Efficient Sequential and Parallel Algorithms for Estimating Higher Order Spectra

Abstract

Higher order spectra (HOS) are a powerful tool in nonlinear time series analysis and they have been extensively used as feature representations in data mining, communications and cosmology domains. However, HOS estimation suffers from high computational cost and memory consumption. Any algorithm for computing the kth order spectra on a dataset of size n needs O(n^k-1 ) time since the output size will be O(n^k-1 ) as well, which makes the direct HOS analysis difficult for long time series, and further prohibits its direct deployment to resource-limited and time-sensitive applications. Existing algorithms for computing HOS are either inefficient or have been implemented on obsolete architectures. Thus it is essential to develop efficient generic algorithms for HOS estimations. In this paper, we present a package of generic sequential and parallel algorithms for computationally and memory efficient HOS estimations which can be employed on any parallel machine or platform. Our proposed algorithms largely reduce the HOS' computational cost and memory usage in spectrum multiplication and smoothing steps through carefully designed prefix sum operations. Moreover, we employ a matrix partitioning technique and design algorithms with optimal memory usage and present the parallel approaches on the PRAM and the mesh models. Furthermore, we implement our algorithms for both bispectrum and trispectrum estimations. We conduct extensive experiments and cross-compare the proposed algorithms' performance. Results show that our algorithms achieve state-of-the-art computational and memory efficiency, and our parallel algorithms achieve close to linear speedups. The code is available at https://github.com/ZigengWang/HOS.