kfoldPredict

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 using the entire data set, which 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');
Mdl1 = CVMdl.Trained{1}

Mdl1 = 
  ClassificationLinear
      ResponseName: 'Y'
        ClassNames: [0 1]
    ScoreTransform: 'none'
              Beta: [34023x1 double]
              Bias: -1.0008
            Lambda: 3.5193e-05
           Learner: 'svm'

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.

Predict labels for the observations that fitclinear did not use in training the folds.

label = kfoldPredict(CVMdl);

Because there is one regularization strength in Mdl1, label is a column vector of predictions containing as many rows as observations in X.

Construct a confusion matrix.

ConfusionTrain = confusionchart(Ystats,label);

The model misclassifies 15 'stats' documentation pages as being outside of the Statistics and Machine Learning Toolbox documentation, and misclassifies nine pages as 'stats' pages.

Linear classification models return posterior probabilities for logistic regression learners only.

Load the NLP data set and preprocess it as in Predict k-fold Cross-Validation Labels. Transpose the predictor data matrix.

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

Cross-validate binary, linear classification models using 5-fold cross-validation. 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','GradientTolerance',1e-8);

Predict the posterior class probabilities for observations not used to train each fold.

[~,posterior] = kfoldPredict(CVMdl);
CVMdl.ClassNames

ans = 2x1 logical array

   0
   1

Because there is one regularization strength in CVMdl, posterior is a matrix with 2 columns and rows equal to the number of observations. Column i contains posterior probabilities of Mdl.ClassNames(i) given a particular observation.

Compute the performance metrics (true positive rates and false positive rates) for a ROC curve and find the area under the ROC curve (AUC) value by creating a rocmetrics object.

rocObj = rocmetrics(Ystats,posterior,CVMdl.ClassNames);

Plot the ROC curve for the second class by using the plot function of rocmetrics.

plot(rocObj,ClassNames=CVMdl.ClassNames(2))

The ROC curve indicates that the model classifies the validation observations almost perfectly.

To determine a good lasso-penalty strength for a linear classification model that uses a logistic regression learner, compare cross-validated AUC values.

Load the NLP data set. Preprocess the data as in Estimate k-fold Cross-Validation Posterior Class Probabilities.

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

There are 9471 observations in the test sample.

Create a set of 11 logarithmically-spaced regularization strengths from $$10^{-6}$$ through $$10^{-0.5}$$.

Lambda = logspace(-6,-0.5,11);

Cross-validate a binary, linear classification models that use each of the regularization strengths and 5-fold cross-validation. 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 = 
  ClassificationPartitionedLinear
    CrossValidatedModel: 'Linear'
           ResponseName: 'Y'
        NumObservations: 31572
                  KFold: 5
              Partition: [1x1 cvpartition]
             ClassNames: [0 1]
         ScoreTransform: 'none'

Mdl1 = CVMdl.Trained{1}

Mdl1 = 
  ClassificationLinear
      ResponseName: 'Y'
        ClassNames: [0 1]
    ScoreTransform: 'logit'
              Beta: [34023x11 double]
              Bias: [-13.2936 -13.2936 -13.2936 -13.2936 -13.2936 -6.8954 -5.4359 -4.7170 -3.4108 -3.1566 -2.9792]
            Lambda: [1.0000e-06 3.5481e-06 1.2589e-05 4.4668e-05 1.5849e-04 5.6234e-04 0.0020 0.0071 0.0251 0.0891 0.3162]
           Learner: 'logistic'

Mdl1 is a ClassificationLinear model object. Because Lambda is a sequence of regularization strengths, you can think of Mdl1 as 11 models, one for each regularization strength in Lambda.

Predict the cross-validated labels and posterior class probabilities.

[label,posterior] = kfoldPredict(CVMdl);
CVMdl.ClassNames;
[n,K,L] = size(posterior)

n = 
31572

K = 
2

L = 
11

posterior(3,1,5)

ans = 
1.0000

label is a 31572-by-11 matrix of predicted labels. Each column corresponds to the predicted labels of the model trained using the corresponding regularization strength. posterior is a 31572-by-2-by-11 matrix of posterior class probabilities. Columns correspond to classes and pages correspond to regularization strengths. For example, posterior(3,1,5) indicates that the posterior probability that the first class (label 0) is assigned to observation 3 by the model that uses Lambda(5) as a regularization strength is 1.0000.

For each model, compute the AUC by using rocmetrics.

auc = 1:numel(Lambda);  % Preallocation
for j = 1:numel(Lambda)
    rocObj = rocmetrics(Ystats,posterior(:,:,j),CVMdl.ClassNames);
    auc(j) = rocObj.AUC(1);
end

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

Mdl = fitclinear(X,Ystats,'ObservationsIn','columns', ...
    'Learner','logistic','Solver','sparsa','Regularization','lasso', ...
    'Lambda',Lambda,'GradientTolerance',1e-8);
numNZCoeff = sum(Mdl.Beta~=0);

In the same figure, plot the test-sample error rates and frequency of nonzero coefficients for each regularization strength. Plot all variables on the log scale.

figure
yyaxis left
plot(log10(Lambda),log10(auc),'o-')
ylabel('log_{10} AUC')
yyaxis right
plot(log10(Lambda),log10(numNZCoeff + 1),'o-')
ylabel('log_{10} nonzero-coefficient frequency')
xlabel('log_{10} Lambda')
title('Cross-Validated Statistics')
hold off

Choose the index of the regularization strength that balances predictor variable sparsity and high AUC. In this case, a value between $$10^{-3}$$ to $$10^{-1}$$ should suffice.

idxFinal = 9;

Select the model from Mdl with the chosen regularization strength.

MdlFinal = selectModels(Mdl,idxFinal);

MdlFinal is a ClassificationLinear model containing one regularization strength. To estimate labels for new observations, pass MdlFinal and the new data to predict.