**Class: **RegressionLinear

Predict response of linear regression model

`Mdl`

— Linear regression model`RegressionLinear`

model objectLinear regression model, specified as a `RegressionLinear`

model
object. You can create a `RegressionLinear`

model
object using `fitrlinear`

.

`X`

— Predictor data used to generate responsesfull numeric matrix | sparse numeric matrix | table

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`

`dimension`

— Predictor data observation dimension`'rows'`

(default) | `'columns'`

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.

`YHat`

— Predicted responsesnumeric matrix

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(`

is
the response for observation * i*,

`j`

`i`

`Mdl.Lambda(``j`

)

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

*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`

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: [1x15 double] Lambda: [1x15 double] 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');

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

Usage notes and limitations:

`predict`

does not support tall`table`

data.

For more information, see Tall Arrays.

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

You can generate C/C++ code for both

`predict`

and`update`

by using a coder configurer. Or, generate code only for`predict`

by using`saveLearnerForCoder`

,`loadLearnerForCoder`

, and`codegen`

.Code generation for

`predict`

and`update`

— Create a coder configurer by using`learnerCoderConfigurer`

and then generate code by using`generateCode`

. Then you can update model parameters in the generated code without having to regenerate the code.Code generation for

`predict`

— Save a trained model by using`saveLearnerForCoder`

. Define an entry-point function that loads the saved model by using`loadLearnerForCoder`

and calls the`predict`

function. Then use`codegen`

(MATLAB Coder) to generate code for the entry-point function.

You can also generate single-precision C/C++ code for

`predict`

. For single-precision code generation, specify the name-value pair argument`'DataType','single'`

as an additional input to the`loadLearnerForCoder`

function.This table contains notes about the arguments of

`predict`

. Arguments not included in this table are fully supported.Argument Notes and Limitations `Mdl`

For the usage notes and limitations of the model object, see Code Generation of the

`RegressionLinear`

object.`X`

For general code generation,

`X`

must be a single-precision or double-precision matrix or a table containing`single`

or`double`

predictor variables.If you want to specify

`X`

as a table, then your model must be trained using a table, and you must ensure that your entry-point function for prediction:Accepts data as arrays

Creates a table from the data input arguments and specifies the variable names in the table

Passes the table to

`predict`

For an example of this table workflow, see Generate Code to Classify Numeric Data in Table. For more information on using tables in code generation, see Code Generation for Tables (MATLAB Coder) and Table Limitations for Code Generation (MATLAB Coder).

In the coder configurer workflow,

`X`

must be a`single`

or`double`

matrix.The number of observations in

`X`

can be a variable size, but the number of variables in`X`

must be fixed.

Name-value pair arguments Names in name-value pair arguments must be compile-time constants.

The value for the

`'ObservationsIn'`

name-value pair argument must be a compile-time constant. For example, to use the`'ObservationsIn','columns'`

name-value pair argument in the generated code, include`{coder.Constant('ObservationsIn'),coder.Constant('columns')}`

in the`-args`

value of`codegen`

(MATLAB Coder).

For more information, see Introduction to Code Generation.

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