  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.
