Evaluation of Scaling Invariance Embedded in Short Time Series

Abstract

Scaling invariance of time series has been making great contributions in diverse research fields. But how to evaluate scaling exponent from a real-world series is still an open problem. Finite length of time series may induce unacceptable fluctuation and bias to statistical quantities and consequent invalidation of currently used standard methods. In this paper a new concept called correlation-dependent balanced estimation of diffusion entropy is developed to evaluate scale-invariance in very short time series with length . Calculations with specified Hurst exponent values of show that by using the standard central moving average de-trending procedure this method can evaluate the scaling exponents for short time series with ignorable bias () and sharp confidential interval (standard deviation ). Considering the stride series from ten volunteers along an approximate oval path of a specified length, we observe that though the averages and deviations of scaling exponents are close, their evolutionary behaviors display rich patterns. It has potential use in analyzing physiological signals, detecting early warning signals, and so on. As an emphasis, the our core contribution is that by means of the proposed method one can estimate precisely shannon entropy from limited records.

Evolution of scaling behaviors embedded in sentence series from A Story of the Stone

“…Hence, a correction of the simple estimation in Eq (7) should make the bias and the statistical fluctuation reach simultaneously minima. By a procedure of minimizing the statistical average of the summation of bias and standard deviation of the estimated diffusion entropy, a tedious computation leads to a new estimator, called correlation dependent balanced estimation of diffusion entropy (cBEDE) [ 55 ]. Detailed calculations show that it can give us a reliable estimation of scaling exponent from a very short time series with several hundreds a length (e.g., for a trajectory of the fractional Brownian motion, if the length is 300, the bias is less than 0.03 and the standard deviation is about 0.05).…”

“…Hence, if we select the bin size in the procedure of estimating the PDF to be a certain fraction of the standard deviation of the de-trended displacements at time s (i.e, the size will increase with s ), the diffusion entropy should keep constant. Slight deviations from this constant comes from the algorithm and noises, which have been corrected in the final results of [ 55 ].…”

“…The key idea is to balance the bias and fluctuation to be sure their errors are all acceptable. Replacing the original definition of entropy in the DEA method with the balanced estimator, we proposed a new method called balanced estimator of diffusion entropy (cBEDE) [ 52 – 55 ]. Calculations with a large amount of theoretical and empirical data show that it can give us a reliable estimation of scaling behavior embedded in a very short time series with a length of ∼ 10 2 .…”

Brain Network Constancy and Participant Recognition: an Integrated Approach to Big Data and Complex Network Analysis

“…This study was limited in catching the flexible and dynamic characteristics of EEG signals when calculating the STE (McAuliffe, 2014). Further studies with the STE of short EEG sequences (about 10 2 points) (Zhang et al, 2012;Pan et al, 2014) would be required to avoid excessive reduction of brainwave features. 3.…”

Cellular coupling determines scale-invariant behavior of neurons in suprachiasmatic nucleus

“…With only ~200 data points in the bioluminescence time series, the application of DFA may lead to misleading results, because DFA requires the length of the time series to be larger than ~1000 (Ren et al 2018). Accordingly, a method which is tailored to measure scale-invariance in shorter time series (with a minimum of only around 100 data points), is correlation-dependent balance estimation of diffusion entropy (cBEDE), which is applied in the present study (Pan et al 2014a(Pan et al , 2014bQi and Yang 2011;Zhang et al 2012).…”