Information measure for long-range correlated sequences: the case of the 24 human chromosomes

Long-Range Dependence in Financial Markets: A Moving Average Cluster Entropy Approach

A perspective is taken on the intangible complexity of economic and social systems by investigating the dynamical processes producing, storing and transmitting information in financial time series. An extensive analysis based on the moving average cluster entropy approach has evidenced market and horizon dependence in highest-frequency data of real world financial assets. The behavior is scrutinized by applying the moving average cluster entropy approach to long-range correlated stochastic processes as the Autoregressive Fractionally Integrated Moving Average (ARFIMA) and Fractional Brownian motion (FBM). An extensive set of series is generated with a broad range of values of the Hurst exponent H and of the autoregressive, differencing and moving average parameters p,d,q. A systematic relation between moving average cluster entropy and long-range correlation parameters H, d is observed. This study shows that the characteristic behaviour exhibited by the horizon dependence of the cluster entropy is related to long-range positive correlation in financial markets. Specifically, long range positively correlated ARFIMA processes with differencing parameter d≃0.05, d≃0.15 and d≃0.25 are consistent with moving average cluster entropy results obtained in time series of DJIA, S&P500 and NASDAQ. The findings clearly point to a variability of price returns, consistently with a price dynamics involving multiple temporal scales and, thus, short- and long-run volatility components. An important aspect of the proposed approach is the ability to capture detailed horizon dependence over relatively short horizons (one to twelve months) and thus its relevance to define risk analysis indices.

Some comments on Bitcoin market (in)efficiency

In this paper, we explore the (in)efficiency of the continuum Bitcoin-USD market in the period ranging from mid 2010 to early 2019. To deal with, we dynamically analyse the evolution of the self-similarity exponent of Bitcoin-USD daily returns via accurate FD4 approach by a 512 day sliding window with overlapping data. Further, we define the memory indicator by the difference between the self-similarity exponent of Bitcoin-USD series and the self-similarity index of its shuffled series. We also carry out additional analyses via FD4 approach by sliding windows of sizes equal to 64, 128, 256, and 1024 days, and also via FD algorithm for values of q equal to 1 and 2 (and sliding windows equal to 512 days). Moreover, we explored the evolution of the self-similarity exponent of actual S&P500 series via FD4 algorithm by sliding windows of sizes equal to 256 and 512 days. In all the cases, the obtained results were found to be similar to our first analysis. We conclude that the self-similarity exponent of the BTC-USD (resp., S&P500) series stands above 0.5. However, this is not due to the presence of significant memory in the series but to its underlying distribution. In fact, it holds that the self-similarity exponent of BTC-USD (resp., S&P500) series is similar or lower than the self-similarity index of a random series with the same distribution. As such, several periods with significant antipersistent memory in BTC-USD (resp., S&P500) series are distinguished.

Analysis of human DNA through power-law statistics

We report an analysis of Homo sapiens DNA through the formalism of κ statistics, which encompasses power-law correlations and provides an optimization principle that permits us to model distinct physical systems; i.e., the power-law distribution of the length of DNA bases is calculated from a general model which follows arguments similar to those proposed in Maxwell's deduction of statistical distributions. The viability of the model is tested using a data set from a catalog of proteins collected from the Ensembl Project. The results indicate that the short-range correlations, always present in coding DNA sequences, are appropriately captured through the Kaniadakis power-law distribution, adequately describing the cumulative length distribution of DNA bases, in contrast with the case of the traditional exponential statistical model.

An efficient algorithm for extracting the magnitude of the measurement error for fractional dynamics

Modern live-imaging fluorescent microscopy techniques following the stochastic motion of labeled tracer particles, i.e. single particle tracking (SPT) experiments, have uncovered significant deviations from the laws of Brownian motion in a variety of biological systems. Accurately characterizing the anomalous diffusion for SPT experiments has become a central issue in biophysics. However, measurement errors raise difficulty in the analysis of single trajectories. In this paper, we introduce a novel surface calibration method based on a fractionally integrated moving average (FIMA) process as an effective tool for extracting both the magnitude of the measurement error and the anomalous exponent for autocorrelated processes of various origins. This method is developed using a toy model - fractional Brownian motion disturbed by independent Gaussian white noise - and is illustrated on both simulated and experimental biological data. We also compare this new method with the mean-squared displacement (MSD) technique, extended to capture the measurement noise in the toy model, which shows inferior results. The introduced procedure is expected to allow for more accurate analysis of fractional anomalous diffusion trajectories with measurement errors across different experimental fields and without the need for any calibration measurements.

Detrended fluctuation analysis and the difference between external drifts and intrinsic diffusionlike nonstationarity

Detrended fluctuation analysis (DFA) has been shown to be an effective method to study long-range correlation of nonstationary series. In principle, DFA considers F_{DFA}^{2}(s), the mean of variance around the local polynomial fit in segments with length s, and then estimates the scaling exponent α_{DFA} in F_{DFA}(s)∼s^{α_{DFA}} with varying s. Usually, the methodological studies of DFA focus on its effect on removing the drift due to the external trends. Only few paid attention to nonstationary series without drift, such as fractional Brownian motion (FBM) with nonstationarity due to its intrinsic dynamics. Both of these types of nonstationarity can shift the local mean by drift or diffusion and can be treated as the additive nonstationarity eliminable by the additive decomposition. In this study, we limit our discussion to such additive nonstationarity and furthermore specifically distinguish these two types of nonstationarity, namely the drift and the intrinsic diffusionlike nonstationarity. To understand how DFA works for the intrinsic diffusionlike nonstationarity, we take FBM as the example and seek for the answers to two fundamental questions: (1) what DFA removes from FBM; and (2) why DFA can handle such intrinsic diffusionlike nonstationarity, in contrast to methods only applicable to stationary series such as the fluctuation analysis. A crucial condition, i.e., statistical equivalence among all segments, is proposed and checked in the fluctuation analysis and DFA. As shown, the crucial condition is a natural requirement for the connection between DFA and autocorrelation function. With the help of the crucial condition, our study analytically and numerically demonstrates for the intrinsic diffusionlike nonstationary series that (1) rather than the nonstationarity as thought, DFA actually removes the difference among all segments; (2) the detrended segments fulfill the crucial condition so that the average over segments becomes equivalent to the ensemble average over realizations. These answers are also true for series with a drift. Thus, we provide a unified perspective to refresh the understanding of how DFA works on nonstationarity and underpin the mathematical ground of DFA.

Detrending moving average algorithm: Frequency response and scaling performances

The Detrending Moving Average (DMA) algorithm has been widely used in its several variants for characterizing long-range correlations of random signals and sets (one-dimensional sequences or high-dimensional arrays) over either time or space. In this paper, mainly based on analytical arguments, the scaling performances of the centered DMA, including higher-order ones, are investigated by means of a continuous time approximation and a frequency response approach. Our results are also confirmed by numerical tests. The study is carried out for higher-order DMA operating with moving average polynomials of different degree. In particular, detrending power degree, frequency response, asymptotic scaling, upper limit of the detectable scaling exponent, and finite scale range behavior will be discussed.

Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors - An alternative approach

Accurately characterizing the anomalous diffusion of a tracer particle has become a central issue in biophysics. However, measurement errors raise difficulty in the characterization of single trajectories, which is usually performed through the time-averaged mean square displacement (TAMSD). In this paper, we study a fractionally integrated moving average (FIMA) process as an appropriate model for anomalous diffusion data with measurement errors. We compare FIMA and traditional TAMSD estimators for the anomalous diffusion exponent. The ability of the FIMA framework to characterize dynamics in a wide range of anomalous exponents and noise levels through the simulation of a toy model (fractional Brownian motion disturbed by Gaussian white noise) is discussed. Comparison to the TAMSD technique, shows that FIMA estimation is superior in many scenarios. This is expected to enable new measurement regimes for single particle tracking (SPT) experiments even in the presence of high measurement errors.