# A New Learning Strategy for Classification Problems with Different Training and Test Distributions

## Abstract

Standard machine learning techniques assume that the statistical structure of the training and test datasets are the same (i.e. same attribute distribution *p*(*x*), and same class distribution *p*(*c*|*x*)). However, in real prediction problems this is not usually the case for different reasons. For example, the training set is not usually representative of the whole problem due to sample selection biases during its acquisition. In addition, the measurement biases in training could be different than in test (for example, when the measurement devices are different). Another reason is that in real prediction tasks the statistical structure of the classes is not usually static but evolves in time, and there is usually a time lag between training and test sets. Due to these different problems, the performance of a learning algorithm can severely degrade. Here we present a new learning strategy that constructs a classifier in two steps. First, the labeled examples of the training set are used for constructing a statistical model of the problem. In the second step, the model is improved using the unlabeled patterns of the test set by means of a novel extension of the Expectation-Maximization (EM) algorithm presented here. 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.

## Keywords

Linear Discriminant Analysis Test Pattern Concept Drift Sample Selection Bias Covariate Shift## Preview

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## References

- 1.Black, M., Hickey, R.J.: Maintaining the performance of a learned classifier under concept drift. Intelligent Data Analysis 3, 453–474 (1999)CrossRefGoogle Scholar
- 2.Chen, Y., Wang, G., Dong, S.: Learning with progressive transductive support vector machine. Pattern Recognition Letters 24, 1845–1855 (2003)CrossRefGoogle Scholar
- 3.Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum-likelihood from incomplete data via the EM algorithm. J. Royal Statist. Soc. Ser. B. 39 (1977)Google Scholar
- 4.Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. John Wiley and Sons, New York (2001)zbMATHGoogle Scholar
- 5.Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, Heidelberg (2001)zbMATHGoogle Scholar
- 6.Heckman, J.J.: Sample selection bias as a specification error. Econometrica 47, 153–162 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
- 7.Newman, D.J., Hettich, S., Blake, C.L., Merz, C.J.: UCI Repository of machine learning databases. University of California, Irvine, CA, Department of Information and Computer Science (1998), http://www.ics.uci.edu/~mlearn/MLRepository.html
- 8.Salganicoff, M.: Tolerating concept and sampling shift in lazy learning using prediction error context switching. Artificial Intelligence Review 11, 133–155 (1997)CrossRefGoogle Scholar
- 9.Shimodaira, H.: Improving predictive inference under covariate shift by weighting the log-likelihood function. Journal of Statistical Planning and Inference 90, 227–244 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
- 10.Sugiyama, M., Müller, K.-R.: Input-dependent estimation of generalization error under covariate shift. Statistics and Decisions 23, 249–279 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
- 11.Vapnik, V.: Statistical Learning Theory. Wiley Interscience, New York (1998)zbMATHGoogle Scholar
- 12.Wang, H., Fan, W., Yu, P.S., Han, J.: Minining concept-drifting data streams using ensemble classifiers. In: Proc. 9th Int. Conf. on Knowledge Discovery and Data Mining, KDD (2003)Google Scholar
- 13.Widmer, G., Kubat, M.: Learning in the presence of concept drift and hidden contexts. Machine Learning 23, 69–101 (1996)Google Scholar
- 14.Wiens, D.P.: Robust weights and designs for biased regression models: Least Squares and Generalized M-Estimation. Journal of Statistical Planning and Inference 12, 412–483 (2000)Google Scholar
- 15.Wu, D., Bennett, K., Cristianini, N., Shawe-Taylor, J.: Large Margin Trees for Induction and Transduction. In: Proc. 16th International Conf. on Machine Learning (1999)Google Scholar