SALSA: Sequential Approximate Leverage-Score Algorithm with Application in Analyzing Big Time Series Data
This provides an efficient solution for analyzing big time series data, with incremental improvements in computational complexity and accuracy.
The paper tackles the problem of computing leverage scores for large matrices and fitting ARMA models to big time series data, resulting in an algorithm (SALSA) with high-probability accuracy within (1 + O(ε)) of true scores and worst-case running time significantly improved over state-of-the-art alternatives.
We develop a new efficient sequential approximate leverage score algorithm, SALSA, using methods from randomized numerical linear algebra (RandNLA) for large matrices. We demonstrate that, with high probability, the accuracy of SALSA's approximations is within $(1 + O({\varepsilon}))$ of the true leverage scores. In addition, we show that the theoretical computational complexity and numerical accuracy of SALSA surpass existing approximations. These theoretical results are subsequently utilized to develop an efficient algorithm, named LSARMA, for fitting an appropriate ARMA model to large-scale time series data. Our proposed algorithm is, with high probability, guaranteed to find the maximum likelihood estimates of the parameters for the true underlying ARMA model. Furthermore, it has a worst-case running time that significantly improves those of the state-of-the-art alternatives in big data regimes. Empirical results on large-scale data strongly support these theoretical results and underscore the efficacy of our new approach.