A Recursive Recomputation Approach for Smoothing in Nonlinear State–Space Modeling: An Attempt for Reducing Space Complexity

In this paper, we develop a new generic implementation scheme for numerical smoothing in nonlinear and Bayesian state-space modeling. Our new generic implementation scheme, which we call recursive recomputation scheme, reduces the space complexity from O(MT) to O(M log T), at the cost of O(log T) times computation of filtering distributions in time complexity. This reduction is accomplished by employing carefully designed recursive recomputation. The Japanese stock market price time-series data with T = 956 is taken up as an instance to demonstrate advantage of the proposed scheme. The path-sampling particle smoother is implemented with the scheme to smooth the whole interval estimating the change of volatility. The number of particles is 3 000 000, and the whole interval is smoothed with 5.3-GB storage, accomplishing saving of storage by a factor of 1/20. The computed smoothing distribution is compared with the ones computed with the existing two other well-known smoothers, the forward-backward smoother and the smoother based on two-filter formula. It turns out that, among the three, ours is the only method which succeeded in computing a reliable and plausible smoothing distribution in the situation.