Embedding Phase Space Research Articles

Estimation of Carleman operator from a univariate time series.

Reconstructing a nonlinear dynamical system from empirical time series is a fundamental task in data-driven analysis. One of the main challenges is the existence of hidden variables; we only have records for some variables, and those for hidden variables are unavailable. In this work, the techniques for Carleman linearization, phase-space embedding, and dynamic mode decomposition are integrated to rebuild an optimal dynamical system from time series for one specific variable. Using the Takens theorem, the embedding dimension is determined, which is adopted as the dynamical system's dimension. The Carleman linearization is then used to transform this finite nonlinear system into an infinite linear system, which is further truncated into a finite linear system using the dynamic mode decomposition technique. We illustrate the performance of this integrated technique using data generated by the well-known Lorenz model, the Duffing oscillator, and empirical records of electrocardiogram, electroencephalogram, and measles outbreaks. The results show that this solution accurately estimates the operators of the nonlinear dynamical systems. This work provides a new data-driven method to estimate the Carleman operator of nonlinear dynamical systems.

Unveiling dynamics changes: Singular spectrum analysis-based method for detecting concept drift in industrial data streams

Industrial data streams frequently experience concept drifts. Current drift detection methods, focusing on prediction performance or data distribution, often neglect temporal dependencies and require prior distribution assumptions. These limitations have catalyzed the development of dynamic theory-based approaches that identify drifts by discerning variations in data dynamics. However, the majority of these exhibit high complexity in dynamics characterization and still underperform in industrial drift detection. To overcome these challenges, we propose a singular spectrum analysis-based drift detection method for industry data streams, comprising three modules: common dynamics extraction, noise-robust drift detection, and threshold adaptation. Our approach uses singular spectrum analysis (SSA) as the dynamics representation method instead of traditional high-dimensional phase space embedding and Fourier transform. SSA allows for the efficient description of principal dynamics by intelligently mixing the Fourier basis functions according to given data. The first two modules are specifically engineered to address the common dynamics and stochastic noise inherent in industrial data streams through a well-designed online common dynamics extraction algorithm and a noise-robust detection index respectively. Threshold adaptation module provides a more reasonable way of updating the drift threshold based on the extreme value theory, ensuring global consistency for parameter estimation of extreme value distribution. Our proposed method was tested on simulated examples and a real-world sintering process, exhibiting enhanced drift detection performance.

Lean blowout detection using topological data analysis.

Modern lean premixed combustors are operated in ultra-lean mode to conform to strict emission norms. However, this causes the combustors to become prone to lean blowout (LBO). Online monitoring of combustion dynamics may help to avoid LBO and help the combustor run more safely and reliably. Previous studies have suggested various techniques to early predict LBO in single-burner combustors. In contrast, early detection of LBO in multi-burner combustors has been little explored to date. Recent studies have discovered significantly different combustion dynamics between multi-burner combustors and single-burner combustors. In the present paper, we show that some well-established early LBO detection techniques suitable for single-burner combustor are less effective in early detecting LBO in multi-burner combustors. To resolve this, we propose a novel tool, topological data analysis (TDA), for real-time LBO prediction in a wide range of combustor configurations. We find that the TDA metrics are computationally cheap and follow monotonic trends during the transition to LBO. This indicates that the TDA metrics can be used to fine-tune the LBO safety margin, which is a desirable feature from practical implementation point of view. Furthermore, we show that the sublevel set TDA metrics show approximately monotonic changes during the transition to LBO even with low sampling-rate signals. Sublevel set TDA is computationally inexpensive and does not require phase-space embedding. Therefore, TDA can potentially be used for real-time monitoring of combustor dynamics with simple, low-cost, and low sampling-rate sensors.

Mapping nonlinear brain dynamics by phase space embedding with fMRI data

The human brain is a complex neurobiological system exhibiting complex nonlinear spatiotemporal dynamics. While functional magnetic resonance imaging (fMRI) has been widely used to study brain activity, whole-brain nonlinear dynamics in fMRI data have not been extensively examined. The present study applied phase space embedding on resting-state fMRI data and characterized their phase space dynamics with the sum of lengths of portrait edges (SE) in the reconstructed phase portrait. The effects of repetition time (TR), bandpass filtering and the added noise power of BOLD signals on the optimal embedding parameters (embedding time delay τ and embedding dimension m) were examined with experimental or simulated fMRI data. Our results show that τ and m vary with the three acquisition parameters. The present method was applied to the autism spectrum disorder dataset from Autism Imaging Data Exchange I to demonstrate its capability in the characterization of abnormal brain dynamics. The resultant SE maps were statistically compared between patients and controls, and the significant differences in SE were fed into a support vector machine (SVM) for classification. A significant increase in SE in the default mode network (DMN) and salience network (SN), as well as the visual network, was found in autistic patients. With the SE features of these regions, our SVM classifier achieved superior accuracy (74.55% with 10-folds cross validation) compared with prior studies, indicating that phase space embedding and SE mapping are promising in characterizing the nonlinear dynamics of the BOLD signal and might be useful for brain biomarker discovery in clinical psychiatry.

Data driven structural damage assessment using phase space embedding and Koopman operator under stochastic excitations

To address the issue faced by phase space trajectory (PST) based methods for identifying structural damage using high dimensional dynamic responses of structures under stochastic excitations, a novel data driven approach for structural damage assessment based on phase space embedding in conjunction with stochastic Koopman operator, is proposed in this study. Vibration acceleration responses are used to reconstruct the phase space representation of system dynamic attractor using embedding strategy, followed by subspace dynamic mode decomposition (subspace DMD) to estimate the unbiased eigenvalues of the corresponding stochastic Koopman operator. Under the hypothesis that the Koopman operator eigenpairs will vary with structural condition change, the Mahalanobis distance of Koopman operator eigenvalues approximated from heathy state and current testing state is introduced as a damage sensitive feature (DSF) to detect structural damage. A planar truss model is adopted in numerical studies to demonstrate the feasibility and applicability of the proposed approach. The robustness of the proposed approach under operational and environmental variations is tested by considering 10% white noise in vibration measurement and six different ambient loading scenarios. Numerical results show that the proposed approach is sensitive to the occurrence and severity of structural damage, and is insensitive to the measurement noise and the variation of stochastic excitation amplitudes. Furthermore, the proposed method is applied to identify damage introduced as the artificially applied settlements of pier in the Z24 benchmark bridge. Results demonstrate that the bridge condition under the reference state and the damage scenarios with different levels of pier settlement are well identified by using the proposed approach with in-field measurement data including different test environment. The defined DSF value can be used to reflect the damage severity in these damage scenarios.

Mapping Nonlinear Brain Dynamics by Phase Space Embedding with Fmri Data

Mapping Nonlinear Brain Dynamics by Phase Space Embedding with Fmri Data

Spatio-temporal connections in streamflow: a complex networks-based approach

This study introduces a new complex networks-based method to examine the spatio-temporal connections in streamflow. The method involves reconstruction of two (or more) time series jointly in a multi-dimensional phase space using a nonlinear phase space embedding procedure and construction of the spatio-temporal streamflow network of nodes and links based on the reconstructed vectors. After this spatio-temporal network construction, the clustering property of the network is measured using the clustering coefficient, which quantifies the tendency of a network to cluster. The approach is applied to monthly streamflow time series observed at each of 639 streamflow stations in the United States. Different distance threshold values are used to identify the presence/absence of links in the streamflow network and, hence, to calculate the clustering coefficient. The clustering coefficient results help to identify the critical distance threshold and optimal embedding dimension of each streamflow time series using different distance threshold values. The optimal embedding dimensions and the clustering coefficient values of the 639 streamflow time series are also discussed in terms of the role of catchment and flow properties (drainage area, elevation, flow mean, and flow coefficient of variation). The dimensions for the 639 streamflow time series are generally found to range from 2 to 18 (but even up to 30 for a few stations), indicating a wide range of complexity in the spatio-temporal connections in streamflow across the United States. The clustering coefficient values for the 639 stations are found to be in the range of 0.53–0.99, which suggest generally strong connections. The outcomes of this study clearly indicate the usefulness of the networks-based approach for examining the spatio-temporal connections in streamflow.

Q-learning and LSTM based deep active learning strategy for malware defense in industrial IoT applications

Edge devices are extensively used as intermediaries between the device and the service layer in an industrial Internet of things (IIoT) environment. These devices are quite vulnerable to malware attacks. Existing studies have worked on designing complex learning algorithms or deep architectures to accurately classify malware assuming that a sufficient number of labeled examples are provided. In the real world, getting labeled examples is one of the major issues for training any classification algorithm. Recent advances have allowed researchers to use active learning strategies that are trained on a handful of labeled examples to perform the classification task, but they are based on the selection of informative instances. This study integrates the Q-learning characteristics into an active learning framework, which allows the network to either request or predict a label during the training process. We proposed the use of phase space embedding, sparse autoencoder, and LSTM with the action-value function to classify malware applications while using a handful of labeled examples. The network relies on its uncertainty to either request or predict a label. The experimental results show that the proposed method can achieve better accuracy than the supervised learning strategy while using few labeled requests. The results also show that the trained network is resilient to the adversarial attacks, which proves the robustness of the proposed method. Additionally, this study explores the tradeoff between classification accuracy and number of label requests via the choice of rewards and the use of decision-level fusion strategies to boost the classification performance. Furthermore, we also provide a hypothetical framework as an implication of the proposed method.

Advanced Homological Analysis for Biometric Identification Using Accelerometer

Biometric information acquired by mobile devices has been explored for various emerging applications in recent years. One of these new applications is advanced authentication. On-line user-identification based on biometrics collected by smartphones has been drawing tremendous research interest lately. The accelerometer data are available pervasively on cellular phones and therefore there incurs no extra cost to facilitate accelerometer-based biometric systems. Individual time-series constituted by accelerometer data are quite complex and therefore impose challenges to the user-identification research. In this work, we propose a novel on-line user-identification approach using accelerometer data by exploiting new homological analysis. The inherent walking patterns of different users in the accelerometer data are monitored in the embedded phase space. Then the expected persistence diagrams (EPDs) are formed and transformed into the corresponding probability distribution functions. Thus, the discrepancy in walking patterns can be measured by the Kullback-Leibler divergence (KLD) of those ultimate distribution functions. Users can be identified according to the KLDs resulting from the sampled data and pre-stored training data. Experimental results from the real-world accelerometer data have demonstrated that the accuracy of our proposed new scheme reaches up to 94.5% and it greatly outperforms other existing methods.

Influence of characteristic parameters of signal on fault feature extraction of singular value method

The detection of mechanical fault signals by singular value decomposition is a commonly used method in fault diagnosis. The delay time of the fault signal time series and the rationality of the value of the phase space embedding dimension, as well as the fluctuation of the characteristic parameters of the fault signal, will cause the singular value decomposition method to have a greater impact on the accuracy of fault feature identification and diagnosis. In this article, the simulation model of the similarity signal is established by the combination of the autocorrelation function method and the Cao’s algorithm. Then, the delay time of the signal sequence and the optimal value of the embedded dimension are obtained through simulation. Next, using this method to study the fluctuation of the characteristic parameters such as the frequency, amplitude and initial phase of the signal, the relationship between the characteristic parameters of the signal and the singular value of the signal is obtained. Finally, through the experimental study of the pitting corrosion of the gear tooth surface, the vibration of the fault feature is obtained. The research shows that the combination of autocorrelation function method and Cao's algorithm can calculate the optimal characteristic parameters for the singular value decomposition method and improve the ability of the method to identify fault features.

Interplay between myosin II and actin dynamics in chemotactic amoeba

The actin cytoskeleton and its response to external chemical stimuli is fundamental to the mechano-biology of eukaryotic cells and their functions. One of the key players that governs the dynamics of the actin network is the motor protein myosin II. Based on a phase space embedding we have identified from experiments three phases in the cytoskeletal dynamics of starved Dictyostelium discoideum in response to a precisely controlled chemotactic stimulation. In the first two phases the dynamics of actin and myosin II in the cortex is uncoupled, while in the third phase the time scale for the recovery of cortical actin is determined by the myosin II dynamics. We report a theoretical model that captures the experimental observations quantitatively. The model predicts an increase in the optimal response time of actin with decreasing myosin II-actin coupling strength highlighting the role of myosin II in the robust control of cell contraction.

Assessing Hydrograph Similarity and Rare Runoff Dynamics by Cross Recurrence Plots

AbstractThis paper introduces a novel measure to assess similarity between event hydrographs. It is based on cross recurrence plots (CRP) and recurrence quantification analysis (RQA), which have recently gained attention in a range of disciplines when dealing with complex systems. The method attempts to quantify the event runoff dynamics and is based on the time delay embedded phase space representation of discharge hydrographs. A phase space trajectory is reconstructed from the event hydrograph, and pairs of hydrographs are compared to each other based on the distance of their phase space trajectories. Time delay embedding allows considering the multidimensional relationships between different points in time within the event. Hence, the temporal succession of discharge values is taken into account, such as the impact of the initial conditions on the runoff event. We provide an introduction to cross recurrence plots and discuss their parameterization. An application example based on flood time series demonstrates how the method can be used to measure the similarity or dissimilarity of events, and how it can be used to detect events with rare runoff dynamics. It is argued that this methods provides a more comprehensive approach to quantify hydrograph similarity compared to conventional hydrological signatures.

The Chaotic Long-term X-ray Variability of 4U 1705-44.

The low-mass X-ray binary 4U1705-44 exhibits dramatic long-term X-ray time variability with a timescale of several hundred days. The All-Sky Monitor (ASM) aboard the Rossi X-ray Timing Explorer (RXTE) and the Japanese Monitor of All-sky X-ray Image (MAXI) aboard the International Space Station together have continuously observed the source from December 1995 through May 2014. The combined ASM-MAXI data provide a continuous time series over fifty times the length of the timescale of interest. Topological analysis can help us identify fingerprints in the phase-space of a system unique to its equations of motion. The Birman-Williams theorem postulates that if such fingerprints are the same between two systems, then their equations of motion must be closely related. The phase-space embedding of the source light curve shows a strong resemblance to the double-welled nonlinear Duffing oscillator. We explore a range of parameters for which the Duffing oscillator closely mirrors the time evolution of 4U1705-44. We extract low period, unstable periodic orbits from the 4U1705-44 and Duffing time series and compare their topological information. The Duffing and 4U1705-44 topological properties are identical, providing strong evidence that they share the same underlying template. This suggests that we can look to the Duffing equation to help guide the development of a physical model to describe the long-term X-ray variability of this and other similarly behaved X-ray binary systems.

Data-adaptive harmonic spectra and multilayer Stuart-Landau models.

Harmonic decompositions of multivariate time series are considered for which we adopt an integral operator approach with periodic semigroup kernels. Spectral decomposition theorems are derived that cover the important cases of two-time statistics drawn from a mixing invariant measure. The corresponding eigenvalues can be grouped per Fourier frequency and are actually given, at each frequency, as the singular values of a cross-spectral matrix depending on the data. These eigenvalues obey, furthermore, a variational principle that allows us to define naturally a multidimensional power spectrum. The eigenmodes, as far as they are concerned, exhibit a data-adaptive character manifested in their phase which allows us in turn to define a multidimensional phase spectrum. The resulting data-adaptive harmonic (DAH) modes allow for reducing the data-driven modeling effort to elemental models stacked per frequency, only coupled at different frequencies by the same noise realization. In particular, the DAH decomposition extracts time-dependent coefficients stacked by Fourier frequency which can be efficiently modeled-provided the decay of temporal correlations is sufficiently well-resolved-within a class of multilayer stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators. Applications to the Lorenz 96 model and to a stochastic heat equation driven by a space-time white noise are considered. In both cases, the DAH decomposition allows for an extraction of spatio-temporal modes revealing key features of the dynamics in the embedded phase space. The multilayer Stuart-Landau models (MSLMs) are shown to successfully model the typical patterns of the corresponding time-evolving fields, as well as their statistics of occurrence.

Psychotherapy Is Chaotic—(Not Only) in a Computational World

Objective: The aim of this article is to outline the role of chaotic dynamics in psychotherapy. Besides some empirical findings of chaos at different time scales, the focus is on theoretical modeling of change processes explaining and simulating chaotic dynamics. It will be illustrated how some common factors of psychotherapeutic change and psychological hypotheses on motivation, emotion regulation, and information processing of the client's functioning can be integrated into a comprehensive nonlinear model of human change processes.Methods: The model combines 5 variables (intensity of emotions, problem intensity, motivation to change, insight and new perspectives, therapeutic success) and 4 parameters into a set of 5 coupled nonlinear difference equations. The results of these simulations are presented as time series, as phase space embedding of these time series (i.e., attractors), and as bifurcation diagrams.Results: The model creates chaotic dynamics, phase transition-like phenomena, bi- or multi-stability, and sensibility of the dynamic patterns on parameter drift. These features are predicted by chaos theory and by Synergetics and correspond to empirical findings. The spectrum of these behaviors illustrates the complexity of psychotherapeutic processes.Conclusion: The model contributes to the development of an integrative conceptualization of psychotherapy. It is consistent with the state of scientific knowledge of common factors, as well as other psychological topics, such as: motivation, emotion regulation, and cognitive processing. The role of chaos theory is underpinned, not only in the world of computer simulations, but also in practice. In practice, chaos demands technologies capable of real-time monitoring and reporting on the nonlinear features of the ongoing process (e.g., its stability or instability). Based on this monitoring, a client-centered, continuous, and cooperative process of feedback and control becomes possible. By contrast, restricted predictability and spontaneous changes challenge the usefulness of prescriptive treatment manuals or other predefined programs of psychotherapy.

Structural interrogation using phase space topology of the wind-induced responses

The applicability of the phase space interrogation (PSI) methodology for structural health monitoring (SHM) is limited on account of the fact that the structure needs to be excited by a low dimensional chaotic signal. The present study demonstrates that the phase space interrogation can still be applied to structures subjected to ambient/moderate wind excitations. Key to this extension is the relative low dimensionality of the wind-induced structural responses, amenable to phase space embedding by virtue of Takens’ embedding theorem. The so-formed pseudo-attractor is shown to sufficiently reflect the changes in system dynamics induced by structural damage(s). A widely employed damage feature, namely, the changes in phase space topology (CPST) is subsequently employed to the reconstructed attractor to link it with the presence, severity, as well as localization of damage(s). The CPST is established as a legitimate damage-sensitive feature by studying its variability with alternative damage scenarios in a multistoried frame building subjected to wind excitations. The performance of the methodology is demonstrated under different degrees of noise contamination in the measured responses as well as varying intensity of wind speed. The statistical robustness of the procedure is also assessed. The numerical findings are supported by the evidence from a limited number of experimental investigations carried out on a model building with inflicted damage scenarios. The wind loadings for the tests are simulated using a wind tunnel testing facility. Finally, a simple analysis is presented that establish the viability of the phase space analysis analytically.

Application of first-order canonical perturbation method with dissipative Hori-like kernel

Lie-Hori canonical perturbation theory provides asymptotic solutions for conservative Hamiltonian systems. This restriction prevents the canonical method from being applied directly to dissipative mechanical systems. There are, however, two main alternatives to overcome this difficulty, enabling the application of canonical perturbation methods. The first one consists in constructing a time-dependent Hamiltonian, through a generating function, related to the energy dissipation pattern of the system. The second embeds the original phase space into a double dimensional one where the dynamics of the system can be formulated in a Hamiltonian way. In this paper, a modified Lie-Hori method that avoid the disadvantages of the former approaches is proposed. Namely, it is not necessary to find out a time-dependent generating function, nor doubling the number of the canonical variables of the original problem. The new algorithm provides first order analytical solutions for a certain set of dissipative non-linear dynamical systems. It is based on a suitable modification of the Hori kernel in the double-dimensional embedding phase space, allowing the inclusion of the dissipative (or generalized) forces. By means of this redefined auxiliary system, the path-integrals of the method can be performed in a domain of the phase space with the same dimensionality as the original problem.

Using chaotic features of leakage current for monitoring dynamic behavior of surface discharges on an ice-covered insulator

Based on AC flashover tests on an ice-covered insulator, leakage current (LC) was measured to check the chaotic features through estimation of the largest Lyapunov exponent. Based on the embedding space theory and phase space reconstruction method, the LC attractor phase portrait and related correlation dimension were obtained in relation with the insulator surface performance. It was found that the largest Lyapunov exponents are all positive demonstrating that LC has chaotic features. The variation in chaos attractors can graphically indicate an increasing regularity in the surface discharges. The correlation dimension shows a decreasing tendency with the development of arcing discharges, which reveals a decreasing complexity in the surface discharges, especially just prior to the flashover occurrence. Therefore, chaotic features of LC are sensitive to dynamic discharge behavior on ice-covered insulators, which can be applied as a monitoring method for pre-warning flashover occurrence of outdoor insulators under icing conditions.

Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems

We investigate a generalised version of the recently proposed ordinal partition time series to network transformation algorithm. First, we introduce a fixed time lag for the elements of each partition that is selected using techniques from traditional time delay embedding. The resulting partitions define regions in the embedding phase space that are mapped to nodes in the network space. Edges are allocated between nodes based on temporal succession thus creating a Markov chain representation of the time series. We then apply this new transformation algorithm to time series generated by the Rössler system and find that periodic dynamics translate to ring structures whereas chaotic time series translate to band or tube-like structures-thereby indicating that our algorithm generates networks whose structure is sensitive to system dynamics. Furthermore, we demonstrate that simple network measures including the mean out degree and variance of out degrees can track changes in the dynamical behaviour in a manner comparable to the largest Lyapunov exponent. We also apply the same analysis to experimental time series generated by a diode resonator circuit and show that the network size, mean shortest path length, and network diameter are highly sensitive to the interior crisis captured in this particular data set.

A dynamic marker of very short-term heartbeat under pathological states via network analysis

We developed a novel network to probe pathological states with the aid of heartbeat time series, by regarding each vector in the embedding phase space as one node of the network and using a permutation-based measure to determine links between nodes. The entropy of the degree distribution of the network shows a general and significant reduction under pathological conditions, even when there are only ultra short-term heartbeats available. The reduction of is possibly a dynamic marker of cardiac disorders. Our results reveal that a comparatively strong “memory” should usually exist in the healthy cardiovascular system whereas it dramatically declines when a cardiac disease is arising. The proposed method shows great promise in screening cardiac diseases and monitoring dynamic changes of the autonomic nervous system. Besides, as a universal method for analyzing the time series, the proposed approach seems to be promising also for other research disciplines.