Nonlinear Time Series Analysis Research Articles

Deciphering air quality index through sample entropy: A nonlinear time series analysis

Pollution and its impacts on human health have become a crisis in regions with poor air quality index (AQI), which is an indicator of concentrations of pollutants, prompting the United Nations (UN) to set sustainable development goals (SDG). The present study proposes a surrogate sample entropy-based method in tune with UN’s SDG, to assess AQI from the time series of any of the pollutants. New Delhi is one of the world's most polluted state capital, with a higher level of particulate matter (PM). The temporal data of the pollutants in New Delhi, recorded in the one-hour interval during the years 2016 and 2017, are subjected to time series analysis. The data collected from the Central Pollution Control Board of India are analyzed with special reference to PM and compared with the World Air Quality Report 2021 and University of Washington data. The dependence of PM2.5 concentration on humidity and rain is also studied. The study reveals the increase in complexity with the concentration of pollutants through the phase portrait. The sample entropy analysis of the nonlinear time series of the pollutants exhibits a linear relation with AQI suggesting the possibility of using sample entropy as a surrogate measure of AQI.

Characterizing the Solar Cycle Variability Using Nonlinear Time Series Analysis at Different Amounts of Dynamo Supercriticality: Solar Dynamo is Not Highly Supercritical

The solar dynamo is essentially a cyclic process in which the toroidal component of the magnetic field is converted into the poloidal one and vice versa. This cyclic loop is disturbed by some nonlinear and stochastic processes mainly operating in the toroidal to poloidal part. Hence, the memory of the polar field decreases in every cycle. On the other hand, the dynamo efficiency and, thus, the supercriticality of the dynamo decreases with the Sun’s age. Previous studies have shown that the memory of the polar magnetic field decreases with the increase of supercriticality of the dynamo. In this study, we employ popular techniques of time series analysis, namely, compute Higuchi’s fractal dimension, Hurst exponent, and Multi-Fractal Detrended Fluctuation Analysis to the amplitude of the solar magnetic cycle obtained from dynamo models operating at near-critical and supercritical regimes. We show that the magnetic field in the near-critical regime is governed by strong memory, less stochasticity, intermittency, and breakdown of self-similarity. On the contrary, the magnetic field in the supercritical region has less memory, strong stochasticity, and shows a good amount of self-similarity. Finally, applying the same time series analysis techniques in the reconstructed sunspot data of 85 cycles and comparing their results with that from models, we conclude that the solar dynamo is possibly operating near the critical regime and not too much supercritical regime. Thus the Sun may not be too far from the critical dynamo transition.

Machine learning approach to detect dynamical states from recurrence measures.

We integrate machine learning approaches with nonlinear time series analysis, specifically utilizing recurrence measures to classify various dynamical states emerging from time series. We implement three machine learning algorithms: Logistic Regression, Random Forest, and Support Vector Machine for this study. The input features are derived from the recurrence quantification of nonlinear time series and characteristic measures of the corresponding recurrence networks. For training and testing, we generate synthetic data from standard nonlinear dynamical systems and evaluate the efficiency and performance of the machine learning algorithms in classifying time series into periodic, chaotic, hyperchaotic, or noisy categories. Additionally, we explore the significance of input features in the classification scheme and find that the features quantifying the density of recurrence points are the most relevant. Furthermore, we illustrate how the trained algorithms can successfully predict the dynamical states of two variable stars, SX Her and AC Her, from the data of their light curves. We also indicate how the algorithms can be trained to classify data from discrete systems.

Uncertain nonlinear time series analysis with applications to motion analysis and epidemic spreading

Uncertain nonlinear time series analysis with applications to motion analysis and epidemic spreading

Spatio-temporal variations of climate along possible African-Arabian routes of H. sapiens expansion

Eastern Africa and Arabia were major hominin hotspots and critical crossroads for migrating towards Asia during the late Pleistocene. To decipher the role of spatiotemporal environmental change on human occupation and migration patterns, we remeasured the marine core from Meteor Site KL 15 in the Gulf of Aden and reanalyzed its data together with the aridity index from ICDP Site Chew Bahir in eastern Africa and the wet-dry index from ODP Site 967 in the eastern Mediterranean Sea using linear and nonlinear time series analysis. These analyses show major changes in the spatiotemporal paleoclimate dynamics at 400 and 150 ka BP (thousand years before 1950), presumably driven by changes in the amplitude of the orbital eccentricity. From 400 to 150 ka BP, eastern Africa and Arabia show synchronized wet-dry shifts, which changed drastically at 150 ka BP. After 150 ka BP, an overall trend to dry climate states is observable, and the hydroclimate dynamics between eastern Africa and Arabia are negatively correlated. Those spatio-temporal variations and interrelationships of climate potentially influenced the availability of spatial links for human expansion along those vertices. We observe positively correlated network links during the supposed out-of-Africa migration phases of H. sapiens. Furthermore, our data do not suggest hominin occupation phases during specific time intervals of humid or stable climates but provide evidence of the so far underestimated potential role of climate predictability as an important factor of hominin ecological competitiveness.

Ground air pollutants explanation based on multiple visibility graph of complex network by temporal community division.

In air pollution studies, the correlation analysis of environmental variables has usually been challenged by parametric diversity. Such variable variations are not only from the extrinsic meteorological conditions and industrial activities but also from the interactive influences between the multiple parameters. A promising solution has been motivated by the recent development of visibility graph (VG) on multi-variable data analysis, especially for the characterization of pollutants' correlation in the temporal domain, the multiple visibility graph (MVG) for nonlinear multivariate time series analysis has been verified effectively in different realistic scenarios. To comprehensively study the correlation between pollutant data and season, in this work, we propose a multi-layer complex network with a community division strategy based on the joint analysis of the atmospheric pollutants. Compared to the single-layer-based complex networks, our proposed method can integrate multiple different atmospheric pollutants for analysis, and combine them with multivariate time series data to obtain higher temporary community division for ground air pollutants interpretation. Substantial experiments have shown that this method effectively utilizes air pollution data from multiple representative indicators. By mining community information in the data, it successfully achieves reasonable and strong interpretive analysis of air pollution data.

Dispersion complexity-entropy curves: An effective method to characterize the structures of nonlinear time series.

The complexity-entropy curve (CEC) is a valuable tool for characterizing the structure of time series and finds broad application across various research fields. Despite its widespread usage, the original permutation complexity-entropy curve (PCEC), which is founded on permutation entropy (PE), exhibits a notable limitation: its inability to take the means and amplitudes of time series into considerations. This oversight can lead to inaccuracies in differentiating time series. In this paper, drawing inspiration from dispersion entropy (DE), we propose the dispersion complexity-entropy curve (DCEC) to enhance the capability of CEC in uncovering the concealed structures within nonlinear time series. Our approach initiates with simulated data including the logistic map, color noises, and various chaotic systems. The outcomes of our simulated experiments consistently showcase the effectiveness of DCEC in distinguishing nonlinear time series with diverse characteristics. Furthermore, we extend the application of DCEC to real-world data, thereby asserting its practical utility. A novel approach is proposed, wherein DCEC-based feature extraction is combined with multivariate support vector machine for the diagnosis of various types of bearing faults. This combination achieved a high accuracy rate in our experiments. Additionally, we employ DCEC to assess stock indices from different countries and periods, thereby facilitating an analysis of the complexity inherent in financial markets. Our findings reveal significant insights into the dynamic regularities and distinct structures of these indices, offering a novel perspective for analyzing financial time series. Collectively, these applications underscore the potential of DCEC as an effective tool for the nonlinear time series analysis.

Stacking-based neural network for nonlinear time series analysis

Stacking-based neural network for nonlinear time series analysis

Fourier phase index for extracting signatures of determinism and nonlinear features in time series.

Detecting determinism and nonlinear properties from empirical time series is highly nontrivial. Traditionally, nonlinear time series analysis is based on an error-prone phase space reconstruction that is only applicable for stationary, largely noise-free data from a low-dimensional system and requires the nontrivial adjustment of various parameters. We present a data-driven index based on Fourier phases that detects determinism at a well-defined significance level, without using Fourier transform surrogate data. It extracts nonlinear features, is robust to noise, provides time-frequency resolution by a double running window approach, and potentially distinguishes regular and chaotic dynamics. We test this method on data derived from dynamical models as well as on real-world data, namely, intracranial recordings of an epileptic patient and a series of density related variations of sediments of a paleolake in Tlaxcala, Mexico.

Nonlinear Time Series Analysis for Anomaly Detection in Cybersecurity using Cloud IoT

The CloudIoT is an extensive network that hosts several IoT gadgets and apps. This comprises the servers and the storage space required for real-time processing and operations. For commercial purposes, data gathered by IoT devices may be safely saved and handled on the cloud. CloudIoT allows for the scalable distribution of software. IoT devices capture information using built-in sensors and monitors. After collection, it is sent to a central location where data from other Internet of Things devices is kept. Cloud storage offers a scalable, cost-effective alternative for maintaining and administering IoT data—the capacity to grow or change quickly. More and more Internet of Things devices means a greater need for data storage and processing capacity. The cloud computing solution is scalable and adaptable enough to meet these shifting requirements. Hence, this paper proposes a nonlinear time series analysis for enhancing anomaly detection in cybersecurity based on Cloud-Internet of Things (CloudIoT-CS). Statistics like precision and positive and negative results are considered part of assessing the suggested methodology by comparing it against conventional anomaly detection techniques. The study also looks at the system's resistance to hostile assaults, guaranteeing its efficacy in practical cybersecurity settings. This study's findings provide a holistic and flexible approach that accounts for the unique challenges posed by CloudIoT settings. As well as improving the precision of anomaly identification, integrating nonlinear time series evaluation with CloudIoT creates a scalable and robust framework for protecting vital systems from ever-evolving cyber threats.

Impacts of non-locality and memory kernel of fractional derivative, awareness and treatment strategies on HIV/AIDS prevalence

We propose a Liouville–Caputo non-linear fractional order α(0<α≤1) epidemic model to understand the transmission dynamics of infectious diseases, like HIV/AIDS, and explore the presence of memory kernel of fractional derivative for the precautionary measure against the transmission of disease to control HIV/AIDS prevalence. A proper sex education, which depends on number of HIV and AIDS, to unaware susceptible as well as fear of infection among these aware susceptible are employed here. Treatment compartment is incorporated to treat the symptomatic HIV and AIDS infected individuals.Using the Banach fixed point theorem, we first ensure the positivity, existence and uniqueness of solution, and biologically feasible region. Next after defining the basic reproduction number (R0), its uses on stability (local and global) and direction of bifurcation are manifested here. A quantitative assessment is conducted onR0and then we explore the sensitive parameter/s, which control the HIV/AIDS prevalence.Numerical experiment is conducted based on the Adams–Bashforth–Moulton predictor corrector method to obtain the approximate solution of the proposed model. Impacts of disease transmission rate, awareness, psychological fear of infection and treatment strategies on the HIV/AIDS prevalence are analysed based on non-linear dynamical tools, like, time series, phase diagram and spectra analysis. An effort is made to see how the order and kernel of the fractional derivative play a crucial role on the dynamics of the system and in the precautionary measure against the transmission of disease during transition as well as post-transition phases. We employ intervention strategies for controlling the spread as well as possibility of eradicating the HIV/AIDS prevalence. It is seen that the reduction ofαis comparable to awareness, psychological fear of infection and treatment strategies to control disease. Finally, it is shown that the use of fractional derivative in disease transmission dynamics is one of the most efficient applied mathematics content compatible with the natural characteristics of human memory.

Identifying quasi-periodic variability using multivariate empirical mode decomposition: a case of the tropical Pacific

Abstract. A variety of statistical tools have been used in climate science to gain a better understanding of the climate system's variability on various temporal and spatial scales. However, these tools are mostly linear, stationary, or both. In this study, we use a recently developed nonlinear and nonstationary multivariate time series analysis tool – multivariate empirical mode decomposition (MEMD). MEMD is a powerful tool for objectively identifying (intrinsic) timescales of variability within a given spatio-temporal system without any timescale pre-selection. Additionally, a red noise significance test is developed to robustly extract quasi-periodic modes of variability. We apply these tools to reanalysis and observational data of the tropical Pacific. This reveals a quasi-periodic variability in the tropical Pacific on timescales ∼ 1.5–4.5 years, which is consistent with El Niño–Southern Oscillation (ENSO) – one of the most prominent quasi-periodic modes of variability in the Earth’s climate system. The approach successfully confirms the well-known out-of-phase relationship of the tropical Pacific mean thermocline depth with sea surface temperature in the eastern tropical Pacific (recharge–discharge process). Furthermore, we find a co-variability between zonal wind stress in the western tropical Pacific and the tropical Pacific mean thermocline depth, which only occurs on the quasi-periodic timescale. MEMD coupled with a red noise test can therefore successfully extract (nonstationary) quasi-periodic variability from the spatio-temporal data and could be used in the future for identifying potential (new) relationships between different variables in the climate system.

Experimental investigation of the period-adding bifurcation route to chaos in plasma.

Since the characteristic timescales of the various transport processes inside the discharge plasma span several orders of magnitude, it can be regarded as a typical fast-slow system. Interestingly, in this work, a special kind of complex oscillatory dynamics composed of a series of large-amplitude relaxation oscillations and small-amplitude near-harmonic oscillations, namely, mixed-mode oscillations (MMOs), was observed. By using the ballast resistance as the control parameter, a period-adding bifurcation sequence of the MMOs, i.e., from L^{s} to L^{s+1}, was obtained in a low-pressure DC glow discharge system. Meanwhile, a series of intermittently chaotic regions caused by inverse saddle-node bifurcation was embedded between the two adjacent periodic windows. The formation mechanism of MMOs was analyzed, and the results indicated that the competition between electron production and electron loss plays an important role. Meanwhile, the nonlinear time series analysis technique was used to study the dynamic behavior quantitatively. The attractor in the reconstructed phase space indicated the existence of the homoclinic orbits of type Γ^{-}. In addition, by calculating the largest Lyapunov exponent (LLE), the chaotic nature of these states was confirmed and quantitatively characterized. With the decrease in the ballast resistance, the return map of the chaotic state gradually changed from the nearly one-dimensional single-peak structure to the multibranch structure, which indicates that the dissipation of the system decreased. By further calculating the correlation dimension, it was shown that the complexity of the strange attractors increased for higher-order chaotic states.

Nonlinear time series analysis of coastal temperatures and El Niño–Southern Oscillation events in the eastern South Pacific

Abstract. We carry out a nonlinear time series analysis motivated by dynamical systems theory to investigate the links between temperatures on the eastern South Pacific coast, influenced by the Humboldt Current System, and El Niño–Southern Oscillation (ENSO) events. To this aim, we use a set of 16 oceanic and atmospheric temperature time series from Chilean coastal stations distributed between 18 and 45∘ S. The spectral analysis indicates periodicities that can be related to both internal and external forcing, involving not only ENSO, but also the Pacific Decadal Oscillation, the Southern Annual Mode, the Quasi-Biennial Oscillation and the lunar nodal cycle. The asymptotic neural network test for chaos based on the largest global Lyapunov exponent indicates that the temperature dynamics along the Chilean coast is not chaotic. We use local Lyapunov exponents to characterize the short-term stability of the series. Using a cross-entropy test, we find that two stations in northern Chile, one oceanic (Iquique) and one atmospheric (Arica), present a significant positive cross-dependence between local Lyapunov exponents and ENSO. Iquique is the station that presents the greater number of regional characteristics and correlates with ENSO differently from the rest. The unique large-scale study area, combined with time series from hitherto unused sources (Chilean naval records), reveals the nonlinear dynamics of climate variability in Chile.

Modelling of noise due to impulsive excitation using nonlinear time series analysis

Early acoustic machine design phases should target safety and comfort requirements, minimising the risk of hearing damage and noise exposure and ensuring a pleasant machine sound. Therefore, physical mechanisms of sound generation, propagation and radiation need to be modelled. Today, nonlinear physical relations in vibroacoustic models such as contact problems are mostly linearised. The aim of this paper is to increase the prediction quality of vibroacoustic models so that physical mechanisms behind sound generation due to complex physical, nonlinear phenomena can be better understood. The focus is on sound generation due to impulsive excitation, which is a significant sound generation mechanism in machinery, e.g. considering gears, forming processes (e.g. punching), or road loads of vehicles. We describe a model setup to simulate impulsive excitation on a laboratory scale, where structural vibrations can be analysed. Physical correlations including nonlinear correlations could be identified using nonlinear time series analysis. Recurrence plots will be used to identify features such as phase space contraction, e.g. to identify damping. These features form the basis to deepen the physical understanding of machinery noise, to contribute to enhanced acoustic machine designs and to enhance data-driven modelling using sparse regression in future research.

Directed recurrence networks.

Complex network approaches have attracted a growing interest in the analysis of nonlinear time series. Among other reconstruction methods, it has been shown that the recurrence plot can be used as the adjacency matrix for recurrence networks, expanding the applications of the already successful recurrence analysis. We study here the potential benefits of a directed formulation of recurrence networks through a simple modification of the recurrence plot. As it is directly related to the recurrence analysis field, this approach takes advantage of the progresses regarding the creation and treatment of the recurrence plot. It appears that directed recurrence networks provide more robust results than their undirected counterpart for transitions detection as well as temporal patterns discovery and clustering. New applications for network cleaning and data modeling are also demonstrated.

Spiral bevel gears: Bifurcation and chaos analyses of pure torsional system

The goal of this paper is to provide a detailed analysis of the complex dynamic scenario of spiral bevel gears by considering the torsional shaft stiffness. The dynamic model takes into account: time-dependent stiffness and non-smooth nonlinearity due to the backlash, i.e., teeth contact loss. The time-varying meshing stiffness is evaluated by means of a nonlinear finite element model, which allowed an accurate evaluation of the global and local teeth deformation in both directions: forward and reverse motions. Due to intentional tooth profile errors, e.g., profile modification, manufacturing error, or assembly error, the loaded and unloaded tooth contact analyses were conducted. The dynamic model was validated by comparisons with a verified SDOF model in terms of linear natural frequencies and nonlinear dynamic response. The present study provides amplitude frequency diagrams and bifurcation diagrams; for specific regimes, periodic, quasiperiodic, and chaotic responses were found. The dynamic behavior of the systems was evaluated using various tools such as modal analysis, nonlinear time series analysis, spectra, 3D-phase diagrams, and Poincaré maps. This study proves that decreasing the DOFs of the system can lose some dynamics. Different phenomena have been found such as trapping and boom-and-bust cycle. The dynamic response of different cases is evaluated by estimating the largest Lyapunov exponent and the correlation dimension. It is found that the system undergoes complex dynamic phenomena including nT-periodic, trapping, and aperiodic motions, which are evidenced in 3D-bifurcation diagrams, spectra, Poincaré maps, and 3D-phase diagrams. The chaotic motion found in this study is an undesirable behavior from a practical point of view, which can be avoided by adjusting the designed parameters. The results provide a basis for parameter design and dynamic characteristic control of the spiral bevel gear drive system.

Characterization of homogeneous nucleation of superheated liquid droplets by nonlinear analysis

The homogeneous bubble nucleation in superheated liquid droplets can be initiated by energetic particles or radiation. The nonlinearity in the heat transfer introduces chaos in the temperature time series of a boiling system. In the present study, the acoustical pulses from the homogeneous nucleation of superheated liquid droplets by neutrons and gamma-rays have been collected. Nonlinear time series analysis methods like 3D attractor reconstruction and Fast Fourier transformation (FFT) have been performed on the collected pulses. The analysis predicts the possibility of the existence of chaos, and the positive value of the largest Lyapunov exponent confirms it. The nonlinearity in the energy deposition by both the recoil nuclei and electrons from the neutrons and gamma-rays respectively, introduce chaos. A qualitative difference has been observed in the 3D attractor and Rescaled Range (R/S) plots between the neutron and gamma-ray-induced pulses, and the present differences are useful for the identification of these pulses.

Seismic Analysis of the Soil Behavior and Titling of the Building

The mechanism of this yielding and its behavior under seismic loading of soil are the primary focus of this investigation. Nonlinear analysis is taken into account to see the real behavior of soil. In order to analyze soil, we employ both its solid mass and its lumped mass. In this study, the Finite Element Model (FEM) forms the basis for the mathematical formulas. Soil analysis in the case of a lumped mass takes into account the soil's one DOF, two DOF, and multi DOF degrees of freedom. In order to determine soil characteristics for MDOF, a soil bore log must be employed. In the instance of MDOF, the soil is composed of 12 distinct layers. SAP 2000 is used to do linear and nonlinear analysis of time series for this research. According to the findings, solid and lumped soil mass displacements are almost identical. Therefore, it is possible to get insight into the behavior of soil mass during an earthquake by studying lumped soil mass. The soil's nonlinear behavior is investigated using a variety of linear completely plastic hysteretic loops. Soil characteristics are shown to be crucial in this regard. Inadequate soil stiffness may result in persistent deformation, which in turn can lead structures to lean out of alignment. It is also noted that near the soil's surface, amplification is greatest.

Nonlinear Dynamic Analysis of Seafloor Pressure Time Series with and without Internal Waves

Ocean internal wave as a nonlinear process oscillates within ocean interior rather than the surface. Internal waves have significant influences on the transport of material, energy and even global climate. The generation, transmission and attenuation of internal wave play a significant role in the ocean dynamics owing to the exchange of the energy within each process. In this paper, reduced complexity of seafloor pressure time series induced by internal wave in the southern Taiwan bank has been investigated using nonlinear dynamic analysis. Cao method has been used to distinguish between the pressure signal with internal wave and the random signal without internal wave. The reconstructed phase spaces and correlation dimensions of the time series with and without internal wave are compared. Internal wave leads the seafloor pressure time series to produce a more regular phase space reconstruction and more low dimensional dynamics. Our study suggests that nonlinear dynamic analysis could be a valuable and promising method in real-time ocean internal wave detection.