Hurst Exponent Analysis of Financial Time Series

Statistical properties of stock market time series and the implication of their Hurst exponents are discussed. Hurst exponents of DJIA (Dow Jones Industrial Average) components are tested using re scaled range analysis. In addition to the original stock return series, the linear prediction errors of the daily returns are also tested. Numerical results show that the Hurst exponent analysis can provide some information about the statistical properties of the financial time series.

Decrease in Hurst exponent of human gait with aging and neurodegenerative diseases

In this paper the decrease in the Hurst exponent of human gait with aging and neurodegenerative diseases was observed by using an improved rescaled range （R/S） analysis method. It indicates that the long-range correlations of gait rhythm from young healthy people are stronger than those from the healthy elderly and the diseased. The result further implies that fractal dynamics in human gait will be altered due to weakening or impairment of neural control on locomotion resulting from aging and neurodegenerative diseases. Due to analysing short-term data sequences rather than long datasets required by most nonlinear methods, the algorithm has the characteristics of simplicity and sensitivity, most importantly, fast calculation as well as powerful anti-noise capacities. These findings have implications for modelling locomotor control and also for quantifying gait dynamics in varying physiologic and pathologic states.

Improvement in Hurst exponent estimation and its application to financial markets

This research aims to improve the efficiency in estimating the Hurst exponent in financial time series.A new procedure is developed based on equality in distribution and is applicable to the estimation methods of the Hurst exponent.We show how to use this new procedure with three of the most popular algorithms(generalized Hurst exponet,total triangles area,and fractal dimension)in the literature.Findings show that this new approach improves the accuracy of the original methods,mainly for longer series.The second contribution of this study is that we show how to use this methodology to test whether the series is self-similar,constructing a confidence interval for the Hurst exponent for which the series satisfies this property.Finally,we present an empirical application of this new procedure to stocks of the S&P500 index.Similar to previous contributions,we consider this to be relevant to financial literature,as it helps to avoid inappropriate interpretations of market efficiency that can lead to erroneous decisions not only by market participants but also by policymakers.

Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Multifractal detrended fluctuation analysis （MF-DFA） is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis （DFA）, which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression r（q） = qh（q） - 1 stipulating the relationship between the multifractal exponent T（q） and the generalized Hurst exponent h（q）. As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as .t-（q） = qh（q） - qH - 1, where H is the nonconservation parameter in the universal multifractal formalism. The singular spectra, a and f（a）, are also derived according to this new relationship.

Spatial-temporal changes and driving factors of eco-environmental quality in the Three-North region of China

Eco-environmental quality is a measure of the suitability of the ecological environment for human survival and socioeconomic development.Understanding the spatial-temporal distribution and variation trend of eco-environmental quality is essential for environmental protection and ecological balance.The remote sensing ecological index(RSEI)can quickly and objectively quantify eco-environmental quality and has been extensively utilized in regional ecological environment assessment.In this paper,Moderate Resolution Imaging Spectroradiometer(MODIS)images during the growing period(July-September)from 2000 to 2020 were obtained from the Google Earth Engine(GEE)platform to calculate the RSEI in the three northern regions of China(the Three-North region).The Theil-Sen median trend method combined with the Mann-Kendall test was used to analyze the spatial-temporal variation trend of eco-environmental quality,and the Hurst exponent and the Theil-Sen median trend were superimposed to predict the future evolution trend of eco-environmental quality.In addition,ten variables from two categories of natural and anthropogenic factors were analyzed to determine the drivers of the spatial differentiation of eco-environmental quality by the geographical detector.The results showed that from 2000 to 2020,the RSEI in the Three-North region exhibited obvious regional characteristics:the RSEI values in Northwest China were generally between 0.2 and 0.4;the RSEI values in North China gradually increased from north to south,ranging from 0.2 to 0.8;and the RSEI values in Northeast China were mostly above 0.6.The average RSEI value in the Three-North region increased at an average growth rate of 0.0016/a,showing the spatial distribution characteristics of overall improvement and local degradation in eco-environmental quality,of which the areas with improved,basically stable and degraded eco-environmental quality accounted for 65.39%,26.82%and 7.79%of the total study area,respectively.The Hurst exponent of the RSEI ranged from 0.20 to 0.76 and the future trend of eco-environmental quality was generally consistent with the trend over the past 21 years.However,the areas exhibiting an improvement trend in eco-environmental quality mainly had weak persistence,and there was a possibility of degradation in eco-environmental quality without strengthening ecological protection.Average relative humidity,accumulated precipitation and land use type were the dominant factors driving the spatial distribution of eco-environmental quality in the Three-North region,and two-factor interaction also had a greater influence on eco-environmental quality than single factors.The explanatory power of meteorological factors on the spatial distribution of eco-environmental quality was stronger than that of topographic factors.The effect of anthropogenic factors(such as population density and land use type)on eco-environmental quality gradually increased over time.This study can serve as a reference to protect the ecological environment in arid and semi-arid regions.

A time fractional model to represent rainfall process

This paper deals with a stochastic representation of the rainfall process. The analysis of a rainfall time series shows that cumulative representation of a rainfall time series can be modeled as a non-Gaussian random walk with a log-normal jump distribution and a time-waiting distribution following a tempered a-stable probability law. Based on the random walk model, a fractional Fokker-Planck equation （FFPE） with tempered a-stable waiting times was obtained. Through the comparison of observed data and simulated results from the random walk model and FFPE model with tempered a-stable waiting times, it can be concluded that the behavior of the rainfall process is globally reproduced, and the FFPE model with tempered a-stable waiting times is more efficient in reproducing the observed behavior.

Analysis and Predictability of Dry Spell Lengths Observed in Synoptic Stations of Benin Republic (West Africa)

The complex behavior and predictability of the Dry Spell Lengths (DSL) series obtained in Benin synoptic stations, from 1951 to 2010 are analyzed in this paper using a fractal approach. The synoptic stations are located in Cotonou, Bohicon, Save (subequatorial climate), and Parakou, Natitingou, Kandi (Sudanian climate). The DSLs are computed by considering four thresholds level, R0 = {1.0, 1.5, 2.0 and 5.0} mm/day. The fractal trace is estimated for dry spell density by the mean of the “Dry Spell Spell” (DSS) n-index. The rescaled range method is used to determine the predictability of DSL. By analyzing the DSS, results show that low DSS n-index values (n-index < 0.4) are more favored in the northern part of Benin than in the southern region, whereas, high values of DSS n-index (n-index > 0.4) occur preferentially in the southern part. Therefore, during 1951-2010, the Sudanian region presents frequent wet spells, alternated with short dry spells than in the subequatorial one. However, a high degree of long dry spell persistence, followed by short dry events is observed in the subequatorial region than in the Sudanian one. The longest DSL is observed in the subequatorial region, especially in Cotonou. Except for the Kandi station, the DSLs series obtained at synoptic stations are characterized by persistence. Therefore, autoregressive processes could be applied to the DSL series. Generally, the physical process governing dry spells observed at Save, Natitingou, and Bohicon are consistently predictable than the process governing the stations of the Cotonou and Parakou. However, at Kandi station, the DSL process approximates the usual Brownian motion, and it is, therefore, unpredictable or difficult to predict.

A Vicenary Analysis of SARS-CoV-2 Genomes

Coronaviruses are responsible for various diseases ranging from the common cold to severe infections like the Middle East syndromes and the severe acute respiratory syndrome.However,a new coronavirus strain known as COVID-19 developed into a pandemic resulting in an ongoing global public health crisis.Therefore,there is a need to understand the genomic transformations that occur within this family of viruses in order to limit disease spread and develop new therapeutic targets.The nucleotide sequences of SARS-CoV-2 are consist of several bases.These bases can be classified into purines and pyrimidines according to their chemical composition.Purines include adenine(A)and guanine(G),while pyrimidines include cytosine(C)and tyrosine(T).There is a need to understand the spatial distribution of these bases on the nucleotide sequence to facilitate the development of antivirals(including neutralizing antibodies)and epitomes necessary for vaccine development.This study aimed to evaluate all the purine and pyrimidine associations within the SARS-CoV-2 genome sequence by measuring mathematical parameters including;Shannon entropy,Hurst exponent,and the nucleotide guanine-cytosine content.The Shannon entropy is used to identify closely associated sequences.Whereas Hurst exponent is used to identifying the auto-correlation of purine-pyrimidine bases even if their organization differs.Different frequency patterns can be used to determine the distribution of all four proteins and the density of each base.The GC-content is used to understand the stability of the DNA.The relevant genome sequences were extracted from the National Center for Biotechnology Information(NCBI)virus database.Furthermore,the phylogenetic properties of the COVID-19 virus were characterized to compare the closeness of the COVID-19 virus with other coronaviruses by evaluating the purine and pyrimidine distribution.

A New Mean Reversion Model of Close-End Fund

On the basis of fractal theory, one of the nonlinear theories, this paper studies the validity of Chinese fund market fractal time sequence through Hurst exponent, calculates the H value and proposes a new close-end fund mean reversion model. Meanwhile, this paper validates the mean reversion time sequence for consecutive 54 week data of fund market. The result indicates that this model can effectively prove that Chinese close-end fund market follows the biased random walk. The research also proves that the fund discount does have mean reversion tendency and averagely the fund with high discount has a higher excess yield than that of the fund with low discount. The mean excess yield and the ratio between discount rate deviation and standard deviation demonstrate a descending relationship. The optimum investment period based on ＂mean reversion＂ is one month. Consequently this model provides a new arbitrage method through the discount of close-end fund.

Evidence for self-organized criticality in unscheduled downtime data of equipment

How to reduce downtime and improve availability of the complex equipment is very important. Although the unscheduled downtime（USDT） issues of the equipment are very complex, the self-organized criticality（SOC） is the right theory to study complex systems evolution and opens up a new window to the investigation of disasters, such as the sudden failure of the equipment. Firstly,SOC theory and its validation method are introduced. Then an SOC validation method for USDT of the equipment is proposed based on the above theory. Case study is done on bottleneck equipment in a factory and corresponding data pre-process work is done. The rescaled-range（R/S） analysis method is used to calculate the Hurst exponent of USDT time-series data in order to determine the long-range correlation of USDT data on time scale;at the same time the spatial power-law characteristic of USDT time series data is studied. The result shows that the characteristics of SOC are revealed in USDT data of the equipment according to the criterion of SOC. In addition, based on the characteristics of SOC,the overall framework of the prediction method for major sudden failure of the equipment is proposed based on SOC.

Self-similar segmentation and multifractality of post-stack seismic data

Layering detection is an important step in petroleum engineering.Time series of post-stack seismic data and wire-line log data belong to subsurface layering.They exhibit multifractal properties with complex patterns because of the heterogeneity and different genetic properties in the earth layers.In a multifractal configuration,any piece of a series has a distinct Hurst exponent that reflects its nature and can be used for zone detection.Time series are post-stack seismic traces and wire-line log data near the well-bores.Self-similar Autoregressive Exogenous(SAE)model is a modified method which can place self-similar post-stack seismic and wire-line log segments across layers with the same lithology.The results satisfy the capability of layering identification from seismic data by SAE model.

The Compound Indexing of Human Self-Similar Behavioural Patterns

Applications of a constitutive framework providing compound complexity analysis and indexing of coarse-grained self-similar time series representing behavioural data are presented. A notion of behavioural entropy and hysteresis is introduced as two different forms of compound measures. These measures provide clinically applicable complexity analysis of behavioural patterns yielding scalar characterisation of time-varying behaviours registered over an extended period of time. The behavioural data are obtained using body attached sensors providing non-invasive readings of heart rate, skin blood perfusion, blood oxygenation, skin temperature, movement and steps frequency. The results using compound measures of behavioural patterns of fifteen healthy individuals are presented. The application of the compound measures is shown to correlate with complexity analysis. The correlation is demonstrated using two healthy subjects compared against a control group. This indicates a possibility to use these measures in place of fractional dimensions to provide a finer characterisation of behavioural patterns observed using sensory data acquired over a long period of time.

Scaling Properties of Optokinetic Nystagmus Amplitude Sequence

Optokinetic nystagmus (OKN) is rhythmic eye movements, back and forth, with a slow and fast phase when the eyes are presented for full-field visual stimulus. OKN was recorded in a healthy subject for four conditions, stripes moving 30°/s left and right and 60°/s left and right. In this paper, surrogate data analysis was applied to test the Hurst exponents for increasing time horizons for the integrated OKN amplitude sequences statistically against 100 shuffled sequences (i.e., the serial dependency of the original data sequences is broken by changing the order of the sequence). The result shows that the pattern of the OKN amplitude sequence scales statistically different (p < 0.01) compared to random permutations of the same numbers for scaling shorter than 16 nystagmus components (slow and fast phases) for all four test conditions.

基于Hurst指数的1982–2013年蒙古高原植被动态变化趋势及其影响因素分析（英文）

This study analyzed the spatial and temporal variations in the Normalized Difference Vegetation Index(NDVI) on the Mongolian Plateau from 1982–2013 using Global Inventory Modeling and Mapping Studies(GIMMS) NDVI3 g data and explored the effects of climate factors and human activities on vegetation. The results indicate that NDVI has slight upward trend in the Mongolian Plateau over the last 32 years. The area in which NDVI increased was much larger than that in which it decreased. Increased NDVI was primarily distributed in the southern part of the plateau, especially in the agro-pastoral ecotone of Inner Mongolia. Improvement in the vegetative cover is predicted for a larger area compared to that in which degradation is predicted based on Hurst exponent analysis. The NDVI-indicated vegetation growth in the Mongolian Plateau is a combined result of climate variations and human activities. Specifically, the precipitation has been the dominant factor and the recent human effort in protecting the ecological environments has left readily detectable imprints in the NDVI data series.

ANALYSIS OF THE PREDICTION CAPABILITY OF WEB SEARCH DATA BASED ON THE HE-TDC METHOD - PREDICTION OF THE VOLUME OF DAILY TOURISM VISITORS

Web search query data are obtained to reflect social spots and serve as novel economic indicators. When faced with high-dimensional query data, selecting keywords that have plausible predictive ability and can reduce dimensionality is critical. This paper presents a new integrative method that combines Hurst Exponent （HE） and Time Difference Correlation （TDC） analysis to select keywords with powerful predictive ability. The method is called the HE-TDC screening method and requires keywords with predictive ability to satisfy two characteristics, namely, high correlation and fluctuation memorability similar to the predicting target series. An empirical study is employed to predict the volume of tourism visitors in the Jiuzhai Valley scenic area. The study shows that keywords selected using HE-TDC method produce a model with better robustness and predictive ability.

Evaluating the accuracy of two satellite-based Quantitative Precipitation Estimation products and their application for meteorological drought monitoring over the Lake Victoria Basin,East Africa

This study evaluates the high-resolution satellite estimated long-term precipitation data for monitor-ing the drought condition over the Lake Victoria Basin(LVB)from 1984 to 2020.Standardized Precipitation Indices(SPI)were used to capture the short,medium and long-term meteorological drought conditions at multiple time scales(i.e.3,6,and 12).For these,the following two primaries Quantitative Precipitation Estimation(QPEs)products were employed-1)Climate Hazards group Infra-Red Precipitation with Station(CHIRPS),and 2)the Precipitation Estimation from Remotely Sensed Information using Artificial Neural Network-Climate Data Record(PERSIANN-CDR).This dataset was compared based on the observation data obtained from the Climate Research Unit(CRU)over the nine selected regions surrounding lake basins.The performance of these two QPEs products was evaluated using seven statistical metrics.The findings of this study indicate that the CHIRPS and PERSIANN-CDR datasets could capture the behavior of drought magnitude based on the time scale of SPI-3,SPI-6,SPI-12.The results indicate that 2012 and 2017 are significant severe drought years in the recent decade over LVB.However,the CHIRPS datasets provide good agreement(Correlation Coefficient(CC)=0.65)with observation,whereas PERSIANN-CDR present satisfactory results(CC=0.54).In addition,Hurst(H)exponent was used to predict the future drought trend and found that the CHIRPS performed well to predict the degree of drought trend.Therefore,this study considers the CHIRPS product for near-real-time drought monitoring and PERSIANN-CDR for historical drought assessment.Moreover,the outcome from the H values is greater than 0.5,which indicates the future drought trend would be decreased over LVB.These results are useful for developing the strategies for drought hazards and water resource management in LVB.

Finding Chaos in Finnish GDP

The goal of this paper is to analyze the Finnish gross domestic product(GDP) and to find chaos in the Finnish GDP. We chose Finland where data has been available since 1975, because we needed the longest time series possible. At first we estimated the time delay and the embedding dimension, which is needed for the Lyapunov exponent estimation and for the phase space reconstruction.Subsequently, we computed the largest Lyapunov exponent, which is one of the important indicators of chaos. Then we calculated the 0-1 test for chaos. Finally we computed the Hurst exponent by rescaled range analysis and by dispersional analysis. The Hurst exponent is a numerical estimate of the predictability of a time series. In the end, we executed a recurrent analysis and displayed recurrence plots of detrended GDP time series. The results indicated that chaotic behaviors obviously exist in GDP.

CHAOTIC HYDRODYNAMICS IN HORIZONTAL GAS-LIQUID BUBBLY FLOW

To investigate the non-linear hydrodynamics in horizontal gas-liquid bubblyflow, time-series data of local void fraction were obtained using a double sensor probe. Bothdeterministic chaos and stochastic analyses were used to diagnose the dynamic behaviors in thesystem. The power spectra of the local void fraction exhibited no single distinct peak, indicatingthe aperiodic nature of the variation of the local void fraction. The kolmogorov entropy andcorrelation dimension of the attractor reconstructed from the time-series data by the embeddingmethod were to be positive and very large, the Hurst exponent computed from the rescaled rangeanalysis was found to be larger than 0. 5, indicating the non-linear chaotic feature of the system.The Kolmogorov entropy and the correlation dimension decrease with increasing liquid superficialvelocity, and the correlation dimension changes less if the gas superficial velocity is high. On theother hand, the fluctuation of local void fraction in horizontal gas-liquid bubbly flow could beconsidered to take place due to the zigzag motion of bubbles with different shape, size andvelocity.

R/S Analysis of China Securities Markets

This paper presents the law of changes of indices and stocks in the Shanghai and Shenzhen Stock Exchanges by using rescaled range analysis in nonlinear time series analysis. The Hurst exponents of the stock indices and of all stocks listed in the Shanghai and Shenzhen Stock Exchanges are estimated. The results show that the changes of indices and stocks in the last period have positive impact in the next period in the short run, but this impact disappears for long time.