Fast and Accurate Partial Fourier Transform for Time Series Data

Given a time-series vector, how can we efficiently detect anomalies? A widely used method is to use Fast Fourier transform (FFT) to compute Fourier coefficients, take first few coefficients while discarding the remaining small coefficients, and reconstruct the original time series to find points with large errors. Despite the pervasive use, the method requires to compute all of the Fourier coefficients which can be cumbersome if the input length is large or when we need to perform many FFT operations.

In this paper, we propose Partial Fourier Transform (PFT), an efficient and accurate algorithm for computing only a part of Fourier coefficients. PFT approximates a part of twiddle factors (trigonometric constants) using polynomials, thereby reducing the computational complexity due to the mixture of many twiddle factors. We derive the asymptotic time complexity of PFT with respect to input and output sizes, and tolerance. We also show that PFT provides an option to set an arbitrary approximation error bound, which is useful especially when the fast evaluation is of utmost importance. Experimental results show that PFT outperforms the current state-of-the-art algorithms, with an order of magnitude of speedup for sufficiently small output sizes without sacrificing accuracy. In addition, we demonstrate the accuracy and efficacy of PFT on real-world anomaly detection, with interpretations of anomalies in stock price data.