Smooth estimates of multiple quantiles in dynamically varying data streams

In this paper, we investigate the problem of estimating multiple quantiles when samples are received online (data stream). We assume a dynamical system, i.e., the distribution of the samples from the data stream changes with time. A major challenge of using incremental quantile estimators to track multiple quantiles is that we are not guaranteed that the monotone property of quantiles will be satisfied, i.e, an estimate of a lower quantile might erroneously overpass that of a higher quantile estimate. Surprisingly, we have only found two papers in the literature that attempt to counter these challenges, namely the works of Cao et al. (Proceedings of the first ACM workshop on mobile internet through cellular networks, ACM, 2009) and Hammer and Yazidi (Proceedings of the 30th international conference on industrial engineering and other applications of applied intelligent systems (IEA/AIE), France, Springer, 2017) where the latter is a preliminary version of the work in this paper. Furthermore, the state-of-the-art incremental quantile estimator called deterministic update-based multiplicative incremental quantile estimator (DUMIQE), due to Yazidi and Hammer (IEEE Trans Cybernet, 2017), fails to guarantee the monotone property when estimating multiple quantiles. A challenge with the solutions, in Cao et al. (2009) and Hammer and Yazidi (2017), is that even though the estimates satisfy the monotone property of quantiles, the estimates can be highly irregular relative to each other which usually is unrealistic from a practical point of view. In this paper, we suggest to generate the quantile estimates by inserting the quantile probabilities (e.g., \(0.1, 0.2, \ldots , 0.9\)) into a monotonically increasing and infinitely smooth function (can be differentiated infinitely many times). The function is incrementally updated from the data stream. The monotonicity and smoothness of the function ensure that both the monotone property and regularity requirement of the quantile estimates are satisfied. The experimental results show that the method performs very well and estimates multiple quantiles more precisely than the original DUMIQE (Yazidi and Hammer 2017), and the approaches reported in Hammer and Yazidi (2017) and Cao et al. (2009).