Symbolic Time Series Analysis Research Articles

State Identification Via Symbolic Time Series Analysis for Reinforcement Learning Control

Abstract This technical brief makes use of the concept of symbolic time-series analysis (STSA) for identifying discrete states from the nonlinear time response of a chaotic dynamical system for model-free reinforcement learning (RL) control. Along this line, a projection-based method is adopted to construct probabilistic finite state automata (PFSA) for identification of the current state (i.e., operational regime) of the Lorenz system; and a simple Q-map-based (and model-free) RL control strategy is formulated to reach the target state from the (identified) current state. A synergistic combination of PFSA-based state identification and RL control is demonstrated by the simulation of a numeric model of the Lorenz system, which yields very satisfactory performance to reach the target states from the current states in real-time.

Data Fusion and Pattern Classification in Dynamical Systems Via Symbolic Time Series Analysis

Abstract Symbolic time series analysis (STSA) plays an important role in the investigation of continuously evolving dynamical systems, where the capability to interpret the joint effects of multiple sensor signals is essential for adequate representation of the embedded knowledge. This technical brief develops and validates, by simulation, an STSA-based algorithm to make timely decisions on dynamical systems for information fusion and pattern classification from ensembles of multisensor time series data. In this context, one of the most commonly used methods has been neural networks (NN) in their various configurations; however, these NN-based methods may require large-volume data and prolonged computational time for training. An alternative feasible method is the STSA-based probabilistic finite state automata (PFSA), which has been shown in recent literature to require significantly less training data and to be much faster than NN for training and, to some extent, for testing. This technical brief reports a modification of the current PFSA methods to accommodate (possibly heterogeneous and not necessarily tightly synchronized) multisensor data fusion and (supervised learning-based) pattern classification in real-time. Efficacy of the proposed method is demonstrated by fusion of time series of position and velocity sensor data, generated from a simulation model of the forced Duffing equation.

Spontaneous Symmetry Breaking in Systems Obeying the Dynamics of On–Off Intermittency and Presenting Bimodal Amplitude Distributions

In this work, first, it is confirmed that a recently introduced symbolic time-series-analysis method based on the prime-numbers-based algorithm (PNA), referred to as the “PNA-based symbolic time-series analysis method” (PNA-STSM), can accurately determine the exponent of the distribution of waiting times in the symbolic dynamics of two symbols produced by the 3D Ising model in its critical state. After this numerical verification of the reliability of PNA-STSM, three examples of how PNA-STSM can be applied to the category of systems that obey the dynamics of the on–off intermittency are presented. Usually, such time series, with on–off intermittency, present bimodal amplitude distributions (i.e., with two lobes). As has recently been found, the phenomenon of on–off intermittency is associated with the spontaneous symmetry breaking (SSB) of the second-order phase transition. Thus, the revelation that a system is close to SSB supports a deeper understanding of its dynamics in terms of criticality, which is quite useful in applications such as the analysis of pre-earthquake fracture-induced electromagnetic emission (also known as fracture-induced electromagnetic radiation) (FEME/FEMR) signals. Beyond the case of on–off intermittency, PNA-STSM can provide credible results for the dynamics of any two-symbol symbolic dynamics, even in cases in which there is an imbalance in the probability of the appearance of the two respective symbols since the two symbols are not considered separately but, instead, simultaneously, considering the information from both branches of the symbolic dynamics.

A New Symbolic Time Series Analysis Method Based on Time-to-Space Mapping, through a Symmetric Magnetic Field, Quantized by Prime Numbers

This work presents a new analysis method for two-symbol symbolic time series based on the time-to-space mapping achieved through a device of current carrying circular rings. An algorithm based on the theory of prime numbers is proposed for the approximate estimation of the stratified magnetic field produced by the aforementioned device. The main property of the specific algorithm is that it quantizes the stratified magnetic field. If a two-symbol symbolic time series is used to determine the flow directions of the rings’ currents, a time-to-space mapping of the dynamics of the system producing the time series is observed. A unique “fingerprint” of the symbolic dynamics is shaped by the spatial allocation of the values of the six-valued symmetric quantized magnetic field produced by the device. This allows for the quantitative evaluation of the original system’s dynamics by analyzing the resultant quantized magnetic field values space allocation, in a spectrum ranging from the lack of dynamics (randomness) to the presence of dynamics at all scales (criticality). Two examples of application–corresponding to the extremes of the dynamics spectrum, specifically, for symbolic time series resulting from (a) a random numbers generator and (b) the spin alternation of 2D-Ising in its critical state, verify the reliable time-to-space mapping of the involved symbolic dynamics. Moreover, an application to the symbolic sequence produced by the DNA of the GAPDH (Glyceraldehyde-3-Phosphate Dehydrogenase) human gene is presented as a real-world, intermediate dynamics case. The proposed symbolic time series analysis method presents the advantage that can take into account information related to both symbols, which is particularly useful in analyzing two-symbol time series of relatively short length where the probabilities of occurrence of the two symbols are not equal. By inferring the universality class of an artificial-neural-network-based hybrid spin model through the value of the critical exponent δ, it is shown that for such time series, the proposed method provides a unique way to expose the real dynamics of the underlying complex system, in contrast to the analysis of waiting times in the time domain that leads to an ambiguous quantitative result.

Comparative analysis of time irreversibility and amplitude irreversibility based on joint permutation

Although time irreversibility (TIR) and amplitude irreversibility (AIR) are relevant concepts for nonequilibrium analysis, their association has received little attention. This paper conducts a systematic comparative analysis of the relationship between TIR and AIR based on statistical descriptions and numerical simulations. To simplify the quantification of TIR and AIR, the amplitude permutation and global information of the associated vector are combined to produce a joint probability estimation. Chaotic logistic, Henon, and Lorenz series are generated to evaluate TIR and AIR according to surrogate theory, and the distributions of joint permutations for these model series are measured to compare the degree of TIR and AIR. The joint permutation TIR and AIR are then used to investigate nonequilibrium pathological features in epileptic electroencephalography data. Test results suggest that epileptic brain electrical activities, particular those during the onset of seizure, manifest higher nonequilibrium characteristics. According to the statistical definitions and targeted pairs of joint permutations in the chaotic model data, TIR and AIR are fundamentally different nonequilibrium descriptors from time- and amplitude-reversibility, respectively, and thus require different forms of numerical analysis. At the same time, TIR and AIR both provide measures for fluctuation theorems associated with nonequilibrium processes, and have similar probabilistic differences in the pairs of joint permutations and consistent outcomes when used to analyze both the model series and real-world signals. Overall, comparative analysis of TIR and AIR contributes to our understanding of nonequilibrium features and broadens the scope of quantitative nonequilibrium measures. Additionally, the construction of joint permutations contributes to the development of symbolic time series analysis.

Numerical Demonstration of the Transmission of Low Frequency Fluctuation Dynamics Generated by a Semiconductor Laser with Optical Feedback

In this paper, the transmission mechanism of the spike information embedded in the low frequency fluctuation (LFF) dynamic in a cascaded laser system is numerically demonstrated. In the cascaded laser system, the LFF waveform is first generated by a drive laser with optical feedback and is then injected into a response laser. The range of crucial system parameters that can make the response laser generate the LFF dynamic is studied, and the effect of parameter mismatch on the transmission of LFF dynamics is explored through a method of symbolic time-series analysis and the index, such as the spike rate and the cross-correlation coefficient. The results show that the mismatch of the pump current has a more significant influence on the transmission of LFF waveforms than that of the internal physical parameter of the laser, such as the linewidth enhancement factor. Moreover, increasing the injection strength can enhance the robustness of LFF transmission. As spikes of the LFF dynamic generated by lasers with optical feedback is similar to the spike of neurons, the results of this paper can help understanding the information transporting and processing inside the photonic neurons.

A Spherical Phase Space Partitioning Based Symbolic Time Series Analysis (SPSP-STSA) for Emotion Recognition Using EEG Signals.

Emotion recognition systems have been of interest to researchers for a long time. Improvement of brain-computer interface systems currently makes EEG-based emotion recognition more attractive. These systems try to develop strategies that are capable of recognizing emotions automatically. There are many approaches due to different features extractions methods for analyzing the EEG signals. Still, Since the brain is supposed to be a nonlinear dynamic system, it seems a nonlinear dynamic analysis tool may yield more convenient results. A novel approach in Symbolic Time Series Analysis (STSA) for signal phase space partitioning and symbol sequence generating is introduced in this study. Symbolic sequences have been produced by means of spherical partitioning of phase space; then, they have been compared and classified based on the maximum value of a similarity index. Obtaining the automatic independent emotion recognition EEG-based system has always been discussed because of the subject-dependent content of emotion. Here we introduce a subject-independent protocol to solve the generalization problem. To prove our method’s effectiveness, we used the DEAP dataset, and we reached an accuracy of 98.44% for classifying happiness from sadness (two- emotion groups). It was 93.75% for three (happiness, sadness, and joy), 89.06% for four (happiness, sadness, joy, and terrible), and 85% for five emotional groups (happiness, sadness, joy, terrible and mellow). According to these results, it is evident that our subject-independent method is more accurate rather than many other methods in different studies. In addition, a subject-independent method has been proposed in this study, which is not considered in most of the studies in this field.

Point-kinetics neutron noise modeling and analysis via probabilistic finite state automata

This paper addresses two issues for safety & performance analysis of nuclear reactors; these issues are: (i) Modeling of the neutron noise process, which is constructed as stochastic differential equations by augmenting deterministic dynamic models of point kinetics, thermal hydraulics, and mechanical vibrations in the reactor core, and (ii) Time-series analysis of nuclear reactor internals for early-stage anomaly detection by making use of the neutron noise model. The model is used to generate an ensemble of time series of neutron noise under both normal and anomalous operating conditions, where an anomaly in the postulated probability distribution of neutron noise is caused by abnormal vibration of the fuel assembly. A symbolic time series analysis (STSA) tool, called probabilisitc finite state automata (PFSA), has been selected to detect the anomalies, if any. The PFSA method symbolizes a time-series and then encodes the resulting symbol string into a state transition probability matrix, which is both stochastic and ergodic by construction. The discrimination function for anomaly detection is built upon the residual error of the principal (left) eigenvector of the state transition probability matrix, which is also the state probability vector of the PFSA. The accuracy of the proposed analysis and its capability for online anomaly detection is demonstrated by simulation on a low-order dynamic model of a 2,500 MWt PWR of the Three-Mile-Island type.

A novel crude oil price trend prediction method: Machine learning classification algorithm based on multi-modal data features

Reliable forecasting of crude oil price has received a prodigious attention by both investment companies and governments. Motivated by this issue, this paper seeks to propose a new hybrid forecasting model for crude oil price trend prediction. For this purpose, the crude oil price series is initially decomposed by variational mode decomposition algorithm, and the multi-modal data features are extracted based on the decomposed modes. The volatility of crude oil prices is simultaneously converted into trend symbols through symbolic time series analysis. Machine learning multi-classifier are then trained with multi-modal data features and historical volatility as input and trend symbols as output. The well-trained models are used to predict the trend symbols of West Texas Intermediate crude oil future price. Empirical results demonstrate that the proposed hybrid forecasting model outperforms its counterparts. Among the classifiers used, the hybrid prediction model using support vector machine classifier exhibits superior predictive ability. The accuracy of the proposed model for predicting high volatility of crude oil prices is evidenced to be better than that of low volatility.

Impact of China‘s ban on Korean wave on the Korean stock market: Evidence from the entertainment sector

This study investigates the impact of China’s ban on the Korean wave (BKW) on domestic entertainment stock prices. The ban was triggered by the Korean government’s decision to deploy a terminal high-altitude area defence in Korea in 2016. We explore the statistical equilibrium of stock returns using the entropy applied with symbolic time series analysis. Moreover, we confirmed that the BKW harmed the pricing complexity of Korean entertainment stocks. Our finding can be understood as a monotonous stock price pattern because the market information was not reflected efficiently due to the decrease in transactions among investors concerned about the negative impact on the entertainment sector.

Transfer Learning for Detection of Combustion Instability Via Symbolic Time-Series Analysis

Abstract Transfer learning (TL) is a machine learning (ML) tool where the knowledge, acquired from a source domain, is “transferred” to perform a task in a target domain that has (to some extent) a similar setting. The underlying concept does not require the ML method to analyze a new problem from the beginning, and thereby both the learning time and the amount of required target-domain data are reduced for training. An example is the occurrence of thermoacoustic instability (TAI) in combustors, which may cause pressure oscillations, possibly leading to flame extinction as well as undesirable vibrations in the mechanical structures. In this situation, it is difficult to collect useful data from industrial combustion systems, due to the transient nature of TAI phenomena. A feasible solution is the usage of prototypes or emulators, like a Rijke tube, to produce largely similar phenomena. This paper proposes symbolic time-series analysis (STSA)-based TL, where the key idea is to develop a capability of discrimination between stable and unstable operations of a combustor, based on the time-series of pressure oscillations from a data source that contains sufficient information, even if it is not the target regime, and then transfer the learnt models to the target regime. The proposed STSA-based pattern classifier is trained on a previously validated numerical model of a Rijke-tube apparatus. The knowledge of this trained classifier is transferred to classify similar operational regimes in: (i) an experimental Rijke-tube apparatus and (ii) an experimental combustion system apparatus. Results of the proposed TL have been validated by comparison with those of two shallow neural networks (NNs)-based TL and another NN having an additional long short-term memory (LSTM) layer, which serve as benchmarks, in terms of classification accuracy and computational complexity.

Measure invariance of ergodic symbolic systems for low-delay detection of anomalous events

This paper investigates the spectral properties of ergodic measure of symbolic systems for applications to signal processing, pattern recognition, and anomaly detection in uncertain dynamical systems. The underlying algorithm is built upon the concept of ergodic sequences of measure-preserving transformations (MPT) on probability spaces, where non-stationary probabilistic finite state automata (PFSA) are constructed using short-length windows of time-series measurements. The resulting PFSA are non-homogeneous Markov, in which spectral properties of the MPT depend on several parameters that include the window length. The paper also develops an MPT-based metric to quantify the divergence of the evolving PFSA from that of the nominal PFSA; this information is then used for pattern classification and anomaly detection with low-delay tolerance. The MPT-based methodology has been validated with experimental data generated from a laboratory apparatus that deals with detection of thermoacoustic instabilities in combustion processes, which are known to have chaotic characteristics on a time scale of milliseconds. In this application, the concepts of MPT and ergodicity have been used to develop a novel symbolic time series analysis (STSA)-based detection method, whose performance is validated by comparison with two well-known techniques, namely, hidden Markov model (HMM) and cumulative sum (CUSUM), on the same experimental data. The results consistently show superior performance of the proposed MPT-based STSA.

Identification of Long-Term Behavior of Natural Circulation Loops: A Thresholdless Approach from an Initial Response

Natural circulation loop (NCL) systems are buoyancy-driven heat exchangers that are used in various industrial applications. The concept of passive heat exchange in NCL systems is attractive, because there is no need for an externally driven equipment (e.g., a pump) to maintain the fluid circulation. However, relying on buoyancy as the sole driving force may lead to several potential difficulties, one of which is generation of (possibly) time-varying nonlinearities in the dynamical system, where a difference in the time scales of heat transfer and fluid flow causes the flow to change from a steady-state regime to either an oscillatory regime or a flow-reversal regime, both of which are undesirable. In this paper, an algorithm is developed using tools of symbolic time-series analysis (e.g., probabilistic finite state automata (PFSA)) for the purpose of identifying selected regimes of operation in NCL systems using only data from the early transient operation, where the underlying principle is built upon the concept of pattern classification from measurements of fluid-flow dynamics. The proposed method is shown to be capable of identifying the current regime of operation from the initial time response under a given set of operational parameters. The efficacy of regime classification is demonstrated by testing on two datasets, generated from numerical simulation of a MATLAB SimuLink model that has previously been validated with experimental data. The results of the proposed PFSA-based classification are compared with those of a hidden Markov model (HMM) that serves as the baseline.

Data-driven Detection and Early Prediction of Thermoacoustic Instability in a Multi-nozzle Combustor

ABSTRACT Thermoacoustic instability (TAI) is a critical issue in modern lean-burn gas-turbine combustors, which is induced by a strong coupling between the resonant combustor acoustics and fluctuations in the heat release rate. This instability may lead to high-amplitude pressure waves that generate undesirable noise levels as well as fatigue stresses in mechanical structures of the combustor. The intense pressure fluctuations due to TAI may also cause large flow perturbations and possibly flow reversal that may lead to flame oscillations, flame liftoff, and even flame blow-out. Hence, there is a strong need for exercising control actions in a timely fashion to mitigate the TAI phenomena. Anomaly detection is an essential prerequisite to the design of a good controller and such a detector must be able to reliably predict a forthcoming TAI. To detect and predict the onset of a TAI from an ensemble of pressure time series, this paper investigates three data-driven methods: Fast Fourier transform (FFT), symbolic time series analysis (STSA), and hidden Markov modeling (HMM). The main focus of the paper is to make a comparative evaluation of these three anomaly detection methods for classification of the current regime of operation into stable and unstable categories as well as for real-time identification of precursors to impending instabilities with short-length time series of measured variables (e.g., pressure oscillations). The results, generated on experimental data from a multi-nozzle combustor apparatus, have been compared to evaluate the performance of FFT, STSA, and HMM methods for TAI analysis.

Flow reversal prediction of a single-phase square natural circulation loop using symbolic time series analysis

Flow reversal prediction of a single-phase square natural circulation loop using symbolic time series analysis

HMM conditional-likelihood based change detection with strict delay tolerance

Hidden Markov models (HMMs) have been widely used for anomaly and change point detection due to their representation power and computational efficiency in capturing statistical dependencies in time series. However, often information is integrated over relatively long observation windows, with detections made when the observed sequence’s likelihood under the (null) HMM deviates significantly from its typical range. Three related limitations are: i) use of long windows entails large decision delay, which may e.g. fail to prevent machine failure/damage; ii) typical approaches do not narrowly identify an interval within which the change point occurred. Such information could be useful e.g. for process control, where one wants to know how long it takes for control inputs to induce desired change points; iii) The decision statistic is usually the likelihood of the data in the current window, without consideration of past observations. This is suboptimal – this likelihood should be conditioned on past observations to optimally account for statistical dependency in the time series. In this paper, we propose a framework for change point detection which overcomes all of these limitations: i) it applies a standard HMM Forward recursion, but used to evaluate the likelihood of an observation subsequence conditioned on the subsequence’s entire past. This approach is used to efficiently evaluate the conditional likelihoods of all intervals of fixed length (hence with fixed delay, d), until a change point is first detected. Here d is a design parameter whose proper value (needed to have a quick response/mitigate damage) may be known for a given application domain; ii) the algorithm narrowly estimates the interval within which a detected change point lies; iii) we propose a novel performance criterion well-matched to low-delay, narrowly localized change point detection – the true detection interval rate (TDIR) – and also evaluate the false positive rate (FPR) and the bias and variance of the estimated change point, all as a function of d. The proposed method is shown to outperform a CUSUM algorithm, symbolic time series analysis (STSA) methods, and a standard HMM method (evaluating the unconditioned likelihood) for instability onset in combustion systems and fatigue failure initiation in a material.

Online Discovery and Classification of Operational Regimes From an Ensemble of Time Series Data

Abstract One of the pertinent problems in decision and control of dynamical systems is to identify the current operational regime of the physical process under consideration. To this end, there has been an upsurge in (data-driven) machine learning methods, such as symbolic time series analysis, hidden Markov modeling, and artificial neural networks, which often rely on some form of supervised learning based on preclassified data to construct the classifier. However, this approach may not be adequate for dynamical systems with a variety of operational regimes and possible anomalous/failure conditions. To address this issue, the technical brief proposes a methodology, built upon the concept of symbolic time series analysis, wherein the classifier learns to discover the patterns so that the algorithms can train themselves online while simultaneously functioning as a classifier. The efficacy of the methodology is demonstrated on time series of: (i) synthetic data from an unforced Van der Pol equation and (ii) pressure oscillation data from an experimental Rijke tube apparatus that emulates the thermoacoustics in real-life combustors where the process dynamics undergoes changes from the stable regime to an unstable regime and vice versa via transition to transient regimes. The underlying algorithms are capable of accurately learning and capturing the various regimes online in a (primarily) unsupervised manner.

Symbolic Time Series Analysis for Anomaly Detection in Measure-Invariant Ergodic Systems

Abstract This paper presents a novel framework of symbolic time series analysis (STSA) for anomaly detection in dynamical systems. The core concept is built upon a property of measure-preserving transformation (MPT) sequence, acting on a probability space with ergodic measure, that the eigenfunctions of these transformations would be time-invariant. As a result, unlike a standard STSA that is required to generate time-homogeneous Markov chains, the proposed MPT-based STSA is allowed to have time-inhomogeneous Markov chains, where the (possibly time-varying) state transition probability matrices have time-invariant eigenvectors. Such a time-invariance facilitates analysis of the dynamical system by using short-length time series of measurements. This is particularly important in applications, where the underlying dynamics and process anomalies need fast monitoring and control actions in order to mitigate any potential structural damage and/or to avoid catastrophic failures. The MPT-based STSA has been applied for low-delay detection of fatigue damage, which is a common source of failures in mechanical structures and which is known to have uncertain dynamical characteristics. The underlying algorithm has been validated with experimental data generated from a laboratory apparatus that uses ultrasonic sensors to detect fatigue damage in polycrystalline–alloy specimens. The performance of the proposed MPT-based STSA is evaluated by comparison with those of a standard STSA and a hidden Markov model (HMM) on the same experimental data. The results consistently show superior performance of the MPT-based STSA.

Predicting Onset of Combustion Instability in Gas Turbines Using Fiber Optic Sensors

A data driven method based on cross-entropy derived from symbolic time series analysis of the chemiluminescence and pressure variations has recently been found to be robust and computationally efficient among existing methods for predicting instability. In this article, we report the development and testing of compact and field-deployable optical fiber-based modules for sensing chemiluminescence and pressure variations in the combustor. The time-series data obtained from the above sensor modules are used to deduce the D-Markov cross-entropy for different protocols corresponding to typical combustor operation conditions. Such cross-entropy data is compared with similar data obtained from standard high speed camera and PZT-based pressure sensors. It is found that the fiber optic sensor modules compare favorably with the conventional sensors thereby providing an exciting new pathway for field-deployable solutions to monitor the onset of thermo-acoustic instability in combustors.

Comparing the dynamics of periodically forced lasers and neurons

Neuromorphic photonics is a new paradigm for ultra-fast neuro-inspired optical computing that can revolutionize information processing and artificial intelligence systems. To implement practical photonic neural networks is crucial to identify low-cost energy-efficient laser systems that can mimic neuronal activity. Here we study experimentally the spiking dynamics of a semiconductor laser with optical feedback under periodic modulation of the pump current, and compare with the dynamics of a neuron that is simulated with the stochastic FitzHugh–Nagumo model, with an applied periodic signal whose waveform is the same as that used to modulate the laser current. Sinusoidal and pulse-down waveforms are tested. We find that the laser response and the neuronal response to the periodic forcing, quantified in terms of the variation of the spike rate with the amplitude and with the frequency of the forcing signal, is qualitatively similar. We also compare the laser and neuron dynamics using symbolic time series analysis. The characterization of the statistical properties of the relative timing of the spikes in terms of ordinal patterns unveils similarities, and also some differences. Our results indicate that semiconductor lasers with optical feedback can be used as low-cost, energy-efficient photonic neurons, the building blocks of all-optical signal processing systems; however, the length of the external cavity prevents optical feedback on the chip.