficient estimates, summary statistics, fitting method, and input data. Use the object functions to predict responses and to evaluate and visualize the nonlinear regression model. You can predict model responses using the predict or feval object function.

Creation

Create a NonLinearModel object using fitnlm.

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 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 coefficients

positive integer

This property is read-only.

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

Data Types: double

NumEstimatedCoefficients — Number of estimated coefficients

positive integer

This property is read-only.

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

Summary Statistics

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

Diagnostics — Diagnostic information

table

This property is read-only.

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

Field	Meaning	Utility
Leverage	Diagonal elements of HatMatrix	Leverage for each observation indicates to what extent the fit is determined by the observed predictor values. A value close to 1 indicates that the fit is largely determined by that observation, with little contribution from the other observations. A value close to 0 indicates that 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. A Leverage value greater than 2*P/N indicates high leverage.
CooksDistance	Cook's distance	CooksDistance is a measure of scaled change in fitted values. An observation with CooksDistance greater than three times the mean Cook's distance can be an outlier.
HatMatrix	Projection matrix to compute fitted from observed responses	HatMatrix 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.

Diagnostics contains information that is helpful in finding outliers and influential observations. For more details, see Hat Matrix and Leverage and Cook’s Distance.

Use plotDiagnostics to plot observation diagnostics.

Rows not used in the fit because of missing values (in ObservationInfo.Missing) or excluded values (in ObservationInfo.Excluded) contain NaN values in the CooksDistance columns and zeros in the Leverage and HatMatrix columns.

To obtain any of these columns as an array, index into the property using dot notation. For example, obtain the Cook's distance vector in the model mdl:

CooksDistance = mdl.Diagnostics.CooksDistance;

Data Types: table

Fitted — Fitted response values based on input data

numeric vector

This property is read-only.

Fitted (predicted) response values based on input data, specified as an n-by-1 numeric vector. n is the number of observations in the input data. Use predict to compute predictions for other predictor values, or to compute confidence bounds on Fitted.

Data Types: single | double

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

ModelFitVsNullModel — F-statistic of regression model

structure

This property is read-only.

F-statistic of the regression model, specified as a structure containing these fields:

• Fstats — F-statistic of the fitted model versus the null model

• Pvalue — p-value for the F-statistic

• NullModel — Null model type, specified as 'constant' or 'zero'

If the fitted model contains an intercept term, NullModel is 'constant' (contains only an intercept term). Otherwise, NullModel is 'zero' (contains no terms).

Data Types: struct

MSE — Mean squared error

numeric value

This property is read-only.

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

Residuals — Residuals for fitted model

table

This property is read-only.

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

Column	Description
Raw	Observed minus fitted values
Pearson	Raw residuals divided by the root mean squared error (RMSE)
Standardized	Raw residuals divided by their estimated standard deviation
Studentized	Raw 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

RMSE — Root mean squared error

numeric value

This property is read-only.

Root mean squared error (residuals), specified as a numeric value.

RMSE = sqrt(MSE),

where RMSE is the root mean squared error and MSE is the mean squared error. RMSE is an estimate of the standard deviation of the error term in the model.

Data Types: single | double

Rsquared — R-squared value for model

structure

This property is read-only.

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

• Ordinary — Ordinary (unadjusted) R-squared

• Adjusted — R-squared adjusted for the number of coefficients

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,

where SST is the total sum of squares, 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:

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

Fitting Method

Robust — Robust fit information

structure

This property is read-only.

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

Field	Description
WgtFun	Robust weighting function, such as 'bisquare' (see Robust Options)
Tune	Value specified for tuning parameter (can be [])
Weights	Vector of weights used in final iteration of robust fit

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

Data Types: struct

Input Data

Formula — Model information

NonLinearFormula object

This property is read-only.

Model information, specified as a NonLinearFormula object.

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

mdl.Formula

Iterative — Information about fitting process

structure

This property is read-only.

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

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

ObservationInfo — Observation information

table

This property is read-only.

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

Column	Description
Weights	Observation weights, specified as a numeric value. The default value is 1.
Excluded	Indicator of excluded observations, 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.
Missing	Indicator of missing observations, specified as a logical value. The value is true if the observation is missing.
Subset	Indicator of whether or not the fitting function uses the observation, specified as a logical value. The value is true if the observation is not excluded or missing, meaning the 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

ObservationNames — Observation names

cell array of character vectors

This property is read-only.

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

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

Variables — Input data

table

This property is read-only.

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 the response vector y.

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

Data Types: table

Object Functions

Predict Responses

feval	Evaluate nonlinear regression model prediction
predict	Predict response of nonlinear regression model
random	Simulate responses for nonlinear regression model

Evaluate Nonlinear Model

coefCI	Confidence intervals of coefficient estimates of nonlinear regression model
coefTest	Linear hypothesis test on nonlinear regression model coefficients
partialDependence	Compute partial dependence

Visualize Nonlinear Model and Summary Statistics

plotDiagnostics	Plot diagnostics of nonlinear regression model
plotPartialDependence	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
plotResiduals	Plot residuals of nonlinear regression model
plotSlice	Plot of slices through fitted nonlinear regression surface

Examples

Fit a Nonlinear Regression Model

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).

load carbig
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
R-Squared: 0.745,  Adjusted R-Squared 0.743
F-statistic vs. constant model: 283, p-value = 1.79e-113

Find the predicted mileage of an average car. Because the sample data contains some missing (NaN) observations, compute the mean using mean with the 'omitnan' option.

Xnew = mean(X,'omitnan')  

Xnew = 1×2
103 ×

    0.1051    2.9794

MPGnew = predict(mdl,Xnew)

MPGnew = 
21.8073

More About

Hat Matrix

The hat matrix H is defined in terms of the data matrix X and the Jacobian matrix J:

$$J_{i,j}={\frac{\partial f}{\partial {\beta }_j}|}_{x_i,\beta }$$

Here f is the nonlinear model function, and β is the vector of model coefficients.

The Hat Matrix H is

H = J(J^TJ)^–1J^T.

The diagonal elements H_ii satisfy

$$\begin{array}{l}0\le h_{ii}\le 1\\ {\displaystyle \sum _{i=1}^n{h}_{ii}}=p,\end{array}$$

where n is the number of observations (rows of X), and p is the number of coefficients in the regression model.

Leverage

Leverage is a measure of the effect of a particular observation on the regression predictions due to the position of that observation in the space of the inputs.

The leverage of observation i is the value of the ith diagonal term h_ii of the hat matrix H. Because the sum of the leverage values is p (the number of coefficients in the regression model), an observation i can be considered an outlier if its leverage substantially exceeds p/n, where n is the number of observations.

Cook’s Distance

The Cook’s distance D_i of observation i is

$$D_i=\frac{{\displaystyle \sum _{j=1}^n{\left({\widehat{y}}_j-{\widehat{y}}_{j(i)}\right)}^2}}{pMSE},$$

where

• $${\widehat{y}}_j$$ is the jth fitted response value.

• $${\widehat{y}}_{j(i)}$$ is the jth fitted response value, where the fit does not include observation i.

• MSE is the mean squared error.

• p is the number of coefficients in the regression model.

Cook’s distance is algebraically equivalent to the following expression:

$$D_i=\frac{r_i^2}{pMSE}\left(\frac{h_{ii}}{{\left(1-h_{ii}\right)}^2}\right),$$

where e_i is the ith residual.

Version History

Introduced in R2012a