Estimating Tukey Depth Using Incremental Quantile Estimators

Abstract

The concept of depth measures how deep an arbitrary point is positioned in a dataset and can be seen as the opposite of outlyingness. A wide range of approaches within pattern recognition and computational statistics build on this concept.

To address the well-known computational challenges associated with the depth concept, we suggest to estimate Tukey depth contours using recently developed incremental quantile estimators. The suggested algorithm can not only estimate depth contours when the dataset is known in advance, but also can track Tukey depth contours for dynamically varying data stream distributions using recursive update. The tracking ability of the algorithm was demonstrated based on a real-life application associated with detecting changes in human activity from real-time accelerometer observations. Given the flexibility of the suggested approach, it can detect virtually any kind of changes in the distributional patterns of the observations, and thus outperforms detection approaches based on the Mahalanobis distance.