Linear Regression

Prepare Data

To begin fitting a regression, put your data into a form that fitting functions expect. All regression techniques begin with input data in an array X and response data in a separate vector y, or input data in a table or dataset array tbl and response data as a column in tbl. Each row of the input data represents one observation. Each column represents one predictor (variable).

For a table or dataset array tbl, indicate the response variable with the 'ResponseVar' name-value pair:

mdl = fitlm(tbl,'ResponseVar','BloodPressure');

The response variable is the last column by default.

You can use numeric categorical predictors. A categorical predictor is one that takes values from a fixed set of possibilities.

Represent missing numeric data as NaN. To represent missing data for other data types, see Missing Group Values.

Dataset Array for Input and Response Data

To create a dataset array from an Excel^® spreadsheet:

ds = dataset('XLSFile','hospital.xls', ...
    'ReadObsNames',true);

To create a dataset array from workspace variables:

load carsmall
ds = dataset(MPG,Weight);
ds.Year = categorical(Model_Year);

Table for Input and Response Data

To create a table from an Excel spreadsheet:

tbl = readtable('hospital.xls', ...
    'ReadRowNames',true);

To create a table from workspace variables:

load carsmall
tbl = table(MPG,Weight);
tbl.Year = categorical(Model_Year);

Numeric Matrix for Input Data, Numeric Vector for Response

For example, to create numeric arrays from workspace variables:

load carsmall
X = [Weight Horsepower Cylinders Model_Year];
y = MPG;

To create numeric arrays from an Excel spreadsheet:

[X, Xnames] = xlsread('hospital.xls');
y = X(:,4); % response y is systolic pressure
X(:,4) = []; % remove y from the X matrix

Notice that the nonnumeric entries, such as sex, do not appear in X.

Choose a Fitting Method

There are three ways to fit a model to data:

• Least-Squares Fit

• Robust Fit

• Stepwise Fit

Least-Squares Fit

Use fitlm to construct a least-squares fit of a model to the data. This method is best when you are reasonably certain of the model’s form, and mainly need to find its parameters. This method is also useful when you want to explore a few models. The method requires you to examine the data manually to discard outliers, though there are techniques to help (see Examine Quality and Adjust Fitted Model).

Robust Fit

Use fitlm with the RobustOpts name-value pair to create a model that is little affected by outliers. Robust fitting saves you the trouble of manually discarding outliers. However, step does not work with robust fitting. This means that when you use robust fitting, you cannot search stepwise for a good model.

Stepwise Fit

Use stepwiselm to find a model, and fit parameters to the model. stepwiselm starts from one model, such as a constant, and adds or subtracts terms one at a time, choosing an optimal term each time in a greedy fashion, until it cannot improve further. Use stepwise fitting to find a good model, which is one that has only relevant terms.

The result depends on the starting model. Usually, starting with a constant model leads to a small model. Starting with more terms can lead to a more complex model, but one that has lower mean squared error. See Compare large and small stepwise models.

You cannot use robust options along with stepwise fitting. So after a stepwise fit, examine your model for outliers (see Examine Quality and Adjust Fitted Model).

Choose a Model or Range of Models

There are several ways of specifying a model for linear regression. Use whichever you find most convenient.

• Brief Name

• Terms Matrix

• Formula

For fitlm, the model specification you give is the model that is fit. If you do not give a model specification, the default is 'linear'.

For stepwiselm, the model specification you give is the starting model, which the stepwise procedure tries to improve. If you do not give a model specification, the default starting model is 'constant', and the default upper bounding model is 'interactions'. Change the upper bounding model using the Upper name-value pair.

Brief Name

For example, to specify an interaction model using fitlm with matrix predictors:

mdl = fitlm(X,y,'interactions');

To specify a model using stepwiselm and a table or dataset array tbl of predictors, suppose you want to start from a constant and have a linear model upper bound. Assume the response variable in tbl is in the third column.

mdl2 = stepwiselm(tbl,'constant', ...
    'Upper','linear','ResponseVar',3);

Terms Matrix

A terms matrix T 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 +1 accounts for the response variable. The value of T(i,j) is the exponent of variable j in term i.

For example, suppose that an input includes three predictor variables x1, x2, and x3 and the response variable y in the order x1, x2, x3, and y. Each row of T represents one term:

The 0 at the end of each term represents the response variable. In general, a column vector of zeros in a terms matrix represents the position of the response variable. If you have the predictor and response variables in a matrix and column vector, then you must include 0 for the response variable in the last column of each row.

Formula

A formula for a model specification is a character vector or string scalar of the form

Examples:

For example, to specify an interaction model using fitlm with matrix predictors:

mdl = fitlm(X,y,'y ~ x1*x2*x3 - x1:x2:x3');

To specify a model using stepwiselm and a table or dataset array tbl of predictors, suppose you want to start from a constant and have a linear model upper bound. Assume the response variable in tbl is named 'y', and the predictor variables are named 'x1', 'x2', and 'x3'.

mdl2 = stepwiselm(tbl,'y ~ 1','Upper','y ~ x1 + x2 + x3');

Fit Model to Data

The most common optional arguments for fitting:

For example,

mdl = fitlm(X,y,'linear', ...
    'RobustOpts','on','CategoricalVars',3);
mdl2 = stepwiselm(tbl,'constant', ...
    'ResponseVar','MPG','Upper','quadratic');