# kfoldLoss

Regression loss for observations not used in training

## Syntax

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

## Description

Description

example

````L = kfoldLoss(CVMdl)` returns the cross-validated mean squared error (MSE) obtained by the cross-validated, linear regression model `CVMdl`. That is, for every fold, `kfoldLoss` estimates the regression loss for observations that it holds out when it trains using all other observations.`L` contains a regression loss for each regularization strength in the linear regression models that compose `CVMdl`.```

example

````L = kfoldLoss(CVMdl,Name,Value)` uses additional options specified by one or more `Name,Value` pair arguments. For example, indicate which folds to use for the loss calculation or specify the regression-loss function.```

## Input Arguments

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Cross-validated, linear regression model, specified as a `RegressionPartitionedLinear` model object. You can create a `RegressionPartitionedLinear` model using `fitrlinear` and specifying any of the one of the cross-validation, name-value pair arguments, for example, `CrossVal`.

To obtain estimates, kfoldLoss applies the same data used to cross-validate the linear regression model (`X` and `Y`).

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Fold indices to use for response prediction, specified as the comma-separated pair consisting of `'Folds'` and a numeric vector of positive integers. The elements of `Folds` must range from `1` through `CVMdl.KFold`.

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

Data Types: `single` | `double`

Loss function, specified as the comma-separated pair consisting of `'LossFun'` and a built-in, loss-function name or function handle.

• The following table lists the available loss functions. Specify one using its corresponding character vector or string scalar. Also, in the table, $f\left(x\right)=x\beta +b.$

• β is a vector of p coefficients.

• x is an observation from p predictor variables.

• b is the scalar bias.

ValueDescription
`'epsiloninsensitive'`Epsilon-insensitive loss: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,|y-f\left(x\right)|-\epsilon \right]$
`'mse'`MSE: $\ell \left[y,f\left(x\right)\right]={\left[y-f\left(x\right)\right]}^{2}$

`'epsiloninsensitive'` is appropriate for SVM learners only.

• Specify your own function using function handle notation.

Let `n` be the number of observations in `X`. Your function must have this signature

``lossvalue = lossfun(Y,Yhat,W)``
where:

• The output argument `lossvalue` is a scalar.

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

• `Y` is an `n`-dimensional vector of observed responses. `kfoldLoss` passes the input argument `Y` in for `Y`.

• `Yhat` is an `n`-dimensional vector of predicted responses, which is similar to the output of `predict`.

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

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

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

Loss aggregation level, specified as the comma-separated pair consisting of `'Mode'` and `'average'` or `'individual'`.

ValueDescription
`'average'`Returns losses averaged over all folds
`'individual'`Returns losses for each fold

Example: `'Mode','individual'`

## Output Arguments

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Cross-validated regression losses, returned as a numeric scalar, vector, or matrix. The interpretation of `L` depends on `LossFun`.

Let `R` be the number of regularizations strengths is the cross-validated models (stored in `numel(CVMdl.Trained{1}.Lambda)`) and `F` be the number of folds (stored in `CVMdl.KFold`).

• If `Mode` is `'average'`, then `L` is a 1-by-`R` vector. `L(j)` is the average regression loss over all folds of the cross-validated model that uses regularization strength `j`.

• Otherwise, `L` is an `F`-by-`R` matrix. `L(i,j)` is the regression loss for fold `i` of the cross-validated model that uses regularization strength `j`.

To estimate `L`, `kfoldLoss` uses the data that created `CVMdl` (see `X` and `Y`).

## Examples

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Simulate 10000 observations from this model

`$y={x}_{100}+2{x}_{200}+e.$`

• $X=\left\{{x}_{1},...,{x}_{1000}\right\}$ is a 10000-by-1000 sparse matrix with 10% nonzero standard normal elements.

• e is random normal error with mean 0 and standard deviation 0.3.

```rng(1) % For reproducibility n = 1e4; d = 1e3; nz = 0.1; X = sprandn(n,d,nz); Y = X(:,100) + 2*X(:,200) + 0.3*randn(n,1);```

Cross-validate a linear regression model using SVM learners.

```rng(1); % For reproducibility CVMdl = fitrlinear(X,Y,'CrossVal','on');```

`CVMdl` is a `RegressionPartitionedLinear` model. By default, the software implements 10-fold cross validation. You can alter the number of folds using the `'KFold'` name-value pair argument.

Estimate the average of the test-sample MSEs.

`mse = kfoldLoss(CVMdl)`
```mse = 0.1735 ```

Alternatively, you can obtain the per-fold MSEs by specifying the name-value pair `'Mode','individual'` in `kfoldLoss`.

Simulate data as in Estimate k-Fold Mean Squared Error.

```rng(1) % For reproducibility n = 1e4; d = 1e3; nz = 0.1; X = sprandn(n,d,nz); Y = X(:,100) + 2*X(:,200) + 0.3*randn(n,1); X = X'; % Put observations in columns for faster training ```

Cross-validate a linear regression model using 10-fold cross-validation. Optimize the objective function using SpaRSA.

```CVMdl = fitrlinear(X,Y,'CrossVal','on','ObservationsIn','columns',... 'Solver','sparsa');```

`CVMdl` is a `RegressionPartitionedLinear` model. It contains the property `Trained`, which is a 10-by-1 cell array holding `RegressionLinear` models that the software trained using the training set.

Create an anonymous function that measures Huber loss ($\delta$ = 1), that is,

`$L=\frac{1}{\sum {w}_{j}}\sum _{j=1}^{n}{w}_{j}{\ell }_{j},$`

where

`$\begin{array}{l}\\ {\ell }_{j}=\left\{\begin{array}{c}0.5{\underset{}{\overset{ˆ}{{e}_{j}}}}^{2};\\ |\underset{}{\overset{ˆ}{{e}_{j}}}|-0.5;\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\end{array}\begin{array}{c}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}|\underset{}{\overset{ˆ}{{e}_{j}}}|\le 1\\ \phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}|\underset{}{\overset{ˆ}{{e}_{j}}}|>1\end{array}.\end{array}$`

$\underset{}{\overset{ˆ}{{e}_{j}}}$ is the residual for observation j. Custom loss functions must be written in a particular form. For rules on writing a custom loss function, see the `'LossFun'` name-value pair argument.

```huberloss = @(Y,Yhat,W)sum(W.*((0.5*(abs(Y-Yhat)<=1).*(Y-Yhat).^2) + ... ((abs(Y-Yhat)>1).*abs(Y-Yhat)-0.5)))/sum(W);```

Estimate the average Huber loss over the folds. Also, obtain the Huber loss for each fold.

`mseAve = kfoldLoss(CVMdl,'LossFun',huberloss)`
```mseAve = -0.4447 ```
`mseFold = kfoldLoss(CVMdl,'LossFun',huberloss,'Mode','individual')`
```mseFold = 10×1 -0.4454 -0.4473 -0.4452 -0.4469 -0.4434 -0.4427 -0.4471 -0.4430 -0.4438 -0.4426 ```

To determine a good lasso-penalty strength for a linear regression model that uses least squares, implement 5-fold cross-validation.

Simulate 10000 observations from this model

`$y={x}_{100}+2{x}_{200}+e.$`

• $X=\left\{{x}_{1},...,{x}_{1000}\right\}$ is a 10000-by-1000 sparse matrix with 10% nonzero standard normal elements.

• e is random normal error with mean 0 and standard deviation 0.3.

```rng(1) % For reproducibility n = 1e4; d = 1e3; nz = 0.1; X = sprandn(n,d,nz); Y = X(:,100) + 2*X(:,200) + 0.3*randn(n,1);```

Create a set of 15 logarithmically-spaced regularization strengths from $1{0}^{-5}$ through $1{0}^{-1}$.

`Lambda = logspace(-5,-1,15);`

Cross-validate the models. To increase execution speed, transpose the predictor data and specify that the observations are in columns. Optimize the objective function using SpaRSA.

```X = X'; CVMdl = fitrlinear(X,Y,'ObservationsIn','columns','KFold',5,'Lambda',Lambda,... 'Learner','leastsquares','Solver','sparsa','Regularization','lasso'); numCLModels = numel(CVMdl.Trained)```
```numCLModels = 5 ```

`CVMdl` is a `RegressionPartitionedLinear` model. Because `fitrlinear` implements 5-fold cross-validation, `CVMdl` contains 5 `RegressionLinear` models that the software trains on each fold.

Display the first trained linear regression model.

`Mdl1 = CVMdl.Trained{1}`
```Mdl1 = RegressionLinear ResponseName: 'Y' ResponseTransform: 'none' Beta: [1000x15 double] Bias: [1x15 double] Lambda: [1x15 double] Learner: 'leastsquares' Properties, Methods ```

`Mdl1` is a `RegressionLinear` model object. `fitrlinear` constructed `Mdl1` by training on the first four folds. Because `Lambda` is a sequence of regularization strengths, you can think of `Mdl1` as 15 models, one for each regularization strength in `Lambda`.

Estimate the cross-validated MSE.

`mse = kfoldLoss(CVMdl);`

Higher values of `Lambda` lead to predictor variable sparsity, which is a good quality of a regression model. For each regularization strength, train a linear regression model using the entire data set and the same options as when you cross-validated the models. Determine the number of nonzero coefficients per model.

```Mdl = fitrlinear(X,Y,'ObservationsIn','columns','Lambda',Lambda,... 'Learner','leastsquares','Solver','sparsa','Regularization','lasso'); numNZCoeff = sum(Mdl.Beta~=0);```

In the same figure, plot the cross-validated MSE and frequency of nonzero coefficients for each regularization strength. Plot all variables on the log scale.

```figure [h,hL1,hL2] = plotyy(log10(Lambda),log10(mse),... log10(Lambda),log10(numNZCoeff)); hL1.Marker = 'o'; hL2.Marker = 'o'; ylabel(h(1),'log_{10} MSE') ylabel(h(2),'log_{10} nonzero-coefficient frequency') xlabel('log_{10} Lambda') hold off```

Choose the index of the regularization strength that balances predictor variable sparsity and low MSE (for example, `Lambda(10)`).

`idxFinal = 10;`

Extract the model with corresponding to the minimal MSE.

`MdlFinal = selectModels(Mdl,idxFinal)`
```MdlFinal = RegressionLinear ResponseName: 'Y' ResponseTransform: 'none' Beta: [1000x1 double] Bias: -0.0050 Lambda: 0.0037 Learner: 'leastsquares' Properties, Methods ```
`idxNZCoeff = find(MdlFinal.Beta~=0)`
```idxNZCoeff = 2×1 100 200 ```
`EstCoeff = Mdl.Beta(idxNZCoeff)`
```EstCoeff = 2×1 1.0051 1.9965 ```

`MdlFinal` is a `RegressionLinear` model with one regularization strength. The nonzero coefficients `EstCoeff` are close to the coefficients that simulated the data.