Common scaling patterns in intertrade times of U. S. stocks

Multifractal Analysis of the Influence of Indole-3-Acetic Acid on Fast-Activating Vacuolar (FV) Channels of Beta vulgaris L. Taproot Cells

In this paper, the multifractal properties of the ion current time series in the fast-activating vacuolar (FV) channels of Beta vulgaris L. taproot cells were investigated. These channels are permeable for only monovalent cations and mediate K+ at very low concentrations of cytosolic Ca2+ and large voltages of either polarity. Using the patch clamp technique, the currents of the FV channels in red beet taproot vacuoles were recorded and analysed by using the multifractal detrended fluctuation analysis (MFDFA) method. The activity of the FV channels depended on the external potential and was sensitive to the auxin. It was also shown that the singularity spectrum of the ion current in the FV channels is non-singular, and the multifractal parameters, i.e., the generalised Hurst exponent and the singularity spectrum, were modified in the presence of IAA. Taking into account the obtained results, it can be suggested that the multifractal properties of fast-activating vacuolar (FV) K+ channels, indicating the existence of long-term memory, should be taken into account in the molecular mechanism of the auxin-induced growth of plant cells.

A master equation for power laws

We propose a new mechanism for generating power laws. Starting from a random walk, we first outline a simple derivation of the Fokker-Planck equation. By analogy, starting from a certain Markov chain, we derive a master equation for power laws that describes how the number of cascades changes over time (cascades are consecutive transitions that end when the initial state is reached). The partial differential equation has a closed form solution which gives an explicit dependence of the number of cascades on their size and on time. Furthermore, the power law solution has a natural cut-off, a feature often seen in empirical data. This is due to the finite size a cascade can have in a finite time horizon. The derivation of the equation provides a justification for an exponent equal to 2, which agrees well with several empirical distributions, including Richardson's Law on the size and frequency of deadly conflicts. Nevertheless, the equation can be solved for any exponent value. In addition, we propose an urn model where the number of consecutive ball extractions follows a power law. In all cases, the power law is manifest over the entire range of cascade sizes, as shown through log-log plots in the frequency and rank distributions.

Analysis of inter-transaction time fluctuations in the cryptocurrency market

We analyze tick-by-tick data representing major cryptocurrencies traded on some different cryptocurrency trading platforms. We focus on such quantities like the inter-transaction times, the number of transactions in time unit, the traded volume, and volatility. We show that the inter-transaction times show long-range power-law autocorrelations. These lead to multifractality expressed by the right-side asymmetry of the singularity spectra f ( α ) indicating that the periods of increased market activity are characterized by richer multifractality compared to the periods of quiet market. We also show that neither the stretched exponential distribution nor the power-law-tail distribution is able to model universally the cumulative distribution functions of the quantities considered in this work. For each quantity, some data sets can be modeled by the former and some data sets by the latter, while both fail in other cases. An interesting, yet difficult to account for, observation is that parallel data sets from different trading platforms can show disparate statistical properties.

Multifractal Cross-Correlations of Bitcoin and Ether Trading Characteristics in the Post-COVID-19 Time

Unlike price fluctuations, the temporal structure of cryptocurrency trading has seldom been a subject of systematic study. In order to fill this gap, we analyse detrended correlations of the price returns, the average number of trades in time unit, and the traded volume based on high-frequency data representing two major cryptocurrencies: bitcoin and ether. We apply the multifractal detrended cross-correlation analysis, which is considered the most reliable method for identifying nonlinear correlations in time series. We find that all the quantities considered in our study show an unambiguous multifractal structure from both the univariate (auto-correlation) and bivariate (cross-correlation) perspectives. We looked at the bitcoin–ether cross-correlations in simultaneously recorded signals, as well as in time-lagged signals, in which a time series for one of the cryptocurrencies is shifted with respect to the other. Such a shift suppresses the cross-correlations partially for short time scales, but does not remove them completely. We did not observe any qualitative asymmetry in the results for the two choices of a leading asset. The cross-correlations for the simultaneous and lagged time series became the same in magnitude for the sufficiently long scales.

Fractional Dynamics Identification via Intelligent Unpacking of the Sample Autocovariance Function by Neural Networks

Many single-particle tracking data related to the motion in crowded environments exhibit anomalous diffusion behavior. This phenomenon can be described by different theoretical models. In this paper, fractional Brownian motion (FBM) was examined as the exemplary Gaussian process with fractional dynamics. The autocovariance function (ACVF) is a function that determines completely the Gaussian process. In the case of experimental data with anomalous dynamics, the main problem is first to recognize the type of anomaly and then to reconstruct properly the physical rules governing such a phenomenon. The challenge is to identify the process from short trajectory inputs. Various approaches to address this problem can be found in the literature, e.g., theoretical properties of the sample ACVF for a given process. This method is effective; however, it does not utilize all of the information contained in the sample ACVF for a given trajectory, i.e., only values of statistics for selected lags are used for identification. An evolution of this approach is proposed in this paper, where the process is determined based on the knowledge extracted from the ACVF. The designed method is intuitive and it uses information directly available in a new fashion. Moreover, the knowledge retrieval from the sample ACVF vector is enhanced with a learning-based scheme operating on the most informative subset of available lags, which is proven to be an effective encoder of the properties inherited in complex data. Finally, the robustness of the proposed algorithm for FBM is demonstrated with the use of Monte Carlo simulations.

Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution

The observable outputs of many complex dynamical systems consist of time series exhibiting autocorrelation functions of great diversity of behaviors, including long-range power-law autocorrelation functions, as a signature of interactions operating at many temporal or spatial scales. Often, numerical algorithms able to generate correlated noises reproducing the properties of real time series are used to study and characterize such systems. Typically, many of those algorithms produce a Gaussian time series. However, the real, experimentally observed time series are often non-Gaussian and may follow distributions with a diversity of behaviors concerning the support, the symmetry, or the tail properties. It is always possible to transform a correlated Gaussian time series into a time series with a different marginal distribution, but the question is how this transformation affects the behavior of the autocorrelation function. Here, we study analytically and numerically how the Pearson's correlation of two Gaussian variables changes when the variables are transformed to follow a different destination distribution. Specifically, we consider bounded and unbounded distributions, symmetric and non-symmetric distributions, and distributions with different tail properties from decays faster than exponential to heavy-tail cases including power laws, and we find how these properties affect the correlation of the final variables. We extend these results to a Gaussian time series, which are transformed to have a different marginal distribution, and show how the autocorrelation function of the final non-Gaussian time series depends on the Gaussian correlations and on the final marginal distribution. As an application of our results, we propose how to generalize standard algorithms producing a Gaussian power-law correlated time series in order to create a synthetic time series with an arbitrary distribution and controlled power-law correlations. Finally, we show a practical example of this algorithm by generating time series mimicking the marginal distribution and the power-law tail of the autocorrelation function of real time series: the absolute returns of stock prices.

The dynamics of the aggressive order during a crisis

We investigate the dynamics of aggressive order in the financial market to further understand volatility. To analyze aggressive order, market orders in the order book are scrutinized. The market orders have different degrees of aggressiveness; therefore, we categorize market orders into four types: types Zero, One, A, and B, of which type B is the most aggressive. To examine the dynamics and impacts of each type of order, we use both macro- and micro-level approaches. From the macroscopic perspective, the burstiness and memory of type B is highly correlated with volatility. When traders face a financial crisis, they place bursty aggressive orders, and the orders are more predictable than usual. From the microscopic perspective, we additionally focus on the influence of the orders, particularly the price impact and resilience. The aggressive order has a greater impact than others, even when the price change of the aggressive order is smaller. Moreover, the aggressive order delivers more information on price because the aggressive order has a higher price impact than the execution cost.

Traceability and dynamical resistance of precursor of extreme events

Extreme events occur in a variety of natural, technical, and societal systems and often have catastrophic consequences. Their low-probability, high-impact nature has recently triggered research into improving our understanding of generating mechanisms, providing early warnings as well as developing control strategies. For the latter to be effective, knowledge about dynamical resistance of a system prior to an extreme event is of utmost importance. Here we introduce a novel time-series-based and non-perturbative approach to efficiently monitor dynamical resistance and apply it to high-resolution observations of brain activities from 43 subjects with uncontrollable epileptic seizures. We gain surprising insights into pre-seizure dynamical resistance of brains that also provide important clues for success or failure of measures for seizure prevention. The novel resistance monitoring perspective advances our understanding of precursor dynamics in complex spatio-temporal systems with potential applications in refining control strategies.

Multifractality and cross-correlation analysis of streamflow and sediment fluctuation at the apex of the Pearl River Delta

The fluctuation and distribution of hydrological signals are highly related to the fluvial and geophysical regime at estuarine regions. Based on the long daily streamflow and sediment data of Makou (MK) and Sanshui (SS) stations at the apex of the Pearl River Delta, the scaling behavior of the streamflow and sediment is explored by multifractal detrended fluctuation analysis (MF-DFA). The results indicated that there was significant multifractal structure present in the fluctuations of streamflow and sediment. Meanwhile, the multifractal degree and complexity of sediment were much stronger than streamflow. Although the scaling exponents of streamflow were larger than sediment at both MK and SS, no evident differences have been found on the scaling properties of streamflow and sediment for the ratios MK/SS. Moreover, the cross-correlation between streamflow and sediment is further detected by Multifractal Detrended Cross-Correlation Analysis (MF-DXA). The multifractal response between streamflow and sediment at small timescale is characterized by long-range correlations whereas it exhibits random behavior at large timescale. The interaction of the broadness of probability density function and the long-range correlations should be responsible for the multifractal properties of hydrological time series as the multifractal degree of surrogate and shuffled data was significantly undermined.

Modeling correlated bursts by the bursty-get-burstier mechanism

Temporal correlations of time series or event sequences in natural and social phenomena have been characterized by power-law decaying autocorrelation functions with decaying exponent γ. Such temporal correlations can be understood in terms of power-law distributed interevent times with exponent α and/or correlations between interevent times. The latter, often called correlated bursts, has recently been studied by measuring power-law distributed bursty trains with exponent β. A scaling relation between α and γ has been established for the uncorrelated interevent times, while little is known about the effects of correlated interevent times on temporal correlations. In order to study these effects, we devise the bursty-get-burstier model for correlated bursts, by which one can tune the degree of correlations between interevent times, while keeping the same interevent time distribution. We numerically find that sufficiently strong correlations between interevent times could violate the scaling relation between α and γ for the uncorrelated case. A nontrivial dependence of γ on β is also found for some range of α. The implication of our results is discussed in terms of the hierarchical organization of bursty trains at various time scales.

Dual-induced multifractality in online viewing activity

Although recent studies have found that the long-term correlations relating to the fat-tailed distribution of inter-event times exist in human activity and that these correlations indicate the presence of fractality, the property of fractality and its origin have not been analyzed. We use both detrended fluctuation analysis and multifractal detrended fluctuation analysis to analyze the time series in online viewing activity separating from Movielens and Netflix. We find long-term correlations at both the individual and communal levels and that the extent of correlation at the individual level is determined by the activity level. These long-term correlations also indicate that there is fractality in the pattern of online viewing. We first find a multifractality that results from the combined effect of the fat-tailed distribution of inter-event times (i.e., the times between successive viewing actions of individuals) and the long-term correlations in online viewing activity and verify this finding using three synthesized series. Therefore, it can be concluded that the multifractality in online viewing activity is caused by both the fat-tailed distribution of inter-event times and the long-term correlations and that this enlarges the generic property of human activity to include not just physical space but also cyberspace.

Entropy measures, entropy estimators, and their performance in quantifying complex dynamics: Effects of artifacts, nonstationarity, and long-range correlations

Entropy measures are widely applied to quantify the complexity of dynamical systems in diverse fields. However, the practical application of entropy methods is challenging, due to the variety of entropy measures and estimators and the complexity of real-world time series, including nonstationarities and long-range correlations (LRC). We conduct a systematic study on the performance, bias, and limitations of three basic measures (entropy, conditional entropy, information storage) and three traditionally used estimators (linear, kernel, nearest neighbor). We investigate the dependence of entropy measures on estimator- and process-specific parameters, and we show the effects of three types of nonstationarities due to artifacts (trends, spikes, local variance change) in simulations of stochastic autoregressive processes. We also analyze the impact of LRC on the theoretical and estimated values of entropy measures. Finally, we apply entropy methods on heart rate variability data from subjects in different physiological states and clinical conditions. We find that entropy measures can only differentiate changes of specific types in cardiac dynamics and that appropriate preprocessing is vital for correct estimation and interpretation. Demonstrating the limitations of entropy methods and shedding light on how to mitigate bias and provide correct interpretations of results, this work can serve as a comprehensive reference for the application of entropy methods and the evaluation of existing studies.

Universality of market superstatistics

We use a key concept of the continuous-time random walk formalism, i.e., continuous and fluctuating interevent times in which mutual dependence is taken into account, to model market fluctuation data when traders experience excessive (or superthreshold) losses or excessive (or superthreshold) profits. We analytically derive a class of "superstatistics" that accurately model empirical market activity data supplied by Bogachev, Ludescher, Tsallis, and Bunde that exhibit transition thresholds. We measure the interevent times between excessive losses and excessive profits and use the mean interevent discrete (or step) time as a control variable to derive a universal description of empirical data collapse. Our dominant superstatistic value is a power-law corrected by the lower incomplete gamma function, which asymptotically tends toward robustness but initially gives an exponential. We find that the scaling shape exponent that drives our superstatistics subordinates itself and a "superscaling" configuration emerges. Thanks to the Weibull copula function, our approach reproduces the empirically proven dependence between successive interevent times. We also use the approach to calculate a dynamic risk function and hence the dynamic VaR, which is significant in financial risk analysis. Our results indicate that there is a functional (but not literal) balance between excessive profits and excessive losses that can be described using the same body of superstatistics but different calibration values and driving parameters. We also extend our original approach to cover empirical seismic activity data (e.g., given by Corral), the interevent times of which range from minutes to years. Superpositioned superstatistics is another class of superstatistics that protects power-law behavior both for short- and long-time behaviors. These behaviors describe well the collapse of seismic activity data and capture so-called volatility clustering phenomena.

Surrogate-assisted network analysis of nonlinear time series

The performance of recurrence networks and symbolic networks to detect weak nonlinearities in time series is compared to the nonlinear prediction error. For the synthetic data of the Lorenz system, the network measures show a comparable performance. In the case of relatively short and noisy real-world data from active galactic nuclei, the nonlinear prediction error yields more robust results than the network measures. The tests are based on surrogate data sets. The correlations in the Fourier phases of data sets from some surrogate generating algorithms are also examined. The phase correlations are shown to have an impact on the performance of the tests for nonlinearity.

Impact of stock market structure on intertrade time and price dynamics

We analyse times between consecutive transactions for a diverse group of stocks registered on the NYSE and NASDAQ markets, and we relate the dynamical properties of the intertrade times with those of the corresponding price fluctuations. We report that market structure strongly impacts the scale-invariant temporal organisation in the transaction timing of stocks, which we have observed to have long-range power-law correlations. Specifically, we find that, compared to NYSE stocks, stocks registered on the NASDAQ exhibit significantly stronger correlations in their transaction timing on scales within a trading day. Further, we find that companies that transfer from the NASDAQ to the NYSE show a reduction in the correlation strength of transaction timing on scales within a trading day, indicating influences of market structure. We also report a persistent decrease in correlation strength of intertrade times with increasing average intertrade time and with corresponding decrease in companies' market capitalization–a trend which is less pronounced for NASDAQ stocks. Surprisingly, we observe that stronger power-law correlations in intertrade times are coupled with stronger power-law correlations in absolute price returns and higher price volatility, suggesting a strong link between the dynamical properties of intertrade times and the corresponding price fluctuations over a broad range of time scales. Comparing the NYSE and NASDAQ markets, we demonstrate that the stronger correlations we find in intertrade times for NASDAQ stocks are associated with stronger correlations in absolute price returns and with higher volatility, suggesting that market structure may affect price behavior through information contained in transaction timing. These findings do not support the hypothesis of universal scaling behavior in stock dynamics that is independent of company characteristics and stock market structure. Further, our results have implications for utilising transaction timing patterns in price prediction and risk management optimization on different stock markets.

Size effects on correlation measures

The detection and quantification of long-range correlations in time series is a fundamental tool to characterize the properties of different dynamical systems, and is applied in many different fields, including physics, biology or engineering. Due to the diversity of applications, many techniques for measuring correlations have been designed. Here, we study systematically the influence of the length of a time series on the results obtained from several techniques commonly used to detect and quantify long-range correlations: the autocorrelation analysis, Hurst's analysis, and detrended fluctuation analysis (DFA). Using the Fourier filtering method, we generate artificial time series with known and controlled long-range correlations and with a broad range of lengths, and apply on them the different correlation measures we have studied. Our results indicate that while the DFA method is practically unaffected by the length of the time series, and almost always provides accurate results, the results from Hurst's analysis and the autocorrelation analysis strongly depend on the length of the time series.

Calling patterns in human communication dynamics

Modern technologies not only provide a variety of communication modes (e.g., texting, cell phone conversation, and online instant messaging), but also detailed electronic traces of these communications between individuals. These electronic traces indicate that the interactions occur in temporal bursts. Here, we study intercall duration of communications of the 100,000 most active cell phone users of a Chinese mobile phone operator. We confirm that the intercall durations follow a power-law distribution with an exponential cutoff at the population level but find differences when focusing on individual users. We apply statistical tests at the individual level and find that the intercall durations follow a power-law distribution for only 3,460 individuals (3.46%). The intercall durations for the majority (73.34%) follow a Weibull distribution. We quantify individual users using three measures: out-degree, percentage of outgoing calls, and communication diversity. We find that the cell phone users with a power-law duration distribution fall into three anomalous clusters: robot-based callers, telecom fraud, and telephone sales. This information is of interest to both academics and practitioners, mobile telecom operators in particular. In contrast, the individual users with a Weibull duration distribution form the fourth cluster of ordinary cell phone users. We also discover more information about the calling patterns of these four clusters (e.g., the probability that a user will call the c(r)-th most contact and the probability distribution of burst sizes). Our findings may enable a more detailed analysis of the huge body of data contained in the logs of massive users.

Scaling of seismic memory with earthquake size

It has been observed that discrete earthquake events possess memory, i.e., that events occurring in a particular location are dependent on the history of that location. We conduct an analysis to see whether continuous real-time data also display a similar memory and, if so, whether such autocorrelations depend on the size of earthquakes within close spatiotemporal proximity. We analyze the seismic wave form database recorded by 64 stations in Japan, including the 2011 "Great East Japan Earthquake," one of the five most powerful earthquakes ever recorded, which resulted in a tsunami and devastating nuclear accidents. We explore the question of seismic memory through use of mean conditional intervals and detrended fluctuation analysis (DFA). We find that the wave form sign series show power-law anticorrelations while the interval series show power-law correlations. We find size dependence in earthquake autocorrelations: as the earthquake size increases, both of these correlation behaviors strengthen. We also find that the DFA scaling exponent α has no dependence on the earthquake hypocenter depth or epicentral distance.

Quantifying reflexivity in financial markets: toward a prediction of flash crashes

We introduce a measure of activity of financial markets that provides a direct access to their level of endogeneity. This measure quantifies how much of price changes is due to endogenous feedback processes, as opposed to exogenous news. For this, we calibrate the self-excited conditional Poisson Hawkes model, which combines in a natural and parsimonious way exogenous influences with self-excited dynamics, to the E-mini S&P 500 futures contracts traded in the Chicago Mercantile Exchange from 1998 to 2010. We find that the level of endogeneity has increased significantly from 1998 to 2010, with only 70% in 1998 to less than 30% since 2007 of the price changes resulting from some revealed exogenous information. Analogous to nuclear plant safety measures concerned with avoiding "criticality," our measure provides a direct quantification of the distance of the financial market from a critical state defined precisely as the limit of diverging trading activity in the absence of any external driving.

Ivanov P, Carpena P. Phase transitions in the first-passage time of scale-invariant correlated processes

A key quantity describing the dynamics of complex systems is the first-passage time (FPT). The statistical properties of FPT depend on the specifics of the underlying system dynamics. We present a unified approach to account for the diversity of statistical behaviors of FPT observed in real-world systems. We find three distinct regimes, separated by two transition points, with fundamentally different behavior for FPT as a function of increasing strength of the correlations in the system dynamics: stretched exponential, power-law, and saturation regimes. In the saturation regime, the average length of FPT diverges proportionally to the system size, with important implications for understanding electronic delocalization in one-dimensional correlated-disordered systems.

Effects of coarse-graining on the scaling behavior of long-range correlated and anti-correlated signals

We investigate how various coarse-graining (signal quantization) methods affect the scaling properties of long-range power-law correlated and anti-correlated signals, quantified by the detrended fluctuation analysis. Specifically, for coarse-graining in the magnitude of a signal, we consider (i) the Floor, (ii) the Symmetry and (iii) the Centro-Symmetry coarse-graining methods. We find that for anti-correlated signals coarse-graining in the magnitude leads to a crossover to random behavior at large scales, and that with increasing the width of the coarse-graining partition interval Δ, this crossover moves to intermediate and small scales. In contrast, the scaling of positively correlated signals is less affected by the coarse-graining, with no observable changes when Δ < 1, while for Δ > 1 a crossover appears at small scales and moves to intermediate and large scales with increasing Δ. For very rough coarse-graining (Δ > 3) based on the Floor and Symmetry methods, the position of the crossover stabilizes, in contrast to the Centro-Symmetry method where the crossover continuously moves across scales and leads to a random behavior at all scales; thus indicating a much stronger effect of the Centro-Symmetry compared to the Floor and the Symmetry method. For coarse-graining in time, where data points are averaged in non-overlapping time windows, we find that the scaling for both anti-correlated and positively correlated signals is practically preserved. The results of our simulations are useful for the correct interpretation of the correlation and scaling properties of symbolic sequences.

Switching processes in financial markets

For an intriguing variety of switching processes in nature, the underlying complex system abruptly changes from one state to another in a highly discontinuous fashion. Financial market fluctuations are characterized by many abrupt switchings creating upward trends and downward trends, on time scales ranging from macroscopic trends persisting for hundreds of days to microscopic trends persisting for a few minutes. The question arises whether these ubiquitous switching processes have quantifiable features independent of the time horizon studied. We find striking scale-free behavior of the transaction volume after each switching. Our findings can be interpreted as being consistent with time-dependent collective behavior of financial market participants. We test the possible universality of our result by performing a parallel analysis of fluctuations in time intervals between transactions. We suggest that the well known catastrophic bubbles that occur on large time scales--such as the most recent financial crisis--may not be outliers but single dramatic representatives caused by the formation of increasing and decreasing trends on time scales varying over nine orders of magnitude from very large down to very small.

Recurrence interval analysis of trading volumes

We study the statistical properties of the recurrence intervals τ between successive trading volumes exceeding a certain threshold q. The recurrence interval analysis is carried out for the 20 liquid Chinese stocks covering a period from January 2000 to May 2009, and two Chinese indices from January 2003 to April 2009. Similar to the recurrence interval distribution of the price returns, the tail of the recurrence interval distribution of the trading volumes follows a power-law scaling, and the results are verified by the goodness-of-fit tests using the Kolmogorov-Smirnov (KS) statistic, the weighted KS statistic and the Cramér-von Mises criterion. The measurements of the conditional probability distribution and the detrended fluctuation function show that both short-term and long-term memory effects exist in the recurrence intervals between trading volumes. We further study the relationship between trading volumes and price returns based on the recurrence interval analysis method. It is found that large trading volumes are more likely to occur following large price returns, and the comovement between trading volumes and price returns is more pronounced for large trading volumes.

Effect of extreme data loss on long-range correlated and anticorrelated signals quantified by detrended fluctuation analysis

Detrended fluctuation analysis (DFA) is an improved method of classical fluctuation analysis for nonstationary signals where embedded polynomial trends mask the intrinsic correlation properties of the fluctuations. To better identify the intrinsic correlation properties of real-world signals where a large amount of data is missing or removed due to artifacts, we investigate how extreme data loss affects the scaling behavior of long-range power-law correlated and anticorrelated signals. We introduce a segmentation approach to generate surrogate signals by randomly removing data segments from stationary signals with different types of long-range correlations. The surrogate signals we generate are characterized by four parameters: (i) the DFA scaling exponent alpha of the original correlated signal u(i) , (ii) the percentage p of the data removed from u(i) , (iii) the average length mu of the removed (or remaining) data segments, and (iv) the functional form P(l) of the distribution of the length l of the removed (or remaining) data segments. We find that the global scaling exponent of positively correlated signals remains practically unchanged even for extreme data loss of up to 90%. In contrast, the global scaling of anticorrelated signals changes to uncorrelated behavior even when a very small fraction of the data is lost. These observations are confirmed on two examples of real-world signals: human gait and commodity price fluctuations. We further systematically study the local scaling behavior of surrogate signals with missing data to reveal subtle deviations across scales. We find that for anticorrelated signals even 10% of data loss leads to significant monotonic deviations in the local scaling at large scales from the original anticorrelated to uncorrelated behavior. In contrast, positively correlated signals show no observable changes in the local scaling for up to 65% of data loss, while for larger percentage of data loss, the local scaling shows overestimated regions (with higher local exponent) at small scales, followed by underestimated regions (with lower local exponent) at large scales. Finally, we investigate how the scaling is affected by the average length, probability distribution, and percentage of the remaining data segments in comparison to the removed segments. We find that the average length mu_{r} of the remaining segments is the key parameter which determines the scales at which the local scaling exponent has a maximum deviation from its original value. Interestingly, the scales where the maximum deviation occurs follow a power-law relationship with mu_{r} . Whereas the percentage of data loss determines the extent of the deviation. The results presented in this paper are useful to correctly interpret the scaling properties obtained from signals with extreme data loss.

Cross-correlations between volume change and price change

In finance, one usually deals not with prices but with growth rates R, defined as the difference in logarithm between two consecutive prices. Here we consider not the trading volume, but rather the volume growth rate R, the difference in logarithm between two consecutive values of trading volume. To this end, we use several methods to analyze the properties of volume changes |R|, and their relationship to price changes |R|. We analyze 14,981 daily recordings of the Standard and Poor's (S & P) 500 Index over the 59-year period 1950-2009, and find power-law cross-correlations between |R| and |R| by using detrended cross-correlation analysis (DCCA). We introduce a joint stochastic process that models these cross-correlations. Motivated by the relationship between |R| and |R|, we estimate the tail exponent alpha of the probability density function P(|R|) approximately |R|(-1-alpha) for both the S & P 500 Index as well as the collection of 1819 constituents of the New York Stock Exchange Composite Index on 17 July 2009. As a new method to estimate alpha, we calculate the time intervals tau(q) between events where R > q. We demonstrate that tau(q), the average of tau(q), obeys tau(q) approximately q(alpha). We find alpha approximately 3. Furthermore, by aggregating all tau(q) values of 28 global financial indices, we also observe an approximate inverse cubic law.

Analysis of the ciliary/flagellar beating of Chlamydomonas

Eukaryotic flagella and cilia are alternative names, for the slender cylindrical protrusions of a cell (240nm diameter, approximately 12,800nm-long in Chlamydomonas reinhardtii) that propel a cell or move fluid. Cilia are extraordinarily successful complex organelles abundantly found in animals performing many tasks. They play a direct or developmental role in the sensors of fluid flow, light, sound, gravity, smells, touch, temperature, and taste in mammals. The failure of cilia can lead to hydrocephalus, infertility, and blindness. However, in spite of their large role in human function and pathology, there is as yet no consensus on how cilia beat and perform their many functions, such as moving fluids in brain ventricles and lungs and propelling and steering sperm, larvae, and many microorganisms. One needs to understand and analyze ciliary beating and its hydrodynamic interactions. This chapter provides a guide for measuring, analyzing, and interpreting ciliary behavior in various contexts studied in the model system of Chlamydomonas. It describes: (1) how cilia work as self-organized beating structures (SOBSs), (2) the overlaid control in the cilia that optimizes the SOBS to achieve cell dispersal, phototaxis steering, and avoidance of obstacles, (3) the assay of a model intracellular signal processing system that responds to multiple external and internal inputs, choosing mode of behavior and then controlling the cilia, (4) how cilia sense their environment, and (5) potentially an assay of ciliary performance for toxicology or medical assessment.

Multifactor analysis of multiscaling in volatility return intervals

We study the volatility time series of 1137 most traded stocks in the U.S. stock markets for the two-year period 2001-2002 and analyze their return intervals tau , which are time intervals between volatilities above a given threshold q . We explore the probability density function of tau , P_(q)(tau) , assuming a stretched exponential function, P_(q)(tau) approximately e;(-tau;(gamma)) . We find that the exponent gamma depends on the threshold in the range between q=1 and 6 standard deviations of the volatility. This finding supports the multiscaling nature of the return interval distribution. To better understand the multiscaling origin, we study how gamma depends on four essential factors, capitalization, risk, number of trades, and return. We show that gamma depends on the capitalization, risk, and return but almost does not depend on the number of trades. This suggests that gamma relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of tau , mu_(m) identical with(tautau);(m);(1m) , in the range of 10<tau< or =100 by a power law, micro_(m) approximately tau;(delta). The exponent delta is found also to depend on the capitalization, risk, and return but not on the number of trades, and its tendency is opposite to that of gamma . Moreover, we show that delta decreases with increasing gamma approximately by a linear relation. The return intervals demonstrate the temporal structure of volatilities and our findings suggest that their multiscaling features may be helpful for portfolio optimization.

Indication of multiscaling in the volatility return intervals of stock markets

The distribution of the return intervals tau between price volatilities above a threshold height q for financial records has been approximated by a scaling behavior. To explore how accurate is the scaling and therefore understand the underlined nonlinear mechanism, we investigate intraday data sets of 500 stocks which consist of Standard & Poor's 500 index. We show that the cumulative distribution of return intervals has systematic deviations from scaling. We support this finding by studying the m -th moment micro_{m} identical with(tau/tau);{m};{1/m} , which show a certain trend with the mean interval tau . We generate surrogate records using the Schreiber method, and find that their cumulative distributions almost collapse to a single curve and moments are almost constant for most ranges of tau . Those substantial differences suggest that nonlinear correlations in the original volatility sequence account for the deviations from a single scaling law. We also find that the original and surrogate records exhibit slight tendencies for short and long tau , due to the discreteness and finite size effects of the records, respectively. To avoid as possible those effects for testing the multiscaling behavior, we investigate the moments in the range 10<tau< or =100 , and find that the exponent alpha from the power law fitting micro_{m} approximately tau;{alpha} has a narrow distribution around alpha not equal0 which depends on m for the 500 stocks. The distribution of alpha for the surrogate records are very narrow and centered around alpha=0 . This suggests that the return interval distribution exhibits multiscaling behavior due to the nonlinear correlations in the original volatility.

Scaling theory of temporal correlations and size-dependent fluctuations in the traded value of stocks

Records of the traded value of fi stocks display fluctuation scaling, a proportionality between the standard deviation sigma(i) and the average <fi>: sigma(i) is proportional to <fi> alpha, with a strong time scale dependence alpha(Delta(t)). The nontrivial (i.e., neither 0.5 nor 1) value of alpha may have different origins and provides information about the microscopic dynamics. We present a set of stylized facts and then show their connection to such behavior. The functional form alpha(Delta(t)) originates from two aspects of the dynamics: Stocks of larger companies both tend to be traded in larger packages and also display stronger correlations of traded value. The results are integrated into a general framework that can be applied to a wide range of complex systems.

Scaling and memory of intraday volatility return intervals in stock markets

We study the return interval tau between price volatilities that are above a certain threshold q for 31 intraday data sets, including the Standard and Poor's 500 index and the 30 stocks that form the Dow Jones Industrial index. For different threshold q, the probability density function Pq(tau)scales with the mean interval tau as [Formula: see text], similar to that found in daily volatilities. Since the intraday records have significantly more data points compared to the daily records, we could probe for much higher thresholds and still obtain good statistics. We find that the scaling function f(x)is consistent for all 31 intraday data sets in various time resolutions, and the function is well-approximated by the stretched exponential, f(x) similar to e(-ax)(gamma), with gamma=0.38+/-0.05 and a=3.9+/-0.5, which indicates the existence of correlations. We analyze the conditional probability distribution Pq(tau/tau0) for tau following a certain interval tau0, and find Pq(tau/tau0) depends on tau0, which demonstrates memory in intraday return intervals. Also, we find that the mean conditional interval (tau/tau0) increases with tau0, consistent with the memory found for Pq(tau/tau0). Moreover, we find that return interval records, in addition to having short-term correlations as demonstrated by Pq(tau/tau0), have long-term correlations with correlation exponents similar to that of volatility records.

Scaling and data collapse for the mean exit time of asset prices

We study theoretical and empirical aspects of the mean exit time (MET) of financial time series. The theoretical modeling is done within the framework of continuous time random walk. We empirically verify that the mean exit time follows a quadratic scaling law and it has associated a prefactor which is specific to the analyzed stock. We perform a series of statistical tests to determine which kind of correlation are responsible for this specificity. The main contribution is associated with the autocorrelation property of stock returns. We introduce and solve analytically both two-state and three-state Markov chain models. The analytical results obtained with the two-state Markov chain model allows us to obtain a data collapse of the 20 measured MET profiles in a single master curve.

Fractionally integrated process with power-law correlations in variables and magnitudes

Motivated by the fact that many empirical time series--including changes of heartbeat intervals, physical activity levels, intertrade times in finance, and river flux values--exhibit power-law anticorrelations in the variables and power-law correlations in their magnitudes, we propose a simple stochastic process that can account for both types of correlations. The process depends on only two parameters, where one controls the correlations in the variables and the other controls the correlations in their magnitudes. We apply the process to time series of heartbeat interval changes and air temperature changes and find that the statistical properties of the modeled time series are in agreement with those observed in the data.

Quantifying signals with power-law correlations: a comparative study of detrended fluctuation analysis and detrended moving average techniques

Detrended fluctuation analysis (DFA) and detrended moving average (DMA) are two scaling analysis methods designed to quantify correlations in noisy nonstationary signals. We systematically study the performance of different variants of the DMA method when applied to artificially generated long-range power-law correlated signals with an a priori known scaling exponent alpha(0) and compare them with the DFA method. We find that the scaling results obtained from different variants of the DMA method strongly depend on the type of the moving average filter. Further, we investigate the optimal scaling regime where the DFA and DMA methods accurately quantify the scaling exponent alpha(0) , and how this regime depends on the correlations in the signal. Finally, we develop a three-dimensional representation to determine how the stability of the scaling curves obtained from the DFA and DMA methods depends on the scale of analysis, the order of detrending, and the order of the moving average we use, as well as on the type of correlations in the signal.