Analysis of non-stationary data for heart-rate fluctuations in terms of drift and diffusion coefficients

Physics-informed and data-driven discovery of governing equations for complex phenomena in heterogeneous media

Rapid evolution of sensor technology, advances in instrumentation, and progress in devising data-acquisition software and hardware are providing vast amounts of data for various complex phenomena that occur in heterogeneous media, ranging from those in atmospheric environment, to large-scale porous formations, and biological systems. The tremendous increase in the speed of scientific computing has also made it possible to emulate diverse multiscale and multiphysics phenomena that contain elements of stochasticity or heterogeneity, and to generate large volumes of numerical data for them. Thus, given a heterogeneous system with annealed or quenched disorder in which a complex phenomenon occurs, how should one analyze and model the system and phenomenon, explain the data, and make predictions for length and time scales much larger than those over which the data were collected? We divide such systems into three distinct classes. (i) Those for which the governing equations for the physical phenomena of interest, as well as data, are known, but solving the equations over large length scales and long times is very difficult. (ii) Those for which data are available, but the governing equations are only partially known, in the sense that they either contain various coefficients that must be evaluated based on the data, or that the number of degrees of freedom of the system is so large that deriving the complete equations is very difficult, if not impossible, as a result of which one must develop the governing equations with reduced dimensionality. (iii) In the third class are systems for which large amounts of data are available, but the governing equations for the phenomena of interest are not known. Several classes of physics-informed and data-driven approaches for analyzing and modeling of the three classes of systems have been emerging, which are based on machine learning, symbolic regression, the Koopman operator, the Mori-Zwanzig projection operator formulation, sparse identification of nonlinear dynamics, data assimilation combined with a neural network, and stochastic optimization and analysis. This perspective describes such methods and the latest developments in this highly important and rapidly expanding area and discusses possible future directions.

Introducing a new approach for modeling a given time series based on attributing any random variation to a jump event: jump-jump modeling

When analyzing the data sampled at discrete times, one encounters successive discontinuities in the trajectory of the sampled time series, even if the underlying path is continuous. On the other hand, the distinction between discontinuities caused by finite sampling of continuous stochastic process and real discontinuities in the sample path is one of the main problems. Clues like these led us to the question: Is it possible to provide a model that treats any random variation in the data set as a jump event, regardless of whether the given time series is classified as diffusion or jump-diffusion processes? To address this question, we wrote a new stochastic dynamical equation, which includes a drift term and a combination of Poisson jump processes with different distributed sizes. In this article, we first introduce this equation in its simplest form including a drift term and a jump process, and show that such a jump-drift equation is able to describe the discrete time evolution of a diffusion process. Afterwards, we extend the modeling by considering more jump processes in the equation, which can be used to model complex systems with various distributed amplitudes. At each step, we also show that all the unknown functions and parameters required for modeling can be obtained non-parametrically from the measured time series.

Ameliorated New Media Literacy Model Based on an Esthetic Model: The Ability of a College Student Audience to Enter the Field of Digital Art

In the current digital environment, people can visit every corner of the world without leaving their homes. New media technology compresses distance and time, but it also subverts the traditional mode of audience presence. Many traditional, offline content expression modes are also moving toward the digital field, and digital art is among them. Digital new media is a new art form that requires its audience to have a new media literacy; this literacy is necessary for esthetic experience and for audience participation. At present, the relatively lack of objective methodology for scientific research on aesthetic and media literacy has limited our current understanding. Therefore, we need to develop a new model and conduct empirical research with college students as the audience. Empirical research was conducted with an audience of college students. The study had the following purposes: (1) to add a new dimension to the esthetic model, namely new media literacy, to align the model with the current digital environment, and (2) to test the moderating effect of new media literacy on esthetic emotion as represented by interest and confusion. The experiment verified the study’s hypothesis that higher new media literacy was associated with higher esthetic interest and lower confusion. By contrast, has a substantial influence on the cognitive processes in humans, lower new media literacy was associated with lower esthetic interest and higher confusion. New media literacy is an essential quality for contemporary audiences. This knowledge may be useful for effective design. It provides a traditional and favorable learning environment and empirical reference for the subsequent improvement of digital aesthetics and media literacy.

Testing Jump-Diffusion in Epileptic Brain Dynamics: Impact of Daily Rhythms

Stochastic approaches to complex dynamical systems have recently provided broader insights into spatial-temporal aspects of epileptic brain dynamics. Stochastic qualifiers based on higher-order Kramers-Moyal coefficients derived directly from time series data indicate improved differentiability between physiological and pathophysiological brain dynamics. It remains unclear, however, to what extent stochastic qualifiers of brain dynamics are affected by other endogenous and/or exogenous influencing factors. Addressing this issue, we investigate multi-day, multi-channel electroencephalographic recordings from a subject with epilepsy. We apply a recently proposed criterion to differentiate between Langevin-type and jump-diffusion processes and observe the type of process most qualified to describe brain dynamics to change with time. Stochastic qualifiers of brain dynamics are strongly affected by endogenous and exogenous rhythms acting on various time scales—ranging from hours to days. Such influences would need to be taken into account when constructing evolution equations for the epileptic brain or other complex dynamical systems subject to external forcings.

Effect of heterogeneity and spatial correlations on the structure of a tumor invasion front in cellular environments

Analysis of invasion front has been widely used to decipher biological properties, as well as the growth dynamics of the corresponding populations. Likewise, the invasion front of tumors has been investigated, from which insights into the biological mechanisms of tumor growth have been gained. We develop a model to study how tumors' invasion front depends on the relevant properties of a cellular environment. To do so, we develop a model based on a nonlinear reaction-diffusion equation, the Fisher-Kolmogorov-Petrovsky-Piskunov equation, to model tumor growth. Our study aims to understand how heterogeneity in the cellular environment's stiffness, as well as spatial correlations in its morphology, the existence of both of which has been demonstrated by experiments, affects the properties of tumor invasion front. It is demonstrated that three important factors affect the properties of the front, namely the spatial distribution of the local diffusion coefficients, the spatial correlations between them, and the ratio of the cells' duplication rate and their average diffusion coefficient. Analyzing the scaling properties of tumor invasion front computed by solving the governing equation, we show that, contrary to several previous claims, the invasion front of tumors and cancerous cell colonies cannot be described by the well-known models of kinetic growth, such as the Kardar-Parisi-Zhang equation.

Regulation of migration of chemotactic tumor cells by the spatial distribution of collagen fiber orientation

Collagen fibers, an important component of the extracellular matrix (ECM), can both inhibit and promote cellular migration. In vitro studies have revealed that the fibers' orientations are crucial to cellular invasion, while in vivo investigations have led to the development of tumor-associated collagen signatures (TACS) as an important prognostic factor. Studying biophysical regulation of cell invasion and the effect of the fibers' orientation not only deepens our understanding of the phenomenon, but also helps classify the TACSs precisely, which is currently lacking. We present a stochastic model for random or chemotactic migration of cells in fibrous ECM, and study the role of the various factors in it. The model provides a framework for quantitative classification of the TACSs, and reproduces quantitatively recent experimental data for cell motility. It also indicates that the spatial distribution of the fibers' orientations and extended correlations between them, hitherto ignored, as well as dynamics of cellular motion all contribute to regulation of the cells' invasion length, which represents a measure of metastatic risk. Although the fibers' orientations trivially affect randomly moving cells, their effect on chemotactic cells is completely nontrivial and unexplored, which we study in this paper.

CRSIDLab: A Toolbox for Multivariate Autonomic Nervous System Analysis Using Cardiorespiratory Identification

This paper presents the Cardiorespiratory System Identification Lab (CRSIDLab), a MATLAB-based software tool for multivariate autonomic nervous system (ANS) evaluation through heart rate variability (HRV) analysis and cardiorespiratory system identification. Based on a graphical user interface, CRSIDLab provides a complete set of tools including pre-processing cardiorespiratory data (electrocardiogram, continuous blood pressure, airflow, and instantaneous lung volume), power spectral density estimation, and multivariable cardiorespiratory system model identification. Parametrized multivariate models can assess both HRV and baroreflex sensitivity (BRS) by considering the causal relationship from respiration to heart rate (or its reciprocal, R-to-R interval - RRI) and from systolic blood pressure to RRI, for instance. The impulse response, estimated from the model, is used as a mathematical tool to effectively open the inherently closed-loop nature of the cardiorespiratory system, allowing the investigation of the dynamic response between pairs of cardiorespiratory variables. This system modeling approach provides information on gain and temporal behavior regarding dynamics, such as the baroreflex, complementing traditional HRV, and BRS indices. The toolbox is presented and used to investigate autonomic function in sleep apnea. The results show that, while traditional HRV indices were unable to differentiate between apneic and non-apneic subjects, the autonomic descriptors obtained from the multivariate system identification techniques were able to show vagal impairment in apneic compared to non-apneic subjects. Thus, CRSIDLab can help promote the use of cardiorespiratory system identification as a potentially more sensitive measure of ANS activity than classical HRV analysis.

Empirical Fokker-Planck-based test of stationarity for time series

We propose a test of stationarity based on the drift coefficients of the Langevin type and the associated Fokker-Planck equations. The test relies on the estimation of the drift coefficients of the underlying probability densities and posits that a time series is nonstationary if the estimated drift term is a nonlinear function of the random variable of the observed time series and the Markov property holds. We provide ample empirical evidence that demonstrates that well- known stationary systems give rise to linear estimates of the drift coefficients, whereas nonstationary time series exhibit nonlinear estimates of the drift term. This does not, indeed, imply that a nonlinear drift term in the Fokker-Planck equation of a dynamic stochastic process causes nonstationarity.

Quantifying memory in complex physiological time-series

In a time-series, memory is a statistical feature that lasts for a period of time and distinguishes the time-series from a random, or memory-less, process. In the present study, the concept of “memory length” was used to define the time period, or scale over which rare events within a physiological time-series do not appear randomly. The method is based on inverse statistical analysis and provides empiric evidence that rare fluctuations in cardio-respiratory time-series are ‘forgotten’ quickly in healthy subjects while the memory for such events is significantly prolonged in pathological conditions such as asthma (respiratory time-series) and liver cirrhosis (heart-beat time-series). The memory length was significantly higher in patients with uncontrolled asthma compared to healthy volunteers. Likewise, it was significantly higher in patients with decompensated cirrhosis compared to those with compensated cirrhosis and healthy volunteers. We also observed that the cardio-respiratory system has simple low order dynamics and short memory around its average, and high order dynamics around rare fluctuations.

Extraction of stochastic dynamics from time series

We present a method for the reconstruction of the dynamics of processes with discrete time. The time series from such a system is described by a stochastic recurrence equation, the continuous form of which is known as the Langevin equation. The deterministic f and stochastic g components of the stochastic equation are directly extracted from the measurement data with the assumption that the noise has finite moments and has a zero mean and a unit variance. No other information about the noise distribution is needed. This is contrary to the usual Langevin description, in which the additional assumption that the noise is Gaussian (δ-correlated) distributed as necessary. We test the method using one dimensional deterministic systems (the tent and logistic maps) with Gaussian and with Gumbel noise. In addition, results for human heart rate variability are presented as an example of the application of our method to real data. The differences between cardiological cases can be observed in the properties of the deterministic part f and of the reconstructed noise distribution.

Analysis of porosity distribution of large-scale porous media and their reconstruction by Langevin equation

Several methods have been developed in the past for analyzing the porosity and other types of well logs for large-scale porous media, such as oil reservoirs, as well as their permeability distributions. We developed a method for analyzing the porosity logs ϕ(h) (where h is the depth) and similar data that are often nonstationary stochastic series. In this method one first generates a new stationary series based on the original data, and then analyzes the resulting series. It is shown that the series based on the successive increments of the log y(h)=ϕ(h+δh)-ϕ(h) is a stationary and Markov process, characterized by a Markov length scale h(M). The coefficients of the Kramers-Moyal expansion for the conditional probability density function (PDF) P(y,h|y(0),h(0)) are then computed. The resulting PDFs satisfy a Fokker-Planck (FP) equation, which is equivalent to a Langevin equation for y(h) that provides probabilistic predictions for the porosity logs. We also show that the Hurst exponent H of the self-affine distributions, which have been used in the past to describe the porosity logs, is directly linked to the drift and diffusion coefficients that we compute for the FP equation. Also computed are the level-crossing probabilities that provide insight into identifying the high or low values of the porosity beyond the depth interval in which the data have been measured.

Exact moment scaling from multiplicative noise

For a general class of diffusion processes with multiplicative noise, describing a variety of physical as well as financial phenomena, mostly typical of complex systems, we obtain the analytical solution for the moments at all times. We allow for a nontrivial time dependence of the microscopic dynamics and we analytically characterize the process evolution, possibly toward a stationary state, and the direct relationship existing between the drift and diffusion coefficients and the time scaling of the moments.

Improved estimation of Fokker-Planck equations through optimization

An improved method for the description of hierarchical complex systems by means of a Fokker-Planck equation is presented. In particular the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm for constraint problems is used to minimize the distance between the numerical solutions of the Fokker-Planck equation and the empirical probability density functions and thus to estimate properly the drift and diffusion term of the Fokker-Planck equation. The optimization routine is applied to a time series of velocity measurements obtained from a turbulent helium gas jet in order to demonstrate the benefits and to quantify the improvements of this optimization routine.

Nonstationary Langevin equation: statistical properties and application to explain effects observed in cardiological time series

Using the Langevin equation we develop the model of a stochastic process subject to a given time-dependent regulatory mechanism. The effects of this nonstationarity on the statistical properties of the time series, i.e., on global and conditional probability densities and on the moments of the distribution, are derived. Application of these results on simple model trends allows one to approximate cardiological data and thus to explain effects recently observed in the reconstruction of the deterministic part of the Langevin equation for time series of heart rate.

Markov analysis and Kramers-Moyal expansion of nonstationary stochastic processes with application to the fluctuations in the oil price

We describe a general method for analyzing a nonstationary stochastic process X(t) which, unlike many of the previous analysis methods, does not require X(t) to have any scaling feature. The method is used to study the fluctuations in the daily price of oil. It is shown that the returns time series, y(t)=ln[X(t+1)X(t)] , is a stationary and Markov process, characterized by a Markov time scale t_{M} . The coefficients of the Kramers-Moyal expansion for the probability density function P(y,tmid R:y_{0},t_{0}) are computed. P(y,tmid R:,y_{0},t_{0}) satisfies a Fokker-Planck equation, which is equivalent to a Langevin equation for y(t) that provides quantitative predictions for the oil price over times that are of the order of t_{M}. Also studied is the average frequency of positive-slope crossings, nu_{alpha};{+}=P(y_{i}>alpha,y_{i-1}<alpha) , for the returns, where T(alpha)=1nu_{alpha};{+} is the average waiting time for observing y(t)=alpha again.