Possibilistic KNN Regression Using Tolerance Intervals

Abstract

Abstract. By employing regression methods minimizing predictive risk, we are usually looking for precise values which tends to their true response value. However, in some situations, it may be more reasonable to predict intervals rather than precise values. In this paper, we focus to find such intervals for the K-nearest neighbors (KNN) method with precise values for inputs and output. In KNN, the prediction intervals are usually built by considering the local probability distribution of the neighborhood. In situations where we do not dispose of enough data in the neighborhood to obtain statistically significant distributions, we would rather wish to build intervals which takes into account such distribution uncertainties. For this latter we suggest to use tolerance intervals to build the maximal specific possibility distribution that bounds each population quantiles of the true distribution (with a fixed confidence level) that might have generated our sample set. Next we propose a new interval regression method based on KNN which take advantage of our possibility distribution in order to choose, for each instance, the value of K which will be a good trade-off between precision and uncertainty due to the limited sample size. Finally we apply our method on an aircraft trajectory prediction problem.

Naive possibilistic classifiers for imprecise or uncertain numerical data

“…Moreover, G(x, µ, σ 2 ) is the cumulated distribution function of the N (µ, σ 2 ). Possibility distributions encoding a family of probability distributions have been successfully applied to regression [35] where possibilistic k−NN regression consists in predicting intervals rather than precise values. For a detailed discussion about encoding probability distributions by possibility distributions, see [1,36].…”