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

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. (i) Many methods have been proposed to assess long-range correlation in time series. Under what circumstances do they yield consistent results? (ii) 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? (iii) 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 Levy 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.