Multifractality of Traffic Time Series

Abundant observations have shown that road traffic flow contains complex fluctuations. One frequently mentioned question is: whether traffic flow belongs to 1/f fluctuation and in which time-scale region the 1/f fluctuation appears? The answer to this problem is useful, since it may help to distinguish the implicit short-range/long-range correlation and self-similarity of traffic flow time series. Notice that traffic flow yields different behaviors at different time scale, some previous studies applies Detrended Fluctuation Analysis (DFA) method to filter out the polynomial trends. In this paper, the urban traffic flow data collected from Beijing,China and the freeway traffic flow data collected from Washington State, USA are analyzed using Multi-Fractal Detrended Fluctuation Analysis (MF-DFA). The Hurst exponents of these two kinds of traffic flow are larger than that of 1/f noises and smaller than Brownian noises, which indicates the existence of the long-range correlation in urban traffic flow time series. The multi-fractal spectra of these two flows are carefully studied and the differences are highlighted. Results indicate that a multi-fractal description is required for traffic flow, since multi-fractal behaviors are found. This result also partly answers another attractive question: What difference can we observe between freeway and urban transportation network traffic flow?

Air pollution is a complex phenomenon caused by humans causing certain substances to enter the atmosphere during production activities or natural processes. Its formation and evolution have long-term sustainability, macroscopicity, and integrity. The concentration of each pollutant is high or low, and it will last for a period of time. So, what are the macro and overall characteristics of the self-evolution of the concentration of each pollutant? This article will use the detrended fluctuation analysis (DFA) method to analyze the long-term sustainability of each pollutant concentration sequence at nine sites. At the same time, in order to describe the nonlinear characteristics of each pollutant concentration sequence in more detail, use the multifractal detrended fluctuation analysis (MF-DFA) method to analyze the internal local structure. The MF-DFA method can describe the unique mode of the pollution process during the haze period, record the detailed information of the pollutants on different time scales during the haze period, provide probability estimates for the pollutant concentration, and display the pollutant concentration. The MF-DFA method can also describe the characteristics of time series in a more detailed, precise, and comprehensive manner and quantitatively describe the long-term sustainability of time series evolution. The experimental analysis results of the MF-DFA method on the concentration of each pollutant at nine monitoring points during the haze period have achieved extraordinary results.

Environmental changes have brought about significant changes and challenges to water resources and management in the world; these include increasing climate variability, land use change, intensive agriculture, and rapid urbanization and industrial development, especially much more frequency extreme precipitation events. All of which greatly affect water resource and the development of social economy. In this study, we take extreme precipitation events in the Midwest of Jilin Province as an example; daily precipitation data during 1960–2014 are used. The threshold of extreme precipitation events is defined by multifractal detrended fluctuation analysis (MF-DFA) method. Extreme precipitation (EP), extreme precipitation ratio (EPR), and intensity of extreme precipitation (EPI) are selected as the extreme precipitation indicators, and then the Kolmogorov-Smirnov (K-S) test is employed to determine the optimal probability distribution function of extreme precipitation indicators. On this basis, copulas connect nonparametric estimation method and the Akaike Information Criterion (AIC) method is adopted to determine the bivariate copula function. Finally, we analyze the characteristics of single variable extremum and bivariate joint probability distribution of the extreme precipitation events. The results show that the threshold of extreme precipitation events in semi-arid areas is far less than that in subhumid areas. The extreme precipitation frequency shows a significant decline while the extreme precipitation intensity shows a trend of growth; there are significant differences in spatiotemporal of extreme precipitation events. The spatial variation trend of the joint return period gets shorter from the west to the east. The spatial distribution of co-occurrence return period takes on contrary changes and it is longer than the joint return period.

Many types of financial time series, most notably market returns, have been found to exhibit long-range memory as well as dramatic day-to-day swings that cannot be adequately represented by light-tailed distributions such as the normal distribution. In particular, this means that for such time series, the usual variance parameter (i.e., the sum of covariances at all time lags) is not defined because the covariance function does not converge to zero fast enough as the time lag increases. Moreover in such time series, often the tails of the marginal density converge to zero so slowly that higher-order marginal moments such as skewness and kurtosis fail to exist. Therefore conventional methods for analyzing simulation-generated time series cannot generally be applied to high-fidelity simulations of financial markets.Building on earlier work in fractal geometry and fractal time series, Mandelbrot et al. proposed the multifractal model of asset returns (MMAR) as an alternative to the ARCH models for analyzing time series exhibiting volatility clustering, long-range dependence, and heavy-tailed returns (Mandelbrot et al. 1997). They defined the multifractal spectrum as the renormalized probability density function of the Holder exponents observed in the time series; and they used the multifractal spectrum to measure the ability of MMAR to match the statistical properties of real data. In 2002 Kantelhardt et al. formulated multifractal detrended fluctuation analysis (MF-DFA), an algorithm for extracting the multifractal spectrum from a time series (Kantelhardt et al. 2002).Many economists have recently adopted agent-based simulation for modeling financial markets. Fads based on new products, shocks related to world events, scandals involving company leaders, or outright criminal activity can drastically change the processes and relationships governing a market. Already there is evidence that agent-based models yield more accurate approximations to the true or observed behavior in financial markets. Agent-based models exhibit emergent behaviors that have been linked to non-Gaussian interaction metrics and singularities in the time series they generate (Chan 2011). Farmer et al. constructed a agent-based model that is capable of exhibiting many of the statistical properties observed in real financial data simply due to the rules governing a double auction order book (Farmer et al. 2005).During the 20th century, economists relied heavily on Brownian motion to construct theoretical models of finance. The advantages of using Brownian motion stem from its foundation in the Gaussian distribution; and with advances in stochastic calculus and differential equations, it became a staple in models of empirical finance. The well-known Black-Scholes model for options pricing is a prime example of this application. Multifractal analysis has the ability to illuminate the underlying difference between a monofractal like Brownian motion and a considerably more diverse construction like the multifractal binomial measure. Thus, combining multifractal analysis with an agent-based model has the potential to provide insight into the underlying processes and behaviors of a dynamic economic system that is constantly changing and evolving.To analyze market-return time series exhibiting volatility clustering, long-range dependence, or heavy-tailed marginals, we exploit multifractal analysis and agent-based simulation. We develop a robust, automated software tool for extracting the multifractal spectrum of a time series based on MF-DFA. Guidelines are given for setting MF-DFA's parameters in practice. The software is tested on simulated data with closed-form monofractal and multifractal spectra as well as on observed data, and the results are analyzed. We also present a prototype agent-based financial market model based on the zero-intelligence model of Farmer et al. and analyze its output using MF-DFA. The figure below compares the multifractal spectrum of the output from the prototype model with that of the multifractal binomial measure. The ultimate objective is to expand this model to study the effects of microlevel agent behaviors on the macrolevel time series output as analyzed by MF-DFA. Finally we explore the potential for validating agent-based models using MF-DFA and thus being able to tune these models to the multifractal spectrum of empirical data.

This paper presents a fault diagnosis method for nonlinear analog circuit based on multifractal detrended fluctuation analysis (MFDFA) method. The MFDFA method is applied to analysis fault signal and extracts the multifractal features from the raw signal. The selected features are given to SVM classifier for further classification. The data required to develop the classifier are generated by simulating various faults using Pspice software. The simulation results show that the proposed method provides a robust and accurate method for nonlinear circuit fault diagnosis.

Multifractal detrended fluctuation analysis (MFDFA) method can examine higher-dimensional fractal and multifractal characteristics hidden in time series. However, removal of local trends in MFDFA is based on discontinuous polynomial fitting, resulting in pseudo-fluctuation errors. In this paper, we propose a two-stage modified MFDFA for multifractal analysis. First, an overlap moving window (OMW) algorithm is introduced to divide time series of the classic MFDFA method. Second, detrending by polynomial fitting local trend in traditional MFDFA is replaced by ensemble empirical mode decomposition (EEMD)-based local trends. The modified MFDFA is named OMW-EEMD-MFDFA. Then, the performance of the OMW-EEMD-MFDFA method is assessed by extensive numeric simulation experiments based on a p-model of multiplicative cascading process. The results show that the modified OMW-EEMD-MFDFA method performs better than conventional MFDFA and OMW-MFDFA methods. Lastly, the modified OMW-EEMD-MFDFA method is applied to explore multifractal characteristics and multifractal sources of daily precipitation time series data at the Mapoling and Zhijiang stations in Dongting Lake Basin. Our results showed that the scaling properties of the daily precipitation time series at the two stations presented a long-range correlation, showing a long-term persistence of the previous state. The strong q-dependence of H ( q ) and τ ( q ) indicated strong multifractal characteristics in daily precipitation time series data at the two stations. Positive Δ f values demonstrate that precipitation may have a local increasing trend. Comparing the generalized Hurst exponent and the multifractal strength of the original precipitation time series data with its shuffled and surrogate time series data, we found that the multifractal characteristics of the daily precipitation time series data were caused by both long-range correlations between small and large fluctuations and broad probability density function, but the broad probability density function was dominant. This study may be of practical and scientific importance in regional precipitation forecasting, extreme precipitation regulation, and water resource management in Dongting Lake Basin.

Multifractal characteristics analysis of daily reference evapotranspiration in different climate zones of China

Uncertain systems are those wherein some variability is observed, meaning that different observations of the system will produce different measurements. Studying such systems demands the use of statistical methods over multiple measurements, which allows overcoming the uncertainty, based on the premise that a single measurement is not representative of the system’s behavior. In such cases, the current multifractal detrended fluctuation analysis (MFDFA) method cannot offer confident conclusions. This work presents multi-signal MFDFA (MS-MFDFA), a novel methodology for accurately characterizing uncertain systems using the MFDFA algorithm, which enables overcoming the uncertainty of the system by simultaneously considering a large set of signals. As a case study, we consider the problem of characterizing software (Sw) consumption. The difficulty of the problem mainly comes from the complexity of the interactions between Sw and hardware (Hw), as well as from the high uncertainty level of the consumption measurements, which are affected by concurrent Sw services, the Hw, and external factors such as ambient temperature. We apply MS-MFDFA to generate a signature of the Sw consumption profile, regardless of the execution time, the consumption levels, and uncertainty. Multiple consumption signals (or time series) are built from different Sw runs, obtaining a high frequency sampling of the instant input current for each of them while running the Sw. A benchmark of eight Sw programs for analysis is also proposed. Moreover, a fully functional application to automatically perform MS-MFDFA analysis has been made freely available. The results showed that the proposed methodology is a suitable approximation for the multifractal analysis of a large number of time series obtained from uncertain systems. Moreover, analysis of the multifractal properties showed that this approach was able to differentiate between the eight Sw programs studied, showing differences in the temporal scaling range where multifractal behavior is found.

Climate dynamics and trends have significant environmental and socioeconomic impacts; however, in the Benin Republic, they are generally studied with diverse statistical methods ignoring the nonstationarity, nonlinearity, and self‐similarity characteristics contained in precipitation time series. This can lead to erroneous conclusions and an unclear understanding of climatic dynamics. Based on daily precipitation data observed in the six synoptic stations of Benin Republic, in the period from 1951 to 2010, we have proposed (i) determining the local trends of precipitations, (ii) investigating precipitation nonlinear dynamics, and (iii) assessing climatic shift in the study period by Ensemble Empirical Mode Decomposition (EEMD) and Multifractal Detrended Fluctuation Analysis (MFDFA) method. To overcome the detrending issue in the standard MFDFA method, the EEMD algorithm is embedded into the MFDFA. The study period is subdivided into three subperiods: 1951–1970, 1971–1990, and 1991–2010. Intrinsic Mode Functions (IMFs) are obtained according to the climatic region in which the stations are located. Results show that precipitation variation is significantly governed by the five first IMFs, in which oscillation periods vary from 1 to 25 days. The trend curves decrease at all the synoptic stations, and their slope values vary accordingly to the subperiods. Referring to the values of the multifractal spectrum parameters, α0, and the width of the spectrum w, consistent changes are observed regardless of the subperiods and the concerned stations. The spatial and temporal variability of precipitation indicates that the multifractal properties are good indicators for assessing changes in precipitation dynamics, and the changes in their features could be explained by the global change than by the local climate variation (climatic zones). Despite the observed differences in multifractal spectra properties from the three subperiods, it is not possible to verify the subdivision of 1951–2010 in three subperiods as it is done by previous studies in West Africa. Our findings can be used in the validation of global and regional climate models since a valid model should explain empirically detected scaling properties in observed data.

An improved multifractal detrended fluctuation analysis(MF-DFA) method is applied to analyze the long-term monthly runoff records of a hydrological station in the Yangtze River with seasonal trend eliminated, through which the long-range correlation and the multifractal characteristics have been found. The multifractal spectrum has been fitted by a generalized expression of the multiplicative cascade model, and the results show that the monthly runoff series has strong multifractal characteristics. Comparing the results for the original runoff series with those of shuffled and surrogate series, it concludes that the multifractal characteristics of the monthly runoff time series is due to the broadness of both the probability density function and long-range correlation, and the broadness of the probability density function is dominant.

Long-range correlation and fluctuation in the gold market time series of the worldʼs two leading gold consuming countries, namely China and India, are studied. For both the market series during the period 1985–2013 we observe a long-range persistence of memory in the sequences of maxima (minima) of returns in successive time windows of fixed length, but the series, as a whole, are found to be uncorrelated. Multifractal analysis for these series as well as for the sequences of maxima (minima) is carried out in terms of the multifractal detrended fluctuation analysis (MF-DFA) method. We observe a weak multifractal structure for the original series that mainly originates from the fat-tailed probability distribution function of the values, and the multifractal nature of the original time series is enriched into their sequences of maximal (minimal) returns. A quantitative measure of multifractality is provided by using a set of ‘complexity parameters’.

In the past few years, multifractal detrended fluctuation analysis (MF-DFA) method has been widely applied in the field of agricultural image processing. However, this image analysis involves a great deal of iterative processes and matrix calculations, which require massive computational capacity. In order to overcome this problem, we first develop an MF-DFA program that involves image preprocessing, image segmentation, local area accumulation matrix calculation, local area trend fitting, local area trend elimination, a global qth order fluctuation function, and the Hurst index. We then compare and analyze the MF-DFA module's performance characteristics and propose a parallel optimization scheme of the MF-DFA based on OpenMP. The results of our rigorous performance evaluation study results clearly demonstrate that our proposed parallel optimization scheme is a highly efficient method in extracting rape leaf image texture characteristics.

Surge prediction in a gas-liquid mixed-flow pump based on multifractal detrended fluctuation analysis

China’s soybean crush spread: Nonlinear analysis based on MF-DCCA

Multiscale multifractal detrended fluctuation analysis of multivariate time series