Detecting the relationships among multivariate time series using reduced auto-regressive modeling.

An information theoretic reduction of auto-regressive modeling called the Reduced Auto-Regressive (RAR) modeling is applied to several multivariate time series as a method to detect the relationships among the components in the time series. The results are compared with the results of the transfer entropy, one of the common techniques for detecting causal relationships. These common techniques are pairwise by definition and could be inappropriate in detecting the relationships in highly complicated dynamical systems. When the relationships between the dynamics of the components are linear and the time scales in the fluctuations of each component are in the same order of magnitude, the results of the RAR model and the transfer entropy are consistent. When the time series contain components that have large differences in the amplitude and the time scales of fluctuation, however, the transfer entropy fails to detect the correct relationships between the components, while the results of the RAR modeling are still correct. For a highly complicated dynamics such as human brain activity observed by electroencephalography measurements, the results of the transfer entropy are drastically different from those of the RAR modeling.

Long-range correlation properties of stationary linear models with mixed periodicities.

We consider the problem of (stationary and linear) source systems which generate time series data with long-range correlations. We use the discrete Fourier transform (DFT) and build stationary linear models using artificial time series data exhibiting a 1/f spectrum, where the models can include only terms that contribute significantly to the model as assessed by information criteria. The result is that the optimal (best) model is only composed of mixed periodicities [that is, the model does not include all (continuous) periodicities] and the time series data generated by the model exhibit a clear 1/f spectrum in a wide frequency range. It is considered that as the 1/f spectrum is a consequence of the contributions of all periods, consecutive periods are indispensable to generate such data by stationary linear models. However, the results indicate that there are cases where this expectation is not always met. These results also imply that although we can know linear features of time series data using the DFT, we always cannot substantially infer the type of the source system, even if the system is stationary linear.

Constructing networks from a dynamical system perspective for multivariate nonlinear time series.

We describe a method for constructing networks for multivariate nonlinear time series. We approach the interaction between the various scalar time series from a deterministic dynamical system perspective and provide a generic and algorithmic test for whether the interaction between two measured time series is statistically significant. The method can be applied even when the data exhibit no obvious qualitative similarity: a situation in which the naive method utilizing the cross correlation function directly cannot correctly identify connectivity. To establish the connectivity between nodes we apply the previously proposed small-shuffle surrogate (SSS) method, which can investigate whether there are correlation structures in short-term variabilities (irregular fluctuations) between two data sets from the viewpoint of deterministic dynamical systems. The procedure to construct networks based on this idea is composed of three steps: (i) each time series is considered as a basic node of a network, (ii) the SSS method is applied to verify the connectivity between each pair of time series taken from the whole multivariate time series, and (iii) the pair of nodes is connected with an undirected edge when the null hypothesis cannot be rejected. The network constructed by the proposed method indicates the intrinsic (essential) connectivity of the elements included in the system or the underlying (assumed) system. The method is demonstrated for numerical data sets generated by known systems and applied to several experimental time series.

From phase space to frequency domain: a time-frequency analysis for chaotic time series.

Time-frequency analysis is performed for chaotic flow with a power spectrum estimator based on the phase-space neighborhood. The relation between the reference phase point and its nearest neighbors is demonstrated. The nearest neighbors, representing the state recurrences in the phase space reconstructed by time delay embedding, actually cover data segments with similar wave forms and thus possess redundant information, but recur with no obvious temporal regularity. To utilize this redundant recurrence information, a neighborhood-based spectrum estimator is devised. Then time-frequency analysis with this estimator is performed for the Lorenz time series, the Rössler time series, experimental laser data, and colored noise. Features revealed by the spectrogram can be used to distinguish noisy chaotic flow from colored noise.

Detecting temporal and spatial correlations in pseudoperiodic time series.

Recently there has been much attention devoted to exploring the complicated possibly chaotic dynamics in pseudoperiodic time series. Two methods [Zhang, Phys. Rev. E 73, 016216 (2006); Zhang and Small, Phys. Rev. Lett. 96, 238701 (2006)] have been forwarded to reveal the chaotic temporal and spatial correlations, respectively, among the cycles in the time series. Both these methods treat the cycle as the basic unit and design specific statistics that indicate the presence of chaotic dynamics. In this paper, we verify the validity of these statistics to capture the chaotic correlation among cycles by using the surrogate data method. In particular, the statistics computed for the original time series are compared with those from its surrogates. The surrogate data we generate is pseudoperiodic type (PPS), which preserves the inherent periodic components while destroying the subtle nonlinear (chaotic) structure. Since the inherent chaotic correlations among cycles, either spatial or temporal (which are suitably characterized by the proposed statistics), are eliminated through the surrogate generation process, we expect the statistics from the surrogate to take significantly different values than those from the original time series. Hence the ability of the statistics to capture the chaotic correlation in the time series can be validated. Application of this procedure to both chaotic time series and real world data clearly demonstrates the effectiveness of the statistics. We have found clear evidence of chaotic correlations among cycles in human electrocardiogram and vowel time series. Furthermore, we show that this framework is more sensitive to examine the subtle changes in the dynamics of the time series due to the match between PPS surrogate and the statistics adopted. It offers a more reliable tool to reveal the possible correlations among cycles intrinsic to the chaotic nature of the pseudoperiodic time series.

Testing for correlation structures in short-term variabilities with long-term trends of multivariate time series.

We describe a method for identifying correlation structures in irregular fluctuations (short-term variabilities) of multivariate time series, even if they exhibit long-term trends. This method is based on the previously proposed small shuffle surrogate method. The null hypothesis addressed by this method is that there is no short-term correlation structure among data or that the irregular fluctuations are independent. The method is demonstrated for numerical data generated by known systems and applied to several experimental time series.

Testing for nonlinearity in irregular fluctuations with long-term trends.

We describe a method for investigating nonlinearity in irregular fluctuations (short-term variability) of time series even if the data exhibit long-term trends (periodicities). Such situations are theoretically incompatible with the assumption of previously proposed methods. The null hypothesis addressed by our algorithm is that irregular fluctuations are generated by a stationary linear system. The method is demonstrated for numerical data generated by known systems and applied to several actual time series.

Degeneracy of time series models: the best model is not always the correct model.

There are a number of good techniques for finding, in some sense, the best model of a deterministic system given a time series of observations. We examine a problem called model degeneracy, which has the consequence that even when a perfect model of a system exists, one does not find it using the best techniques currently available. The problem is illustrated using global polynomial models and the theory of Grobner bases.

Testing for nonlinearity in time series without the Fourier transform.

A method to test for nonlinearity in time series, without the need to apply the Fourier transform, is proposed. This method therefore avoids the drawbacks of previously proposed surrogate techniques associated with the estimation of the signal's power spectrum. The test addressed by this algorithm is that the data are generated by a stationary linear system. To achieve this, the algorithm takes advantage of the fundamentally different structure of linear and nonlinear systems.

Small-shuffle surrogate data: testing for dynamics in fluctuating data with trends.

We describe a method for identifying dynamics in irregular time series (short term variability). The method we propose focuses attention on the flow of information in the data. We can apply the method even for irregular fluctuations which exhibit long term trends (periodicities): situations in which previously proposed surrogate methods would give erroneous results. The null hypothesis addressed by our algorithm is that irregular fluctuations are independently distributed random variables (in other words, there is no short term dynamics). The method is demonstrated for numerical data generated by known systems, and applied to several actual time series.

Surrogate test to distinguish between chaotic and pseudoperiodic time series.

In this paper a different algorithm is proposed to produce surrogates for pseudoperiodic time series. By imposing a few constraints on the noise components of pseudoperiodic data sets, we devise an effective method to generate surrogates. Unlike other algorithms, this method properly copes with pseudoperiodic orbits contaminated with linear colored observational noise. We will demonstrate the ability of this algorithm to distinguish chaotic orbits from pseudoperiodic orbits through simulation data sets from the Rössler system. As an example of application of this algorithm, we will also employ it to investigate a human electrocardiogram record.