Maximum-likelihood period estimation from sparse, noisy timing data

Abstract

The problem of estimating the period of a periodic point process is considered when the observations are sparse and noisy. There is a class of estimators that operate by maximizing an objective function over an interval of possible periods, notably the periodogram estimator of Fogel & Gavish [1] and the line-search algorithms of Sidiropoulos et al. [2] and Clarkson [3]. For numerical calculation, the interval is sampled. However, it is not known how fine the sampling must be in order to ensure statistically accurate results. In this paper, a new estimator is proposed which eliminates the need for sampling. For the proposed statistical model, it calculates a maximumlikelihood estimate. It is shown that the expected arithmetic complexity of the algorithm is O(n 3 log n) where n is the number of observations. Numerical simulations demonstrate the superior statistical performance of the new estimator.

Fast approximate maximum likelihood period estimation from incomplete timing data

“…However, it was artificially determined without any theoretic basis previously in Refs. [8][9][10] . In Section 3, a formula for automatically calculating the GS is derived.…”

“…(3), it is easy to see that the MLE can be obtained by searching the peak of SðfÞ. Usually a two-step numerical search procedure is adopted [8][9][10][11] as follows.…”

“…10 Here the GS determination problem is demonstrated based on the periodogram estimator. 7 Fogel and Gavish proposed the periodogram algorithm, 7 which was the earliest estimator for period estimation from incomplete observations.…”

“…9 Recently, McKilliam and Clarkson have shown that the performance of these estimators is determined by the chosen grid spacing (GS). 10 In fact, too large a GS may lead to an estimate with large bias. Moreover, they also pointed out that no theory had been proposed for GS determination.…”

“…Moreover, they also pointed out that no theory had been proposed for GS determination. Most recently, they proposed an ML estimator called integer lattice line search (ZnLLS) 10 to circumvent grid search by enumerating all the points in the so-called Bresenham set. However, this manner leads to a high computational complexity of Oðn 3 lg nÞ.…”