fitnlm
Fit nonlinear regression model
Syntax
Description
fits
a nonlinear regression model with additional options specified by
one or more mdl
= fitnlm(___,modelfun
,beta0
,Name,Value
)Name,Value
pair arguments.
Examples
Nonlinear Model from Table
Create a nonlinear model for auto mileage based on the carbig
data.
Load the data and create a nonlinear model.
load carbig tbl = table(Horsepower,Weight,MPG); 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(tbl,modelfun,beta0)
mdl = Nonlinear regression model: MPG ~ b1 + b2*Horsepower^b3 + b4*Weight^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
Nonlinear Model from Matrix Data
Create a nonlinear model for auto mileage based on the carbig
data.
Load the data and create a nonlinear model.
load carbig X = [Horsepower,Weight]; y = MPG; 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
Adjust Fitting Options in Nonlinear Model
Create a nonlinear model for auto mileage based on the carbig
data. Strive for more accuracy by lowering the TolFun
option, and observe the iterations by setting the Display
option.
Load the data and create a nonlinear model.
load carbig X = [Horsepower,Weight]; y = MPG; modelfun = @(b,x)b(1) + b(2)*x(:,1).^b(3) + ... b(4)*x(:,2).^b(5); beta0 = [-50 500 -1 500 -1];
Create options to lower TolFun
and to report iterative display, and create a model using the options.
opts = statset('Display','iter','TolFun',1e-10); mdl = fitnlm(X,y,modelfun,beta0,'Options',opts);
Norm of Norm of Iteration SSE Gradient Step ----------------------------------------------------------- 0 1.82248e+06 1 678600 788810 1691.07 2 616716 6.12739e+06 45.4738 3 249831 3.9532e+06 293.557 4 17675 361544 369.284 5 11746.6 69670.5 169.079 6 7242.22 343738 394.822 7 6250.32 159719 452.941 8 6172.87 91622.9 268.674 9 6077 6957.44 100.208 10 6076.34 6370.39 88.1905 11 6075.75 5199.08 77.9694 12 6075.3 4646.61 69.764 13 6074.91 4235.96 62.9114 14 6074.55 3885.28 57.0647 15 6074.23 3571.1 52.0036 16 6073.93 3286.48 47.5795 17 6073.66 3028.34 43.6844 18 6073.4 2794.31 40.2352 19 6073.17 2582.15 37.1663 20 6072.95 2389.68 34.4243 21 6072.74 2214.84 31.9651 22 6072.55 2055.78 29.7516 23 6072.37 1910.83 27.753 24 6072.21 1778.51 25.9428 25 6072.05 1657.5 24.2986 26 6071.9 1546.65 22.8011 27 6071.76 1444.93 21.4338 28 6071.63 1351.44 20.1822 29 6071.51 1265.39 19.0339 30 6071.39 1186.06 17.978 31 6071.28 1112.83 17.0052 32 6071.17 1045.13 16.107 33 6071.07 982.465 15.2762 34 6070.98 924.389 14.5063 35 6070.89 870.498 13.7916 36 6070.8 820.434 13.127 37 6070.72 773.872 12.5081 38 6070.64 730.521 11.9307 39 6070.57 690.117 11.3914 40 6070.5 652.422 10.887 41 6070.43 617.219 10.4144 42 6070.37 584.315 9.97115 43 6070.31 553.53 9.55489 44 6070.25 524.703 9.1635 45 6070.19 497.686 8.79506 46 6070.14 472.345 8.44785 47 6070.08 448.557 8.12028 48 6070.03 426.21 7.81092 49 6069.99 405.201 7.51845 50 6069.94 385.435 7.2417 51 6069.9 366.825 6.97956 52 6069.85 349.293 6.73104 53 6069.81 332.764 6.49523 54 6069.77 317.171 6.27127 55 6069.74 302.453 6.0584 56 6069.7 288.55 5.85591 57 6069.66 275.411 5.66315 58 6069.63 262.986 5.47949 59 6069.6 251.23 5.3044 60 6069.57 240.1 5.13734 61 6069.54 229.558 4.97784 62 6069.51 219.567 4.82545 63 6069.48 210.094 4.67977 64 6069.45 201.108 4.5404 65 6069.43 192.578 4.407 66 6069.4 184.479 4.27923 67 6069.38 176.785 4.15677 68 6069.35 169.472 4.03935 69 6069.33 162.518 3.9267 70 6069.31 155.903 3.81855 71 6069.29 149.608 3.71468 72 6069.26 143.615 3.61486 73 6069.24 137.907 3.5189 74 6069.22 132.468 3.42658 75 6069.21 127.283 3.33774 76 6069.19 122.339 3.25221 77 6069.17 117.623 3.16981 78 6069.15 113.123 3.09041 79 6069.14 108.827 3.01386 80 6069.12 104.725 2.94002 81 6069.1 100.806 2.86877 82 6069.09 97.0611 2.8 83 6069.07 93.4814 2.73358 84 6069.06 90.0583 2.66942 85 6069.05 86.7842 2.60741 86 6069.03 83.6513 2.54745 87 6069.02 80.6528 2.48947 88 6069.01 77.7821 2.43338 89 6068.99 75.0328 2.37908 90 6068.98 72.399 2.32652 91 6068.97 69.8752 2.27561 92 6068.96 67.4561 2.22629 93 6068.95 65.1367 2.17849 94 6068.94 62.9122 2.13216 95 6068.93 60.7784 2.08723 96 6068.92 58.7308 2.04364 97 6068.91 56.7655 2.00135 98 6068.9 54.8787 1.9603 99 6068.89 4349.28 18.1917 100 6068.77 2416.27 14.4439 101 6068.71 1721.26 12.1305 102 6068.66 1228.78 10.289 103 6068.63 884.002 8.82019 104 6068.6 639.615 7.62745 105 6068.58 464.84 6.64627 106 6068.56 338.878 5.82964 107 6068.55 247.508 5.14297 108 6068.54 180.878 4.56032 109 6068.53 132.084 4.06194 110 6068.52 96.2342 3.63255 111 6068.51 69.8362 3.26019 112 6068.51 50.3734 2.93541 113 6068.5 36.0205 2.65062 114 6068.5 25.4451 2.39969 115 6068.49 17.6693 2.17764 116 6068.49 1027.39 14.0164 117 6068.48 544.039 5.3137 118 6068.48 94.0569 2.86662 119 6068.48 113.637 3.73503 120 6068.48 0.51834 1.37051 121 6068.48 4.59439 0.912827 122 6068.48 1.56359 0.629276 123 6068.48 1.13825 0.432567 124 6068.48 0.296021 0.297532 Iterations terminated: relative change in SSE less than OPTIONS.TolFun
Specify Nonlinear Regression Using Model Name Syntax
Specify a nonlinear regression model for estimation using a function handle or model syntax.
Load sample data.
S = load('reaction');
X = S.reactants;
y = S.rate;
beta0 = S.beta;
Use a function handle to specify the Hougen-Watson model for the rate data.
mdl = fitnlm(X,y,@hougen,beta0)
mdl = Nonlinear regression model: y ~ hougen(b,X) Estimated Coefficients: Estimate SE tStat pValue ________ ________ ______ _______ b1 1.2526 0.86701 1.4447 0.18654 b2 0.062776 0.043561 1.4411 0.18753 b3 0.040048 0.030885 1.2967 0.23089 b4 0.11242 0.075157 1.4957 0.17309 b5 1.1914 0.83671 1.4239 0.1923 Number of observations: 13, Error degrees of freedom: 8 Root Mean Squared Error: 0.193 R-Squared: 0.999, Adjusted R-Squared 0.998 F-statistic vs. zero model: 3.91e+03, p-value = 2.54e-13
Alternatively, you can use an expression to specify the Hougen-Watson model for the rate data.
myfun = 'y~(b1*x2-x3/b5)/(1+b2*x1+b3*x2+b4*x3)';
mdl2 = fitnlm(X,y,myfun,beta0)
mdl2 = Nonlinear regression model: y ~ (b1*x2 - x3/b5)/(1 + b2*x1 + b3*x2 + b4*x3) Estimated Coefficients: Estimate SE tStat pValue ________ ________ ______ _______ b1 1.2526 0.86701 1.4447 0.18654 b2 0.062776 0.043561 1.4411 0.18753 b3 0.040048 0.030885 1.2967 0.23089 b4 0.11242 0.075157 1.4957 0.17309 b5 1.1914 0.83671 1.4239 0.1923 Number of observations: 13, Error degrees of freedom: 8 Root Mean Squared Error: 0.193 R-Squared: 0.999, Adjusted R-Squared 0.998 F-statistic vs. zero model: 3.91e+03, p-value = 2.54e-13
Estimate Nonlinear Regression Using Robust Fitting Options
Generate sample data from the nonlinear regression model
where , , and are coefficients, and the error term is normally distributed with mean 0 and standard deviation 0.5.
modelfun = @(b,x)(b(1)+b(2)*exp(-b(3)*x)); rng('default') % for reproducibility b = [1;3;2]; x = exprnd(2,100,1); y = modelfun(b,x) + normrnd(0,0.5,100,1);
Set robust fitting options.
opts = statset('nlinfit'); opts.RobustWgtFun = 'bisquare';
Fit the nonlinear model using the robust fitting options. Here, use an expression to specify the model.
b0 = [2;2;2]; modelstr = 'y ~ b1 + b2*exp(-b3*x)'; mdl = fitnlm(x,y,modelstr,b0,'Options',opts)
mdl = Nonlinear regression model (robust fit): y ~ b1 + b2*exp( - b3*x) Estimated Coefficients: Estimate SE tStat pValue ________ _______ ______ __________ b1 1.0218 0.07202 14.188 2.1344e-25 b2 3.6619 0.25429 14.401 7.974e-26 b3 2.9732 0.38496 7.7232 1.0346e-11 Number of observations: 100, Error degrees of freedom: 97 Root Mean Squared Error: 0.501 R-Squared: 0.807, Adjusted R-Squared 0.803 F-statistic vs. constant model: 203, p-value = 2.34e-35
Fit Nonlinear Regression Model Using Weights Function Handle
Load sample data.
S = load('reaction');
X = S.reactants;
y = S.rate;
beta0 = S.beta;
Specify a function handle for observation weights. The function accepts the model fitted values as input, and returns a vector of weights.
a = 1; b = 1; weights = @(yhat) 1./((a + b*abs(yhat)).^2);
Fit the Hougen-Watson model to the rate data using the specified observation weights function.
mdl = fitnlm(X,y,@hougen,beta0,'Weights',weights)
mdl = Nonlinear regression model: y ~ hougen(b,X) Estimated Coefficients: Estimate SE tStat pValue ________ ________ ______ _______ b1 0.83085 0.58224 1.427 0.19142 b2 0.04095 0.029663 1.3805 0.20477 b3 0.025063 0.019673 1.274 0.23842 b4 0.080053 0.057812 1.3847 0.20353 b5 1.8261 1.281 1.4256 0.19183 Number of observations: 13, Error degrees of freedom: 8 Root Mean Squared Error: 0.037 R-Squared: 0.998, Adjusted R-Squared 0.998 F-statistic vs. zero model: 1.14e+03, p-value = 3.49e-11
Nonlinear Regression Model Using Nonconstant Error Model
Load sample data.
S = load('reaction');
X = S.reactants;
y = S.rate;
beta0 = S.beta;
Fit the Hougen-Watson model to the rate data using the combined error variance model.
mdl = fitnlm(X,y,@hougen,beta0,'ErrorModel','combined')
mdl = Nonlinear regression model: y ~ hougen(b,X) Estimated Coefficients: Estimate SE tStat pValue ________ ________ ______ _______ b1 1.2526 0.86702 1.4447 0.18654 b2 0.062776 0.043561 1.4411 0.18753 b3 0.040048 0.030885 1.2967 0.23089 b4 0.11242 0.075158 1.4957 0.17309 b5 1.1914 0.83671 1.4239 0.1923 Number of observations: 13, Error degrees of freedom: 8 Root Mean Squared Error: 1.27 R-Squared: 0.999, Adjusted R-Squared 0.998 F-statistic vs. zero model: 3.91e+03, p-value = 2.54e-13
Input Arguments
tbl
— Input data
table | dataset array
Input data including predictor and response variables, specified as a table or dataset array. The predictor variables and response variable must be numeric.
If you specify
modelfun
using a formula, the model specification in the formula specifies the predictor and response variables.If you specify
modelfun
using a function handle, the last variable is the response variable and the others are the predictor variables, by default. You can set a different column as the response variable by using theResponseVar
name-value pair argument. To select a subset of the columns as predictors, use thePredictorVars
name-value pair argument.
The variable names in a table do not have to be valid MATLAB® identifiers, but the names must not contain leading or trailing blanks. If
the names are not valid, you cannot specify modelfun
using a
formula.
You can verify the variable names in tbl
by using the isvarname
function. If the variable names are
not valid, then you can convert them by using the matlab.lang.makeValidName
function.
Data Types: table
X
— Predictor variables
matrix
Predictor variables, specified as an n-by-p matrix,
where n is the number of observations and p is
the number of predictor variables. Each column of X
represents
one variable, and each row represents one observation.
Data Types: single
| double
y
— Response variable
vector
Response variable, specified as an
n-by-1 vector, where
n is the number of
observations. Each entry in y
is the response for the corresponding row of
X
.
Data Types: single
| double
modelfun
— Functional form of the model
function handle | character vector or string scalar formula in the form
'y
~
f
(b1,b2,...,bj,x1,x2,...,xk)'
y
f
(b1,b2,...,bj,x1,x2,...,xk)'Functional form of the model, specified as either of the following.
Function handle
@
ormodelfun
@(b,x)
, wheremodelfun
b
is a coefficient vector with the same number of elements asbeta0
.x
is a matrix with the same number of columns asX
or the number of predictor variable columns oftbl
.
modelfun
(b,x)
returns a column vector that contains the same number of rows asx
. Each row of the vector is the result of evaluatingmodelfun
on the corresponding row ofx
. In other words,modelfun
is a vectorized function, one that operates on all data rows and returns all evaluations in one function call.modelfun
should return real numbers to obtain meaningful coefficients.Character vector or string scalar formula in the form
'
, wherey
~f
(b1,b2,...,bj,x1,x2,...,xk)'f
represents a scalar function of the scalar coefficient variablesb1
,...,bj
and the scalar data variablesx1
,...,xk
. The variable names in the formula must be valid MATLAB identifiers.
Data Types: function_handle
| char
| string
beta0
— Initial coefficient values
numeric vector
Initial coefficient values for the nonlinear model, specified as a numeric vector.
NonLinearModel
starts its search for optimal
coefficients from beta0
.
Data Types: single
| double
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'ErrorModel','combined','Exclude',2,'Options',opt
specifies
the error model as the combined model, excludes the second observation
from the fit, and uses the options defined in the structure opt
to
control the iterative fitting procedure.
CoefficientNames
— Names of the model coefficients
{'b1','b2',...,'bk
'}
(default) | string array | cell array of character vectors
k
'}Names of the model coefficients, specified as a string array or cell array of character vectors.
Data Types: string
| cell
ErrorModel
— Form of the error variance model
'constant'
(default) | 'proportional'
| 'combined'
Form of the error variance model, specified as one of the following. Each model defines the error using a standard mean-zero and unit-variance variable e in combination with independent components: the function value f, and one or two parameters a and b
'constant' (default) | |
'proportional' | |
'combined' |
The only allowed error model when using Weights
is 'constant'
.
Note
options.RobustWgtFun
must have value []
when
using an error model other than 'constant'
.
Example: 'ErrorModel','proportional'
ErrorParameters
— Initial estimates of the error model parameters
numeric array
Initial estimates of the error model parameters for the chosen ErrorModel
,
specified as a numeric array.
Error Model | Parameters | Default Values |
---|---|---|
'constant' | a | 1 |
'proportional' | b | 1 |
'combined' | a, b | [1,1] |
You can only use the 'constant'
error model
when using Weights
.
Note
options.RobustWgtFun
must have value []
when
using an error model other than 'constant'
.
For example, if 'ErrorModel'
has the value 'combined'
,
you can specify the starting value 1 for a and
the starting value 2 for b as follows.
Example: 'ErrorParameters',[1,2]
Data Types: single
| double
Exclude
— Observations to exclude
logical or numeric index vector
Observations to exclude from the fit, specified as the comma-separated
pair consisting of 'Exclude'
and a logical or numeric
index vector indicating which observations to exclude from the fit.
For example, you can exclude observations 2 and 3 out of 6 using either of the following examples.
Example: 'Exclude',[2,3]
Example: 'Exclude',logical([0 1 1 0 0 0])
Data Types: single
| double
| logical
Options
— Options for controlling the iterative fitting procedure
[ ] (default) | structure
Options for controlling the iterative fitting procedure, specified as a structure created by
statset
. The relevant fields are the
nonempty fields in the structure returned by the
call statset('fitnlm')
.
Option | Meaning | Default |
---|---|---|
DerivStep | Relative difference used in finite difference derivative calculations. A positive scalar, or a vector of positive scalars the same size as the vector of parameters estimated by the Statistics and Machine Learning Toolbox™ function using the options structure. | eps^(1/3) |
Display |
Amount of information displayed by the fitting algorithm.
| 'off' |
FunValCheck | Character vector or string scalar indicating to check for invalid values, such as
NaN or Inf ,
from the model function. | 'on' |
MaxIter | Maximum number of iterations allowed. Positive integer. | 200 |
RobustWgtFun | Weight function for robust fitting. Can also be a function
handle that accepts a normalized residual as input and returns the
robust weights as output. If you use a function handle, give a Tune constant.
See Robust Options | [] |
Tune | Tuning constant used in robust fitting to normalize the residuals before applying the weight function. A positive scalar. Required if the weight function is specified as a function handle. | See Robust Options for the
default, which depends on RobustWgtFun . |
TolFun | Termination tolerance for the objective function value. Positive scalar. | 1e-8 |
TolX | Termination tolerance for the parameters. Positive scalar. | 1e-8 |
Data Types: struct
PredictorVars
— Predictor variables
string array | cell array of character vectors | logical or numeric index vector
Predictor variables to use in the fit, specified as the
comma-separated pair consisting of 'PredictorVars'
and either a string array or cell array of character vectors of the
variable names in the table or dataset array tbl
, or
a logical or numeric index vector indicating which columns are predictor
variables.
The string values or character vectors should be among the names in
tbl
, or the names you specify using the
'VarNames'
name-value pair argument.
The default is all variables in X
, or all variables
in tbl
except for
ResponseVar
.
For example, you can specify the second and third variables as the predictor variables using either of the following examples.
Example: 'PredictorVars',[2,3]
Example: 'PredictorVars',logical([0 1 1 0 0
0])
Data Types: single
| double
| logical
| string
| cell
ResponseVar
— Response variable
last column of tbl
(default) | variable name | logical or numeric index vector
Response variable to use in the fit, specified as the comma-separated
pair consisting of 'ResponseVar'
and either a variable
name in the table or dataset array tbl
, or a logical
or numeric index vector indicating which column is the response variable.
If you specify a model, it specifies the response variable.
Otherwise, when fitting a table or dataset array, 'ResponseVar'
indicates
which variable fitnlm
should use as the response.
For example, you can specify the fourth variable, say yield
,
as the response out of six variables, in one of the following ways.
Example: 'ResponseVar','yield'
Example: 'ResponseVar',[4]
Example: 'ResponseVar',logical([0 0 0 1 0 0])
Data Types: single
| double
| logical
| char
| string
VarNames
— Names of variables
{'x1','x2',...,'xn','y'}
(default) | string array | cell array of character vectors
Names of variables, specified as the comma-separated pair consisting of
'VarNames'
and a string array or cell array of character vectors
including the names for the columns of X
first, and the name for the
response variable y
last.
'VarNames'
is not applicable to variables in a table or dataset
array, because those variables already have names.
Example: 'VarNames',{'Horsepower','Acceleration','Model_Year','MPG'}
Data Types: string
| cell
Weights
— Observation weights
ones(n,1)
(default) | vector of nonnegative scalar values | function handle
Observation weights, specified as a vector of nonnegative scalar values or function handle.
If you specify a vector, then it must have n elements, where n is the number of rows in
tbl
ory
.If you specify a function handle, then the function must accept a vector of predicted response values as input, and return a vector of real positive weights as output.
Given weights, W
, NonLinearModel
estimates
the error variance at observation i
by MSE*(1/W(i))
,
where MSE is the mean squared error.
Data Types: single
| double
| function_handle
Output Arguments
mdl
— Nonlinear model
NonLinearModel
object
Nonlinear model representing a least-squares fit of the response
to the data, returned as a NonLinearModel
object.
If the Options
structure contains a nonempty RobustWgtFun
field,
the model is not a least-squares fit, but uses the RobustWgtFun
robust
fitting function.
For properties and methods of the nonlinear model object, mdl
,
see the NonLinearModel
class
page.
More About
Robust Options
Weight Function | Equation | Default Tuning Constant |
---|---|---|
"andrews" | w = (abs(r)<pi) .* sin(r) ./ r | 1.339 |
"bisquare" (default) | w = (abs(r)<1) .* (1 - r.^2).^2 | 4.685 |
"cauchy" | w = 1 ./ (1 + r.^2) | 2.385 |
"fair" | w = 1 ./ (1 + abs(r)) | 1.400 |
"huber" | w = 1 ./ max(1, abs(r)) | 1.345 |
"logistic" | w = tanh(r) ./ r | 1.205 |
"talwar" | w = 1 * (abs(r)<1) | 2.795 |
"welsch" | w = exp(-(r.^2)) | 2.985 |
Algorithms
fitnlm
uses the same fitting algorithm asnlinfit
.fitnlm
considersNaN
values intbl
,X
, andy
to be missing values. When fitting a model,fitnlm
does not use observations with missing values or observations at whichmodelfun
returnsNaN
values. TheObservationInfo
property of a fitted model contains information regarding whether or notfitnlm
uses each observation in the fit.
References
[1] Seber, G. A. F., and C. J. Wild. Nonlinear Regression. Hoboken, NJ: Wiley-Interscience, 2003.
[2] DuMouchel, W. H., and F. L. O'Brien. “Integrating a Robust Option into a Multiple Regression Computing Environment.” Computer Science and Statistics: Proceedings of the 21st Symposium on the Interface. Alexandria, VA: American Statistical Association, 1989.
[3] Holland, P. W., and R. E. Welsch. “Robust Regression Using Iteratively Reweighted Least-Squares.” Communications in Statistics: Theory and Methods, A6, 1977, pp. 813–827.
Version History
Introduced in R2013b
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