predict

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';

Train 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 
Mdl = fitclinear(X,Ystats);

Mdl is a ClassificationLinear model.

Predict the training-sample, or resubstitution, labels.

label = predict(Mdl,X);

Because there is one regularization strength in Mdl, label is column vectors with lengths equal to the number of observations.

Construct a confusion matrix.

ConfusionTrain = confusionchart(Ystats,label);

The model misclassifies only one 'stats' documentation page as being outside of the Statistics and Machine Learning Toolbox documentation.

Load the NLP data set and preprocess it as in Predict Training-Sample Labels. Transpose the predictor data matrix.

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

Train 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. Specify to hold out 30% of the observations. Optimize the objective function using SpaRSA.

rng(1) % For reproducibility 
CVMdl = fitclinear(X,Ystats,'Solver','sparsa','Holdout',0.30,...
    'ObservationsIn','columns');
Mdl = CVMdl.Trained{1};

CVMdl is a ClassificationPartitionedLinear model. It contains the property Trained, which is a 1-by-1 cell array holding a ClassificationLinear model that the software trained using the training set.

Extract the training and test data from the partition definition.

trainIdx = training(CVMdl.Partition);
testIdx = test(CVMdl.Partition);

Predict the training- and test-sample labels.

labelTrain = predict(Mdl,X(:,trainIdx),'ObservationsIn','columns');
labelTest = predict(Mdl,X(:,testIdx),'ObservationsIn','columns');

Because there is one regularization strength in Mdl, labelTrain and labelTest are column vectors with lengths equal to the number of training and test observations, respectively.

Construct a confusion matrix for the training data.

ConfusionTrain = confusionchart(Ystats(trainIdx),labelTrain);

The model misclassifies only three documentation pages as being outside of Statistics and Machine Learning Toolbox documentation.

Construct a confusion matrix for the test data.

ConfusionTest = confusionchart(Ystats(testIdx),labelTest);

The model misclassifies three documentation pages as being outside the Statistics and Machine Learning Toolbox, and two pages as being inside.

Estimate test-sample, posterior class probabilities, and determine the quality of the model by plotting a receiver operating characteristic (ROC) curve. Linear classification models return posterior probabilities for logistic regression learners only.

Load the NLP data set and preprocess it as in Predict Test-Sample Labels.

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

Randomly partition the data into training and test sets by specifying a 30% holdout sample. Identify the test-set indices.

cvp = cvpartition(Ystats,'Holdout',0.30);
idxTest = test(cvp);

Train a binary linear classification model. Fit logistic regression learners using SpaRSA. To hold out the test set, specify the partitioned model.

CVMdl = fitclinear(X,Ystats,'ObservationsIn','columns','CVPartition',cvp,...
    'Learner','logistic','Solver','sparsa');
Mdl = CVMdl.Trained{1};

Mdl is a ClassificationLinear model trained using the training set specified in the partition cvp only.

Predict the test-sample posterior class probabilities.

[~,posterior] = predict(Mdl,X(:,idxTest),'ObservationsIn','columns');

Because there is one regularization strength in Mdl, posterior is a matrix with 2 columns and rows equal to the number of test-set 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(idxTest),posterior,Mdl.ClassNames);

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

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

The ROC curve indicates that the model classifies the test-sample observations almost perfectly.

To determine a good lasso-penalty strength for a linear classification model that uses a logistic regression learner, compare test-sample values of the AUC.

Load the NLP data set. Preprocess the data as in Predict Test-Sample Labels.

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

Create a data partition that specifies to holdout 10% of the observations. Extract test-sample indices.

rng(10); % For reproducibility
Partition = cvpartition(Ystats,'Holdout',0.10);
testIdx = test(Partition);
XTest = X(:,testIdx);
n = sum(testIdx)

n = 
3157

YTest = Ystats(testIdx);

There are 3157 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);

Train binary, linear classification models that use each of the regularization strengths. Optimize the objective function using SpaRSA. Lower the tolerance on the gradient of the objective function to 1e-8.

CVMdl = fitclinear(X,Ystats,'ObservationsIn','columns',...
    'CVPartition',Partition,'Learner','logistic','Solver','sparsa',...
    'Regularization','lasso','Lambda',Lambda,'GradientTolerance',1e-8)

CVMdl = 
  ClassificationPartitionedLinear
    CrossValidatedModel: 'Linear'
           ResponseName: 'Y'
        NumObservations: 31572
                  KFold: 1
              Partition: [1x1 cvpartition]
             ClassNames: [0 1]
         ScoreTransform: 'none'

Extract the trained linear classification model.

Mdl1 = CVMdl.Trained{1}

Mdl1 = 
  ClassificationLinear
      ResponseName: 'Y'
        ClassNames: [0 1]
    ScoreTransform: 'logit'
              Beta: [34023x11 double]
              Bias: [-11.9079 -11.9079 -11.9079 -11.9079 -9.3362 -6.4290 -5.1424 -4.4991 -3.5732 -3.1742 -2.9839]
            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'

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

Estimate the test-sample predicted labels and posterior class probabilities.

[label,posterior] = predict(Mdl1,XTest,'ObservationsIn','columns');
Mdl1.ClassNames;
posterior(3,1,5)

ans = 
1.0000

label is a 3157-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 3157-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(YTest,posterior(:,:,j),Mdl1.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('Test-Sample Statistics')
hold off

Choose the index of the regularization strength that balances predictor variable sparsity and high AUC. In this case, a value between $$10^{-2}$$ 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.