Assessment of long-range correlation in time series: How to avoid pitfalls

Abstract

Due to the ubiquity of time series with long-range correlation in many areas of science and engineering, analysis and modeling of such data is an important problem. While the field seems to be mature, three major issues have not been satisfactorily resolved. ͑i͒ Many methods have been proposed to assess long-range correlation in time series. Under what circumstances do they yield consistent results? ͑ii͒ The mathematical theory of long-range correlation concerns the behavior of the correlation of the time series for very large times. A measured time series is finite, however. How can we relate the fractal scaling break at a specific time scale to important parameters of the data? ͑iii͒ An important technique in assessing long-range correlation in a time series is to construct a random walk process from the data, under the assumption that the data are like a stationary noise process. Due to the difficulty in determining whether a time series is stationary or not, however, one cannot be 100% sure whether the data should be treated as a noise or a random walk process. Is there any penalty if the data are interpreted as a noise process while in fact they are a random walk process, and vice versa? In this paper, we seek to gain important insights into these issues by examining three model systems, the autoregressive process of order 1, on-off intermittency, and Lévy motions, and considering an important engineering problem, target detection within sea-clutter radar returns. We also provide a few rules of thumb to safeguard against misinterpretations of long-range correlation in a time series, and discuss relevance of this study to pattern recognition.

A nonlinear dynamics model for simulating long range correlations of cognitive bistability

“…This relationship is valid for exactly second-order self-similar processes and for lim m → ∞. In Gao et al (2006b), the authors investigate multifractal processes and show the equivalence of Eq. 8 with the fluctuation analysis (FA) of a random walk process derived from a zero mean fractional Gaussian noise process.…”

“…The three methods used in this analysis for determining H -values provide consistent results concerning the H dependence on memory time constant τ M . They differ, however, with regard to the magnitude of the effect and have to be discussed with regard to problems mentioned in Gao et al (2006b).…”

“…The power spectrum of the dwell time series could in fact be (only) evaluated in a low frequency range, providing significantly lower H -values. In the present analysis, I follow the recommendation of Gao et al (2006b) and use the spectral method not alone but for cross-checking only. The results can be treated as upper limit due to the distortion of the spectrum by the broad mean reversal rate (1/ m ) frequency peak originating from the self-oscillator dynamics.…”

“…For example, H = 0.5 describes a completely stochastic (random walk) system governed by the laws of Brownian motion without any memory effects, whereas a system with H = 1 is completely deterministic and the initial conditions completely determine the future development. Difficulties arise from the requirements on the amount and quality of the data to be evaluated by sophisticated methods of nonlinear time series analysis for the quantification of the deterministic content (Gao et al 2006b(Gao et al ,c, 2007.…”

“…. , n} on the number of data points m used for averaging non-overlapping blocks of size m of the original data (variance-time (VT) method; e.g., Beran 1992; Gao et al 2006b), via …”