Load Sample Data

This example uses the 1994 census data stored in census1994.mat. The data set consists of demographic data from the US Census Bureau to predict whether an individual makes over $50,000 per year. The classification task is to fit a model that predicts the salary category of people given their age, working class, education level, marital status, race, and so on.

Load the sample data census1994 and display the variables in the data set.

load census1994
whos

  Name                 Size              Bytes  Class    Attributes

  Description         20x74               2960  char               
  adultdata        32561x15            1872566  table              
  adulttest        16281x15             944466  table              

census1994 contains the training data set adultdata and the test data set adulttest. For this example, to reduce the running time, subsample 5000 training and test observations each, from the original tables adultdata and adulttest, by using the datasample function. (You can skip this step if you want to use the complete data sets.)

NumSamples = 5000;
s = RandStream('mlfg6331_64'); % For reproducibility
adultdata = datasample(s,adultdata,NumSamples,'Replace',false);
adulttest = datasample(s,adulttest,NumSamples,'Replace',false);

Preview the first few rows of the training data set.

head(adultdata)

    age     workClass       fnlwgt       education      education_num      marital_status         occupation         relationship     race      sex      capital_gain    capital_loss    hours_per_week    native_country    salary
    ___    ___________    __________    ____________    _____________    __________________    _________________    ______________    _____    ______    ____________    ____________    ______________    ______________    ______

    39     Private          4.91e+05    Bachelors            13          Never-married         Exec-managerial      Other-relative    Black    Male           0               0                45          United-States     <=50K 
    25     Private        2.2022e+05    11th                  7          Never-married         Handlers-cleaners    Own-child         White    Male           0               0                45          United-States     <=50K 
    24     Private        2.2761e+05    10th                  6          Divorced              Handlers-cleaners    Unmarried         White    Female         0               0                58          United-States     <=50K 
    51     Private        1.7329e+05    HS-grad               9          Divorced              Other-service        Not-in-family     White    Female         0               0                40          United-States     <=50K 
    54     Private        2.8029e+05    Some-college         10          Married-civ-spouse    Sales                Husband           White    Male           0               0                32          United-States     <=50K 
    53     Federal-gov         39643    HS-grad               9          Widowed               Exec-managerial      Not-in-family     White    Female         0               0                58          United-States     <=50K 
    52     Private             81859    HS-grad               9          Married-civ-spouse    Machine-op-inspct    Husband           White    Male           0               0                48          United-States     >50K  
    37     Private        1.2429e+05    Some-college         10          Married-civ-spouse    Adm-clerical         Husband           White    Male           0               0                50          United-States     <=50K 

Each row represents the attributes of one adult, such as age, education, and occupation. The last column salary shows whether a person has a salary less than or equal to $50,000 per year or greater than $50,000 per year.

Understand Data and Choose Classification Models

Statistics and Machine Learning Toolbox™ provides several options for classification, including classification trees, discriminant analysis, naive Bayes, nearest neighbors, support vector machines (SVMs), and classification ensembles. For the complete list of algorithms, see Classification.

Before choosing the algorithms to use for your problem, inspect your data set. The census data has several noteworthy characteristics:

Without making any assumptions or using prior knowledge of algorithms that you expect to work well on your data, you simply train all the algorithms that support tabular data and binary classification. Error-correcting output codes (ECOC) models are used for data with more than two classes. Discriminant analysis and nearest neighbor algorithms do not analyze data that contains both numeric and categorical variables. Therefore, the algorithms appropriate for this example are SVMs, a decision tree, an ensemble of decision trees, and a naive Bayes model. Some of these models, like decision tree and naive Bayes models, are better at handling data with missing values; that is, they return non-NaN predicted scores for observations with missing values.

Build Models and Tune Hyperparameters

To speed up the process, customize the hyperparameter optimization options. Specify 'ShowPlots' as false and 'Verbose' as 0 to disable plot and message displays, respectively. Also, specify 'UseParallel' as true to run Bayesian optimization in parallel, which requires Parallel Computing Toolbox™. Due to the nonreproducibility of parallel timing, parallel Bayesian optimization does not necessarily yield reproducible results.

hypopts = struct('ShowPlots',false,'Verbose',0,'UseParallel',true);

Start a parallel pool.

poolobj = gcp;

Starting parallel pool (parpool) using the 'Processes' profile ...
Connected to parallel pool with 8 workers.

You can fit the training data set and tune parameters easily by calling each fitting function and setting its 'OptimizeHyperparameters' name-value pair argument to 'auto'. Create the classification models.

% SVMs: SVM with polynomial kernel & SVM with Gaussian kernel
mdls{1} = fitcsvm(adultdata,'salary','KernelFunction','polynomial','Standardize','on', ...
    'OptimizeHyperparameters','auto','HyperparameterOptimizationOptions', hypopts);
mdls{2} = fitcsvm(adultdata,'salary','KernelFunction','gaussian','Standardize','on', ...
    'OptimizeHyperparameters','auto','HyperparameterOptimizationOptions', hypopts);

% Decision tree
mdls{3} = fitctree(adultdata,'salary', ...
    'OptimizeHyperparameters','auto','HyperparameterOptimizationOptions', hypopts);

% Ensemble of Decision trees
mdls{4} = fitcensemble(adultdata,'salary','Learners','tree', ...
    'OptimizeHyperparameters','auto','HyperparameterOptimizationOptions', hypopts);

% Naive Bayes
mdls{5} = fitcnb(adultdata,'salary', ...
    'OptimizeHyperparameters','auto','HyperparameterOptimizationOptions', hypopts);

Plot Minimum Objective Curves

Extract the Bayesian optimization results from each model and plot the minimum observed value of the objective function for each model over every iteration of the hyperparameter optimization. The objective function value corresponds to the misclassification rate measured by five-fold cross-validation using the training data set. The plot compares the performance of each model.

figure
hold on
N = length(mdls);
for i = 1:N
    mdl = mdls{i};
    results = mdls{i}.HyperparameterOptimizationResults;
    plot(results.ObjectiveMinimumTrace,'Marker','o','MarkerSize',5);
end
names = {'SVM-Polynomial','SVM-Gaussian','Decision Tree','Ensemble-Trees','Naive Bayes'};
legend(names,'Location','northeast')
title('Bayesian Optimization')
xlabel('Number of Iterations')
ylabel('Minimum Objective Value')

Using Bayesian optimization to find better hyperparameter sets improves the performance of models over several iterations. In this case, the plot indicates that the ensemble of decision trees has the best prediction accuracy for the data. This model performs well consistently over several iterations and different sets of Bayesian optimization hyperparameters.

Check Performance with Test Set

Check the classifier performance with the test data set by using the confusion matrix and the receiver operating characteristic (ROC) curve.

Find the predicted labels and the score values of the test data set.

label = cell(N,1);
score = cell(N,1);
for i = 1:N
    [label{i},score{i}] = predict(mdls{i},adulttest);
end

Confusion Matrix

Obtain the most likely class for each test observation by using the predict function of each model. Then compute the confusion matrix with the predicted classes and the known (true) classes of the test data set by using the confusionchart function.

figure
c = cell(N,1);
for i = 1:N
    subplot(2,3,i)
    c{i} = confusionchart(adulttest.salary,label{i});
    title(names{i})
end

The diagonal elements indicate the number of correctly classified instances of a given class. The off-diagonal elements are instances of misclassified observations.

ROC Curve

Inspect the classifier performance more closely by plotting a ROC curve for each classifier and computing the area under the ROC curve (AUC). A ROC curve shows the true positive rate versus the false positive rate for different thresholds of classification scores. For a perfect classifier, whose true positive rate is always 1 regardless of the threshold, AUC = 1. For a binary classifier that randomly assigns observations to classes, AUC = 0.5. A large AUC value (close to 1) indicates good classifier performance.

Compute the metrics for a ROC curve and find the AUC value by creating a rocmetrics object for each classifier. Plot the ROC curves for the label '<=50K' by using the plot function of rocmetrics.

figure
AUC = zeros(1,N);
for i = 1:N
    rocObj = rocmetrics(adulttest.salary,score{i},mdls{i}.ClassNames);
    [r,g] = plot(rocObj,'ClassNames','<=50K');
    r.DisplayName = replace(r.DisplayName,'<=50K',names{i});
    g(1).DisplayName = join([names{i},' Model Operating Point']);
    AUC(i) = rocObj.AUC(1);
    hold on
end
title('ROC Curves for Class <=50K')
hold off

A ROC curve shows the true positive rate versus the false positive rate (or, sensitivity versus 1–specificity) for different thresholds of the classifier output.

Now plot the AUC values using a bar graph. For a perfect classifier, whose true positive rate is always 1 regardless of the thresholds, AUC = 1. For a classifier that randomly assigns observations to classes, AUC = 0.5. Larger AUC values indicate better classifier performance.

figure
bar(AUC)
title('Area Under the Curve')
xlabel('Model')
ylabel('AUC')
xticklabels(names)
xtickangle(30)
ylim([0.85,0.925])

Based on the confusion matrix and the AUC bar graph, the ensemble of decision trees and SVM models achieve better accuracy than the decision tree and naive Bayes models.

Resume Optimization of Most Promising Models

Running Bayesian optimization on all models for further iterations can be computationally expensive. Instead, select a subset of models that have performed well so far and continue the optimization for 30 more iterations by using the resume function. Plot the minimum observed values of the objective function for each iteration of Bayesian optimization.

figure
hold on
selectedMdls = mdls([1,2,4]);
newresults = cell(1,length(selectedMdls));
for i = 1:length(selectedMdls)
    newresults{i} = resume(selectedMdls{i}.HyperparameterOptimizationResults,'MaxObjectiveEvaluations',30);
    plot(newresults{i}.ObjectiveMinimumTrace,'Marker','o','MarkerSize',5)
end
title('Bayesian Optimization with resume')
xlabel('Number of Iterations')
ylabel('Minimum Objective Value')
legend({'SVM-Polynomial','SVM-Gaussian','Ensemble-Trees'},'Location','northeast')

The first 30 iterations correspond to the first round of Bayesian optimization. The next 30 iterations correspond to the results of the resume function. Resuming optimization is useful because the loss continues to reduce further after the first 30 iterations.