Visibility graph similarity: A new measure of generalized synchronization in coupled dynamic systems

Synchronization is defined as interdependencies among coupled dynamic systems. In most coupled systems the intrinsic and internal variants, and the interdependencies among their subsystems are not accessible. Therefore, in order to quantify the interdependencies among the coupled systems, attempts have been made through measuring the synchronization between their outputs represented mostly as time series. In this paper a new method, called Visibility Graph Similarity (VGS), is presented as a method of measuring Generalized Synchronization. First, each time series is reconstructed as a trajectory in a state space. Next, a Distance Time Series (DTS) is created from a sequence of relative distances of the states to a reference state. Subsequently, a visibility graph (VG) is constructed using DTS. Then, a sequence of degrees of the VG, called Degree Sequence (DS), is obtained. Correlation of the DSs of two coupled systems is called VGS and is presented as a measurement of similarity of dynamics of the coupled systems. The synchronization measurement performance of the VGS is compared with synchronization likelihood (SL) and the classical cross correlation method using two identical and non-identical models of two coupled Henon map over the entire time domain. Also, it is compared with SL for tracing temporal synchronization using both models. It is shown that VGS provides a more accurate measure of the overall synchronization compared with SL. It is more reliable for measuring weak couplings compared with the cross correlation method. Moreover, VGS uses fewer parameters and detects the temporal synchronization sooner than the SL.     

Long memory and volatility clustering: Is the empirical evidence consistent across stock markets?

Long memory and volatility clustering are two stylized facts frequently related to financial markets. Traditionally, these phenomena have been studied based on conditionally heteroscedastic models like ARCH, GARCH, IGARCH and FIGARCH, inter alia. One advantage of these models is their ability to capture nonlinear dynamics. Another interesting manner to study the volatility phenomenon is by using measures based on the concept of entropy. In this paper we investigate the long memory and volatility clustering for the SP 500, NASDAQ 100 and Stoxx 50 indexes in order to compare the US and European Markets. Additionally, we compare the results from conditionally heteroscedastic models with those from the entropy measures. In the latter, we examine Shannon entropy, Renyi entropy and Tsallis entropy. The results corroborate the previous evidence of nonlinear dynamics in the time series considered.     

A new indicator of imminent occurrence of drawdown in the stock market

The crashes in financial markets have caught the attention of many researchers since 1929 and several mathematical models have been proposed to try to forecast the occurrence of these events. The main idea in this work is to use a wavelet transform to detect imminent abrupt changes in a financial time series, which may be eventually related to the possibility of a crash. Case studies are conducted using wavelet approaches with data covering pre-crash and post-crash 1929, as well as more recent Hang Seng and IBOVESPA data. The financial crisis of 2008 also is analyzed using this method. These time series provide useful insights into the behavior of wavelet coefficients under the possibility of short term crashes in stock market. However, it is not a trivial task to infer an imminent drawdown by simply examining the pattern of the wavelet transform coefficients. Hence, an index (a real number between 0 and 1) is proposed to aggregate the information provided by the wavelet coefficients. The new index presented good capability of monitoring crashes and drawdown with small error margins, at least in the studied cases.     

The string prediction models as invariants of time series in the forex market

In this paper we apply a new approach of string theory to the real financial market. The models are constructed with an idea of prediction models based on the string invariants (PMBSI). The performance of PMBSI is compared to support vector machines (SVM) and artificial neural networks (ANN) on an artificial and a financial time series. A brief overview of the results and analysis is given. The first model is based on the correlation function as invariant and the second one is an application based on the deviations from the closed string/pattern form (PMBCS). We found the difference between these two approaches. The first model cannot predict the behavior of the forex market with good efficiency in comparison with the second one which is, in addition, able to make relevant profit per year. The presented string models could be useful for portfolio creation and financial risk management in the banking sector as well as for a nonlinear statistical approach to data optimization.     

Discrete Velocity Models for Mixtures

Models of discrete velocity gases have been used for a long time, but only in the last few years have they become a tool to construct sequences converging to solutions of the Boltzmann equation. It appears that the case of mixtures has been rarely considered and only a couple of models, which are trivial in a sense to be explained in this paper, have been introduced. Here we thoroughly investigate the matter, and supply examples of models with both finitely and infinitely many velocities.     

Jamming percolation and glass transitions in lattice models

A new class of lattice gas models with trivial interactions but constrained dynamics is introduced. These models are proven to exhibit a dynamical glass transition: above a critical density rhoc ergodicity is broken due to the appearance of an infinite spanning cluster of jammed particles. The fraction of jammed particles is discontinuous at the transition, while in the unjammed phase dynamical correlation lengths and time scales diverge as exp[C(rhoc-rho)-mu]. Dynamic correlations display two-step relaxation similar to glass formers and jamming systems.     

Recovery of delay time from time series based on the nearest neighbor method

We propose a method for the recovery of delay time from time series of time-delay systems. The method is based on the nearest neighbor analysis. The method allows one to reconstruct delays in various classes of time-delay systems including systems of high order, systems with several coexisting delays, and nonscalar time-delay systems. It can be applied to time series heavily corrupted by additive and dynamical noise.     

Approximate record length constraints for experimental identification of dynamical fractals

The ambiguity that can exist, for short datasets, between the observational power spectra of dynamical fractals and low‐order linear memory processes is demonstrated and explained. It is argued that it could be broadly useful to have a highly practical rule‐of‐thumb for assessing whether a data record is sufficiently long to permit distinguishing the two types of processes, and if it is not, to produce an approximate estimate of the amount of additional data that would be required to do so. Such an expression is developed using the AR(1) process as a loose benchmark. Various aspects of the technique are successfully tested using synthetic time series generated by a range of prescribed models, and its application and relevance to observational datasets is then demonstrated using examples from mathematical ecology (wild steelhead population size), geophysics (river flow volume), and econophysics (stock price volatility).     

Investigation of determinism in heart rate variability

The article searches for the possible presence of determinism in heart rate variability (HRV) signals by using a new approach based on NARMA (nonlinear autoregressive moving average) modeling and free-run prediction. Thirty-three 256-point HRV time series obtained from Wistar rats submitted to different autonomic blockade protocols are considered, and a collection of surrogate data sets are generated from each one of them. These surrogate sequences are assumed to be nondeterministic and therefore they may not be predictable. The original HRV time series and related surrogates are submitted to NARMA modeling and prediction. Special attention has been paid to the problem of stationarity. The results consistently show that the surrogate data sets cannot be predicted better than the trivial predictor-the mean-while most of the HRV control sequences are predictable to a certain degree. This suggests that the normal HRV signals have a deterministic signature. The HRV time series derived from the autonomic blockade segments of the experimental protocols do not show the same predictability performance, albeit the physiological interpretation is not obvious. These results have important implications to the methodology of HRV analysis, indicating that techniques from nonlinear dynamics and deterministic chaos may be applied to elicit more information about the autonomic modulation of the cardiovascular activity. (c) 2000 American Institute of Physics.     

Characterizing pseudoperiodic time series through the complex network approach

Recently a new framework has been proposed to explore the dynamics of pseudoperiodic time series by constructing a complex network [J. Zhang, M. Small, Phys. Rev. Lett. 96 (2006) 238701]. Essentially, this is a transformation from the time domain to the network domain, which allows for the dynamics of the time series to be studied via organization of the network. In this paper, we focus on the deterministic chaotic Rössler time series and stochastic noisy periodic data that yield substantially different structures of networks. In particular, we test an extensive range of network topology statistics, which have not been discussed in previous work, but which are capable of providing a comprehensive statistical characterization of the dynamics from different angles. Our goal is to find out how they reflect and quantify different aspects of specific dynamics, and how they can be used to distinguish different dynamical regimes. For example, we find that the joint degree distribution appears to fundamentally characterize spatial organizations of cycles in phase space, and this is quantified via an assortativity coefficient. We applied network statistics to electrocardiograms of a healthy individual and an arrythmia patient. Such time series are typically pseudoperiodic, but are noisy and nonstationary and degrade traditional phase-space based methods. These time series are, however, better differentiated by our network-based statistics.     

New level of relativity

In their origins Einstein’s studies of relativity principles called into question the validity of important assumptions that had previously been made in formulating physical theories, assumptions made without investigation into alternatives. Examples of this include notions of absolute time and space, flat Euclidean geometry, and trivial topology. In this paper, we review an intermediate niche, differentiable (smooth) structure, which must be defined between topology and geometry. We now know that this choice need not be trivial. Just as it seemed for centuries to be obvious that space should be flat, so it would seem until recently that standard, trivial, smoothness for spacetime is the only choice. We now know that this is not true. In this paper we review these topics in the light of very surprising and often counter-intuitive mathematical discoveries of the last 20 years or so. Since our regions of observability are necessarily constrained we do not have any a priori justification for extending standard smoothness globally. This opens up the possibility of non-standard extension of solutions to field equations to exotically smooth regions, leading to examples such as exotic black holes and exotic cosmological models.     

Conditional independence graph for nonlinear time series and its application to international financial markets

Conditional independence graphs are proposed for describing the dependence structure of multivariate nonlinear time series, which extend the graphical modeling approach based on partial correlation. The vertexes represent the components of a multivariate time series and edges denote direct dependence between corresponding series. The conditional independence relations between component series are tested efficiently and consistently using conditional mutual information statistics and a bootstrap procedure. Furthermore, a method combining information theory with surrogate data is applied to test the linearity of the conditional dependence. The efficiency of the methods is approved through simulation time series with different linear and nonlinear dependence relations. Finally, we show how the method can be applied to international financial markets to investigate the nonlinear independence structure.     

Solution to the nonlinear boltzmann equation for maxwell models for nonisotropic initial conditions

The known solution to the spatially homogeneous nonlinear Boltzmann equation for Maxwell models in a series of Laguerre polynomials is extended to include nonisotropic initial conditions. Existence proofs for a class of solutions are supplied. The equations for the generalized (nonisotropic Laguerre) moments are derived in explicit form for two- and three-dimensional models. Further it is shown that the ordinary moments satisfy the same set of equations as the (Hermite) polynomial moments.     

Exploiting dynamical coherence: A geometric approach to parameter estimation in nonlinear models

Parameter estimation in nonlinear models is a common task, and one for which there is no general solution at present. In the case of linear models, the distribution of forecast errors provides a reliable guide to parameter estimation, but in nonlinear models the facts that predictability may vary with location in state space, and that the distribution of forecast errors is expected not to be Normal, means that parameter estimation based on least squares methods will result in systematic errors. A new approach to parameter estimation is presented which focuses on the geometry of trajectories of the model rather than the distribution of distances between model forecast and the observation at a given lead time. Specifically, we test a number of candidate trajectories to determine the duration for which they can shadow the observations, rather than evaluating a forecast error statistic at any specific lead time(s). This yields insights into both the parameters of the dynamical model and those of the observational noise model. The advances reported here are made possible by extracting more information from the dynamical equations, and thus improving the balance between information gleaned from the structural form of the equations and that from the observations. The technique is illustrated for both flows and maps, applied in 2-, 3-, and 8-dimensional dynamical systems, and shown to be effective in a case of incomplete observation where some components of the state are not observed at all. While the demonstration of effectiveness is strong, there remain fundamental challenges in the problem of estimating model parameters when the system that generated the observations is not a member of the model class. Parameter estimation appears ill defined in this case.     

Using recurrence plot analysis to distinguish between endogenous and exogenous stock market crashes

Recurrence Plots are graphical tools based on Phase Space Reconstruction. Recurrence Quantification Analysis (RQA) is a statistical quantification of RPs. RP and RQA are good at working with non-stationarity and noisy data, in detecting changes in data behavior, in particular in detecting breaks, like a phase transition and in informing about other dynamic properties of a time series. Endogenous Stock Market Crashes have been modeled as phase changes in recent times. Motivated by this, we have used RP and RQA techniques for detecting critical regimes preceding an endogenous crash seen as a phase transition and hence give an estimation of the initial bubble time. We have used a new method for computing RQA measures with confidence intervals. We have also used the techniques on a known exogenous crash to see if the RP reveals a different story or not. The analysis is made on Nifty, Hong Kong AOI and Dow Jones Industrial Average, taken over a time span of about 3 years for the endogenous crashes. Then the RPs of all time series have been observed, compared and discussed. All the time series have been first transformed into the classical momentum divided by the maximum Xmax of the time series over the time window which is considered in the specific analysis. RPs have been plotted for each time series, and RQA variables have been computed on different epochs. Our studies reveal that, in the case of an endogenous crash, we have been able to identify the bubble, while in the case of exogenous crashes the plots do not show any such pattern, thus helping us in identifying such crashes.     

Witten's invariants of rational homology spheres at prime values ofK and trivial connection contribution

We establish a relation between the coefficients of asymptotic expansion of the trivial connection contribution to Witten's invariant of rational homology spheres and the invariants that T. Ohtsuki extracted from Witten's invariant at prime values ofK. We also rederive the properties of primeK invariants discovered by H. Murakami and T. Ohtsuki. We do this by using the bounds on Taylor series expansion of the Jones polynomial of algebraically split links, studied in our previous paper. These bounds are enough to prove that Ohtsuki's invariants are of finite type. The relation between Ohtsuki's invariants and trivial connection contribution is verified explicitly for lens spaces and Seifert manifolds.Work supported by the National Science Foundation under Grant No. PHY-92 009978.     

动力学变换法探测连续系统不稳定周期轨道

本文利用动力学变换方法和庞加莱截面方法对两种连续混沌动力学系统进行不稳定周期轨道探测研究, 并对Lorenz系统进行了替代数据法检验.结果表明:基于庞加莱截面的动力学变换改进算法 可有效探测连续混沌动力学系统中的不稳定周期轨道.     

Linear and nonlinear time series analysis of the black hole candidate cygnus X-1

We analyze the variability in the x-ray lightcurves of the black hole candidate Cygnus X-1 by linear and nonlinear time series analysis methods. While a linear model describes the overall second order properties of the observed data well, surrogate data analysis reveals a significant deviation from linearity. We discuss the relation between shot noise models usually applied to analyze these data and linear stochastic autoregressive models. We debate statistical and interpretational issues of surrogate data testing for the present context. Finally, we suggest a combination of tools from linear and nonlinear time series analysis methods as a procedure to test the predictions of astrophysical models on observed data.     

Wronskian perturbation theory

We develop a perturbation method that generalizes an approach proposed recently to treat velocity-dependent quantum-mechanical models. In order to test the present approach we apply it to some simple trivial and nontrivial examples.     

一种求解二维辐射流体力学方程组的显隐式格式

构造一种求解二维辐射流体力学方程组的有限体积方法.相较于Euler方程组,辐射流体力学方程组的数值格式设计更为困难.不仅辐射压力与辐射能量的强非线性增加了数值计算的难度,而且求解强激波问题也是一大难点.与此同时,物质量以声速传播,辐射量以光速传播也增加了该系统求解的难度.为了克服这些难点,我们使用MUSCL-Hanco...