An Empirical Study of Moment Estimators for Quantile Approximation

Abstract

We empirically evaluate lightweight moment estimators for the single-pass quantile approximation problem, including maximum entropy methods and orthogonal series with Fourier, Cosine, Legendre, Chebyshev and Hermite basis functions. We show how to apply stable summation formulas to offset numerical precision issues for higher-order moments, leading to reliable single-pass moment estimators up to order 15. Additionally, we provide an algorithm for GPU-accelerated quantile approximation based on parallel tree reduction. Experiments evaluate the accuracy and runtime of moment estimators against the state-of-the-art KLL quantile estimator on 14,072 real-world datasets drawn from the OpenML database. Our analysis highlights the effectiveness of variants of moment-based quantile approximation for highly space efficient summaries: their average performance using as few as five sample moments can approach the performance of a KLL sketch containing 500 elements. Experiments also illustrate the difficulty of applying the method reliably and showcases which moment-based approximations can be expected to fail or perform poorly.