# predict

Class: RegressionLinear

Predict response of linear regression model

## Description

example

YHat = predict(Mdl,X) returns predicted responses for each observation in the predictor data X based on the trained linear regression model Mdl. YHat contains responses for each regularization strength in Mdl.

example

YHat = predict(Mdl,X,'ObservationsIn',dimension) specifies the predictor data observation dimension, either 'rows' (default) or 'columns'. For example, specify 'ObservationsIn','columns' to indicate that columns in the predictor data correspond to observations.

## Input Arguments

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Linear regression model, specified as a RegressionLinear model object. You can create a RegressionLinear model object using fitrlinear.

Predictor data used to generate responses, specified as a full or sparse numeric matrix or a table.

By default, each row of X corresponds to one observation, and each column corresponds to one variable.

• For a numeric matrix:

• The variables in the columns of X must have the same order as the predictor variables that trained Mdl.

• If you train Mdl using a table (for example, Tbl) and Tbl contains only numeric predictor variables, then X can be a numeric matrix. To treat numeric predictors in Tbl as categorical during training, identify categorical predictors by using the CategoricalPredictors name-value pair argument of fitrlinear. If Tbl contains heterogeneous predictor variables (for example, numeric and categorical data types) and X is a numeric matrix, then predict throws an error.

• For a table:

• predict does not support multicolumn variables or cell arrays other than cell arrays of character vectors.

• If you train Mdl using a table (for example, Tbl), then all predictor variables in X must have the same variable names and data types as the variables that trained Mdl (stored in Mdl.PredictorNames). However, the column order of X does not need to correspond to the column order of Tbl. Also, Tbl and X can contain additional variables (response variables, observation weights, and so on), but predict ignores them.

• If you train Mdl using a numeric matrix, then the predictor names in Mdl.PredictorNames must be the same as the corresponding predictor variable names in X. To specify predictor names during training, use the PredictorNames name-value pair argument of fitrlinear. All predictor variables in X must be numeric vectors. X can contain additional variables (response variables, observation weights, and so on), but predict ignores them.

Note

If you orient your predictor matrix so that observations correspond to columns and specify 'ObservationsIn','columns', then you might experience a significant reduction in optimization execution time. You cannot specify 'ObservationsIn','columns' for predictor data in a table.

Data Types: double | single | table

Predictor data observation dimension, specified as 'columns' or 'rows'.

Note

If you orient your predictor matrix so that observations correspond to columns and specify 'ObservationsIn','columns', then you might experience a significant reduction in optimization execution time. You cannot specify 'ObservationsIn','columns' for predictor data in a table.

## Output Arguments

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Predicted responses, returned as a n-by-L numeric matrix. n is the number of observations in X and L is the number of regularization strengths in Mdl.Lambda. YHat(i,j) is the response for observation i using the linear regression model that has regularization strength Mdl.Lambda(j).

The predicted response using the model with regularization strength j is ${\stackrel{^}{y}}_{j}=x{\beta }_{j}+{b}_{j}.$

• x is an observation from the predictor data matrix X, and is row vector.

• ${\beta }_{j}$ is the estimated column vector of coefficients. The software stores this vector in Mdl.Beta(:,j).

• ${b}_{j}$ is the estimated, scalar bias, which the software stores in Mdl.Bias(j).

## Examples

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Simulate 10000 observations from this model

$y={x}_{100}+2{x}_{200}+e.$

• $X={x}_{1},...,{x}_{1000}$ is a 10000-by-1000 sparse matrix with 10% nonzero standard normal elements.

• e is random normal error with mean 0 and standard deviation 0.3.

rng(1) % For reproducibility
n = 1e4;
d = 1e3;
nz = 0.1;
X = sprandn(n,d,nz);
Y = X(:,100) + 2*X(:,200) + 0.3*randn(n,1);

Train a linear regression model. Reserve 30% of the observations as a holdout sample.

CVMdl = fitrlinear(X,Y,'Holdout',0.3);
Mdl = CVMdl.Trained{1}
Mdl =
RegressionLinear
ResponseName: 'Y'
ResponseTransform: 'none'
Beta: [1000x1 double]
Bias: -0.0066
Lambda: 1.4286e-04
Learner: 'svm'

Properties, Methods

CVMdl is a RegressionPartitionedLinear model. It contains the property Trained, which is a 1-by-1 cell array holding a RegressionLinear model that the software trained using the training set.

Extract the training and test data from the partition definition.

trainIdx = training(CVMdl.Partition);
testIdx = test(CVMdl.Partition);

Predict the training- and test-sample responses.

yHatTrain = predict(Mdl,X(trainIdx,:));
yHatTest = predict(Mdl,X(testIdx,:));

Because there is one regularization strength in Mdl, yHatTrain and yHatTest are numeric vectors.

Predict responses from the best-performing, linear regression model that uses a lasso-penalty and least squares.

Simulate 10000 observations as in Predict Test-Sample Responses.

rng(1) % For reproducibility
n = 1e4;
d = 1e3;
nz = 0.1;
X = sprandn(n,d,nz);
Y = X(:,100) + 2*X(:,200) + 0.3*randn(n,1);

Create a set of 15 logarithmically-spaced regularization strengths from $1{0}^{-5}$ through $1{0}^{-1}$.

Lambda = logspace(-5,-1,15);

Cross-validate the models. To increase execution speed, transpose the predictor data and specify that the observations are in columns. Optimize the objective function using SpaRSA.

X = X';
CVMdl = fitrlinear(X,Y,'ObservationsIn','columns','KFold',5,'Lambda',Lambda,...
'Learner','leastsquares','Solver','sparsa','Regularization','lasso');

numCLModels = numel(CVMdl.Trained)
numCLModels = 5

CVMdl is a RegressionPartitionedLinear model. Because fitrlinear implements 5-fold cross-validation, CVMdl contains 5 RegressionLinear models that the software trains on each fold.

Display the first trained linear regression model.

Mdl1 = CVMdl.Trained{1}
Mdl1 =
RegressionLinear
ResponseName: 'Y'
ResponseTransform: 'none'
Beta: [1000x15 double]
Bias: [-0.0049 -0.0049 -0.0049 -0.0049 -0.0049 -0.0048 ... ]
Lambda: [1.0000e-05 1.9307e-05 3.7276e-05 7.1969e-05 ... ]
Learner: 'leastsquares'

Properties, Methods

Mdl1 is a RegressionLinear model object. fitrlinear constructed Mdl1 by training on the first four folds. Because Lambda is a sequence of regularization strengths, you can think of Mdl1 as 11 models, one for each regularization strength in Lambda.

Estimate the cross-validated MSE.

mse = kfoldLoss(CVMdl);

Higher values of Lambda lead to predictor variable sparsity, which is a good quality of a regression model. For each regularization strength, train a linear regression model using the entire data set and the same options as when you cross-validated the models. Determine the number of nonzero coefficients per model.

Mdl = fitrlinear(X,Y,'ObservationsIn','columns','Lambda',Lambda,...
'Learner','leastsquares','Solver','sparsa','Regularization','lasso');
numNZCoeff = sum(Mdl.Beta~=0);

In the same figure, plot the cross-validated MSE and frequency of nonzero coefficients for each regularization strength. Plot all variables on the log scale.

figure;
[h,hL1,hL2] = plotyy(log10(Lambda),log10(mse),...
log10(Lambda),log10(numNZCoeff));
hL1.Marker = 'o';
hL2.Marker = 'o';
ylabel(h(1),'log_{10} MSE')
ylabel(h(2),'log_{10} nonzero-coefficient frequency')
xlabel('log_{10} Lambda')
hold off

Choose the index of the regularization strength that balances predictor variable sparsity and low MSE (for example, Lambda(10)).

idxFinal = 10;

Extract the model with corresponding to the minimal MSE.

MdlFinal = selectModels(Mdl,idxFinal)
MdlFinal =
RegressionLinear
ResponseName: 'Y'
ResponseTransform: 'none'
Beta: [1000x1 double]
Bias: -0.0050
Lambda: 0.0037
Learner: 'leastsquares'

Properties, Methods

idxNZCoeff = find(MdlFinal.Beta~=0)
idxNZCoeff = 2×1

100
200

EstCoeff = Mdl.Beta(idxNZCoeff)
EstCoeff = 2×1

1.0051
1.9965

MdlFinal is a RegressionLinear model with one regularization strength. The nonzero coefficients EstCoeff are close to the coefficients that simulated the data.

Simulate 10 new observations, and predict corresponding responses using the best-performing model.

XNew = sprandn(d,10,nz);
YHat = predict(MdlFinal,XNew,'ObservationsIn','columns');

## Version History

Introduced in R2016a