kfoldLoss

Loss for cross-validated partitioned regression model

Syntax

``L = kfoldLoss(CVMdl)``
``L = kfoldLoss(CVMdl,Name,Value)``

Description

example

````L = kfoldLoss(CVMdl)` returns the loss (mean squared error) obtained by the cross-validated regression model `CVMdl`. For every fold, `kfoldLoss` computes the loss for validation-fold observations using a model trained on training-fold observations. `CVMdl.X` and `CVMdl.Y` contain both sets of observations.```

example

````L = kfoldLoss(CVMdl,Name,Value)` returns the loss with additional options specified by one or more name-value arguments. For example, you can specify a custom loss function.```

Examples

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Find the cross-validation loss for a regression ensemble of the `carsmall` data.

Load the `carsmall` data set and select displacement, horsepower, and vehicle weight as predictors.

```load carsmall X = [Displacement Horsepower Weight];```

Train an ensemble of regression trees.

`rens = fitrensemble(X,MPG);`

Create a cross-validated ensemble from `rens` and find the k-fold cross-validation loss.

```rng(10,'twister') % For reproducibility cvrens = crossval(rens); L = kfoldLoss(cvrens)```
```L = 28.7114 ```

The mean squared error (MSE) is a measure of model quality. Examine the MSE for each fold of a cross-validated regression model.

Load the `carsmall` data set. Specify the predictor `X` and the response data `Y`.

```load carsmall X = [Cylinders Displacement Horsepower Weight]; Y = MPG;```

Train a cross-validated regression tree model. By default, the software implements 10-fold cross-validation.

```rng('default') % For reproducibility CVMdl = fitrtree(X,Y,'CrossVal','on');```

Compute the MSE for each fold. Visualize the distribution of the loss values by using a box plot. Notice that none of the values is an outlier.

`losses = kfoldLoss(CVMdl,'Mode','individual')`
```losses = 10×1 42.5072 20.3995 22.3737 34.4255 40.8005 60.2755 19.5562 9.2060 29.0788 16.3386 ```
`boxchart(losses)`

Train a cross-validated generalized additive model (GAM) with 10 folds. Then, use `kfoldLoss` to compute the cumulative cross-validation regression loss (mean squared errors). Use the errors to determine the optimal number of trees per predictor (linear term for predictor) and the optimal number of trees per interaction term.

Alternatively, you can find optimal values of `fitrgam` name-value arguments by using the OptimizeHyperparameters name-value argument. For an example, see Optimize GAM Using OptimizeHyperparameters.

Load the `patients` data set.

`load patients`

Create a table that contains the predictor variables (`Age`, `Diastolic`, `Smoker`, `Weight`, `Gender`, and `SelfAssessedHealthStatus`) and the response variable (`Systolic`).

`tbl = table(Age,Diastolic,Smoker,Weight,Gender,SelfAssessedHealthStatus,Systolic);`

Create a cross-validated GAM by using the default cross-validation option. Specify the `'CrossVal'` name-value argument as `'on'`. Also, specify to include 5 interaction terms.

```rng('default') % For reproducibility CVMdl = fitrgam(tbl,'Systolic','CrossVal','on','Interactions',5);```

If you specify `'Mode'` as `'cumulative'` for `kfoldLoss`, then the function returns cumulative errors, which are the average errors across all folds obtained using the same number of trees for each fold. Display the number of trees for each fold.

`CVMdl.NumTrainedPerFold `
```ans = struct with fields: PredictorTrees: [300 300 300 300 300 300 300 300 300 300] InteractionTrees: [76 100 100 100 100 42 100 100 59 100] ```

`kfoldLoss` can compute cumulative errors using up to 300 predictor trees and 42 interaction trees.

Plot the cumulative, 10-fold cross-validated, mean squared errors. Specify `'IncludeInteractions'` as `false` to exclude interaction terms from the computation.

```L_noInteractions = kfoldLoss(CVMdl,'Mode','cumulative','IncludeInteractions',false); figure plot(0:min(CVMdl.NumTrainedPerFold.PredictorTrees),L_noInteractions)```

The first element of `L_noInteractions` is the average error over all folds obtained using only the intercept (constant) term. The (`J+1`)th element of `L_noInteractions` is the average error obtained using the intercept term and the first `J` predictor trees per linear term. Plotting the cumulative loss allows you to monitor how the error changes as the number of predictor trees in the GAM increases.

Find the minimum error and the number of predictor trees used to achieve the minimum error.

`[M,I] = min(L_noInteractions)`
```M = 28.0506 ```
```I = 6 ```

The GAM achieves the minimum error when it includes 5 predictor trees.

Compute the cumulative mean squared error using both linear terms and interaction terms.

```L = kfoldLoss(CVMdl,'Mode','cumulative'); figure plot(0:min(CVMdl.NumTrainedPerFold.InteractionTrees),L)```

The first element of `L` is the average error over all folds obtained using the intercept (constant) term and all predictor trees per linear term. The (`J+1`)th element of `L` is the average error obtained using the intercept term, all predictor trees per linear term, and the first `J` interaction trees per interaction term. The plot shows that the error increases when interaction terms are added.

If you are satisfied with the error when the number of predictor trees is 5, you can create a predictive model by training the univariate GAM again and specifying `'NumTreesPerPredictor',5` without cross-validation.

Input Arguments

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Cross-validated partitioned regression model, specified as a `RegressionPartitionedModel`, `RegressionPartitionedEnsemble`, `RegressionPartitionedGAM`, `RegressionPartitionedGP`, `RegressionPartitionedNeuralNetwork`, or `RegressionPartitionedSVM` object. You can create the object in two ways:

• Pass a trained regression model listed in the following table to its `crossval` object function.

• Train a regression model using a function listed in the following table and specify one of the cross-validation name-value arguments for the function.

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.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: `kfoldLoss(CVMdl,'Folds',[1 2 3 5])` specifies to use the first, second, third, and fifth folds to compute the mean squared error, but to exclude the fourth fold.

Fold indices to use, specified as a positive integer vector. The elements of `Folds` must be within the range from `1` to `CVMdl.KFold`.

The software uses only the folds specified in `Folds`.

Example: `'Folds',[1 4 10]`

Data Types: `single` | `double`

Flag to include interaction terms of the model, specified as `true` or `false`. This argument is valid only for a generalized additive model (GAM). That is, you can specify this argument only when `CVMdl` is `RegressionPartitionedGAM`.

The default value is `true` if the models in `CVMdl` (`CVMdl.Trained`) contain interaction terms. The value must be `false` if the models do not contain interaction terms.

Example: `'IncludeInteractions',false`

Data Types: `logical`

Loss function, specified as `'mse'` or a function handle.

• Specify the built-in function `'mse'`. In this case, the loss function is the mean squared error.

• Specify your own function using function handle notation.

Assume that n is the number of observations in the training data (`CVMdl.NumObservations`). Your function must have the signature ```lossvalue = lossfun(Y,Yfit,W)```, where:

• The output argument `lossvalue` is a scalar.

• You specify the function name (`lossfun`).

• `Y` is an n-by-1 numeric vector of observed responses.

• `Yfit` is an n-by-1 numeric vector of predicted responses.

• `W` is an n-by-1 numeric vector of observation weights.

Specify your function using `'LossFun',@lossfun`.

Data Types: `char` | `string` | `function_handle`

Aggregation level for the output, specified as `'average'`, `'individual'`, or `'cumulative'`.

ValueDescription
`'average'`The output is a scalar average over all folds.
`'individual'`The output is a vector of length k containing one value per fold, where k is the number of folds.
`'cumulative'`

Note

If you want to specify this value, `CVMdl` must be a `RegressionPartitionedEnsemble` object or `RegressionPartitionedGAM` object.

• If `CVMdl` is `RegressionPartitionedEnsemble`, then the output is a vector of length `min(CVMdl.NumTrainedPerFold)`. Each element `j` is an average over all folds that the function obtains by using ensembles trained with weak learners `1:j`.

• If `CVMdl` is `RegressionPartitionedGAM`, then the output value depends on the `IncludeInteractions` value.

• If `IncludeInteractions` is `false`, then `L` is a `(1 + min(NumTrainedPerFold.PredictorTrees))`-by-1 numeric column vector. The first element of `L` is an average over all folds that is obtained using only the intercept (constant) term. The `(j + 1)`th element of `L` is an average obtained using the intercept term and the first `j` predictor trees per linear term.

• If `IncludeInteractions` is `true`, then `L` is a `(1 + min(NumTrainedPerFold.InteractionTrees))`-by-1 numeric column vector. The first element of `L` is an average over all folds that is obtained using the intercept (constant) term and all predictor trees per linear term. The `(j + 1)`th element of `L` is an average obtained using the intercept term, all predictor trees per linear term, and the first `j` interaction trees per interaction term.

Example: `'Mode','individual'`

Since R2023b

Predicted response value to use for observations with missing predictor values, specified as `"median"`, `"mean"`, `"omitted"`, or a numeric scalar. This argument is valid only for a Gaussian process regression, neural network, or support vector machine model. That is, you can specify this argument only when `CVMdl` is a `RegressionPartitionedGP`, `RegressionPartitionedNeuralNetwork`, or `RegressionPartitionedSVM` object.

ValueDescription
`"median"`

`kfoldLoss` uses the median of the observed response values in the training-fold data as the predicted response value for observations with missing predictor values.

This value is the default when `CVMdl` is a `RegressionPartitionedGP`, `RegressionPartitionedNeuralNetwork`, or `RegressionPartitionedSVM` object.

`"mean"``kfoldLoss` uses the mean of the observed response values in the training-fold data as the predicted response value for observations with missing predictor values.
`"omitted"``kfoldLoss` excludes observations with missing predictor values from the loss computation.
Numeric scalar`kfoldLoss` uses this value as the predicted response value for observations with missing predictor values.

If an observation is missing all predictor values, an observed response value, or an observation weight, then `kfoldLoss` does not use the observation in the loss computation.

Example: `"PredictionForMissingValue","omitted"`

Data Types: `single` | `double` | `char` | `string`

Output Arguments

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Loss, returned as a numeric scalar or numeric column vector.

By default, the loss is the mean squared error between the validation-fold observations and the predictions made with a regression model trained on the training-fold observations.

• If `Mode` is `'average'`, then `L` is the average loss over all folds.

• If `Mode` is `'individual'`, then `L` is a k-by-1 numeric column vector containing the loss for each fold, where k is the number of folds.

• If `Mode` is `'cumulative'` and `CVMdl` is `RegressionPartitionedEnsemble`, then `L` is a `min(CVMdl.NumTrainedPerFold)`-by-1 numeric column vector. Each element `j` is the average loss over all folds that the function obtains using ensembles trained with weak learners `1:j`.

• If `Mode` is `'cumulative'` and `CVMdl` is `RegressionPartitionedGAM`, then the output value depends on the `IncludeInteractions` value.

• If `IncludeInteractions` is `false`, then `L` is a `(1 + min(NumTrainedPerFold.PredictorTrees))`-by-1 numeric column vector. The first element of `L` is the average loss over all folds that is obtained using only the intercept (constant) term. The `(j + 1)`th element of `L` is the average loss obtained using the intercept term and the first `j` predictor trees per linear term.

• If `IncludeInteractions` is `true`, then `L` is a `(1 + min(NumTrainedPerFold.InteractionTrees))`-by-1 numeric column vector. The first element of `L` is the average loss over all folds that is obtained using the intercept (constant) term and all predictor trees per linear term. The `(j + 1)`th element of `L` is the average loss obtained using the intercept term, all predictor trees per linear term, and the first `j` interaction trees per interaction term.

Alternative Functionality

If you want to compute the cross-validated loss of a tree model, you can avoid constructing a `RegressionPartitionedModel` object by calling `cvloss`. Creating a cross-validated tree object can save you time if you plan to examine it more than once.

Version History

Introduced in R2011a

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