Ordinal Pattern Analysis Research Articles

Complexity of couplings in multivariate time series via ordinal persistent homology.

We propose a new measure of the complexity of couplings in multivariate time series by combining the techniques of ordinal pattern analysis and topological data analysis. We construct an increasing sequence of simplicial complexes encoding the information about couplings among the components of a given multivariate time series through the intersection of ordinal patterns. The complexity measure is then defined by making use of the persistent homology groups. We validate the complexity measure both theoretically and numerically.

Machining process condition monitoring based on ordinal pattern analysis and image matching

Machining process condition monitoring based on ordinal pattern analysis and image matching

Ordinal Pattern Analysis: A Tutorial on Assessing the Fit of Hypotheses to Individual Repeated Measures Data.

This article provides a tutorial introduction to ordinal pattern analysis, a statistical analysis method designed to quantify the extent to which hypotheses of relative change across experimental conditions match observed data at the level of individuals. This method may be a useful addition to familiar parametric statistical methods including repeated measures analysis of variance and generalized linear mixed-effects models, particularly when analyzing inherently individual characteristics, such as perceptual processes, and where experimental effects are usefully modeled in relative rather than absolute terms. Three analyses of increasing complexity are demonstrated using ordinal pattern analysis. An initial analysis of a very small data set is designed to explicate the simple mathematical calculations that make up ordinal pattern analysis, which can be performed without the aid of a computer. Analyses of slightly larger data sets are used to demonstrate familiar concepts, including comparison of competing hypotheses, handling missing data, group comparisons, and pairwise tests. All analyses can be reproduced using provided code and data. Ordinal pattern analysis results are presented, along with an analogous linear mixed-effects analysis, to illustrate the similarities and differences in information provided by ordinal pattern analysis in comparison to familiar parametric methods. Although ordinal pattern analysis does not produce familiar numerical effect sizes, it does provide highly interpretable results in terms of the proportion of individuals whose results are consistent with a hypothesis, along with individual and group-level statistics, which quantify hypothesis performance.

A combinatorial view of stochastic processes: White noise.

White noise is a fundamental and fairly well understood stochastic process that conforms to the conceptual basis for many other processes, as well as for the modeling of time series. Here, we push a fresh perspective toward white noise that, grounded on combinatorial considerations, contributes to giving new interesting insights both for modeling and theoretical purposes. To this aim, we incorporate the ordinal pattern analysis approach, which allows us to abstract a time series as a sequence of patterns and their associated permutations, and introduce a simple functional over permutations that partitions them into classes encoding their level of asymmetry. We compute the exact probability mass function (p.m.f.) of this functional over the symmetric group of degree n, thus providing the description for the case of an infinite white noise realization. This p.m.f. can be conveniently approximated by a continuous probability density from an exponential family, the Gaussian, hence providing natural sufficient statistics that render a convenient and simple statistical analysis through ordinal patterns. Such analysis is exemplified on experimental data for the spatial increments from tracks of gold nanoparticles in 3D diffusion.

Performance Evaluation of New Feature based on Ordinal Pattern Analysis for Iris Biometric Recognition

The Iris recognition technique is currently the most efficient biometric identification system and is a common system on the practical front. Though most of the commercial systems use the patented Daugman’s algorithm, which mainly uses wavelet-based features, research is still active in identifying novel features that can provide personal identification. Here the first novel proposal of using ordinal pattern measure based on nonlinear time series analysis is put forth to characterize the unique pattern of the iris of individuals and thereby perform personal identification. Dispersion Entropy is a nonlinear time-series analysis method highly efficient in the characterization of the complexity of any data series with proven effectiveness in the characterization of model system dynamics as well as real-world data series. The results show that dispersion entropy can be used to identify iris images of specific individuals. The efficiency of this method is evaluated by computing correlation and RMSE between dispersion entropy values of normalized iris image rubber sheet data. The experimental results on the popular IRIS database- CASIA v1- demonstrate that the proposed method can effectively perform differential identification of iris images from different individuals. The results specifically indicate that the density of information along the angular direction of iris images which falls along the rows of rubber sheet data. This can be efficiently utilized with the method or ordinal pattern characterization and proves to be having promising potential for being incorporated into biometrics personal identification systems.

The Effect of Register, Direction, and Magnitude on Musicians’ Evaluations of Chamber Ensemble Intonation: A Within-Study Comparison for Analysis of Repeated Measures

This study had two primary purposes: (1) to investigate the effect of register, direction, and magnitude on musicians’ evaluation of chamber ensemble intonation, and (2) determine whether a novel nonparametric technique, ordinal pattern analysis (OPA), was a viable alternative to repeated-measures analysis of variance (rANOVA). I digitally mastered a recording of a string quartet performing a phrase from Capriol Suite by altering the intonation of the violin or cello voice ±20 and 30 cents sharp and flat. Participants ( N = 72) completed a discrimination task and an evaluation task with the recordings, with task order, and within-task item order presented in a random order. Analysis using rANOVA revealed significant differences due to register, direction, and magnitude: Excerpts with cello errors were rated as more in tune than excerpts with violin errors; excerpts with flat errors were rated as more in tune than excerpts with sharp errors, and excerpts with 20-cent magnitude errors were rated as more in tune than excerpts with 30-cent magnitude errors. OPA results were consistent with rANOVA results. Substantive implications for music teaching and learning are discussed alongside methodological considerations and implications for music education research using repeated-measures designs.

A new dissimilarity measure based on ordinal pattern for analyzing physiological signals

Complex physiological signals carry critical information about the underlying dynamics of the complex system that produced them. By analyzing and exploring these signals, we can identify which state the system is in and understand the underlying dynamics of that system. Numerous linear and nonlinear methods have been developed for analyzing physiological signals, but they either rely on certain assumptions or are easily affected by various factors, which limits their applications. In this paper, we propose a novel dissimilarity measure between two signals, which is based on the ordinal pattern representation of signals. Therefore, it naturally inherits the advantages of ordinal pattern analysis, namely simplicity, robustness, and low complexity in computation without further prior assumptions. Finally, we apply this new measure to analyze several physiological signals derived from different physiological and pathological conditions to assess the effectiveness of the proposed measure. Experimental results from heart rate analysis, EEG signals analysis of healthy subjects and patients with epilepsy, and analysis of EEG signals at different sleep stages in healthy subjects demonstrate that the new measure is capable of distinguishing signals from different conditions and is therefore a useful tool for analyzing various physiological signals.

Extracting Robust Biomarkers From Multichannel EEG Time Series Using Nonlinear Dimensionality Reduction Applied to Ordinal Pattern Statistics and Spectral Quantities

In this study, ordinal pattern analysis and classical frequency-based EEG analysis methods are used to differentiate between EEGs of different age groups as well as individuals. As characteristic features, functional connectivity as well as single-channel measures in both the time and frequency domain are considered. We compare the separation power of each feature set after nonlinear dimensionality reduction using t-distributed stochastic neighbor embedding and demonstrate that ordinal pattern-based measures yield results comparable to frequency-based measures applied to preprocessed data, and outperform them if applied to raw data. Our analysis yields no significant differences in performance between single-channel features and functional connectivity features regarding the question of age group separation.

Beyond p values: utilizing multiple methods to evaluate evidence

Null hypothesis significance testing is cited as a threat to validity and reproducibility. While many individuals suggest that we focus on altering the p value at which we deem an effect significant, we believe this suggestion is short-sighted. Alternative procedures (i.e., Bayesian analyses and observation-oriented modeling: OOM) can be more powerful and meaningful to our discipline. However, these methodologies are less frequently utilized and are rarely discussed in combination with NHST. Herein, we discuss three methodologies (NHST, Bayesian Model comparison, and OOM), then compare the possible interpretations of three analyses (ANOVA, Bayes Factor, and an Ordinal Pattern Analysis) in various data environments using a frequentist simulation study. We found that changing significance thresholds had little effect on conclusions. Furthermore, we suggest that evaluating multiple estimates as evidence of an effect allows for more robust and nuanced interpretations of results and implies the need to redefine evidentiary value and reporting practices.

Correlations Preceding High-Intensity Events in the Chaotic Dynamics of a Raman Fiber Laser.

We study the time series of the output intensity of a Raman fiber laser with an ordinal patterns analysis in the laminar-turbulent transition. We look for signatures among consecutive events that indicate when the system changes from triggering low-intensity to high-intensity events. We set two thresholds, a low one and a high one, to distinguish between low intensity versus high-intensity events. We find that when the time series is performing low-intensity events (below the low threshold), it shows some preferred temporal patterns before triggering high-intensity events (above a high threshold). The preferred temporal patterns remain the same all through the pump current range studied, even though two clearly different dynamical regimes are covered (laminar regime for low pump currents and turbulent regime for high pump currents). We also find that the turbulent regime shows clearer signatures of determinism than the laminar regime.

Fusion of Motif- and Spectrum-Related Features for Improved EEG-Based Emotion Recognition.

Emotion recognition is a burgeoning field allowing for more natural human-machine interactions and interfaces. Electroencephalography (EEG) has shown to be a useful modality with which user emotional states can be measured and monitored, particularly primitives such as valence and arousal. In this paper, we propose the use of ordinal pattern analysis, also called motifs, for improved EEG-based emotion recognition. Motifs capture recurring structures in time series and are inherently robust to noise, thus are well suited for the task at hand. Several connectivity, asymmetry, and graph-theoretic features are proposed and extracted from the motifs to be used for affective state recognition. Experiments with a widely used public database are conducted, and results show the proposed features outperforming benchmark spectrum-based features, as well as other more recent nonmotif-based graph-theoretic features and amplitude modulation-based connectivity/asymmetry measures. Feature and score-level fusion suggest complementarity between the proposed and benchmark spectrum-based measures. When combined, the fused models can provide up to 9% improvement relative to benchmark features alone and up to 16% to nonmotif-based graph-theoretic features.

Detection of time reversibility in time series by ordinal patterns analysis.

Time irreversibility is a common signature of nonlinear processes and a fundamental property of non-equilibrium systems driven by non-conservative forces. A time series is said to be reversible if its statistical properties are invariant regardless of the direction of time. Here, we propose the Time Reversibility from Ordinal Patterns method (TiROP) to assess time-reversibility from an observed finite time series. TiROP captures the information of scalar observations in time forward as well as its time-reversed counterpart by means of ordinal patterns. The method compares both underlying information contents by quantifying its (dis)-similarity via the Jensen-Shannon divergence. The statistic is contrasted with a population of divergences coming from a set of surrogates to unveil the temporal nature and its involved time scales. We tested TiROP in different synthetic and real, linear, and non-linear time series, juxtaposed with results from the classical Ramsey's time reversibility test. Our results depict a novel, fast-computation, and fully data-driven methodology to assess time-reversibility with no further assumptions over data. This approach adds new insights into the current non-linear analysis techniques and also could shed light on determining new physiological biomarkers of high reliability and computational efficiency.

Forecasting Events in the Complex Dynamics of a Semiconductor Laser with Optical Feedback

Complex systems performing spiking dynamics are widespread in Nature. They cover from earthquakes, to neurons, variable stars, social networks, or stock markets. Understanding and characterizing their dynamics is relevant in order to detect transitions, or to predict unwanted extreme events. Here we study, under an ordinal patterns analysis, the output intensity of a semiconductor laser with feedback in a regime where it develops a complex spiking behavior. We unveil that, in the transitions towards and from the spiking regime, the complex dynamics presents two competing behaviors that can be distinguished with a thresholding method. Then we use time and intensity correlations to forecast different types of events, and transitions in the dynamics of the system.

Topographic Reconfiguration of Local and Shared Information in Anesthetic-Induced Unconsciousness.

Theoretical consideration predicts that the alteration of local and shared information in the brain is a key element in the mechanism of anesthetic-induced unconsciousness. Ordinal pattern analysis, such as permutation entropy (PE) and symbolic mutual information (SMI), have been successful in quantifying local and shared information in neurophysiological data; however, they have been rarely applied to altered states of consciousness, especially to data obtained with functional magnetic resonance imaging (fMRI). PE and SMI analysis, together with the superb spatial resolution of fMRI recording, enables us to explore the local information of specific brain areas, the shared information between the areas, and the relationship between the two. Given the spatially divergent action of anesthetics on regional brain activity, we hypothesized that anesthesia would differentially influence entropy (PE) and shared information (SMI) across various brain areas, which may represent fundamental, mechanistic indicators of loss of consciousness. FMRI data were collected from 15 healthy participants during four states: wakefulness (W), light (conscious) sedation (L), deep (unconscious) sedation (D), and recovery (R). Sedation was produced by the common, clinically used anesthetic, propofol. Firstly, we found that that global PE decreased from W to D, and increased from D to R. The PE was differentially affected across the brain areas; specifically, the PE in the subcortical network was reduced more than in the cortical networks. Secondly, SMI was also differentially affected in different areas, as revealed by the reconfiguration of its spatial pattern (topographic structure). The topographic structures of SMI in the conscious states W, L, and R were distinctively different from that of the unconscious state D. Thirdly, PE and SMI were positively correlated in W, L, and R, whereas this correlation was disrupted in D. And lastly, PE changes occurred preferentially in highly connected hub regions. These findings advance our understanding of brain dynamics and information exchange, emphasizing the importance of topographic structure and the relationship of local and shared information in anesthetic-induced unconsciousness.

Characterizing Complex Dynamics in the Classical and Semi-Classical Duffing Oscillator Using Ordinal Patterns Analysis.

The driven double-well Duffing oscillator is a well-studied system that manifests a wide variety of dynamics, from periodic behavior to chaos, and describes a diverse array of physical systems. It has been shown to be relevant in understanding chaos in the classical to quantum transition. Here we explore the complexity of its dynamics in the classical and semi-classical regimes, using the technique of ordinal pattern analysis. This is of particular relevance to potential experiments in the semi-classical regime. We unveil different dynamical regimes within the chaotic range, which cannot be detected with more traditional statistical tools. These regimes are characterized by different hierarchies and probabilities of the ordinal patterns. Correlation between the Lyapunov exponent and the permutation entropy is revealed that leads us to interpret dips in the Lyapunov exponent as transitions in the dynamics of the system.

Identifying the scale-dependent motifs in atmospheric surface layer by ordinal pattern analysis

Ramp-like structures in various atmospheric surface layer time series have been long studied, but the presence of motifs with the finer scale embedded within larger scale ramp-like structures has largely been overlooked in the reported literature. Here a novel, objective and well-adapted methodology, the ordinal pattern analysis, is adopted to study the finer-scaled motifs in atmospheric boundary-layer (ABL) time series. The studies show that the motifs represented by different ordinal patterns take clustering properties and 6 dominated motifs out of the whole 24 motifs account for about 45% of the time series under particular scales, which indicates the higher contribution of motifs with the finer scale to the series. Further studies indicate that motif statistics are similar for both stable conditions and unstable conditions at larger scales, but large discrepancies are found at smaller scales, and the frequencies of motifs “1234” and/or “4321” are a bit higher under stable conditions than unstable conditions. Under stable conditions, there are great changes for the occurrence frequencies of motifs “1234” and “4321”, where the occurrence frequencies of motif “1234” decrease from nearly 24% to 4.5% with the scale factor increasing, and the occurrence frequencies of motif “4321” change nonlinearly with the scale increasing. These great differences of dominated motifs change with scale can be taken as an indicator to quantify the flow structure changes under different stability conditions, and motif entropy can be defined just by only 6 dominated motifs to quantify this time-scale independent property of the motifs. All these results suggest that the defined scale of motifs with the finer scale should be carefully taken into consideration in the interpretation of turbulence coherent structures.

Ordinal Pattern Analysis in Comparative Psychology: A Flexible Alternative to Null Hypothesis Significance Testing Using an Observation Oriented Modeling Paradigm

The data of comparative psychology generally differ from the majority of data collected within mainstream psychology in several key respects – most notably in the diversity of forms of measurement and fewer number of subjects. We believe null hypothesis significance testing may not be the most appropriate method of analysis for comparative psychology for these reasons. Comparative psychology has a rich history of performing several analyses on a few subjects due to a philosophical interest in individual subject behavior, along with group assessments. Since first being published in 2011, Observation Oriented Modeling has successfully been used to analyze individual subjects’ responses from honey bees, horses, humans, and rattlesnakes. Observation Oriented Modeling is highly flexible and has allowed comparative researchers to perform a variety of assessments comparable to null hypothesis significance testing’s T-Tests, One-way ANOVA, and Repeated-Measures ANOVA while producing easily-interpretable and, most importantly, relevant results. This paper describes the diverse manners in which comparative psychologists can assess individual and group performances without concerns of statistical assumptions and limitations that complicate assessments when employing Null Hypothesis Significance Testing.

Ordinal Pattern Analysis Method Applied in a P300-based Brain Computer Interface

Ordinal Pattern analysis has been used recently for extracting qualitative information from non-linear time series and it has been applied to usefully track brain dynamics. In this paper, we proposed a novel P300-based BCI system which depends on ordinal time series analysis as a feature extraction method. We have shown that this method can efficiently revel P300 feature, and therefore good classification accuracies and bitrates have been achieved for healthy and disabled subjects.

Efficient detection of the quasi-periodic route to chaos in discrete maps by the three-state test

The three-state test (3ST) is a method based on ordinal pattern analysis for detecting chaos and determining the period in time series. For some well-known chaotic dynamical systems, we showed that the test behaves similar to Lyapunov exponents. However, the 3ST is detecting quasi-periodic motions both as regular and non-regular. In this paper, we propose to use the sensitivity of its chaos indicator $$\lambda $$ on time delay for clear discernment between quasi-periodic and chaotic dynamics. Simulation results obtained using the logistic map and the sine-circle map attest that the sensitivity of $$\lambda $$ on time delay is sufficient for the detection of the periodic and quasi-periodic route to chaos. A comparison with the permutation entropy confirms the effectiveness of the 3ST for the analysis of discrete time series data.

Ordinal pattern and statistical complexity analysis of daily stream flow time series

When calculating the Bandt and Pompe ordinal pattern distribution from given time series at depth D, some of the D! patterns might not appear. This could be a pure finite size effect (missing patterns) or due to dynamical properties of the observed system (forbidden patterns). For pure noise, no forbidden patterns occur, contrary to deterministic chaotic maps. We investigate long time series of river runoff for missing patterns and calculate two global properties of their pattern distributions: the Permutation Entropy and the Permutation Statistical Complexity. This is compared to purely stochastic but long-range correlated processes, the k-noise (noise with power spectrum f −k ), where k is a parameter determining the strength of the correlations. Although these processes closely resemble runoff series in their correlation behavior, the ordinal pattern statistics reveals qualitative differences, which can be phrased in terms of missing patterns behavior or the temporal asymmetry of the observed series. For the latter, an index is developed in the paper, which may be used to quantify the asymmetry of natural processes as opposed to artificially generated data.