Class Prediction in Test Sets with Shifted Distributions

Machine learning has provided powerful algorithms that automatically generate predictive models from experience. One specific technique is supervised learning, where the machine is trained to predict a desired output for each input pattern x. This chapter will focus on classification, that is, supervised learning when the output to predict is a class label. For instance predicting whether a patient in a hospital will develop cancer or not. In this example, the class label c is a variable having two possible values, “cancer” or “no cancer”, and the input pattern x is a vector containing patient data (e.g. age, gender, diet, smoking habits, etc.). In order to construct a proper predictive model, supervised learning methods require a set of examples xi together with their respective labels ci. This dataset is called the “training set”. The constructed model is then used to predict the labels of a set of new cases xj called the “test set”. In the cancer prediction example, this is the phase when the model is used to predict cancer in new patients. One common assumption in supervised learning algorithms is that the statistical structure of the training and test datasets are the same (Hastie, Tibshirani & Friedman, 2001). That is, the test set is assumed to have the same attribute distribution p(x) and same class distribution p(c|x) as the training set. However, this is not usually the case in real applications due to different reasons. For instance, in many problems the training dataset is obtained in a specific manner that differs from the way the test dataset will be generated later. Moreover, the nature of the problem may evolve in time. These phenomena cause pTr(x, c) ≠ pTest(x, c), which can degrade the performance of the model constructed in training. Here we present a new algorithm that allows to re-estimate a model constructed in training using the unlabelled test patterns. We show the convergence properties of the algorithm and illustrate its performance with an artificial problem. Finally we demonstrate its strengths in a heart disease diagnosis problem where the training set is taken from a different hospital than the test set.