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Neural network model for regression

A `RegressionNeuralNetwork`

object is a trained, feedforward, and
fully connected neural network for regression. The first fully connected layer of the neural
network has a connection from the network input (predictor data `X`

), and each
subsequent layer has a connection from the previous layer. Each fully connected layer
multiplies the input by a weight matrix (`LayerWeights`

) and
then adds a bias vector (`LayerBiases`

). An
activation function follows each fully connected layer, excluding the last (`Activations`

and
`OutputLayerActivation`

). The final fully connected layer produces the network's
output, namely predicted response values. For more information, see Neural Network Structure.

Create a `RegressionNeuralNetwork`

object by using `fitrnet`

.

`LayerSizes`

— Sizes of fully connected layerspositive integer vector

This property is read-only.

Sizes of the fully connected layers in the neural network model, returned as a
positive integer vector. The *i*th element of
`LayerSizes`

is the number of outputs in the
*i*th fully connected layer of the neural network model.

`LayerSizes`

does not include the size of the final fully
connected layer. This layer always has one output.

**Data Types: **`single`

| `double`

`LayerWeights`

— Learned layer weightscell array

This property is read-only.

Learned layer weights for fully connected layers, returned as a cell array. The
*i*th entry in the cell array corresponds to the layer weights for
the *i*th fully connected layer. For example,
`Mdl.LayerWeights{1}`

returns the weights for the first fully
connected layer of the model `Mdl`

.

`LayerWeights`

includes the weights for the final fully
connected layer.

**Data Types: **`cell`

`LayerBiases`

— Learned layer biasescell array

This property is read-only.

Learned layer biases for fully connected layers, returned as a cell array. The
*i*th entry in the cell array corresponds to the layer biases for
the *i*th fully connected layer. For example,
`Mdl.LayerBiases{1}`

returns the biases for the first fully
connected layer of the model `Mdl`

.

`LayerBiases`

includes the biases for the final fully connected
layer.

**Data Types: **`cell`

`Activations`

— Activation functions for fully connected layers`'relu'`

| `'tanh'`

| `'sigmoid'`

| `'none'`

| cell array of character vectorsThis property is read-only.

Activation functions for the fully connected layers of the neural network model, returned as a character vector or cell array of character vectors with values from this table.

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

`'relu'` | Rectified linear unit (ReLU) function — Performs a threshold operation on each element of the input, where any value less than zero is set to zero, that is, $$f\left(x\right)=\{\begin{array}{cc}x,& x\ge 0\\ 0,& x<0\end{array}$$ |

`'tanh'` | Hyperbolic tangent (tanh) function — Applies the |

`'sigmoid'` | Sigmoid function — Performs the following operation on each input element: $$f(x)=\frac{1}{1+{e}^{-x}}$$ |

`'none'` | Identity function — Returns each input element without performing any transformation, that is, |

If

`Activations`

contains only one activation function, then it is the activation function for every fully connected layer of the neural network model, excluding the final fully connected layer, which does not have an activation function (`OutputLayerActivation`

).If

`Activations`

is an array of activation functions, then the*i*th element is the activation function for the*i*th layer of the neural network model.

**Data Types: **`char`

| `cell`

`OutputLayerActivation`

— Activation function for final fully connected layer`'none'`

This property is read-only.

Activation function for final fully connected layer, returned as
`'none'`

.

`ModelParameters`

— Parameter values used to train model`NeuralNetworkParams`

objectThis property is read-only.

Parameter values used to train the `RegressionNeuralNetwork`

model,
returned as a `NeuralNetworkParams`

object.
`ModelParameters`

contains parameter values such as the
name-value arguments used to train the regression neural network model.

Access the properties of `ModelParameters`

by using dot
notation. For example, access the function used to initialize the fully connected
layer weights of a model `Mdl`

by using
`Mdl.ModelParameters.LayerWeightsInitializer`

.

`ConvergenceInfo`

— Convergence informationstructure array

This property is read-only.

Convergence information, returned as a structure array.

Field | Description |
---|---|

`Iterations` | Number of training iterations used to train the neural network model |

`TrainingLoss` | Training mean squared error (MSE) for the returned model, or
`resubLoss(Mdl)` for model `Mdl` |

`Gradient` | Gradient of the loss function with respect to the weights and biases at the iteration corresponding to the returned model |

`Step` | Step size at the iteration corresponding to the returned model |

`Time` | Total time spent across all iterations (in seconds) |

`ValidationLoss` | Validation MSE for the returned model |

`ValidationChecks` | Maximum number of times in a row that the validation loss was greater than or equal to the minimum validation loss |

`ConvergenceCriterion` | Criterion for convergence |

`History` | See `TrainingHistory` |

**Data Types: **`struct`

`TrainingHistory`

— Training historytable

This property is read-only.

Training history, returned as a table.

Column | Description |
---|---|

`Iteration` | Training iteration |

`TrainingLoss` | Training mean squared error (MSE) for the model at this iteration |

`Gradient` | Gradient of the loss function with respect to the weights and biases at this iteration |

`Step` | Step size at this iteration |

`Time` | Time spent during this iteration (in seconds) |

`ValidationLoss` | Validation MSE for the model at this iteration |

`ValidationChecks` | Running total of times that the validation loss is greater than or equal to the minimum validation loss |

**Data Types: **`table`

`Solver`

— Solver used to train neural network model`'LBFGS'`

This property is read-only.

Solver used to train the neural network model, returned as
`'LBFGS'`

. To create a `RegressionNeuralNetwork`

model, `fitrnet`

uses a limited-memory
Broyden-Fletcher-Goldfarb-Shanno quasi-Newton algorithm (LBFGS) as its loss function
minimization technique, where the software minimizes the mean squared error
(MSE).

`PredictorNames`

— Predictor variable namescell array of character vectors

This property is read-only.

Predictor variable names, returned as a cell array of character vectors. The order
of the elements of `PredictorNames`

corresponds to the order in which
the predictor names appear in the training data.

**Data Types: **`cell`

`CategoricalPredictors`

— Categorical predictor indicesvector of positive integers |

`[]`

This property is read-only.

Categorical predictor indices, returned as a
vector of positive integers. Assuming that the predictor data contains observations in
rows, `CategoricalPredictors`

contains index values corresponding to
the columns of the predictor data that contain categorical predictors. If none of the
predictors are categorical, then this property is empty
(`[]`

).

**Data Types: **`double`

`ExpandedPredictorNames`

— Expanded predictor namescell array of character vectors

This property is read-only.

Expanded predictor names, returned as a cell array of character vectors. If the
model uses encoding for categorical variables, then
`ExpandedPredictorNames`

includes the names that describe the
expanded variables. Otherwise, `ExpandedPredictorNames`

is the same
as `PredictorNames`

.

**Data Types: **`cell`

`X`

— Unstandardized predictorsnumeric matrix | table

This property is read-only.

Unstandardized predictors used to train the neural network model, returned as a
numeric matrix or table. `X`

retains its original orientation, with
observations in rows or columns depending on the value of the
`ObservationsIn`

name-value argument in the call to
`fitrnet`

.

**Data Types: **`single`

| `double`

| `table`

`ResponseName`

— Response variable namecharacter vector

This property is read-only.

Response variable name, returned as a character vector.

**Data Types: **`char`

`Y`

— Response valuesnumeric vector

This property is read-only.

Response values used to train the model, returned as a numeric vector. Each row of
`Y`

represents the response value of the corresponding observation
in `X`

.

**Data Types: **`single`

| `double`

`ResponseTransform`

— Response transformation function`'none'`

This property is read-only.

Response transformation function, returned as `'none'`

. The
software does not transform the raw response values.

`HyperparameterOptimizationResults`

— Cross-validation optimization of hyperparameters`BayesianOptimization`

object | tableThis property is read-only.

Cross-validation optimization of hyperparameters, specified as a `BayesianOptimization`

object or a table of hyperparameters and associated
values. This property is nonempty if the `'OptimizeHyperparameters'`

name-value pair argument is nonempty when you create the model. The value of
`HyperparameterOptimizationResults`

depends on the setting of the
`Optimizer`

field in the
`HyperparameterOptimizationOptions`

structure when you create the
model.

Value of `Optimizer` Field | Value of `HyperparameterOptimizationResults` |
---|---|

`'bayesopt'` (default) | Object of class `BayesianOptimization` |

`'gridsearch'` or `'randomsearch'` | Table of hyperparameters used, observed objective function values (cross-validation loss), and rank of observations from lowest (best) to highest (worst) |

`NumObservations`

— Number of observationspositive numeric scalar

This property is read-only.

Number of observations in the training data stored in `X`

and
`Y`

, returned as a positive numeric scalar.

**Data Types: **`double`

`RowsUsed`

— Rows used in fitting`[]`

| logical vectorThis property is read-only.

Rows of the original training data used in fitting the model, returned as a logical vector. This property is empty if all rows are used.

**Data Types: **`logical`

`W`

— Observation weightsnumeric vector

This property is read-only.

Observation weights used to train the model, returned as an
*n*-by-1 numeric vector. *n* is the number of
observations (`NumObservations`

).

The software normalizes the observation weights specified in the
`Weights`

name-value argument so that the elements of
`W`

sum up to 1.

**Data Types: **`single`

| `double`

`CompactRegressionNeuralNetwork`

`compact` | Reduce size of machine learning model |

`RegressionPartitionedModel`

`crossval` | Cross-validate machine learning model |

`lime` | Local interpretable model-agnostic explanations (LIME) |

`partialDependence` | Compute partial dependence |

`plotPartialDependence` | Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots |

`shapley` | Shapley values |

`resubLoss` | Resubstitution regression loss |

`resubPredict` | Predict responses for training data using trained regression model |

Train a neural network regression model, and assess the performance of the model on a test set.

Load the `carbig`

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

, `Displacement`

, and so on, as well as the response variable `MPG`

.

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

Partition the data into training and test sets. Use approximately 80% of the observations to train a neural network model, and 20% of the observations to test the performance of the trained model on new data. Use `cvpartition`

to partition the data.

rng("default") % For reproducibility of the data partition c = cvpartition(length(MPG),"Holdout",0.20); trainingIdx = training(c); % Training set indices carsTrain = cars(trainingIdx,:); testIdx = test(c); % Test set indices carsTest = cars(testIdx,:);

Train a 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,"MPG","Standardize",true)

Mdl = RegressionNeuralNetwork PredictorNames: {'Acceleration' 'Displacement' 'Horsepower' 'Model_Year' 'Origin' 'Weight'} ResponseName: 'MPG' CategoricalPredictors: 5 ResponseTransform: 'none' NumObservations: 314 LayerSizes: 10 Activations: 'relu' OutputLayerActivation: 'linear' Solver: 'LBFGS' ConvergenceInfo: [1×1 struct] TrainingHistory: [1000×7 table] Properties, Methods

`Mdl`

is a trained `RegressionNeuralNetwork`

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

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

to get more information about the training history of the neural network model.

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.

`testMSE = loss(Mdl,carsTest,"MPG")`

testMSE = 16.6154

Specify the structure of the neural network regression model, including the size of the fully connected layers.

Load the `carbig`

data set, which contains measurements of cars made in the 1970s and early 1980s. Create a matrix `X`

containing the predictor variables `Acceleration`

, `Cylinders`

, and so on. Store the response variable `MPG`

in the variable `Y`

.

```
load carbig
X = [Acceleration Cylinders Displacement Weight];
Y = MPG;
```

Partition the data into training data (`XTrain`

and `YTrain`

) and test data (`XTest`

and `YTest`

). Reserve approximately 20% of the observations for testing, and use the rest of the observations for training.

rng("default") % For reproducibility of the partition c = cvpartition(length(Y),"Holdout",0.20); trainingIdx = training(c); % Indices for the training set XTrain = X(trainingIdx,:); YTrain = Y(trainingIdx); testIdx = test(c); % Indices for the test set XTest = X(testIdx,:); YTest = Y(testIdx);

Train a neural network regression model. Specify to standardize the predictor data, and to have 30 outputs in the first fully connected layer and 10 outputs in the second fully connected layer. By default, both layers use a rectified linear unit (ReLU) activation function. You can change the activation functions for the fully connected layers by using the `Activations`

name-value argument.

Mdl = fitrnet(XTrain,YTrain,"Standardize",true, ... "LayerSizes",[30 10])

Mdl = RegressionNeuralNetwork ResponseName: 'Y' CategoricalPredictors: [] ResponseTransform: 'none' NumObservations: 318 LayerSizes: [30 10] Activations: 'relu' OutputLayerActivation: 'linear' Solver: 'LBFGS' ConvergenceInfo: [1×1 struct] TrainingHistory: [1000×7 table] Properties, Methods

Access the weights and biases for the fully connected layers of the trained model by using the `LayerWeights`

and `LayerBiases`

properties of `Mdl`

. The first two elements of each property correspond to the values for the first two fully connected layers, and the third element corresponds to the values for the final fully connected layer for regression. For example, display the weights and biases for the first fully connected layer.

Mdl.LayerWeights{1}

`ans = `*30×4*
-1.0617 0.1287 0.0797 0.4648
-0.6497 -1.4565 -2.6026 2.6962
-0.6420 0.2744 -0.0234 -0.0252
-1.9727 -0.4665 -0.5833 0.9371
-0.4373 0.1607 0.3930 0.7859
0.5091 -0.0032 -0.6503 -1.6694
0.0123 -0.2624 -2.2928 -1.0965
-0.1386 1.2747 0.4085 0.5395
-0.1755 1.5641 -3.1896 -1.1336
0.4401 0.4942 1.8957 -1.1617
⋮

Mdl.LayerBiases{1}

`ans = `*30×1*
-1.3086
-1.6205
-0.7815
1.5382
-0.5256
1.2394
-2.3078
-1.0709
-1.8898
1.9443
⋮

The final fully connected layer has one output. The number of layer outputs corresponds to the first dimension of the layer weights and layer biases.

size(Mdl.LayerWeights{end})

`ans = `*1×2*
1 10

size(Mdl.LayerBiases{end})

`ans = `*1×2*
1 1

To estimate the performance of the trained model, compute the test set mean squared error (MSE) for `Mdl`

. Smaller MSE values indicate better performance.

testMSE = loss(Mdl,XTest,YTest)

testMSE = 17.2022

Compare the predicted test set response values to the true response values. Plot the predicted miles per gallon (MPG) along the vertical axis and the true MPG along the horizontal axis. Points on the reference line indicate correct predictions. A good model produces predictions that are scattered near the line.

testPredictions = predict(Mdl,XTest); plot(YTest,testPredictions,".") hold on plot(YTest,YTest) hold off xlabel("True MPG") ylabel("Predicted MPG")

`fitrnet`

| `predict`

| `loss`

| `RegressionPartitionedModel`

| `CompactRegressionNeuralNetwork`

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