fitlm

Create linear regression model

fitlm creates a LinearModel object. Once you create the object, you can see it in the workspace. You can see all the properties the object contains by clicking on it. You can create plots and do further diagnostic analysis by using methods such as plot, plotResiduals, and plotDiagnostics. For a full list of methods for LinearModel, see methods.

Syntax

mdl = fitlm(tbl)

mdl = fitlm(tbl,modelspec)

mdl = fitlm(X,y)

mdl = fitlm(X,y,modelspec)

mdl = fitlm(___,Name,Value)

Description

mdl = fitlm(tbl) returns a linear model fit to variables in the table or dataset array tbl. By default, fitlm takes the last variable as the response variable.

mdl = fitlm(tbl,modelspec) returns a linear model of the type you specify in modelspec fit to variables in the table or dataset array tbl.

mdl = fitlm(X,y) returns a linear model of the responses y, fit to the data matrix X.

mdl = fitlm(X,y,modelspec) returns a linear model of the type you specify in modelspec for the responses y, fit to the data matrix X.

mdl = fitlm(___,Name,Value) returns a linear model with additional options specified by one or more Name,Value pair arguments.

For example, you can specify which variables are categorical, perform robust regression, or use observation weights.

Examples

Fit Linear Regression Using Data in Table

Load the sample data.

load carsmall

Store the variables in a table.

tbl = table(Weight,Acceleration,MPG,'VariableNames',{'Weight','Acceleration','MPG'});

Display the first five rows of the table.

tbl(1:5,:)

ans=5×3 table
    Weight    Acceleration    MPG
    ______    ____________    ___

     3504           12        18 
     3693         11.5        15 
     3436           11        18 
     3433           12        16 
     3449         10.5        17 

Fit a linear regression model for miles per gallon (MPG).

lm = fitlm(tbl,'MPG~Weight+Acceleration')

lm = 
Linear regression model:
    MPG ~ 1 + Weight + Acceleration

Estimated Coefficients:
                     Estimate         SE         tStat       pValue  
                    __________    __________    _______    __________

    (Intercept)         45.155        3.4659     13.028    1.6266e-22
    Weight          -0.0082475    0.00059836    -13.783    5.3165e-24
    Acceleration       0.19694       0.14743     1.3359       0.18493


Number of observations: 94, Error degrees of freedom: 91
Root Mean Squared Error: 4.12
R-squared: 0.743,  Adjusted R-Squared 0.738
F-statistic vs. constant model: 132, p-value = 1.38e-27

This syntax uses Wilkinson notation to specify the modelspec.

The model 'MPG~Weight+Acceleration' in this example is equivalent to fitting the model using 'linear' as modelspec. For example,

lm2 = fitlm(tbl,'linear');

When you use a character vector as modelspec and do not specify the response variable, fitlm by default accepts the last variable in tbl as the response variable and the other variables as the predictor variables.

Fit Linear Regression Using Specified Model Formula

Fit a linear regression model using a model formula specified by Wilkinson notation.

Load the sample data.

load carsmall

Store the variables in a table.

tbl = table(Weight,Acceleration,Model_Year,MPG,'VariableNames',{'Weight','Acceleration','Model_Year','MPG'});

Fit a linear regression model for miles per gallon (MPG) with weight and acceleration as the predictor variables.

lm = fitlm(tbl,'MPG~Weight+Acceleration')

lm = 


Linear regression model:
    MPG ~ 1 + Weight + Acceleration

Estimated Coefficients:
                     Estimate         SE         tStat       pValue  
                    __________    __________    _______    __________

    (Intercept)         45.155        3.4659     13.028    1.6266e-22
    Weight          -0.0082475    0.00059836    -13.783    5.3165e-24
    Acceleration       0.19694       0.14743     1.3359       0.18493


Number of observations: 94, Error degrees of freedom: 91
Root Mean Squared Error: 4.12
R-squared: 0.743,  Adjusted R-Squared 0.738
F-statistic vs. constant model: 132, p-value = 1.38e-27

The -value of 0.18493 indicates that Acceleration does not have a significant impact on MPG.

Remove Acceleration from the model, and try improving the model by adding the predictor variable Model_Year. First define Model_Year as a categorical variable.

tbl.Model_Year = categorical(tbl.Model_Year);
lm = fitlm(tbl,'MPG~Weight+Model_Year')

lm = 


Linear regression model:
    MPG ~ 1 + Weight + Model_Year

Estimated Coefficients:
                      Estimate         SE         tStat       pValue  
                     __________    __________    _______    __________

    (Intercept)           40.11        1.5418     26.016    1.2024e-43
    Weight           -0.0066475    0.00042802    -15.531    3.3639e-27
    Model_Year_76        1.9291       0.74761     2.5804      0.011488
    Model_Year_82        7.9093       0.84975     9.3078    7.8681e-15


Number of observations: 94, Error degrees of freedom: 90
Root Mean Squared Error: 2.92
R-squared: 0.873,  Adjusted R-Squared 0.868
F-statistic vs. constant model: 206, p-value = 3.83e-40

Specifying modelspec using Wilkinson notation enables you to update the model without having to change the design matrix. fitlm uses only the variables that are specified in the formula. It also creates the necessary two dummy indicator variables for the categorical variable Model_Year.

Linear Regression with Categorical Predictor

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, categorical predictor variable Year, and response variable MPG.

tbl = table(MPG,Weight);
tbl.Year = categorical(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 lower-order term of Weight^2. The fitlm function includes Weight 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    2.6648e-19
    Weight          -0.016404     0.0031249    -5.2493    1.0283e-06
    Year_76            2.0887       0.71491     2.9215     0.0044137
    Year_82            8.1864       0.81531     10.041    2.6364e-16
    Weight^2       1.5573e-06    4.9454e-07      3.149     0.0022303


Number of observations: 94, Error degrees of freedom: 89
Root Mean Squared Error: 2.78
R-squared: 0.885,  Adjusted R-Squared 0.88
F-statistic vs. constant model: 172, p-value = 5.52e-41

fitlm creates two dummy (indicator) variables for the categorical 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

Specify Response and Predictor Variables for Linear Model

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     6.821e-30
    Sex_Male               68.336     9.7153      7.0339    3.3386e-10
    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
R-squared: 0.898,  Adjusted R-Squared 0.892
F-statistic vs. constant model: 137, p-value = 6.91e-44

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 Robust Linear Regression Model

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
R-squared: 0.979,  Adjusted R-Squared 0.969
F-statistic vs. constant model: 94.6, p-value = 9.03e-07

Input Arguments

tbl — Input data

table | dataset array

Input 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 name-value pair argument. To use a subset of the columns as predictors, use the PredictorVars name-value pair argument.

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.

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 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 | logical

modelspec — Model specification

'linear' (default) | character vector or string scalar naming the model | t-by-(p + 1) terms matrix | character vector or string scalar of the form 'Y ~ terms'

Model specification, specified as one of the following.

• A character vector or string scalar naming the model.

  Value	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.
  'polyijk'	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.

• t-by-(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 or string scalar representing a formula in the form

  'Y ~ terms',

  where the terms are in Wilkinson Notation.

Example: 'quadratic'

Example: 'y ~ X1 + X2^2 + X1:X2'

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Intercept',false,'PredictorVars',[1,3],'ResponseVar',5,'RobustOpts','logistic' specifies a robust regression model with no constant term, where the algorithm uses the logistic weighting function with the default tuning constant, first and third variables are the predictor variables, and fifth variable is the response variable.

'CategoricalVars' — Categorical variable list

string array | cell array of character vectors | logical or numeric index vector

Categorical variable list, specified as the comma-separated pair consisting of 'CategoricalVars' and either a string array or cell array of character vectors containing categorical variable names 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, by default, fitlm treats all categorical values, logical values, character arrays, string arrays, and cell arrays of character vectors as categorical variables.

• If data is in matrix X, then the default value of 'CategoricalVars' is an empty matrix []. That is, no variable is categorical unless you specify it as categorical.

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 | string | cell

'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

'Intercept' — Indicator for constant term

true (default) | false

Indicator for the constant term (intercept) in the fit, specified as the comma-separated 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 or string scalar, not a formula or matrix.

Example: 'Intercept',false

'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 in tbl (default) | character vector or string scalar containing 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 character vector or string scalar 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 | string

'RobustOpts' — Indicator of robust fitting type

'off' (default) | 'on' | character vector | string scalar | structure with character vector, string scalar, or function handle

Indicator of the robust fitting type to use, specified as the comma-separated 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 or string scalar — 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 or string scalar 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(1-h)),

  where resid is the vector of residuals from the previous iteration, h is the vector of leverage values from a least-squares 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 least-squares 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

{'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.

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: string | cell

'Weights' — Observation weights

ones(n,1) (default) | n-by-1 vector of nonnegative scalar values

Observation weights, specified as the comma-separated pair consisting of 'Weights' and an n-by-1 vector of nonnegative scalar values, where n is the number of observations.

Data Types: single | double

Output Arguments

mdl — Linear model

LinearModel object

Linear model representing a least-squares fit of the response to the data, returned as a LinearModel object.

If the value of the 'RobustOpts' name-value pair is not [] or 'ols', the model is not a least-squares fit, but uses the robust fitting function.

For more information on the properties and methods of the linear model object, mdl, see the LinearModel class page.

More About

Terms Matrix

A terms matrix is a t-by-(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)

The 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 using a table.

  Load the sample data and define a table.

  load hospital
  dsa = table(hospital.Sex,hospital.BloodPressure(:,1), ...
  hospital.Age,hospital.Smoker,'VariableNames', ...
  {'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 table, so the second column of the terms matrix must be a column of 0s for the response variable.

  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 table.

  dsa = table(hospital.BloodPressure(:,1),hospital.Sex, ...
  hospital.Age,hospital.Smoker,'VariableNames', ...
  {'BloodPressure','Sex','Age','Smoker'});
  

  Now, the response variable is the first term in the table. 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 two-way interaction terms for the variables, Acceleration and Weight, and a second-order 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            9.7518e-07    7.5389e-07     1.2935       0.19917
  
  Number of observations: 94, Error degrees of freedom: 89
  Root Mean Squared Error: 4.1
  R-squared: 0.751,  Adjusted R-Squared 0.739
  F-statistic vs. constant model: 67, p-value = 4.99e-26

  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 = 1.643351e-28
  
  mdl = 
  
  Linear regression model:
      y ~ 1 + x2
  
  Estimated Coefficients:
                     Estimate      SE           tStat      pValue    
      (Intercept)        49.238       1.6411     30.002    2.7015e-49
      x2             -0.0086119    0.0005348    -16.103    1.6434e-28
  
  Number of observations: 94, Error degrees of freedom: 92
  Root Mean Squared Error: 4.13
  R-squared: 0.738,  Adjusted R-Squared 0.735
  F-statistic vs. constant model: 259, p-value = 1.64e-28

  The results of the stepwise regression are consistent with the results of fitlm in the previous step.

Formula

A formula for model specification is a character vector or string scalar 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 lower-order 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 -1 in the formula.

For example,

'Y ~ A + B + C' means a three-variable linear model with intercept.
'Y ~ A + B + C - 1' is a three-variable linear model without intercept.
'Y ~ A + B + C + B^2' is a three-variable 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 three-way interaction.
'Y ~ A*(B + C + D)' has all linear terms, plus products of A with each of the other variables.

Wilkinson Notation

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, A2, ..., Ak
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.

Tips

• To access the model properties of the LinearModel object mdl, you can use dot notation. For example, mdl.Residuals returns a table of the raw, Pearson, Studentized, and standardized residual values for the model.

• After training a model, you can generate C/C++ code that predicts responses for new data. Generating C/C++ code requires MATLAB^® Coder™. For details, see Introduction to Code Generation.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function supports tall arrays for out-of-memory data with some limitations.

• If any input argument to fitlm is a tall array, then all of the other inputs must be tall arrays as well. This includes nonempty variables supplied with the 'Weights' and 'Exclude' name-value pairs.

• The 'RobustOpts' name-value pair is not supported with tall arrays.

• For tall data, fitlm returns a CompactLinearModel object that contains most of the same properties as a LinearModel object. The main difference is that the compact object is sensitive to memory requirements. The compact object does not include properties that include the data, or that include an array of the same size as the data. The compact object does not contain these LinearModel properties:

  • Diagnostics

  • Fitted

  • ObservationInfo

  • ObservationNames

  • Residuals

  • Steps

  • Variables

  You can compute the residuals directly from the compact object returned by LM = fitlm(X,Y) using

  RES = Y - predict(LM,X);
  S = LM.RMSE;
  histogram(RES,linspace(-3*S,3*S,51))
  

• If the CompactLinearModel object is missing lower order terms that include categorical factors:

  • The plotEffects and plotInteraction methods are not supported.

  • The anova method with the 'components' option is not supported.

For more information, see Tall Arrays (MATLAB).