RegressionNeuralNetwork

Neural network model for regression

Since R2021a

Description

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.

Creation

Create a RegressionNeuralNetwork object by using fitrnet.

Properties

Neural Network Properties

LayerSizes — Sizes of fully connected layers

positive 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 ith element of LayerSizes is the number of outputs in the ith 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 weights

cell array

This property is read-only.

Learned layer weights for fully connected layers, returned as a cell array. The ith entry in the cell array corresponds to the layer weights for the ith 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 biases

cell array

This property is read-only.

Learned layer biases for fully connected layers, returned as a cell array. The ith entry in the cell array corresponds to the layer biases for the ith 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 vectors

This 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 tanh function to each input element
'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, f(x) = x

• 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 ith element is the activation function for the ith 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 object

This 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.

Convergence Control Properties

ConvergenceInfo — Convergence information

structure 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 history

table

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).

Predictor Properties

PredictorNames — Predictor variable names

cell 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 indices

vector 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 names

cell 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

Mu — Predictor means

numeric vector | []

Since R2023b

This property is read-only.

Predictor means, returned as a numeric vector. If you set Standardize to 1 or true when you train the neural network model, then the length of the Mu vector is equal to the number of expanded predictors (see ExpandedPredictorNames). The vector contains 0 values for dummy variables corresponding to expanded categorical predictors.

If you set Standardize to 0 or false when you train the neural network model, then the Mu value is an empty vector ([]).

Data Types: double

Sigma — Predictor standard deviations

numeric vector | []

Since R2023b

This property is read-only.

Predictor standard deviations, returned as a numeric vector. If you set Standardize to 1 or true when you train the neural network model, then the length of the Sigma vector is equal to the number of expanded predictors (see ExpandedPredictorNames). The vector contains 1 values for dummy variables corresponding to expanded categorical predictors.

If you set Standardize to 0 or false when you train the neural network model, then the Sigma value is an empty vector ([]).

Data Types: double

X — Unstandardized predictors

numeric 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

Response Properties

ResponseName — Response variable name

character vector

This property is read-only.

Response variable name, returned as a character vector.

Data Types: char

Y — Response values

numeric 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' | function handle

Response transformation function, specified as 'none' or a function handle. ResponseTransform describes how the software transforms raw response values.

For a MATLAB^® function or a function that you define, enter its function handle. For example, you can enter Mdl.ResponseTransform = @function, where function accepts a numeric vector of the original responses and returns a numeric vector of the same size containing the transformed responses.

Data Types: char | function_handle

Other Data Properties

HyperparameterOptimizationResults — Cross-validation optimization of hyperparameters

BayesianOptimization object | table

This 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 observations

positive 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 — Observations of original training data stored

logical vector | []

This property is read-only.

Observations of the original training data stored in the model, returned as a logical vector. This property is empty if all observations are stored in X and Y.

Data Types: logical

W — Observation weights

numeric 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

Object Functions

Create CompactRegressionNeuralNetwork

compact

Reduce size of machine learning model

Create RegressionPartitionedNeuralNetwork

crossval

Cross-validate machine learning model

Interpret Prediction

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

Assess Predictive Performance on New Observations

loss	Loss for regression neural network
predict	Predict responses using regression neural network

Assess Predictive Performance on Training Data

resubLoss	Resubstitution regression loss
resubPredict	Predict responses for training data using trained regression model

Examples

Train Neural Network 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);

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 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(height(cars),"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: 'none'
                   Solver: 'LBFGS'
          ConvergenceInfo: [1x1 struct]
          TrainingHistory: [708x7 table]

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 = 7.1092

Specify Neural Network Regression Model Architecture

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;

Delete rows of X and Y where either array has missing values.

R = rmmissing([X Y]);
X = R(:,1:end-1);
Y = R(:,end);

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: 319
               LayerSizes: [30 10]
              Activations: 'relu'
    OutputLayerActivation: 'none'
                   Solver: 'LBFGS'
          ConvergenceInfo: [1x1 struct]
          TrainingHistory: [1000x7 table]

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

    0.0123    0.0117   -0.0094    0.1175
   -0.4081   -0.7849   -0.7201   -2.1720
    0.6041    0.1680   -2.3952    0.0934
   -3.2332   -2.8360   -1.8264   -1.5723
    0.5851    1.5370    1.4623    0.6742
   -0.2106    1.2830   -1.7489   -1.5556
    0.4800    0.1012   -1.0044   -0.7959
    1.8015   -0.5272   -0.7670    0.7496
   -1.1428   -0.9902    0.2436    1.2288
   -0.0833   -2.4265    0.8388    1.8597
      ⋮

Mdl.LayerBiases{1}

ans = 30×1

   -0.4450
   -0.8751
   -0.3872
   -1.1345
    0.4499
   -2.1555
    2.2111
    1.2040
   -1.4595
    0.4639
      ⋮

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 = 18.3681

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")

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• The predict object function supports code generation.

For more information, see Introduction to Code Generation.

Version History

Introduced in R2021a

R2023b: Model stores observations with missing predictor values

Starting in R2023b, training observations with missing predictor values are included in the X, Y, and W data properties. The RowsUsed property indicates the training observations stored in the model, rather than those used for training. Observations with missing predictor values continue to be omitted from the model training process.

In previous releases, the software omitted training observations that contained missing predictor values from the data properties of the model.

R2023b: Neural network models include standardization properties

Neural network models include Mu and Sigma properties that contain the means and standard deviations, respectively, used to standardize the predictors before training. The properties are empty when the fitting function does not perform any standardization.