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Half mean squared error


The half mean squared error operation computes the half mean squared error loss between network predictions and target values for regression tasks.

The loss is calculated using the following formula


where Xi is the network prediction, Ti is the target value, M is the total number of responses in X (across all observations), and N is the total number of observations in X.


This function computes the half mean squared error loss between predictions and targets stored as dlarray data. If you want to calculate the half mean squared error loss within a layerGraph object or Layer array for use with trainNetwork, use the following layer:


loss = mse(dlY,targets) computes the half mean squared error loss between the predictions dlY and the target values targets for regression problems. The input dlY must be a formatted dlarray. The output loss is an unformatted dlarray scalar.

loss = mse(dlY,targets,'DataFormat',FMT) also specifies the dimension format FMT when dlY is not a formatted dlarray.


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The half mean squared error evaluates how well the network predictions correspond to the target values.

Create the input predictions as a single observation of random values with a height and width of six and a single channel.

height = 6;
width = 6;
channels = 1;
observations = 1;

Y = rand(height,width,channels,observations);
dlY = dlarray(Y,'SSCB')

Create the target values as a numeric array with the same dimension order as the input data dlY.

targets = ones(height,width,channels,observations);

Compute the half mean squared error between the predictions and the targets.

loss = mse(dlY,targets)
loss =

  1x1 dlarray


Input Arguments

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Predictions, specified as a formatted dlarray, an unformatted dlarray, or a numeric array. When dlY is not a formatted dlarray, you must specify the dimension format using the DataFormat option.

If dlY is a numeric array, targets must be a dlarray.

Target responses, specified as a formatted or unformatted dlarray or a numeric array.

The size of each dimension of targets must match the size of the corresponding dimension of dlY.

If targets is a formatted dlarray, then its format must be the same as the format of dlY, or the same as DataFormat if dlY is unformatted.

If targets is an unformatted dlarray or a numeric array, then the function applies the format of dlY or the value of DataFormat to targets.


Formatted dlarray objects automatically permute the dimensions of the underlying data to have order "S" (spatial), "C" (channel), "B" (batch), "T" (time), then "U" (unspecified). To ensure that the dimensions of dlY and targets are consistent, when dlY is a formatted dlarray, also specify targets as a formatted dlarray.

Dimension order of unformatted input data, specified as the comma-separated pair consisting of 'DataFormat' and a character array or string FMT that provides a label for each dimension of the data. Each character in FMT must be one of the following:

  • 'S' — Spatial

  • 'C' — Channel

  • 'B' — Batch (for example, samples and observations)

  • 'T' — Time (for example, sequences)

  • 'U' — Unspecified

You can specify multiple dimensions labeled 'S' or 'U'. You can use the labels 'C', 'B', and 'T' at most once.

You must specify 'DataFormat',FMT when the input data is not a formatted dlarray.

Example: 'DataFormat','SSCB'

Data Types: char | string

Output Arguments

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Half mean squared error loss, returned as an unformatted dlarray scalar. The output loss has the same underlying data type as the input dlY.

More About

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Half Mean Squared Error Loss

The mse function computes the half-mean-squared-error loss for regression problems. For more information, see the definition of Regression Output Layer on the RegressionOutputLayer reference page.

Extended Capabilities

Introduced in R2019b