Detecting Periodicity in Short and Noisy Time Series Data

Abstract

A characteristic problem in population ecology is the 5 scarcity of long-term population records. Shortness and noise together comprise a difficult combination for using the methods of time series analysis to reveal the patterns in population records. Here we present a new method, based on Poincare mapping and randomizing techniques, to effectively detect periodicity in population data. The method, phase coherence analysis, is shown to surpass calculation of autocorrelation functions (Box and Jenkins 1976) in efficiency and power, especially with short and noisy data. Although periodic fluctuations in animal numbers exemplify just one possible population dynamic pattern among the others, it has offered an intriguing challenge for theorists for decades (e.g., Elton 1924, Finerty 1980, Turchin 1990, Royama 1992). Naturally, recognizing the patterns in population numbers is a precondition for unveiling the mechanisms producing them. The most commonly used methods for finding periodicity in time series are autocorrelation and spectral analyses (Box and Jenkins 1976, Chatfield 1989). Both of these methods require the data being at least second-order stationary, i.e., having time-invariant mean and variance. Against this background any novel tool for detecting periodicity would be of use in ecological research. Following the rise of chaos theory in the 1980s, some much older ideas, such as Poincar& mapping, experienced a renaissance (Stewart 1990, Peitgen et al. 1992). Periodicity was to some extent shadowed by chaos in the applications of Poincare mapping. It was largely seen only as a way to perceive footprints of chaotic dynamics. However, Poincar6 mapping offers a valuable insight into periodicity as such, which is readily seen in its basic idea. In Poincare mapping the state of a system say, a population at time t is defined by coordinates in n-dimensional space. With different values of t, the state moves tracing out a route in the space. A steady plane is put across this route, and the location of the state on the plane is recorded every time the state perforates the plane. If the solution of the system is periodic, the point perforates exactly the same point of the plane after having completed the next round in its route. Mathematically, a time series {y(t)} is periodic if there exists a period length T so that