# predict

**Class: **RegressionLinear

Predict response of linear regression model

## Description

specifies additional options using one or more name-value arguments. For example,
specify that columns in the predictor data correspond to observations.`YHat`

= predict(`Mdl`

,`X`

,`Name,Value`

)

## Input Arguments

`Mdl`

— Linear regression model

`RegressionLinear`

model object

Linear regression model, specified as a `RegressionLinear`

model
object. You can create a `RegressionLinear`

model
object using `fitrlinear`

.

`X`

— Predictor data used to generate responses

full 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`

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`predict(Mdl,X,"ObservationsIn","columns")`

indicates
that columns in the predictor data correspond to observations.

`ObservationsIn`

— 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.

**Data Types: **`char`

| `string`

`PredictionForMissingValue`

— Predicted response value to use for observations with missing predictor values

`"median"`

(default) | `"mean"`

| numeric scalar

*Since R2023b*

Predicted response value to use for observations with missing predictor values, specified as `"median"`

, `"mean"`

, or a numeric scalar.

Value | Description |
---|---|

`"median"` | `predict` uses the median of the observed response values in the training data as the predicted response value for observations with missing predictor values. |

`"mean"` | `predict` uses the mean of the observed response values in the training data as the predicted response value for observations with missing predictor values. |

Numeric scalar | `predict` uses this value as the predicted response value for observations with missing predictor values. |

**Example: **`"PredictionForMissingValue","mean"`

**Example: **`"PredictionForMissingValue",NaN`

**Data Types: **`single`

| `double`

| `char`

| `string`

## Output Arguments

`YHat`

— Predicted responses

numeric 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`

*using the linear regression model that has regularization strength*

`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`

## Examples

### Predict Test-Sample Responses

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'

`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 from Best-Performing Model

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 -0.0044 -0.0037 -0.0030 -0.0031 -0.0033 -0.0036 -0.0041 -0.0051 -0.0071] Lambda: [1.0000e-05 1.9307e-05 3.7276e-05 7.1969e-05 1.3895e-04 2.6827e-04 5.1795e-04 1.0000e-03 0.0019 0.0037 0.0072 0.0139 0.0268 0.0518 0.1000] Learner: 'leastsquares'

`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'

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');

## Alternative Functionality

### Simulink Block

To integrate the prediction of a linear regression model into Simulink^{®}, you can use the RegressionLinear
Predict block in the Statistics and Machine Learning Toolbox™ library or a MATLAB^{®} Function block with the `predict`

function. For
examples, see Predict Responses Using RegressionLinear Predict Block and Predict Class Labels Using MATLAB Function Block.

When deciding which approach to use, consider the following:

If you use the Statistics and Machine Learning Toolbox library block, you can use the Fixed-Point Tool (Fixed-Point Designer) to convert a floating-point model to fixed point.

Support for variable-size arrays must be enabled for a MATLAB Function block with the

`predict`

function.If you use a MATLAB Function block, you can use MATLAB functions for preprocessing or post-processing before or after predictions in the same MATLAB Function block.

## Extended Capabilities

### Tall Arrays

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.

### C/C++ Code Generation

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.

To generate single-precision C/C++ code for

`predict`

, specify the name-value argument`"DataType","single"`

when you call 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 numeric variables, categorical variables, or both.In the coder configurer workflow,

`X`

must be a single-precision or double-precision matrix.The number of observations in

`X`

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

must be fixed.If you want to specify

`X`

as a table, then your model must be trained using a table, and your entry-point function for prediction must do the following:Accept data as arrays.

Create a table from the data input arguments and specify the variable names in the table.

Pass the table to

`predict`

.

For an example of this table workflow, see Generate Code to Classify 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).

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

The

`ObservationsIn`

value must be a compile-time constant. For example, to use`"ObservationsIn","columns"`

in the generated code, include`{coder.Constant("ObservationsIn"),coder.Constant("columns")}`

in the`-args`

value of`codegen`

(MATLAB Coder).If the value of

`PredictionForMissingValue`

is nonnumeric, then it must be a compile-time constant.

For more information, see Introduction to Code Generation.

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

## Version History

**Introduced in R2016a**

### R2024a: GPU array support

Starting in R2024a, `predict`

accepts GPU array input
arguments with some limitations.

### R2023b: Specify predicted response value to use for observations with missing predictor values

Starting in R2023b, when you predict or compute the loss, some regression models allow you to specify the predicted response value for observations with missing predictor values. Specify the `PredictionForMissingValue`

name-value argument to use a numeric scalar, the training set median, or the training set mean as the predicted value. When computing the loss, you can also specify to omit observations with missing predictor values.

This table lists the object functions that support the
`PredictionForMissingValue`

name-value argument. By default, the
functions use the training set median as the predicted response value for observations with
missing predictor values.

Model Type | Model Objects | Object Functions |
---|---|---|

Gaussian process regression (GPR) model | `RegressionGP` , `CompactRegressionGP` | `loss` , `predict` , `resubLoss` , `resubPredict` |

`RegressionPartitionedGP` | `kfoldLoss` , `kfoldPredict` | |

Gaussian kernel regression model | `RegressionKernel` | `loss` , `predict` |

`RegressionPartitionedKernel` | `kfoldLoss` , `kfoldPredict` | |

Linear regression model | `RegressionLinear` | `loss` , `predict` |

`RegressionPartitionedLinear` | `kfoldLoss` , `kfoldPredict` | |

Neural network regression model | `RegressionNeuralNetwork` , `CompactRegressionNeuralNetwork` | `loss` , `predict` , `resubLoss` , `resubPredict` |

`RegressionPartitionedNeuralNetwork` | `kfoldLoss` , `kfoldPredict` | |

Support vector machine (SVM) regression model | `RegressionSVM` , `CompactRegressionSVM` | `loss` , `predict` , `resubLoss` , `resubPredict` |

`RegressionPartitionedSVM` | `kfoldLoss` , `kfoldPredict` |

In previous releases, the regression model `loss`

and `predict`

functions listed above used `NaN`

predicted response values for observations with missing predictor values. The software omitted observations with missing predictor values from the resubstitution ("resub") and cross-validation ("kfold") computations for prediction and loss.

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