rSquaredMetric

Deep learning R² metric

Since R2025a

Description

Use an RSquaredMetric object to track the R² (also known as the coefficient of determination) when you train or test a deep neural network.

To specify which metrics to use during training, specify the Metrics option of the trainingOptions function. You can use this option only when you train a network using the trainnet function.

To plot the metrics during training, in the training options, specify Plots as "training-progress". If you specify the ValidationData training option, then the software also plots and records the metric values for the validation data. To output the metric values to the Command Window during training, in the training options, set Verbose to true.

You can also access the metrics after training using the TrainingHistory and ValidationHistory fields from the second output of the trainnet function.

To specify which metrics to use when you test a neural network, use the metrics argument of the testnet function.

Creation

Syntax

metric = rSquaredMetric

metric = rSquaredMetric(PropertyName=Value)

Description

metric = rSquaredMetric creates an RSquaredMetric object. You can then specify metric as the Metrics name-value argument in the trainingOptions function or the metrics argument of the testnet function. With no additional options specified, this syntax is equivalent to specifying the metric as "rSquared".

metric = rSquaredMetric(PropertyName=Value) sets the Name and NetworkOutput properties using name-value arguments.

Properties

Name — Metric name

"RSquared" (default) | string scalar | character vector

Metric name, specified as a string scalar or character vector. The metric name appears in the training plot, the verbose output, the training information that you can access as the second output of the trainnet function, and table output of the testnet function.

Data Types: char | string

NetworkOutput — Name of layer to apply metric to

Read-only: [] (default) | string scalar | character vector

This property is read-only.

Name of the layer to apply the metric to, specified as [], a string scalar, or a character vector. When the value is [], the software passes all of the network outputs to the metric.

Note

You can apply the built-in metric to only a single output. If you have a network with multiple outputs, then you must specify the NetworkOutput name-value argument. To apply built-in metrics to multiple outputs, you must create a metric object for each output.

Data Types: char | string

Maximize — Flag to maximize metric

Read-only: 1 (true)

This property is read-only.

Flag to maximize metric, specified as 1 (true) if the optimal value for the metric occurs when the metric is maximized.

For this metric, the Maximize value is always set to 1 (true).

Object Functions

trainingOptions	Options for training deep learning neural network
trainnet	Train deep learning neural network
testnet	Test deep learning neural network

Examples

Plot and Record R-Squared During Training

Plot and record the training and validation R-squared when you train a deep neural network.

Load the training and test data from the DigitsDataTrain and DigitsDataTest MAT files, respectively. To access this data, open the example as a live script. The data set contains synthetic images of handwritten digits and the corresponding angles (in degrees) by which each image is rotated. The anglesTrain and anglesTest variables are the rotation angles in degrees.

load DigitsDataTrain.mat
load DigitsDataTest.mat

You can train a deep learning network to predict the rotation angle of the digit.

Create an image regression network.

layers = [
    imageInputLayer([28 28 1])
    convolution2dLayer(3,8,Padding="same")
    batchNormalizationLayer
    reluLayer
    averagePooling2dLayer(2,Stride=2)
    convolution2dLayer(3,16,Padding="same")
    batchNormalizationLayer
    reluLayer
    averagePooling2dLayer(2,Stride=2)
    convolution2dLayer(3,32,Padding="same")
    batchNormalizationLayer
    reluLayer
    convolution2dLayer(3,32,Padding="same")
    batchNormalizationLayer
    reluLayer
    dropoutLayer(0.2)
    fullyConnectedLayer(1)];

Create an RSquaredMetric object. You can use this object to record and plot the training and validation R-squared value.

metric = rSquaredMetric

metric = 
  RSquaredMetric with properties:

             Name: "RSquared"
    NetworkOutput: []
         Maximize: 1

Specify the R-squared metric in the training options. To plot the R-squared during training, set Plots to "training-progress". To output the values during training, set Verbose to true.

options = trainingOptions("adam", ...
    MaxEpochs=5, ...
    Metrics=metric, ...
    ValidationData={XTest,anglesTest}, ...
    ValidationFrequency=50, ...
    Plots="training-progress", ...
    Verbose=true);

Train the network using the trainnet function.

[net,info] = trainnet(XTrain,anglesTrain,layers,"mse",options);

    Iteration    Epoch    TimeElapsed    LearnRate    TrainingLoss    ValidationLoss    TrainingRSquared    ValidationRSquared
    _________    _____    ___________    _________    ____________    ______________    ________________    __________________
            0        0       00:00:00        0.001                            688.66                               -0.00015879
            1        1       00:00:00        0.001          648.74                              0.003303                      
           50        2       00:00:07        0.001          210.71             335.2             0.70589               0.51317
          100        3       00:00:12        0.001          130.55            131.78             0.84343               0.80862
          150        4       00:00:16        0.001          89.997            126.48             0.86578               0.81631
          195        5       00:00:21        0.001          88.085            93.085             0.87508               0.86481
Training stopped: Max epochs completed

Access the loss and R-squared values for the validation data.

info.ValidationHistory

ans=5×3 table
    Iteration     Loss      RSquared  
    _________    ______    ___________

         0       688.66    -0.00015879
        50        335.2        0.51317
       100       131.78        0.80862
       150       126.48        0.81631
       195       93.085        0.86481

More About

R²

For a set of predictions Y and a set of targets T, the software computes the R² using this equation:

$$R^2=1-\frac{{\displaystyle \sum }_{i=1}^n{(Y_i-T_i)}^2}{{\displaystyle \sum }_{i=1}^n{(T_i-\overline{T})}^2}\text{, where }\overline{T}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{T}_i}$$

and n is the number of observations. Note that if the denominator is zero but the numerator is not, then the software returns a value of zero. If both the numerator and denominator are zero, then the software returns one.

For data with multiple output channels, the software calculates the R² value for each channel and then returns the average R² value across the channels.

Version History

Introduced in R2025a