Long-range Memory Research Articles

Exact Propagators of One-Dimensional Self-Interacting Random Walks.

Self-interacting random walks (SIRWs) show long-range memory effects that result from the interaction of the random walker at time t with the territory already visited at earlier times t^{'}<t. This class of non-Markovian random walks has applications in contexts as diverse as foraging theory, the behavior of living cells, and even machine learning. Despite this importance and numerous theoretical efforts, the propagator, which is the distribution of the walker's position and arguably the most fundamental quantity to characterize the process, has so far remained out of reach for all but a single class of SIRW. Here we fill this gap and provide an exact and explicit expression for the propagator of two important universality classes of SIRWs, namely, the once-reinforced random walk and the polynomially self-repelling walk. These results give access to key observables, such as the diffusion coefficient, which so far had not been determined. We also uncover an inherently non-Markovian mechanism that tends to drive the walker away from its starting point.

Gaussian fluctuations of the elephant random walk with gradually increasing memory

The elephant random walk (ERW) is a discrete-time random walk introduced by Schütz and Trimper (2004) in order to investigate how long-range memory affects the behavior of the random walk. Its particularity is that the next step of the walker depends on its whole past through a parameter p∈[0,1] . In this work, we investigate the validity of the central limit theorem of the ERW when the walker has only a gradually increasing memory. Our contribution provides a positive answer to a conjecture raised in a recent work by Gut and Stadtmüller (2022 Stat. Probab. Lett. 189 109598).

Characteristics of price related fluctuations in non-fungible token (NFT) market.

A non-fungible token (NFT) market is a new trading invention based on the blockchain technology, which parallels the cryptocurrency market. In the present work, we study capitalization, floor price, the number of transactions, the inter-transaction times, and the transaction volume value of a few selected popular token collections. The results show that the fluctuations of all these quantities are characterized by heavy-tailed probability distribution functions, in most cases well described by the stretched exponentials, with a trace of power-law scaling at times, long-range memory, persistence, and in several cases even the fractal organization of fluctuations, mostly restricted to the larger fluctuations, however. We conclude that the NFT market-even though young and governed by somewhat different mechanisms of trading-shares several statistical properties with the regular financial markets. However, some differences are visible in the specific quantitative indicators.

MULTIFRACTAL CHARACTERISTICS OF THE MITOCHONDRIAL BK CHANNEL ACTIVITY IN THE PRESENCE OF THE CHANNEL-ACTIVATING FLAVONOIDS

Over the years, the (multi)fractal nature of signals describing the activity of ion channels has been observed both at the level of single-channel currents and the temporal scaling of the corresponding sojourns in conducting and non-conducting states. The recognized nonlinear and self-similar properties were interpreted regarding the underlying channel gating machinery. Nevertheless, the literature lacks in (multi)fractal description of the biochemical stimulation of ion channels. Therefore, in this work, we provide a multifractal description of the effects exerted by two different flavonoids — naringenin (Nar) and quercetin (Que) — being the regulators of the large-conductance voltage- and Ca[Formula: see text]-activated K[Formula: see text] channels from the inner mitochondrial membrane (mitoBK). Toward this aim, the focus-based multifractal detrended fluctuation analysis (MFDFA) was applied to the patch-clamp data obtained in the presence of Nar and Que in micromolar concentrations and the corresponding controls. Our results show that mitoBK channel activity has well-pronounced multifractal characteristics and long-range memory (measured by the generalized Hurst exponent) under biochemical stimulation by Nar and Que and in the absence of these substances. The spectral properties significantly change after the randomization of the analyzed series. Therefore, multifractality is an inherent feature of the mitoBK channel currents, which stems from their statistical distribution and temporal organization. Both flavonoids have similar structures and exert channel-activating effects, but they differently affect the parameters of the multifractal spectra of the corresponding patch-clamp signals. Coordination of Que leads to more prominent changes in signal complexity than Nar at membrane hyper- and depolarization. It suggests that this flavonoid has a stronger impact on the conformational dynamics of the mitoBK channels in comparison with naringenin.

Decreased long-range temporal correlations in the resting-state functional magnetic resonance imaging blood-oxygen-level-dependent signal reflect motor sequence learning up to 2 weeks following training.

Decreased long-range temporal correlations (LRTC) in brain signals can be used to measure cognitive effort during task execution. Here, we examined how learning a motor sequence affects long-range temporal memory within resting-state functional magnetic resonance imaging signal. Using the Hurst exponent (HE), we estimated voxel-wise LRTC and assessed changes over 5 consecutive days of training, followed by a retention scan 12 days later. The experimental group learned a complex visuomotor sequence while a complementary control group performed tightly matched movements. An interaction analysis revealed that HE decreases were specific to the complex sequence and occurred in well-known motor sequence learning associated regions including left supplementary motor area, left premotor cortex, left M1, left pars opercularis, bilateral thalamus, and right striatum. Five regions exhibited moderate to strong negative correlations with overall behavioral performance improvements. Following learning, HE values returned to pretraining levels in some regions, whereas in others, they remained decreased even 2 weeks after training. Our study presents new evidence of HE's possible relevance for functional plasticity during the resting-state and suggests that a cortical subset of sequence-specific regions may continue to represent a functional signature of learning reflected in decreased long-range temporal dependence after a period of inactivity.

Discrete q-Exponential Limit Order Cancellation Time Distribution

Modeling financial markets based on empirical data poses challenges in selecting the most appropriate models. Despite the abundance of empirical data available, researchers often face difficulties in identifying the best fitting model. Long-range memory and self-similarity estimators, commonly used for this purpose, can yield inconsistent parameter values, as they are tailored to specific time series models. In our previous work, we explored order disbalance time series from the broader perspective of fractional L’evy stable motion, revealing a stable anti-correlation in the financial market order flow. However, a more detailed analysis of empirical data indicates the need for a more specific order flow model that incorporates the power-law distribution of limit order cancellation times. When considering a series in event time, the limit order cancellation times follow a discrete probability mass function derived from the Tsallis q-exponential distribution. The combination of power-law distributions for limit order volumes and cancellation times introduces a novel approach to modeling order disbalance in the financial markets. Moreover, this proposed model has the potential to serve as an example for modeling opinion dynamics in social systems. By tailoring the model to incorporate the unique statistical properties of financial market data, we can improve the accuracy of our predictions and gain deeper insights into the dynamics of these complex systems.

Anomalous diffusion and long-range memory in the scaled voter model.

We analyze the scaled voter model, which is a generalization of the noisy voter model with time-dependent herding behavior. We consider the case when the intensity of herding behavior grows as a power-law function of time. In this case, the scaled voter model reduces to the usual noisy voter model, but it is driven by the scaled Brownian motion. We derive analytical expressions for the time evolution of the first and second moments of the scaled voter model. In addition, we have derived an analytical approximation of the first passage time distribution. By numerical simulation, we confirm our analytical results as well as showing that the model exhibits long-range memory indicators despite being a Markov model. The proposed model has steady-state distribution consistent with the bounded fractional Brownian motion, thus we expect it to be a good substitute model for the bounded fractional Brownian motion.

Response of fiber reinforced concrete while entering into the critical stage: An attempt to detect pre-failure indicators in terms of non-extensive statistical mechanics

The main goal of the present study is to explore the potentialities of Non-Extensive Statistical Mechanics (NESM) to provide signals that could be considered as pre-failure indicators. In this direction concrete beams, either unreinforced or reinforced with short fibers, were submitted to three-point bending. The acoustic signals detected during loading were analyzed in terms of their interevent time intervals and the respective cumulative probability. The experimental data were then properly fitted by means of the q-exponential function providing the temporal evolution of the entropic index q throughout the duration of each experiment. The evolution of q was studied in juxtaposition to that of the load applied as well as to that of some additional parameters used to describe the acoustic activity, namely the b-values and the F-, P- and d-functions. The analysis indicated that at load levels approaching the load-carrying capacity of the specimens, the response of both the unreinforced and the fiber reinforced beams corresponds to q-values approaching 2. It is thus highlighted that, according to NESM principles, the damage processes at the specific load levels are characterized by the existence of a number of non-independent subsystems with long-range mutual interactions and memory effects, which cannot be properly described by the concepts of the traditional Boltzmann-Gibbs Statistical Mechanics. On the contrary, description in terms of NESM provided useful information about the damage evolution during the very last loading stages before macroscopic fracture. In addition, the analysis provided indications that the onset of decrease of the q-values, towards the critical limit of 1, could be considered as a potential pre-failure index, warning about upcoming entrance of the system into its critical stage, namely that of impending fracture.

Functional fluvial landforms of the Pantanal: Hydrologic trends and responses to climate changes

Large-scale fluvial landforms emerge from iterative processes that sculpt Earth's surface. Tectonics, climate variability and erosion are major recurrent processes reshaping fluvial landforms ending up in self-affine patterns. In the case of the Pantanal, the largest wetland in South America, its depositional tract can be subdivided as fluvial megafans, interfans and the main trunk system. Here we provide an outlook of their origins by means of self-affine landforms and on the feedback of climate changes over landform functionalities. Climate variability modulates the magnitude of interannual fluvial discharge and sediment load from highlands to plains, affects groundwater recharge, as well as the subsidence and generation of accommodation in the depositional tract through river avulsion. Scenarios are envisaged by analyzing long-term summer rainfall intensity and the number of dry days in autumn/winter over the Upper Paraguay River Basin, and annual flood peaks measured at fluvial gauge stations in the Paraguay River at Ladário (Brazil) and Asunción (Paraguay). The frequency distributions of the annual flood peaks are found unimodal for Asunción and bimodal for Ladário, which suggests an unknown water supply at Asunción, likely from Pilcomayo River and Pantanal's groundwater. The latter might play a delayed role on Pantanal's hydrologic functioning at 20–40 years timescales. Besides the Paraguay River flood peaks, summer rainfall intensity and the number of dry days in autumn/winter are quasi-periodic at interannual and decadal scales. Cross-correlations analyses indicate a long-range memory between river floods and rainfall intensity, and 15–20 years lag between river floods and the number of dry days in drought seasons. Trends analysis suggests that summer rainfall intensity and the number of dry days in autumn/winter have been consistently increasing by about 0.6 mm/day/decade and 1 day/decade, respectively. Therefore, magnifications of fluvial discharge and sediment load at wet seasons and water deficits at drought seasons are anticipated. Such a scenario indicates extreme dry cycles over all self-affine functional landforms, particularly on abandoned lobes relying exclusively on rainwater, whereas extremes of rainfall intensity at rivers headwaters may amplify the risks of large-scale avulsions at active lobes of the fluvial megafans. In contrast, active lobes of megafans, interfans and the main trunk river system emerge as hotpots for wildlife refuge and ecosystem services.

Is fractional-order chaos theory the new tool to model chaotic pandemics as Covid-19?

The deadly outbreak of the second wave of Covid-19, especially in worst hit lower-middle-income countries like India, and the drastic rise of another growing epidemic of Mucormycosis, call for an efficient mathematical tool to model pandemics, analyse their course of outbreak and help in adopting quicker control strategies to converge to an infection-free equilibrium. This review paper on prominent pandemics reveals that their dispersion is chaotic in nature having long-range memory effects and features which the existing integer-order models fail to capture. This paper thus puts forward the use of fractional-order (FO) chaos theory that has memory capacity and hereditary properties, as a potential tool to model the pandemics with more accuracy and closeness to their real physical dynamics. We investigate eight FO models of Bombay plague, Cancer and Covid-19 pandemics through phase portraits, time series, Lyapunov exponents and bifurcation analysis. FO controllers (FOCs) on the concepts of fuzzy logic, adaptive sliding mode and active backstepping control are designed to stabilise chaos. Also, FOCs based on adaptive sliding mode and active backstepping synchronisation are designed to synchronise a chaotic epidemic with a non-chaotic one, to mitigate the unpredictability due to chaos during transmission. It is found that severity and complexity of the models increase as the memory fades, indicating that FO can be used as a crucial parameter to analyse the progression of a pandemic. To sum it up, this paper will help researchers to have an overview of using fractional calculus in modelling pandemics more precisely and also to approximate, choose, stabilise and synchronise the chaos control parameter that will eliminate the extreme sensitivity and irregularity of the models.

A Monte Carlo Approach to Bitcoin Price Prediction with Fractional Ornstein–Uhlenbeck Lévy Process

Since its inception in 2009, Bitcoin has increasingly gained main stream attention from the general population to institutional investors. Several models, from GARCH type to jump-diffusion type, have been developed to dynamically capture the price movement of this highly volatile asset. While fitting the Gaussian and the Generalized Hyperbolic and the Normal Inverse Gaussian (NIG) distributions to log-returns of Bitcoin, NIG distribution appears to provide the best fit. The time-varying Hurst parameter for Bitcoin price reveals periods of randomness and mean-reverting type of behaviour, motivating the study in this paper through fractional Ornstein–Uhlenbeck driven by a Normal Inverse Gaussian Lévy process. Features such as long-range memory are jump diffusion processes that are well captured with this model. The results present a 95% prediction for the price of Bitcoin for some specific dates. This study contributes to the literature of Bitcoin price forecasts that are useful for Bitcoin options traders.

Self-Interacting Random Walks: Aging, Exploration, and First-Passage Times

Self-interacting random walks are endowed with long-range memory effects that emerge from the interaction of the random walker at time t with the territory that it has visited at earlier times t′<t. This class of non-Markovian random walks has applications in a broad range of examples, from insects to living cells, where a random walker locally modifies its environment—leaving behind footprints along its path and, in turn, responding to its own footprints. Because of their inherent non-Markovian nature, the exploration properties of self-interacting random walks have remained elusive. Here, we show that long-range memory effects can have deep consequences on the dynamics of generic self-interacting random walks; they can induce aging and nontrivial persistence and transience exponents, which we determine quantitatively, in both infinite and confined geometries. Based on this analysis, we quantify the search kinetics of self-interacting random walkers and show that the distribution of the first-passage time to a target site in a confined domain takes universal scaling forms in the large-domain size limit, which we characterize quantitatively. We argue that memory abilities induced by attractive self-interactions provide a decisive advantage for local space exploration, while repulsive self-interactions can significantly accelerate the global exploration of large domains.Received 13 July 2021Revised 8 November 2021Accepted 20 December 2021DOI:https://doi.org/10.1103/PhysRevX.12.011052Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasDiffusionFluctuations & noiseRandom walksStochastic processesStatistical PhysicsInterdisciplinary Physics

Extracting Rules via Markov Chains for Cryptocurrencies Returns Forecasting.

With the growing popularity of digital currencies known as cryptocurrencies, there is a need to develop models capable of robustly analyzing and predicting the value of future returns in these markets. In this article, we extract behavior rules to predict the values of future returns in the Bitcoin, Ethereum, Litecoin, and Ripple closing series. We used categorical data in the analyses and Markov chain models from the first to the tenth order to propose a new way of establishing possible future scenarios, in which we analyze the dependence of memory on the dynamics of the process. We used the measurements of accuracy Mean Quadratic Error, Absolute Error Mean Percentage, and Absolute Standard Deviation for the choice of the best models. Our findings reveal that cryptocurrencies have long-range memory. Bitcoin, Ethereum, and Ripple exposed seven steps of memory, while Litecoin displayed nine memory steps. From the transitions between states that happened the most, we defined decision rules that assisted in the definition of future returns in the series. Our results can support the decisions of traders, investors, crypto-traders, and policy-makers.

Stability and Bifurcation Analysis of Fractional-Order Delayed Prey–Predator System and the Effect of Diffusion

Most biological systems have long-range temporal memory. Such systems can be modeled using fractional-order differential equations. The combination of fractional-order derivative and time delay provides the system more consistency with the reality of the interactions and higher degree of freedom. A fractional-order delayed prey–predator system with the fear effect has been proposed in this work. The time delay is considered in the cost of fear; therefore, there are no dynamical changes observed in the system due to time delay in the absence of fear. The existence and uniqueness of the solutions of the proposed system are studied along with non-negativity and boundedness. The existence of biologically relevant equilibria is discussed, and the conditions for local asymptotic stability are derived. Hopf bifurcation occurs in the system with respect to delay parameter. Further, a spatially extended system is proposed and analyzed. Hopf bifurcation also occurs in the extended system due to the delay parameter. Numerical examples are provided in support of analytical findings. Fractional-order derivative improves the stability and damps the oscillatory behaviors of the solutions of the system. Bistability behavior of the system admits stable dynamics by decreasing the fractional-order. Also, chaotic behavior is destroyed by decreasing fractional-order.

Order flow in the financial markets from the perspective of the Fractional Lévy stable motion

It is a challenging task to identify the best possible models based on given empirical data of observed time series. Though the financial markets provide us with a vast amount of empirical data, the best model selection is still a big challenge for researchers. The widely used long-range memory and self-similarity estimators give varying values of the parameters as these estimators themselves are developed for the specific models of time series. Here we investigate from the general fractional Lévy stable motion perspective the order disbalance time series constructed from the limit order book data of the financial markets. Our results suggest that previous findings of persistence in order flow could be related to the power-law distribution of order sizes and other deviations from the normal distribution. Still, orders have stable estimates of anti-correlation for the 18 randomly selected stocks when Absolute value and Higuchi’s estimators are implemented. Though the burst duration analysis based on the first passage problem of time series and implemented in this research gives slightly higher estimates of the Hurst and memory parameters, it qualitatively supports the importance of the power-law distribution of order sizes.

Understanding the Nature of the Long-Range Memory Phenomenon in Socioeconomic Systems.

In the face of the upcoming 30th anniversary of econophysics, we review our contributions and other related works on the modeling of the long-range memory phenomenon in physical, economic, and other social complex systems. Our group has shown that the long-range memory phenomenon can be reproduced using various Markov processes, such as point processes, stochastic differential equations, and agent-based models—reproduced well enough to match other statistical properties of the financial markets, such as return and trading activity distributions and first-passage time distributions. Research has lead us to question whether the observed long-range memory is a result of the actual long-range memory process or just a consequence of the non-linearity of Markov processes. As our most recent result, we discuss the long-range memory of the order flow data in the financial markets and other social systems from the perspective of the fractional Lèvy stable motion. We test widely used long-range memory estimators on discrete fractional Lèvy stable motion represented by the auto-regressive fractionally integrated moving average (ARFIMA) sample series. Our newly obtained results seem to indicate that new estimators of self-similarity and long-range memory for analyzing systems with non-Gaussian distributions have to be developed.

Change-In-Mean Problem for Long Memory Time Series Model

In this paper, we propose the ratio test to detect the mean change point in the long-range memory. We proved that the limit distribution of the cumulative sum (CUSUM) statistic under the null hypothesis is a functional of fractional Brownian motion. At the same time, the consistency under the alternative hypothesis is also obtained. Simulations demonstrate that our asymptotic results provide good size and power.

US FDA Drug Approvals are Persistent and Polycyclic: Insights into Economic Cycles, Innovation Dynamics, and National Policy.

It is challenging to elucidate the effects of changes in external influences (such as economic or policy) on the rate of US drug approvals. Here, a novel approach—termed the Chronological Hurst Exponent (CHE)—is proposed, which hypothesizes that changes in the long-range memory latent within the dynamics of time series data may be temporally associated with changes in such influences. Using the monthly number FDA’s Center for Drug Evaluation and Research (CDER) approvals from 1939 to 2019 as the data source, it is demonstrated that the CHE has a distinct S-shaped structure demarcated by an 8-year (1939–1947) Stagnation Period, a 27-year (1947–1974) Emergent Period, and a 45-year (1974–2019) Saturation Period. Further, dominant periodicities (resolved via wavelet analyses) are identified during the most recent 45-year CHE Saturation Period at 17, 8 and 4 years; thus, US drug approvals have been following a Juglar/Kuznet mid-term cycle with Kitchin-like bursts. As discussed, this work suggests that (1) changes in extrinsic factors (e.g., of economic and/or policy origin) during the Emergent Period may have led to persistent growth in US drug approvals enjoyed since 1974, (2) the CHE may be a valued method to explore influences on time series data, and (3) innovation-related economic cycles exist (as viewed via the proxy metric of US drug approvals).Supplementary InformationThe online version contains supplementary material available at 10.1007/s43441-021-00279-8.

Temporal Memory Relation Network for Workflow Recognition From Surgical Video.

Automatic surgical workflow recognition is a key component for developing context-aware computer-assisted systems in the operating theatre. Previous works either jointly modeled the spatial features with short fixed-range temporal information, or separately learned visual and long temporal cues. In this paper, we propose a novel end-to-end temporal memory relation network (TMRNet) for relating long-range and multi-scale temporal patterns to augment the present features. We establish a long-range memory bank to serve as a memory cell storing the rich supportive information. Through our designed temporal variation layer, the supportive cues are further enhanced by multi-scale temporal-only convolutions. To effectively incorporate the two types of cues without disturbing the joint learning of spatio-temporal features, we introduce a non-local bank operator to attentively relate the past to the present. In this regard, our TMRNet enables the current feature to view the long-range temporal dependency, as well as tolerate complex temporal extents. We have extensively validated our approach on two benchmark surgical video datasets, M2CAI challenge dataset and Cholec80 dataset. Experimental results demonstrate the outstanding performance of our method, consistently exceeding the state-of-the-art methods by a large margin (e.g., 67.0% v.s. 78.9% Jaccard on Cholec80 dataset).

Comparation of Elman neural network, long short-term memory, and gated recurrent unit in predicting dengue hemorrhagic fever at DKI Jakarta

Dengue hemorrhagic fever (DHF) is a seasonal disease that has quickly spread throughout the world in the past few years. It is caused by the dengue virus that is transmitted by a biological vector in the form of mosquitoes. In DKI Jakarta, the capital of Indonesia, there were 970 DHF cases at the beginning of 2020, thus placing Jakarta in the red zone for the spread of DHF. Besides, DHF can emerge due to various weather and climate factors such as humidity, rainfall, and temperature. With a significant increase in the number of potentially fatal DHF cases in DKI Jakarta, preventing DHF outbreaks is recommended. This research aims to predict the number of DHF cases in DKI Jakarta using weather and DHF case data. Three machine learning models were employed to predict DHF case numbers: Elman neural network (ENN), long short-term memory (LSTM), and gated recurrent unit (GRU). ENNs have a simplified recurrent neural network (RNN) structure, LSTM is a modified RNN with long-range memory and an activation function for deciding which information should be retained or discarded, and GRUs are modified RNNs that are slightly simpler than LSTMs. These methods were implemented in three different data sets as follows: one with 90% training data and 10% testing data, another with 80% training data and 20% testing data, and finally, one with 70% training data and 30% testing data. A grid search is used to determine the best hyperparameter from all three methods. This will determine the best model for predicting the number of DHF cases in DKI Jakarta (excluding Kepulauan Seribu Regency) according to the data used, RMSE values, and the simulation results. Based on these criteria, LSTM is better suited to predicting the number of DHF cases than ENN or GRU in almost every district in DKI Jakarta.