Symbolic Sequence Analysis Research Articles

Multivariate Detrended Fluctuation Analysis of Symbolic Sequences

Sequences formed by symbols are found in diverse fields, including genome sequences, written texts and computer codes. An interesting question is whether a sequence of symbols contains correlated structures. Existing methods to characterize correlations require a numerical representation of the sequence. In this regard, mapping a sequence of text into a sequence of numerical values is a key step for assessing correlation analysis. This work proposes a methodology to study correlations in a sequence of symbols. In the first step, the sequence of symbols is mapped in a multivariate numerical sequence formed by unit vectors in a vectorial space. The main feature of such mapping is that symbols are equally weighted, thus avoiding the numerical overrepresentation of symbols. In the second step, a multivariate version of the detrended fluctuation analysis is used to quantify correlations in the numerical sequence. Genome sequences (first COVID-19), written English texts and comovements between Bitcoin and gold markets were used to illustrate the proposed methodology’s performance. The results showed that the balanced numerical mapping of symbolic sequences and the multivariate DFA provides valuable insights into the correlations in a sequence of symbols.

Towards Learning Discrete Representations via Self-Supervision for Wearables-Based Human Activity Recognition.

Human activity recognition (HAR) in wearable and ubiquitous computing typically involves translating sensor readings into feature representations, either derived through dedicated pre-processing procedures or integrated into end-to-end learning approaches. Independent of their origin, for the vast majority of contemporary HAR methods and applications, those feature representations are typically continuous in nature. That has not always been the case. In the early days of HAR, discretization approaches had been explored-primarily motivated by the desire to minimize computational requirements on HAR, but also with a view on applications beyond mere activity classification, such as, for example, activity discovery, fingerprinting, or large-scale search. Those traditional discretization approaches, however, suffer from substantial loss in precision and resolution in the resulting data representations with detrimental effects on downstream analysis tasks. Times have changed, and in this paper, we propose a return to discretized representations. We adopt and apply recent advancements in vector quantization (VQ) to wearables applications, which enables us to directly learn a mapping between short spans of sensor data and a codebook of vectors, where the index comprises the discrete representation, resulting in recognition performance that is at least on par with their contemporary, continuous counterparts-often surpassing them. Therefore, this work presents a proof of concept for demonstrating how effective discrete representations can be derived, enabling applications beyond mere activity classification but also opening up the field to advanced tools for the analysis of symbolic sequences, as they are known, for example, from domains such as natural language processing. Based on an extensive experimental evaluation of a suite of wearable-based benchmark HAR tasks, we demonstrate the potential of our learned discretization scheme and discuss how discretized sensor data analysis can lead to substantial changes in HAR.

Dynamical analysis of early afterdepolarization patterns in a biophysically detailed cardiac model.

Arrhythmogenic early afterdepolarizations (EADs) are investigated in a biophysically detailed mathematical model of a rabbit ventricular myocyte, providing their location in the parameter phase space and describing their dynamical mechanisms. Simulations using the Sato model, defined by 27 state variables and 177 parameters, are conducted to generate electrical action potentials (APs) for different values of the pacing cycle length and other parameters related to sodium and calcium concentrations. A detailed study of the different AP patterns with or without EADs is carried out, showing the presence of a high variety of temporal AP configurations with chaotic and quasiperiodic behaviors. Regions of bistability are identified and, importantly, linked to transitions between different behaviors. Using sweeping techniques, one-, two-, and three-parameter phase spaces are provided, allowing ascertainment of the role of the selected parameters as well as location of the transition regions. A Devil's staircase, with symbolic sequence analysis, is proposed to describe transitions in the ratio between the number of voltage (EAD and AP) peaks and the number of APs. To conclude, the obtained results are linked to recent studies for low-dimensional models and a conjecture is made for the internal dynamical structure of the transition region from non-EAD to EAD behavior using fold and cusp bifurcations and maximal canards.

Detection of Large-Scale Noisy Multi-Periodic Patterns with Discrete Double Fourier Transform

In many processes, the variations in underlying characteristics can be approximated by noisy multi-periodic patterns. If large-scale patterns are superimposed by a noise with long-range correlations, the detection of multi-periodic patterns becomes especially challenging. To solve this problem, we developed a discrete double Fourier transform (DDFT). DDFT is based on the equidistance property of harmonics generated by multi-periodic patterns in the discrete Fourier transform (DFT) spectra. As the large-scale patterns generate long enough equidistant series, they can be detected by the iteration of the primary DFT. DDFT is defined as Fourier transform of intensity spectral harmonics or of their functions. It comprises widely used cepstrum transform as a particular case. We present also the relevant analytical criteria for the assessment of the statistical significance of peak harmonics in DDFT spectra in the presence of noise. DDFT technique was tested by extensive numerical simulations. The practical applications of the DDFT technique are illustrated by the analysis of variations in solar wind speed related to solar rotation and by the study of large-scale multi-periodic patterns in DNA sequences. The latter application can be considered as a generic example for the general spectral analysis of symbolic sequences. The results are compared with those obtained by the cepstrum transform. The mutual combination of DFT and DDFT provides an efficient technique to search for noisy large-scale multi-periodic patterns.

Generalization of entropy based divergence measures for symbolic sequence analysis.

Entropy based measures have been frequently used in symbolic sequence analysis. A symmetrized and smoothed form of Kullback-Leibler divergence or relative entropy, the Jensen-Shannon divergence (JSD), is of particular interest because of its sharing properties with families of other divergence measures and its interpretability in different domains including statistical physics, information theory and mathematical statistics. The uniqueness and versatility of this measure arise because of a number of attributes including generalization to any number of probability distributions and association of weights to the distributions. Furthermore, its entropic formulation allows its generalization in different statistical frameworks, such as, non-extensive Tsallis statistics and higher order Markovian statistics. We revisit these generalizations and propose a new generalization of JSD in the integrated Tsallis and Markovian statistical framework. We show that this generalization can be interpreted in terms of mutual information. We also investigate the performance of different JSD generalizations in deconstructing chimeric DNA sequences assembled from bacterial genomes including that of E. coli, S. enterica typhi, Y. pestis and H. influenzae. Our results show that the JSD generalizations bring in more pronounced improvements when the sequences being compared are from phylogenetically proximal organisms, which are often difficult to distinguish because of their compositional similarity. While small but noticeable improvements were observed with the Tsallis statistical JSD generalization, relatively large improvements were observed with the Markovian generalization. In contrast, the proposed Tsallis-Markovian generalization yielded more pronounced improvements relative to the Tsallis and Markovian generalizations, specifically when the sequences being compared arose from phylogenetically proximal organisms.

Assessing randomness and complexity in human motion trajectories through analysis of symbolic sequences.

Complexity is a hallmark of intelligent behavior consisting both of regular patterns and random variation. To quantitatively assess the complexity and randomness of human motion, we designed a motor task in which we translated subjects' motion trajectories into strings of symbol sequences. In the first part of the experiment participants were asked to perform self-paced movements to create repetitive patterns, copy pre-specified letter sequences, and generate random movements. To investigate whether the degree of randomness can be manipulated, in the second part of the experiment participants were asked to perform unpredictable movements in the context of a pursuit game, where they received feedback from an online Bayesian predictor guessing their next move. We analyzed symbol sequences representing subjects' motion trajectories with five common complexity measures: predictability, compressibility, approximate entropy, Lempel-Ziv complexity, as well as effective measure complexity. We found that subjects' self-created patterns were the most complex, followed by drawing movements of letters and self-paced random motion. We also found that participants could change the randomness of their behavior depending on context and feedback. Our results suggest that humans can adjust both complexity and regularity in different movement types and contexts and that this can be assessed with information-theoretic measures of the symbolic sequences generated from movement trajectories.

Dynamic analysis of recurrent phase patterns in spontaneous human EEG

Spontaneous human EEG during the resting state is modulating with long lasting high amplitude synchronization and very brief low amplitude de-synchronization states [1]. For the dynamic characterization of modulating alpha rhythm in the resting state EEG, we introduce a method of spontaneous-event related potential (SERP) analysis. Our method consists of three steps: At first, ensemble phase patterns of alpha rhythm at the moment of alpha de-synchronization state are classified using the K-mean clustering algorithm; Secondly, short time evolution of the phase pattern around each de-synchronization event is analyzed using the symbolic sequence dynamics; Finally, a global map of dynamic organization is constructed by integrating the temporal motifs representing the recurrent phase patterns around de-synchronization state. Using the EEG data from seven subjects, very large number of de-synchronization event is collected from spike-like events in the time plot of inverse of alpha amplitude. The classification of the phase patterns of the de-synchronization state produces four different kinds of traveling waves, two propagating from posterior to anterior (PAL and PAR) and two in reverse directions (APL and APR) for C = 8 classification. The presence of two spiral waves, one rotating in clockwise (CS) and the other in count clockwise (CCS), are also observed in addition to two standing waves (STA and STP). For the symbolic sequence analysis, we construct a triad symbol for each de-synchronization event. It is composed of a present pattern and its previous and next patterns, as a sequence of pre-present-post patterns. Then the occurrence rates of all possible triad symbols are compared with those of surrogate data where the sequence of all phase patterns is completely randomized. The triad symbols with very large normalized Z-score could be identified as dominant temporal motifs [2], which include the triad symbols, STP-CCS-STA, STP-CS-STA, CCS-STA-APL, and so on. We could also identify temporal anti-motifs as the triad symbols with very large negative Z-score. The anti-motifs include the triad symbols where strongly inhibited transitions like the transitions between CS and CCS, between STP and two PA patterns, and between SPA and two AP patterns are included. Integrating the information on the temporal motifs and anti-motifs, we could construct a global map of recurrent transition dynamic during the resting state. The global map contains the information on how the transitions among four traveling waves PAL, PAR, APL and APR occur. It is very interesting to notice the role played by the two spiral waves CS and CCS. As the motion of a spiral wave is recorded by tracing the phase singular point of a spiral wave, we observed that a traveling wave could switch its propagation as the spiral wave crosses the traveling wave in a transverse direction. This result indicates that the role of two spiral waves is to switch the propagation of traveling waves systematically. In conclusion, using the SERP analysis of the spontaneous human EEG, we identified the recurrent phase patterns that involve the switching of traveling waves. Very interestingly, it is shown that the propagation of a traveling wave is systematically controlled by a spiral wave which drifts across the traveling wave. It is to be investigated on the function roles of the traveling waves and spiral waves, and also on the neural mechanisms of switching the propagation of the traveling waves [3] and the formation of spiral waves [4].

A Representative Set Method for Symbolic Sequence Clustering

Sequence decomposition into a set of consecutive, distinct subsequences is crucial for symbolic sequence analysis. It reduces significantly the reference base of the recorded sequence for further retrieval and allows for original similarity and membership measures of the sequences. The introduced measures are a start point to a new algorithm for clustering sequences into groups of similar individuals. Algorithms that use the concept of a representative set achieved relatively good clustering results. The representative set that we have introduced is precisely and uniquely defined in contrast to that used in other applications.

Study of the distribution of microvessels in normal and pathologic prostate using an information-based similarity analysis

Categorizing biologic signals by analysis of symbolic sequences was employed in the study of prostate microvessels. The estimates of the volume fraction of the vessels immunostained to Factor-VIII was mapped to binary sequences. The distance between sequences was estimated by comparing the rank and frequency of repetitive elements. These measurements were applied to detect whether there are unique microvascular patterns for each individual, and to search for patterns describing prostate microvessels of different conditions. Normal prostate, benign prostate hyperplasia and prostate carcinoma groups were studied. All the specimens were immunostained to F-VIII and strips formed by adjacent quadrats were explored. At each point of the long axis of the strip, the V(V) F-VIII was calculated. These values were processed with the information-based similarity software to estimate the dissimilarity between two space series. The following comparisons were carried out: intrasubject versus intragroup distances; intragroup distances among the groups studied and intergroup distances. The distance defined between a vessels immunostained to Factor-VIII space series and its randomized surrogate was considered as an index of the nonrandomness of the space series. These indices were compared for all the groups. We conclude that (a) The information-based similarity analysis can be adapted to vessels immunostained to Factor-VIII space series from prostate microvessels. (b) There are no unique microvascular patterns associated with each individual. (c) There are characteristic patterns describing the microvessels from normal prostate, benign prostate hyperplasia and carcinoma. (d) This method is able to account for the differences between prostate cancer and both normal and benign prostate conditions, with respect to the microvessel patterns.

SYMBOLIC SEQUENCE ANALYSIS METHOD OF GAS/LIQUID TWO-PHASE FLOW MEASUREMENT FLUCTUATION SIGNALS

Symbolic sequence analysis method was applied to analyze two-phase flow measurement fluctuating signals and some issues about key parameters which were effect on the symbolic statistical quantities were discussed. Based on the above study, eighty groups differential pressure fluctuating signals of gas/liquid two-phase flow were gathered in vertical upward pipes and we extracted the time irreversibility and chi-square statistics. When the gas superficial velocity is less than 0.02 m/s, the random variable motion feature and the dynamic character of bubble flow become gradually complex with the gas superficial velocity increasing. When gas superficial velocity exceeds 0.02 m/s, the dynamic character of flow pattern evolution becomes gradually simple from bubble flow to slug flow and the dynamic character becomes more and more complex from slug flow to churn flow with increasing gas superficial velocity. This study shows that time irreversibility and chi-square statistics are sensitive feature quantities of characterizing gas/liquid two-phase flow patterns and this can better help for understanding dynamic characteristics of gas/liquid two-phase flow patterns by observing the two statistical quantities changing with the two-phase flow parameters.

Symbolic sequence analysis using approximated partition

A particular attention has been paid to characterizing the dynamical behavior using symbolic sequences when there is no topological criterion to define the partition. In that case, a so-called threshold crossings technique is used. It is shown that maximizing the number of realized sequences provides a partition closer to the topological partition than using a partition leading to equiprobable symbols. After numerical evidences on benchmark maps, an experimental time series from a copper electrodissolution is used to check the applicability of this technique.