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 readonly.
Covariance matrix of coefficient estimates, specified as a
pbyp 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 readonly.
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 readonly.
Coefficient values, specified as a table.
Coefficients
contains one row for each
coefficient and these columns:
Estimate
— Estimated coefficient valueSE
— Standard error of the estimatetStat
— tstatistic for a twosided test with the null hypothesis that the coefficient is zeropValue
— pvalue for the tstatistic
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 readonly.
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 readonly.
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 readonly.
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 readonly.
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 byN 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 readonly.
Fitted (predicted) response values based on input data, specified as
an nby1 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 readonly.
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 readonly.
Criterion for model comparison, specified as a structure with these fields:
AIC
— Akaike information criterion.AIC = –2*logL + 2*m
, wherelogL
is the loglikelihood andm
is the number of estimated parameters.AICc
— Akaike information criterion corrected for the sample size.AICc = AIC + (2*m*(m + 1))/(n – m – 1)
, wheren
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 likelihoodbased 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 bestfitting model. The bestfitting 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
— Fstatistic of regression model
structure
This property is readonly.
Fstatistic of the regression model, specified as a structure containing these fields:
Fstats
— Fstatistic of the fitted model versus the null modelPvalue
— pvalue for the FstatisticNullModel
— 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 readonly.
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 readonly.
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 readonly.
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
— Rsquared value for model
structure
This property is readonly.
Rsquared value for the model, specified as a structure with two fields:
Ordinary
— Ordinary (unadjusted) RsquaredAdjusted
— Rsquared adjusted for the number of coefficients
The Rsquared value is the proportion of the total sum of squares explained by the
model. The ordinary Rsquared 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 (RSquared).
To obtain either of these values as a scalar, index into the property using dot
notation. For example, obtain the adjusted Rsquared value in the model
mdl
:
r2 = mdl.Rsquared.Adjusted
Data Types: struct
SSE
— Sum of squared errors
numeric value
This property is readonly.
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 readonly.
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 readonly.
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 readonly.
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 readonly.
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 readonly.
Information about the fitting process, specified as a structure with the following fields:
InitialCoefs
— Initial coefficient values (thebeta0
vector)IterOpts
— Options included in theOptions
namevalue pair argument forfitnlm
.
Data Types: struct
NumObservations
— Number of observations
positive integer
This property is readonly.
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'
namevalue pair
argument) or rows with missing values.
Data Types: double
NumPredictors
— Number of predictor variables
positive integer
This property is readonly.
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 readonly.
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 readonly.
Observation information, specified as an nby4 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' namevalue 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 readonly.
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 readonly.
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 readonly.
Response variable name, specified as a character vector.
Data Types: char
VariableInfo
— Information about variables
table
This property is readonly.
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

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 readonly.
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'
namevalue 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 readonly.
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
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 RSquared: 0.745, Adjusted RSquared 0.743 Fstatistic vs. constant model: 283, pvalue = 1.79e113
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
10^{3} ×
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^{T}J)^{–1}J^{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
$${D}_{i}=\frac{{\displaystyle \sum _{j=1}^{n}{\left({\widehat{y}}_{j}{\widehat{y}}_{j(i)}\right)}^{2}}}{p\text{\hspace{0.17em}}MSE},$$
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:
$${D}_{i}=\frac{{r}_{i}^{2}}{p\text{\hspace{0.17em}}MSE}\left(\frac{{h}_{ii}}{{\left(1{h}_{ii}\right)}^{2}}\right),$$
where e_{i} is the ith residual.
Version History
Introduced in R2012a
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