# loss

## Description

returns the regression loss for the trained regression neural network
`L`

= loss(`Mdl`

,`Tbl`

,`ResponseVarName`

)`Mdl`

using the predictor data in table `Tbl`

and
the response values in the `ResponseVarName`

table variable. By
default, the regression loss is the mean squared error (MSE).

specifies options using one or more name-value arguments in addition to any of the input
argument combinations in previous syntaxes. For example, you can specify that columns in
the predictor data correspond to observations, specify the loss function, or supply
observation weights.`L`

= loss(___,`Name=Value`

)

## Examples

### Test Set Mean Squared Error of Neural Network

Calculate the test set mean squared error (MSE) of a regression neural network model.

Load the `patients`

data set. Create a table from the data set. Each row corresponds to one patient, and each column corresponds to a diagnostic variable. Use the `Systolic`

variable as the response variable, and the rest of the variables as predictors.

```
load patients
tbl = table(Diastolic,Height,Smoker,Weight,Systolic);
```

Separate the data into a training set `tblTrain`

and a test set `tblTest`

by using a nonstratified holdout partition. The software reserves approximately 30% of the observations for the test data set and uses the rest of the observations for the training data set.

rng("default") % For reproducibility of the partition c = cvpartition(size(tbl,1),"Holdout",0.30); trainingIndices = training(c); testIndices = test(c); tblTrain = tbl(trainingIndices,:); tblTest = tbl(testIndices,:);

Train a regression neural network model using the training set. Specify the `Systolic`

column of `tblTrain`

as the response variable. Specify to standardize the numeric predictors, and set the iteration limit to 50.

Mdl = fitrnet(tblTrain,"Systolic", ... "Standardize",true,"IterationLimit",50);

Calculate the test set MSE. Smaller MSE values indicate better performance.

`testMSE = loss(Mdl,tblTest,"Systolic")`

testMSE = 22.2447

### Select Features to Include in Regression Neural Network

Perform feature selection by comparing test set losses and predictions. Compare the test set metrics for a regression neural network model trained using all the predictors to the test set metrics for a model trained using only a subset of the predictors.

Load the sample file `fisheriris.csv`

, which contains iris data including sepal length, sepal width, petal length, petal width, and species type. Read the file into a table.

`fishertable = readtable('fisheriris.csv');`

Separate the data into a training set `trainTbl`

and a test set `testTbl`

by using a nonstratified holdout partition. The software reserves approximately 30% of the observations for the test data set and uses the rest of the observations for the training data set.

rng("default") c = cvpartition(size(fishertable,1),"Holdout",0.3); trainTbl = fishertable(training(c),:); testTbl = fishertable(test(c),:);

Train one regression neural network model using all the predictors in the training set, and train another model using all the predictors except `PetalWidth`

. For both models, specify `PetalLength`

as the response variable, and standardize the predictors.

allMdl = fitrnet(trainTbl,"PetalLength","Standardize",true); subsetMdl = fitrnet(trainTbl,"PetalLength ~ SepalLength + SepalWidth + Species", ... "Standardize",true);

Compare the test set mean squared error (MSE) of the two models. Smaller MSE values indicate better performance.

allMSE = loss(allMdl,testTbl)

allMSE = 0.0834

subsetMSE = loss(subsetMdl,testTbl)

subsetMSE = 0.0884

For each model, compare the test set predicted petal lengths to the true petal lengths. Plot the predicted petal lengths along the vertical axis and the true petal lengths along the horizontal axis. Points on the reference line indicate correct predictions.

tiledlayout(2,1) % Top axes ax1 = nexttile; allPredictedY = predict(allMdl,testTbl); plot(ax1,testTbl.PetalLength,allPredictedY,".") hold on plot(ax1,testTbl.PetalLength,testTbl.PetalLength) hold off xlabel(ax1,"True Petal Length") ylabel(ax1,"Predicted Petal Length") title(ax1,"All Predictors") % Bottom axes ax2 = nexttile; subsetPredictedY = predict(subsetMdl,testTbl); plot(ax2,testTbl.PetalLength,subsetPredictedY,".") hold on plot(ax2,testTbl.PetalLength,testTbl.PetalLength) hold off xlabel(ax2,"True Petal Length") ylabel(ax2,"Predicted Petal Length") title(ax2,"Subset of Predictors")

Because both models seems to perform well, with predictions scattered near the reference line, consider using the model trained using all predictors except `PetalWidth`

.

### Specify Multiple Response Variables in Neural Network

*Since R2024b*

Create a regression neural network with more than one response variable.

Load the `carbig`

data set, which contains measurements of cars made in the 1970s and early 1980s. Create a table containing the predictor variables `Displacement`

, `Horsepower`

, and so on, as well as the response variables `Acceleration`

and `MPG`

. Display the first eight rows of the table.

load carbig cars = table(Displacement,Horsepower,Model_Year, ... Origin,Weight,Acceleration,MPG); head(cars)

Displacement Horsepower Model_Year Origin Weight Acceleration MPG ____________ __________ __________ _______ ______ ____________ ___ 307 130 70 USA 3504 12 18 350 165 70 USA 3693 11.5 15 318 150 70 USA 3436 11 18 304 150 70 USA 3433 12 16 302 140 70 USA 3449 10.5 17 429 198 70 USA 4341 10 15 454 220 70 USA 4354 9 14 440 215 70 USA 4312 8.5 14

Remove rows of `cars`

where the table has missing values.

cars = rmmissing(cars);

Categorize the cars based on whether they were made in the USA.

cars.Origin = categorical(cellstr(cars.Origin)); cars.Origin = mergecats(cars.Origin,["France","Japan",... "Germany","Sweden","Italy","England"],"NotUSA");

Partition the data into training and test sets. Use approximately 85% of the observations to train a neural network model, and 15% of the observations to test the performance of the trained model on new data. Use `cvpartition`

to partition the data.

rng("default") % For reproducibility c = cvpartition(height(cars),"Holdout",0.15); carsTrain = cars(training(c),:); carsTest = cars(test(c),:);

Train a multiresponse neural network regression model by passing the `carsTrain`

training data to the `fitrnet`

function. For better results, specify to standardize the predictor data.

Mdl = fitrnet(carsTrain,["Acceleration","MPG"], ... Standardize=true)

Mdl = RegressionNeuralNetwork PredictorNames: {'Displacement' 'Horsepower' 'Model_Year' 'Origin' 'Weight'} ResponseName: {'Acceleration' 'MPG'} CategoricalPredictors: 4 ResponseTransform: 'none' NumObservations: 334 LayerSizes: 10 Activations: 'relu' OutputLayerActivation: 'none' Solver: 'LBFGS' ConvergenceInfo: [1x1 struct] TrainingHistory: [1000x7 table]

`Mdl`

is a trained `RegressionNeuralNetwork`

model. You can use dot notation to access the properties of `Mdl`

. For example, you can specify `Mdl.ConvergenceInfo`

to get more information about the model convergence.

Evaluate the performance of the regression model on the test set by computing the test mean squared error (MSE). Smaller MSE values indicate better performance. Return the loss for each response variable separately by setting the `OutputType`

name-value argument to `"per-response"`

.

testMSE = loss(Mdl,carsTest,["Acceleration","MPG"], ... OutputType="per-response")

`testMSE = `*1×2*
1.5341 4.8245

Predict the response values for the observations in the test set. Return the predicted response values as a table.

`predictedY = predict(Mdl,carsTest,OutputType="table")`

`predictedY=`*58×2 table*
Acceleration MPG
____________ ______
9.3612 13.567
15.655 21.406
17.921 17.851
11.139 13.433
12.696 10.32
16.498 17.977
16.227 22.016
12.165 12.926
12.691 12.072
12.424 14.481
16.974 22.152
15.504 24.955
11.068 13.874
11.978 12.664
14.926 10.134
15.638 24.839
⋮

## Input Arguments

`Mdl`

— Trained regression neural network

`RegressionNeuralNetwork`

model object | `CompactRegressionNeuralNetwork`

model object

Trained regression neural network, specified as a `RegressionNeuralNetwork`

model object or `CompactRegressionNeuralNetwork`

model object returned by `fitrnet`

or
`compact`

,
respectively.

`Tbl`

— Sample data

table

Sample data, specified as a table. Each row of `Tbl`

corresponds
to one observation, and each column corresponds to one predictor variable. Optionally,
`Tbl`

can contain additional columns for the response variables and
a column for the observation weights. `Tbl`

must contain all of the
predictors used to train `Mdl`

. Multicolumn variables and cell arrays
other than cell arrays of character vectors are not allowed.

If

`Tbl`

contains the response variables used to train`Mdl`

, then you do not need to specify`ResponseVarName`

or`Y`

.If you trained

`Mdl`

using sample data contained in a table, then the input data for`loss`

must also be in a table.If you set

`Standardize=true`

in`fitrnet`

when training`Mdl`

, then the software standardizes the numeric columns of the predictor data using the corresponding means (`Mdl.Mu`

) and standard deviations (`Mdl.Sigma`

).

**Data Types: **`table`

`ResponseVarName`

— Response variable names

names of variables in `Tbl`

Response variable names, specified as the names of variables in
`Tbl`

. Each response variable must be a numeric vector.

You must specify `ResponseVarName`

as a character vector, string
array, or cell array of character vectors. For example, if `Tbl`

stores
the response variable as `Tbl.Y`

, then specify
`ResponseVarName`

as `"Y"`

. Otherwise, the
software treats the `Y`

column of `Tbl`

as a
predictor.

**Data Types: **`char`

| `string`

| `cell`

`Y`

— Response data

numeric vector | numeric matrix | numeric table

`X`

— Predictor data

numeric matrix

Predictor data, specified as a numeric matrix. By default,
`loss`

assumes that each row of `X`

corresponds to one observation, and each column corresponds to one predictor
variable.

**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 computation time.

`X`

and `Y`

must have the same number of
observations.

If you set `Standardize=true`

in `fitrnet`

when
training `Mdl`

, then the software standardizes the numeric columns of
the predictor data using the corresponding means (`Mdl.Mu`

) and
standard deviations (`Mdl.Sigma`

).

**Data Types: **`single`

| `double`

### 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.

**Example: **`loss(Mdl,Tbl,"Response",Weights="W")`

specifies to use the
`Response`

and `W`

variables in the table
`Tbl`

as the response values and observation weights,
respectively.

`LossFun`

— Loss function

`"mse"`

(default) | function handle

Loss function, specified as `"mse"`

or a function handle.

`"mse"`

— Weighted mean squared error.Function handle — To specify a custom loss function, use a function handle. The function must have this form:

lossval =

*lossfun*(Y,YFit,W)The output argument

`lossval`

is a floating-point scalar.You specify the function name (

).`lossfun`

If

`Mdl`

is a model with one response variable, then`Y`

is a length-*n*numeric vector of observed responses, where*n*is the number of observations in`Tbl`

or`X`

. If`Mdl`

is a model with multiple response variables, then`Y`

is an*n*-by-*k*numeric matrix of observed responses, where*k*is the number of response variables.`YFit`

is a length-*n*numeric vector or an*n*-by-*k*numeric matrix of corresponding predicted responses. The size of`YFit`

must match the size of`Y`

.`W`

is an*n*-by-1 numeric vector of observation weights.

**Example: **`LossFun=@`

`lossfun`

**Data Types: **`char`

| `string`

| `function_handle`

`ObservationsIn`

— Predictor data observation dimension

`"rows"`

(default) | `"columns"`

Predictor data observation dimension, specified as `"rows"`

or
`"columns"`

.

**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 computation time. You cannot specify
`ObservationsIn="columns"`

for predictor data in a table or for
multiresponse regression.

**Data Types: **`char`

| `string`

`OutputType`

— Type of output loss

`"average"`

(default) | `"per-response"`

*Since R2024b*

Type of output loss, specified as `"average"`

or
`"per-response"`

.

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

`"average"` | `loss` averages the loss values across all
response variables and returns a scalar value. |

`"per-response"` | `loss` returns a vector, where each element
is the loss for one response variable. |

**Example: **`OutputType="per-response"`

**Data Types: **`char`

| `string`

`PredictionForMissingValue`

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

`"median"`

(default) | `"mean"`

| `"omitted"`

| numeric scalar

*Since R2023b*

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

, `"mean"`

,
`"omitted"`

, or a numeric scalar.

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

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

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

`"omitted"` | `loss` excludes observations with missing
predictor values from the loss computation. |

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

If an observation is missing an observed response value or an observation weight, then
`loss`

does not use the observation in the loss
computation.

**Example: **`PredictionForMissingValue="omitted"`

**Data Types: **`single`

| `double`

| `char`

| `string`

`StandardizeResponses`

— Flag to standardize response data

`false`

or `0`

(default) | `true`

or `1`

*Since R2024b*

Flag to standardize the response data before computing the loss, specified as a
numeric or logical `0`

(`false`

) or
`1`

(`true`

). If you set
`StandardizeResponses`

to `true`

, then the
software centers and scales each response variable by the corresponding variable mean
and standard deviation in the training data.

Specify `StandardizeResponses`

as `true`

when
you have multiple response variables with very different scales and
`OutputType`

is `"average"`

. Do not standardize
the response data when you have only one response variable.

**Example: **`StandardizeResponses=true`

**Data Types: **`single`

| `double`

| `logical`

`Weights`

— Observation weights

nonnegative numeric vector | name of variable in `Tbl`

Observation weights, specified as a nonnegative numeric vector or the name of a
variable in `Tbl`

. The software weights each observation in
`X`

or `Tbl`

with the corresponding value in
`Weights`

. The length of `Weights`

must equal
the number of observations in `X`

or
`Tbl`

.

If you specify the input data as a table `Tbl`

, then
`Weights`

can be the name of a variable in
`Tbl`

that contains a numeric vector. In this case, you must
specify `Weights`

as a character vector or string scalar. For
example, if the weights vector `W`

is stored as
`Tbl.W`

, then specify it as `"W"`

.

By default, `Weights`

is `ones(n,1)`

, where
`n`

is the number of observations in `X`

or
`Tbl`

.

If you supply weights, then `loss`

computes the weighted
regression loss and normalizes weights to sum to 1.

**Data Types: **`single`

| `double`

| `char`

| `string`

## Output Arguments

`L`

— Regression loss

numeric scalar | numeric vector

Regression loss, returned as a numeric scalar or vector. The type of regression loss
depends on `LossFun`

.

When `Mdl`

is a model with one response variable,
`L`

is a numeric scalar. When `Mdl`

is a model
with multiple response variables, the size and interpretation of `L`

depend on `OutputType`

.

## Extended Capabilities

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™. (since R2024b)

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

## Version History

**Introduced in R2021a**

### R2024b: Compute loss for neural network regression model trained with multiple response variables

You can create a neural network regression model with multiple response variables by
using the `fitrnet`

function.
Regardless of the number of response variables, the function returns a
`RegressionNeuralNetwork`

object. You can use the `loss`

object function to compute the regression loss on new data.

In the call to `loss`

, you can specify to return the average loss or
the loss for each response variable by using the `OutputType`

name-value argument. You can also specify whether to standardize the response data before
computing the loss by using the `StandardizeResponses`

name-value argument.

### R2024b: Specify GPU arrays (requires Parallel Computing Toolbox)

`loss`

fully supports GPU arrays.

### 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.

### R2022a: `loss`

can return NaN for predictor data with missing values

The `loss`

function no longer omits an observation with a
NaN prediction when computing the weighted average regression loss. Therefore,
`loss`

can now return NaN when the predictor data
`X`

or the predictor variables in `Tbl`

contain any missing values. In most cases, if the test set observations do not contain
missing predictors, the `loss`

function does not return
NaN.

This change improves the automatic selection of a regression model when you use
`fitrauto`

.
Before this change, the software might select a model (expected to best predict the
responses for new data) with few non-NaN predictors.

If `loss`

in your code returns NaN, you can update your code
to avoid this result. Remove or replace the missing values by using `rmmissing`

or `fillmissing`

, respectively.

The following table shows the regression models for which the
`loss`

object function might return NaN. For more details,
see the Compatibility Considerations for each `loss`

function.

Model Type | Full or Compact Model Object | `loss` Object Function |
---|---|---|

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

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

Linear regression model | `RegressionLinear` | `loss` |

Neural network regression model | `RegressionNeuralNetwork` , `CompactRegressionNeuralNetwork` | `loss` |

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

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