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Nonlinear regression model class
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.
Create a NonLinearModel
object using fitnlm
.
CoefficientCovariance
— Covariance matrix of coefficient estimatesThis 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.
For details, see Coefficient Standard Errors and Confidence Intervals.
Data Types: single
 double
CoefficientNames
— Coefficient namesThis 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 valuesThis property is readonly.
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
— tstatistic for a test that the
coefficient is zero
pValue
— pvalue for the
tstatistic
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
Diagnostics
— Diagnostic informationThis 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 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. 
CooksDistance  Cook's measure of scaled change in fitted values  CooksDistance 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. 
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. 
Data Types: table
DFE
— Degrees of freedom for errorThis 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
Fitted
— Fitted response values based on input dataThis property is readonly.
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
Formula
— Model informationNonLinearFormula
objectThis 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 processThis property is readonly.
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
namevalue pair argument for
fitnlm
.
Data Types: struct
LogLikelihood
— Log likelihoodThis property is readonly.
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
ModelCriterion
— Criterion for model comparisonThis property is readonly.
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 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
MSE
— Mean squared errorThis 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
NumCoefficients
— Number of model coefficientsThis property is readonly.
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
NumEstimatedCoefficients
— Number of estimated coefficientsThis 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
NumPredictors
— Number of predictor variablesThis property is readonly.
Number of predictor variables used to fit the model, specified as a positive integer.
Data Types: double
NumVariables
— Number of variablesThis 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 informationThis property is readonly.
Observation information, specified as an nby4 table, where
n is equal to the number of rows of input data. The
ObservationInfo
contains the columns described in this
table.
Column  Description 

Weights  Observation weight, specified as a numeric value. The default value
is 1 . 
Excluded  Indicator of excluded observation, 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 observation, specified as a logical value. The
value is true if the observation is missing. 
Subset  Indicator 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
ObservationNames
— Observation namesThis 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 modelThis property is readonly.
Names of predictors used to fit the model, specified as a cell array of character vectors.
Data Types: cell
Residuals
— Residuals for fitted modelThis 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
ResponseName
— Response variable nameThis property is readonly.
Response variable name, specified as a character vector.
Data Types: char
RMSE
— Root mean squared errorThis property is readonly.
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
— Robust fit informationThis property is readonly.
Robust fit information, specified as a structure with the following fields:
Field  Description 

WgtFun  Robust weighting function, such as
'bisquare' (see robustfit ) 
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
Rsquared
— Rsquared value for modelThis property is readonly.
Rsquared value for the model, specified as a structure with two fields:
Ordinary
— Ordinary (unadjusted) Rsquared
Adjusted
— 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 = 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 (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 errorsThis property is readonly.
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
SSR
— Regression sum of squaresThis property is readonly.
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
SST
— Total sum of squaresThis property is readonly.
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
VariableInfo
— Information about variablesThis 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 variablesThis 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 dataThis 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 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
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 
Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).
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 auto. Since the sample data contains some missing (NaN
) observations, compute the mean using nanmean
.
Xnew = nanmean(X)
Xnew = 1×2
10^{3} ×
0.1051 2.9794
MPGnew = predict(mdl,Xnew)
MPGnew = 21.8073
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 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. The hat matrix H is defined in terms of the data matrix X:
H = X(X^{T}X)^{–1}X^{T}.
The hat matrix is also known as the projection matrix because it projects the vector of observations y onto the vector of predictions $$\widehat{y}$$, thus putting the "hat" on y.
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.
For more details, see Hat Matrix and Leverage.
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.
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