Quantitative Data Analysis

Time series in statistics is a series of data points indexed in time order. Most commonly, these data points are considered at successive equally spaced points in time. Thus, we obtain then a sequence of discrete-time data. Examples of time series are daily measures of fever,  yearly counts of an epidemic in a country, and the daily closing value of stock market indices. Such data requires special analysis methods (time-domain methods where we are concentrated and frequency-domain methods) in order to extract meaningful statistics and other characteristics (e.g. Granger causality to detect relationship between time-series, a model from the statistical domain or a model from the machine learning domain to predict future values based on previously observed values, the build of a regression model to measure the impact of factors, autocorrelation for measuring short-term memory, Hurst exponent to measure long-term memory).  

History of development: The Hurst exponent (H) is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. The first studies which used this measure were originally developed in the field of hydrology for the practical matter of determining optimum dam sizing for the Nile river‘s volatile rain and drought conditions and Harold Edwin Hurst was the lead researcher in these studies and this is where the name of this metric come from.

Explanation: H takes on values from 0 to 1. If H=0.5, this fact indicates the absence of long-range dependence. The closer H is to 1, the greater the degree of persistence or long-range dependence. If H is less than 0.5, this fact corresponds to anti-persistency (the opposite of LRD, which indicates strong negative correlation and the process fluctuates violently).

Estimation: Aggregated (or Differenced Aggregated) Variance Method: the time series are divided on blocks and the slope (beta=2*H-2) from the least square fit of the logarithm of the sample variances versus the logarithm of the block sizes provides an estimate for the Hurst exponent , Aggregated Absolute Value/Moment Method: the Hurst exponent from the moments of absolute values of an aggregated FGN or FARIMA time series process. The first moment coincides with the absolute value method, and the second moment with the aggregated variance method, Higuchi or Fractal Dimension Method: Instead of blocks a sliding window is used to compute the aggregated series. The method is very similar with the involves the calculation of the length of a path and, in principle, finding its fractal Dimension D. The slope D=2-H from the least square fit of the logarithm of the expected path lengths versus the logarithm of the block (window) sizes provides an estimate for the Hurst exponent H, Peng or Variance of Residuals Method: In Peng’s variance of residuals method the series is also divided into blocks of size m. Within each block the cumulated sums are computed up to t and a least-squares line a+b*t is fitted to the cumulated sums. Then the sample variance of the residuals is computed which is proportional to m^(2*H). The “mean” or “median” are computed over the blocks. The slope beta=2*H from the least square provides an estimate for the Hurst exponent H, R/S Method: is based on the time-domain analysis of the rescaled adjusted range, Periodogram Method:   estimates the Hurst exponent from the periodogram. In the finite variance case, the periodogram is an estimator of the spectral density of the time series. A series with long range dependence will show a spectral density with a lower law behavior in the frequency. Thus, we expect that a log-log plot of the periodogram versus frequency will display a straight line, and the slopw can be computed as 1-2H, Whittle Estimator:  performs a periodogram analysis. The algorithm is based on the minimization of a likelihood function defined in the frequency domain.                                                                                                                           Details about the estimation methods can be found in the article Taqqu, M. S., Teverovsky, V., & Willinger, W. (1995). Estimators for long-range dependence: an empirical study. Fractals, 3(04), 785-798.

Models which includes the concept of long memory: examples are FARIMA, FIGARCH etc. (discrete-time) and Fractional Brownian Motion etc. (continous-time).

Implementation: Software like R.