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LinearModel.fit
will be removed in a future
release. Use fitlm
instead.
mdl = LinearModel.fit(tbl)
mdl = LinearModel.fit(X,y)
mdl = LinearModel.fit(___,modelspec)
mdl = LinearModel.fit(___,Name,Value)
mdl =
LinearModel.fit(___,modelspec,Name,Value)
creates
a linear model of a table or dataset array mdl
= LinearModel.fit(tbl
)tbl
.
creates
a linear model of the responses mdl
= LinearModel.fit(X
,y
)y
to a data matrix X
.
creates
a linear model of the type specified by mdl
= LinearModel.fit(___,modelspec
)modelspec
,
using any of the previous syntaxes.
or mdl
= LinearModel.fit(___,Name,Value
)
creates
a linear model with additional options specified by one or more mdl
=
LinearModel.fit(___,modelspec
,Name,Value
)Name,Value
pair
arguments. For example, you can specify which predictor variables
to include in the fit or include observation weights.
tbl
— Input dataInput data, specified as a table or dataset array. When modelspec
is
a formula
, it specifies the variables to be used
as the predictors and response. Otherwise, if you do not specify the
predictor and response variables, the last variable is the response
variable and the others are the predictor variables by default.
Predictor variables can be numeric, or any grouping variable type, such as logical or categorical (see Grouping Variables). The response must be numeric or logical.
To set a different column as the response variable, use the ResponseVar
namevalue
pair argument. To use a subset of the columns as predictors, use the PredictorVars
namevalue
pair argument.
Data Types: single
 double
 logical
X
— Predictor variablesPredictor variables, specified as an nbyp 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.
By default, there is a constant term in the model, unless you
explicitly remove it, so do not include a column of 1s in X
.
Data Types: single
 double
 logical
y
— Response variableResponse variable, specified as an nby1
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
modelspec
— Model specification'Y ~ terms'
Model specification, specified as one of the following.
A character vector naming the model.
Character Vector  Model Type 

'constant'  Model contains only a constant (intercept) term. 
'linear'  Model contains an intercept and linear terms for each predictor. 
'interactions'  Model contains an intercept, linear terms, and all products of pairs of distinct predictors (no squared terms). 
'purequadratic'  Model contains an intercept, linear terms, and squared terms. 
'quadratic'  Model contains an intercept, linear terms, interactions, and squared terms. 
'poly  Model is a polynomial with all terms up to degree i in
the first predictor, degree j in the second
predictor, etc. Use numerals 0 through 9 .
For example, 'poly2111' has a constant plus all
linear and product terms, and also contains terms with predictor 1
squared. 
tby(p + 1) matrix, namely terms matrix, specifying terms to include in the model, where t is the number of terms and p is the number of predictor variables, and plus 1 is for the response variable.
A character vector representing a formula in the form
'Y
~ terms'
,
where the terms
are
specified using Wilkinson
Notation.
Example: 'quadratic'
Example: 'y ~ X1 + X2^2 + X1:X2'
Specify optional commaseparated pairs of Name,Value
arguments.
Name
is the argument
name and Value
is the corresponding
value. Name
must appear
inside single quotes (' '
).
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN
.
'CategoricalVars'
— Categorical variablesCategorical variables in the fit, specified as the commaseparated
pair consisting of 'CategoricalVars'
and either
a cell array of character vectors of the names of the categorical
variables in the table or dataset array tbl
, or
a logical or numeric index vector indicating which columns are categorical.
If data is in a table or dataset array tbl
,
then the default is to treat all categorical or logical variables,
character arrays, or cell arrays of character vectors as categorical
variables.
If data is in matrix X
, then the
default value of this namevalue pair argument is an empty matrix []
.
That is, no variable is categorical unless you specify it.
For example, you can specify the observations 2 and 3 out of 6 as categorical using either of the following examples.
Example: 'CategoricalVars',[2,3]
Example: 'CategoricalVars',logical([0 1 1 0 0 0])
Data Types: single
 double
 logical
'Exclude'
— Observations to excludeObservations to exclude from the fit, specified as the commaseparated
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
'Intercept'
— Indicator for constant termtrue
(default)  false
Indicator the for constant term (intercept) in the fit, specified
as the commaseparated pair consisting of 'Intercept'
and
either true
to include or false
to
remove the constant term from the model.
Use 'Intercept'
only when specifying the
model using a character vector, not a formula or matrix.
Example: 'Intercept',false
'PredictorVars'
— Predictor variablesPredictor variables to use in the fit, specified as the commaseparated
pair consisting of 'PredictorVars'
and either a
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 character vectors should be among the names in tbl
,
or the names you specify using the 'VarNames'
namevalue
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
 cell
'ResponseVar'
— Response variabletbl
(default)  character vector containing variable name  logical or numeric index vectorResponse variable to use in the fit, specified as the commaseparated
pair consisting of 'ResponseVar'
and either a character
vector containing the variable name in the table or dataset array tbl
,
or a logical or numeric index vector indicating which column is the
response variable. You typically need to use 'ResponseVar'
when
fitting a table or dataset array tbl
.
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
'RobustOpts'
— Indicator of robust fitting type'off'
(default)  'on'
 character vector  structure with character vector or function handleIndicator of the robust fitting type to use, specified as the
commaseparated pair consisting of 'RobustOpts'
and
one of the following.
'off'
— No robust fitting. fitlm
uses
ordinary least squares.
'on'
— Robust fitting. When
you use robust fitting, 'bisquare'
weight function
is the default.
Character vector — Name of the robust fitting
weight function from the following table. fitlm
uses
the corresponding default tuning constant in the table.
Structure with the character vector RobustWgtFun
containing
the name of the robust fitting weight function from the following
table and optional scalar Tune
fields — fitlm
uses
the RobustWgtFun
weight function and Tune
tuning
constant from the structure. You can choose the name of the robust
fitting weight function from this table. If you do not supply a Tune
field,
the fitting function uses the corresponding default tuning constant.
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 
'ols'  Ordinary least squares (no weighting function)  None 
'talwar'  w = 1 * (abs(r)<1)  2.795 
'welsch'  w = exp((r.^2))  2.985 
The value r in the weight functions is
r = resid/(tune*s*sqrt(1h))
,
where resid
is the vector of residuals from
the previous iteration, h
is the vector of leverage
values from a leastsquares fit, and s
is an estimate
of the standard deviation of the error term given by
s = MAD/0.6745
.
MAD is the median absolute deviation of the residuals from their
median. The constant 0.6745 makes the estimate unbiased for the normal
distribution. If there are p columns in X
,
the smallest p absolute deviations are excluded
when computing the median.
Default tuning constants give coefficient estimates that are approximately 95% as statistically efficient as the ordinary leastsquares estimates, provided the response has a normal distribution with no outliers. Decreasing the tuning constant increases the downweight assigned to large residuals; increasing the tuning constant decreases the downweight assigned to large residuals.
Structure with the function handle RobustWgtFun
and
optional scalar Tune
fields — You can specify
a custom weight function. fitlm
uses the RobustWgtFun
weight
function and Tune
tuning constant from the structure.
Specify RobustWgtFun
as a function handle that
accepts a vector of residuals, and returns a vector of weights the
same size. The fitting function scales the residuals, dividing by
the tuning constant (default 1
) and by an estimate
of the error standard deviation before it calls the weight function.
Example: 'RobustOpts','andrews'
'VarNames'
— Names of variables in fit{'x1','x2',...,'xn','y'}
(default)  cell array of character vectorsNames of variables in fit, specified as the commaseparated
pair consisting of 'VarNames'
and a 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.
For example, if in your data, horsepower, acceleration, and model year of the cars are the predictor variables, and miles per gallon (MPG) is the response variable, then you can name the variables as follows.
Example: 'VarNames',{'Horsepower','Acceleration','Model_Year','MPG'}
Data Types: cell
'Weights'
— Observation weightsones(n,1)
(default)  nby1 vector of nonnegative scalar valuesObservation weights, specified as the commaseparated pair consisting
of 'Weights'
and an nby1 vector
of nonnegative scalar values, where n is the number
of observations.
Data Types: single
 double
mdl
— Linear modelLinearModel
objectLinear model representing a leastsquares fit of the response
to the data, returned as a LinearModel
object.
If the value of the 'RobustOpts'
namevalue
pair is not []
or 'ols'
, the
model is not a leastsquares fit, but uses the robust fitting function.
For properties and methods of the linear model object, see the LinearModel
class page.
Fit a linear model of the Hald data.
Load the data.
load hald X = ingredients; % Predictor variables y = heat; % Response
Fit a default linear model to the data.
mdl = fitlm(X,y)
mdl = Linear regression model: y ~ 1 + x1 + x2 + x3 + x4 Estimated Coefficients: Estimate SE tStat pValue ________ _______ ________ ________ (Intercept) 62.405 70.071 0.8906 0.39913 x1 1.5511 0.74477 2.0827 0.070822 x2 0.51017 0.72379 0.70486 0.5009 x3 0.10191 0.75471 0.13503 0.89592 x4 0.14406 0.70905 0.20317 0.84407 Number of observations: 13, Error degrees of freedom: 8 Root Mean Squared Error: 2.45 Rsquared: 0.982, Adjusted RSquared 0.974 Fstatistic vs. constant model: 111, pvalue = 4.76e07
Fit a model of a table that contains a categorical predictor.
Load the carsmall
data.
load carsmall
Construct a table containing continuous predictor variable Weight
, nominal predictor variable Year
, and response variable MPG
.
tbl = table(MPG,Weight); tbl.Year = nominal(Model_Year);
Create a fitted model of MPG
as a function of Year
, Weight
, and Weight^2
. (You don't have to include Weight
explicitly in your formula because it is a lowerorder term of Weight^2
) and is included automatically.
mdl = fitlm(tbl,'MPG ~ Year + Weight^2')
mdl = Linear regression model: MPG ~ 1 + Weight + Year + Weight^2 Estimated Coefficients: Estimate SE tStat pValue __________ __________ _______ __________ (Intercept) 54.206 4.7117 11.505 2.6648e19 Weight 0.016404 0.0031249 5.2493 1.0283e06 Year_76 2.0887 0.71491 2.9215 0.0044137 Year_82 8.1864 0.81531 10.041 2.6364e16 Weight^2 1.5573e06 4.9454e07 3.149 0.0022303 Number of observations: 94, Error degrees of freedom: 89 Root Mean Squared Error: 2.78 Rsquared: 0.885, Adjusted RSquared 0.88 Fstatistic vs. constant model: 172, pvalue = 5.52e41
fitlm
creates two dummy (indicator) variables for the nominal variate, Year
. The dummy variable Year_76
takes the value 1 if model year is 1976 and takes the value 0 if it is not. The dummy variable Year_82
takes the value 1 if model year is 1982 and takes the value 0 if it is not. And the year 1970 is the reference year. The corresponding model is
Fit a linear regression model to sample data. Specify the response and predictor variables, and include only pairwise interaction terms in the model.
Load sample data.
load hospital
Fit a linear model with interaction terms to the data. Specify weight as the response variable, and sex, age, and smoking status as the predictor variables. Also, specify that sex and smoking status are categorical variables.
mdl = fitlm(hospital,'interactions','ResponseVar','Weight',... 'PredictorVars',{'Sex','Age','Smoker'},... 'CategoricalVar',{'Sex','Smoker'})
mdl = Linear regression model: Weight ~ 1 + Sex*Age + Sex*Smoker + Age*Smoker Estimated Coefficients: Estimate SE tStat pValue ________ _______ ________ __________ (Intercept) 118.7 7.0718 16.785 6.821e30 Sex_Male 68.336 9.7153 7.0339 3.3386e10 Age 0.31068 0.18531 1.6765 0.096991 Smoker_1 3.0425 10.446 0.29127 0.77149 Sex_Male:Age 0.49094 0.24764 1.9825 0.050377 Sex_Male:Smoker_1 0.9509 3.8031 0.25003 0.80312 Age:Smoker_1 0.07288 0.26275 0.27737 0.78211 Number of observations: 100, Error degrees of freedom: 93 Root Mean Squared Error: 8.75 Rsquared: 0.898, Adjusted RSquared 0.892 Fstatistic vs. constant model: 137, pvalue = 6.91e44
The weight of the patients do not seem to differ significantly according to age, or the status of smoking, or interaction of these factors with patient sex at the 5% significance level.
Fit a linear regression model using a robust fitting method.
Load the sample data.
load hald
The hald
data measures the effect of cement composition on its hardening heat. The matrix ingredients
contains the percent composition of four chemicals present in the cement. The array heat
contains the heat of hardening after 180 days for each cement sample.
Fit a robust linear model to the data.
mdl = fitlm(ingredients,heat,'linear','RobustOpts','on')
mdl = Linear regression model (robust fit): y ~ 1 + x1 + x2 + x3 + x4 Estimated Coefficients: Estimate SE tStat pValue ________ _______ ________ ________ (Intercept) 60.09 75.818 0.79256 0.4509 x1 1.5753 0.80585 1.9548 0.086346 x2 0.5322 0.78315 0.67957 0.51596 x3 0.13346 0.8166 0.16343 0.87424 x4 0.12052 0.7672 0.15709 0.87906 Number of observations: 13, Error degrees of freedom: 8 Root Mean Squared Error: 2.65 Rsquared: 0.979, Adjusted RSquared 0.969 Fstatistic vs. constant model: 94.6, pvalue = 9.03e07
A terms matrix is a tby(p + 1) matrix specifying terms in a model, where t is the number of terms, p is the number of predictor variables, and plus one is for the response variable.
The value of T(i,j)
is the exponent of variable j
in
term i
. Suppose there are three predictor variables A
, B
,
and C
:
[0 0 0 0] % Constant term or intercept [0 1 0 0] % B; equivalently, A^0 * B^1 * C^0 [1 0 1 0] % A*C [2 0 0 0] % A^2 [0 1 2 0] % B*(C^2)
0
at
the end of each term represents the response variable. In general,If you have the variables in a table or dataset array,
then 0
must represent the response variable depending
on the position of the response variable. The following example illustrates
this.
Load the sample data and define the dataset array.
load hospital dsa = dataset(hospital.Sex,hospital.BloodPressure(:,1),hospital.Age,... hospital.Smoker,'VarNames',{'Sex','BloodPressure','Age','Smoker'});
Represent the linear model 'BloodPressure ~ 1 + Sex
+ Age + Smoker'
in a terms matrix. The response variable
is in the second column of the dataset array, so there must be a column
of 0s for the response variable in the second column of the terms
matrix.
T = [0 0 0 0;1 0 0 0;0 0 1 0;0 0 0 1]
T = 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1
Redefine the dataset array.
dsa = dataset(hospital.BloodPressure(:,1),hospital.Sex,hospital.Age,... hospital.Smoker,'VarNames',{'BloodPressure','Sex','Age','Smoker'});
Now, the response variable is the first term in the dataset
array. Specify the same linear model, 'BloodPressure ~ 1
+ Sex + Age + Smoker'
, using a terms matrix.
T = [0 0 0 0;0 1 0 0;0 0 1 0;0 0 0 1]
T = 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
If you have the predictor and response variables in
a matrix and column vector, then you must include 0
for
the response variable at the end of each term. The following example
illustrates this.
Load the sample data and define the matrix of predictors.
load carsmall
X = [Acceleration,Weight];
Specify the model 'MPG ~ Acceleration + Weight + Acceleration:Weight
+ Weight^2'
using a term matrix and fit the model to the
data. This model includes the main effect and twoway interaction
terms for the variables, Acceleration
and Weight
,
and a secondorder term for the variable, Weight
.
T = [0 0 0;1 0 0;0 1 0;1 1 0;0 2 0]
T = 0 0 0 1 0 0 0 1 0 1 1 0 0 2 0
Fit a linear model.
mdl = fitlm(X,MPG,T)
mdl = Linear regression model: y ~ 1 + x1*x2 + x2^2 Estimated Coefficients: Estimate SE tStat pValue (Intercept) 48.906 12.589 3.8847 0.00019665 x1 0.54418 0.57125 0.95261 0.34337 x2 0.012781 0.0060312 2.1192 0.036857 x1:x2 0.00010892 0.00017925 0.6076 0.545 x2^2 9.7518e07 7.5389e07 1.2935 0.19917 Number of observations: 94, Error degrees of freedom: 89 Root Mean Squared Error: 4.1 Rsquared: 0.751, Adjusted RSquared 0.739 Fstatistic vs. constant model: 67, pvalue = 4.99e26
Only the intercept and x2
term, which correspond
to the Weight
variable, are significant at the
5% significance level.
Now, perform a stepwise regression with a constant model as the starting model and a linear model with interactions as the upper model.
T = [0 0 0;1 0 0;0 1 0;1 1 0];
mdl = stepwiselm(X,MPG,[0 0 0],'upper',T)
1. Adding x2, FStat = 259.3087, pValue = 1.643351e28 mdl = Linear regression model: y ~ 1 + x2 Estimated Coefficients: Estimate SE tStat pValue (Intercept) 49.238 1.6411 30.002 2.7015e49 x2 0.0086119 0.0005348 16.103 1.6434e28 Number of observations: 94, Error degrees of freedom: 92 Root Mean Squared Error: 4.13 Rsquared: 0.738, Adjusted RSquared 0.735 Fstatistic vs. constant model: 259, pvalue = 1.64e28
The results of the stepwise regression are consistent with the
results of fitlm
in the previous step.
A formula for model specification is a character
vector of the form '
Y
~ terms
'
where
Y
is the response name.
terms
contains
Variable names
+
means include the next variable

means do not include the next
variable
:
defines an interaction, a product
of terms
*
defines an interaction and all lowerorder terms
^
raises the predictor to a power,
exactly as in *
repeated, so ^
includes
lower order terms as well
()
groups terms
Note:
Formulas include a constant (intercept) term by default. To
exclude a constant term from the model, include 
For example,
'Y ~ A + B + C'
means a threevariable
linear model with intercept.'Y ~ A + B +
C  1'
is a threevariable linear model without intercept.'Y ~ A + B + C + B^2'
is a threevariable
model with intercept and a B^2
term.'Y
~ A + B^2 + C'
is the same as the previous example because B^2
includes
a B
term.'Y ~ A + B +
C + A:B'
includes an A*B
term.'Y
~ A*B + C'
is the same as the previous example because A*B
= A + B + A:B
.'Y ~ A*B*C  A:B:C'
has
all interactions among A
, B
,
and C
, except the threeway interaction.'Y
~ A*(B + C + D)'
has all linear terms, plus products of A
with
each of the other variables.
Wilkinson notation describes the factors present in models. The notation relates to factors present in models, not to the multipliers (coefficients) of those factors.
Wilkinson Notation  Factors in Standard Notation 

1  Constant (intercept) term 
A^k , where k is a positive
integer  A , A^{2} ,
..., A^{k} 
A + B  A , B 
A*B  A , B , A*B 
A:B  A*B only 
B  Do not include B 
A*B + C  A , B , C , A*B 
A + B + C + A:B  A , B , C , A*B 
A*B*C  A:B:C  A , B , C , A*B , A*C , B*C 
A*(B + C)  A , B , C , A*B , A*C 
Statistics and Machine Learning Toolbox™ notation always includes a constant term
unless you explicitly remove the term using 1
.
Use robust fitting (RobustOpts
namevalue
pair) to reduce the effect of outliers automatically.
Do not use robust fitting when you want to subsequently
adjust a model using step
.
For other methods or properties of the LinearModel
object,
see LinearModel
.
The main fitting algorithm is QR decomposition. For robust fitting,
the algorithm is robustfit
.
You can also construct a linear model using fitlm
.
You can construct a model in a range of possible models using stepwiselm
. However, you cannot use robust
regression and stepwise regression together.
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