Classification margins for observations not used in training

returns
the cross-validated classification margins obtained
by the cross-validated, binary, linear classification model `m`

= kfoldMargin(`CVMdl`

)`CVMdl`

.
That is, for every fold, `kfoldMargin`

estimates the
classification margins for observations that it holds out when it
trains using all other observations.

`m`

contains classification margins for each
regularization strength in the linear classification models that comprise `CVMdl`

.

`CVMdl`

— Cross-validated, binary, linear classification model`ClassificationPartitionedLinear`

model objectCross-validated, binary, linear classification model, specified as a `ClassificationPartitionedLinear`

model object. You can create a
`ClassificationPartitionedLinear`

model using `fitclinear`

and specifying any one of the cross-validation, name-value
pair arguments, for example, `CrossVal`

.

To obtain estimates, kfoldMargin applies the same data used to cross-validate the linear
classification model (`X`

and `Y`

).

`m`

— Cross-validated classification marginsnumeric vector | numeric matrix

Cross-validated classification margins, returned as a numeric vector or matrix.

`m`

is *n*-by-*L*, where
*n* is the number of observations in the data that
created `CVMdl`

(see `X`

and `Y`

) and
*L* is the number of regularization strengths in
`CVMdl`

(that is,
`numel(CVMdl.Trained{1}.Lambda)`

).

`m(`

is
the cross-validated classification margin of observation * i*,

`j`

`CVMdl.Trained{1}.Lambda(``j`

)

.**Data Types: **`single`

| `double`

Load the NLP data set.

`load nlpdata`

`X`

is a sparse matrix of predictor data, and `Y`

is a categorical vector of class labels. There are more than two classes in the data.

The models should identify whether the word counts in a web page are from the Statistics and Machine Learning Toolbox™ documentation. So, identify the labels that correspond to the Statistics and Machine Learning Toolbox™ documentation web pages.

`Ystats = Y == 'stats';`

Cross-validate a binary, linear classification model that can identify whether the word counts in a documentation web page are from the Statistics and Machine Learning Toolbox™ documentation.

rng(1); % For reproducibility CVMdl = fitclinear(X,Ystats,'CrossVal','on');

`CVMdl`

is a `ClassificationPartitionedLinear`

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 cross-validated margins.

m = kfoldMargin(CVMdl); size(m)

`ans = `*1×2*
31572 1

`m`

is a 31572-by-1 vector. `m(j)`

is the average of the out-of-fold margins for observation `j`

.

Plot the *k*-fold margins using box plots.

```
figure;
boxplot(m);
h = gca;
h.YLim = [-5 30];
title('Distribution of Cross-Validated Margins')
```

One way to perform feature selection is to compare *k*-fold margins from multiple models. Based solely on this criterion, the classifier with the larger margins is the better classifier.

Load the NLP data set. Preprocess the data as in Estimate k-Fold Cross-Validation Margins.

load nlpdata Ystats = Y == 'stats'; X = X';

Create these two data sets:

`fullX`

contains all predictors.`partX`

contains 1/2 of the predictors chosen at random.

rng(1); % For reproducibility p = size(X,1); % Number of predictors halfPredIdx = randsample(p,ceil(0.5*p)); fullX = X; partX = X(halfPredIdx,:);

Cross-validate two binary, linear classification models: one that uses the all of the predictors and one that uses half of the predictors. Optimize the objective function using SpaRSA, and indicate that observations correspond to columns.

CVMdl = fitclinear(fullX,Ystats,'CrossVal','on','Solver','sparsa',... 'ObservationsIn','columns'); PCVMdl = fitclinear(partX,Ystats,'CrossVal','on','Solver','sparsa',... 'ObservationsIn','columns');

`CVMdl`

and `PCVMdl`

are `ClassificationPartitionedLinear`

models.

Estimate the *k*-fold margins for each classifier. Plot the distribution of the *k*-fold margins sets using box plots.

fullMargins = kfoldMargin(CVMdl); partMargins = kfoldMargin(PCVMdl); figure; boxplot([fullMargins partMargins],'Labels',... {'All Predictors','Half of the Predictors'}); h = gca; h.YLim = [-30 60]; title('Distribution of Cross-Validated Margins')

The distributions of the margins of the two classifiers are similar.

To determine a good lasso-penalty strength for a linear classification model that uses a logistic regression learner, compare distributions of *k*-fold margins.

Load the NLP data set. Preprocess the data as in Estimate k-Fold Cross-Validation Margins.

load nlpdata Ystats = Y == 'stats'; X = X';

Create a set of 11 logarithmically-spaced regularization strengths from $$1{0}^{-8}$$ through $$1{0}^{1}$$.

Lambda = logspace(-8,1,11);

Cross-validate a binary, linear classification model using 5-fold cross-validation and that uses each of the regularization strengths. Optimize the objective function using SpaRSA. Lower the tolerance on the gradient of the objective function to `1e-8`

.

rng(10); % For reproducibility CVMdl = fitclinear(X,Ystats,'ObservationsIn','columns','KFold',5, ... 'Learner','logistic','Solver','sparsa','Regularization','lasso', ... 'Lambda',Lambda,'GradientTolerance',1e-8)

CVMdl = classreg.learning.partition.ClassificationPartitionedLinear CrossValidatedModel: 'Linear' ResponseName: 'Y' NumObservations: 31572 KFold: 5 Partition: [1×1 cvpartition] ClassNames: [0 1] ScoreTransform: 'none' Properties, Methods

`CVMdl`

is a `ClassificationPartitionedLinear`

model. Because `fitclinear`

implements 5-fold cross-validation, `CVMdl`

contains 5 `ClassificationLinear`

models that the software trains on each fold.

Estimate the *k*-fold margins for each regularization strength.

m = kfoldMargin(CVMdl); size(m)

`ans = `*1×2*
31572 11

`m`

is a 31572-by-11 matrix of cross-validated margins for each observation. The columns correspond to the regularization strengths.

Plot the *k*-fold margins for each regularization strength. Because logistic regression scores are in [0,1], margins are in [-1,1]. Rescale the margins to help identify the regularization strength that maximizes the margins over the grid.

figure boxplot(10000.^m) ylabel('Exponentiated test-sample margins') xlabel('Lambda indices')

Several values of `Lambda`

yield *k*-fold margin distributions that are compacted near 10000. Higher values of lambda lead to predictor variable sparsity, which is a good quality of a classifier.

Choose the regularization strength that occurs just before the centers of the *k*-fold margin distributions start decreasing.

LambdaFinal = Lambda(5);

Train a linear classification model using the entire data set and specify the desired regularization strength.

MdlFinal = fitclinear(X,Ystats,'ObservationsIn','columns', ... 'Learner','logistic','Solver','sparsa','Regularization','lasso', ... 'Lambda',LambdaFinal);

To estimate labels for new observations, pass `MdlFinal`

and the new data to `predict`

.

The *classification margin* for binary classification
is, for each observation, the difference between the classification score for the true class
and the classification score for the false class.

The software defines the classification margin for binary classification as

$$m=2yf\left(x\right).$$

*x* is an observation. If the true label of
*x* is the positive class, then *y* is 1, and –1
otherwise. *f*(*x*) is the positive-class classification
score for the observation *x*. The classification margin is commonly
defined as *m* =
*y**f*(*x*).

If the margins are on the same scale, then they serve as a classification confidence measure. Among multiple classifiers, those that yield greater margins are better.

For linear classification models, the raw *classification
score* for classifying the observation *x*, a row vector,
into the positive class is defined by

$${f}_{j}(x)=x{\beta}_{j}+{b}_{j}.$$

For the model with regularization strength *j*, $${\beta}_{j}$$ is the estimated column vector of coefficients (the model property
`Beta(:,j)`

) and $${b}_{j}$$ is the estimated, scalar bias (the model property
`Bias(j)`

).

The raw classification score for classifying *x* into
the negative class is –*f*(*x*).
The software classifies observations into the class that yields the
positive score.

If the linear classification model consists of logistic regression learners, then the
software applies the `'logit'`

score transformation to the raw
classification scores (see `ScoreTransform`

).

`ClassificationLinear`

| `ClassificationPartitionedLinear`

| `kfoldEdge`

| `kfoldPredict`

| `margin`

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