Nonlinear Time Series Analysis Research Articles

Determination of the performance of training algorithms and activation functions in meteorological drought index prediction with nonlinear autoregressive neural network

Analysis of long-term meteorological data is critical for monitoring climate trends and understanding the drought situation in a given region. In this study, monthly average precipitation data from the Niğde meteorological station in Turkey covering the period 1950–2020 were used. Within the scope of the study, seven different drought index methods were used for drought analysis, and the number and percentages of drought conditions were calculated according to these indices. For example, according to the Standardized Precipitation Index (SPI) method, the proportion of dry periods was determined as 16.2% and the proportion of humid periods as 83.8%. The Mann-Kendall trend analysis performed to determine the drought trends of the region revealed an increasing trend towards humidity in all indices (e.g., z = 1.299, p = 0.194 for SPI). In the study, 60-month drought forecasts covering the years 2020–2025 were realized using the Nonlinear Autoregressive Neural Network (NARNN) model, and the results were compared with the Autoregressive (AR) model. In the prediction performance analysis, the NARNN model showed superior prediction performance for all indices with lower RMSE values (e.g., NARNN RMSE = 0.977 for SPI; AR RMSE = 1.704). The prediction performances of different training algorithms and activation functions used in the NARNN model were analyzed. The best performance was obtained with the trainbr training algorithm and sigmoid activation function (e.g., RMSE = 0.997 for SPI). Based on these best parameters, more than 70% of the drought conditions during the 2020–2025 period were found to be normal or humid according to NARNN predictions. This study demonstrates the superiority of the NARNN model in nonlinear time series analyses and that it is a reliable tool, especially for future drought forecasts. In addition, comprehensive analyses with different index methods have significantly contributed to understanding the long-term drought trends in the Niğde region.

Nonlinear spatiotemporal and evolution characteristics of extreme precipitation during the Mei-yu period based on a state transition network

The frequent occurrence of extreme precipitation events notably affects the middle and lower reaches of the Yangtze River (MLYR) during the Mei-yu period. Based on meteorological station data from the China Meteorological Administration collected from 1970 to 2019, we employed nonlinear time series analysis and the state transition networks (STN) to identify the spatiotemporal and evolution characteristics of 10 extreme precipitation indices (EPIs). Although previous studies have largely focused on investigating the spatiotemporal characteristics, the suitability of STN for extreme precipitation has not been extensively explored. This study focused on not only the spatiotemporal characteristics of extreme precipitation but also its evolution using the STN. The results showed that the periodic oscillations in extreme precipitation ranged from 2 to 5 years and an abrupt change occurred primarily around 1990. Extreme precipitation displayed similar network parameters, yet different network structures and transition probabilities. The evolution of extreme precipitation exhibited from decreasing to increasing from 1970 to 2019, with an increasing trend after 1990. Similar network structures tended to exhibit higher Pearson correlations. Spatially, high EPIs values were mainly concentrated in the central region of the MLYR. This study could provide new methods and perspectives for studying the characteristics of extreme precipitation.

A Nonlinear Time Series Analysis on the Effect of Foot Injury on Gait Dynamics

The effects of foot injuries regarding bilateral asymmetry and gait dynamics are still poorly understood. Previous work discussed rehabilitation, postural control, and asymmetry, with the models being mainly validated for upper body translations and no or minimal assessment on rotation. The aim of this study was to assess the effect of foot injury on gait dynamics. For this, a wearable sensors system for data collection of the key variables of the of human movement was considered. The dynamics of motion – recorded in the plane of motion using a laser sensor – was assessed using a new projective method which considers the axial rotations, translation, and in-plane rotation patterns for normal human gait vs. simulated gait pathologies. A nonlinear timeseries analysis, along with a Poincare map, phase space, time delay, Lyapunov exponents, and false nearest neighbors (FNN) method have been considered in order to convey the periodicity of the data collected for a healthy individual with and without a simulated injury. The Lyapunov exponents which quantity the degree of separation of nearby trajectories are used to differentiate between the chaotic and non-chaotic behavior. The positive sign of the largest Lyapunov exponents for all data indicated “the exponential separation of nearby trajectories as time evolves”, that is, the chaotic behavior of the system.

Dynamics of Limit Cycle Oscillations in a Multinozzle Lean Direct Injection Combustor

Thermoacoustic oscillations in a high-pressure multinozzle lean direct injection combustor with fuel staging have been examined. The combustor consists of three independently controlled fuel stages, i.e., the pilot, intermediate, and outer fuel stages. Limit-cycle oscillations have been identified for two cases with different fuel supplies to the three stages of the combustor, operating at different equivalence ratios. The oscillation dynamics of the two cases have been characterized and quantified by employing nonlinear time series analysis tools. Phase portraits, recurrence plots, and recurrence quantification methods were used by phase space reconstruction of the scalar pressure measurements. Further comparisons between the two cases were made by correlating the time-resolved OH* chemiluminescence images to the pressure oscillations. Phase averaging and spectral proper orthogonal decomposition were used to understand the flame dynamics between the three fuel stages.

Understanding the temporal variability and predictability of streamflow signatures in the Colorado River Basin

It is well established that streamflow regimes evolve over decadal time scales (i.e., low frequency) leading to long term shifts in distributions. Similar low frequency variations have also been documented in streamflow predictability. Here we explore connections between streamflow distribution attributes and predictability regimes in the Upper Colorado River Basin. We employ nonlinear dynamical time series analysis methods on streamflow timeseries covering the period 762 – 2019 for six locations in the basin. First, a wavelet spectral analysis is performed to obtain the quasi-periodic ‘signal’ of the streamflow. The wavelet analysis also provides the temporal variability of the variance of the signal time series. The signal time series is embedded in a D-dimensional space with appropriate lag to reconstruct the phase space of the dynamics – i.e. the attractor. Overall predictability is assessed by quantifying the average divergence trajectories in the phase space using Global Lyapunov Exponents and the temporal variability of predictability via the Local Lyapunov Exponents. Results show clear oscillations in streamflow predictability with periods of both high and low predictability occurring throughout the study period at all gauges. Comparing predictability timeseries across the stream gauges we find that general consistency in high and low predictability periods, although they do not perfectly align temporally. In general, higher (lower) predictability periods are characterized by lower (higher) streamflow variance. While there is not a clear relationship between streamflow magnitude and predictability in general, modern high predictability epochs are characterized by a slightly greater likelihood of dry years and lower likelihood of wet years than other epochs. These findings indicate the potential for statistically significant differences in streamflow signatures between high and low predictability periods. Exploring these fundings further with potential connections to large-scale climate can be helpful in exploiting them for skillful short and medium term flow projections.

Nonlinear time series analysis of state-wise COVID-19 in Malaysia using wavelet and persistent homology

The nonlinear progression of COVID-19 positive cases, their fluctuations, the correlations in amplitudes and phases across different regions, along with seasonality or periodicity, pose challenges to thoroughly examining the data for revealing similarities or detecting anomalous trajectories. To address this, we conducted a nonlinear time series analysis combining wavelet and persistent homology to detect the qualitative properties underlying COVID-19 daily infection numbers at the state level from the pandemic’s onset to June 2024 in Malaysia. The first phase involved investigating the evolution of daily confirmed cases by state in the time-frequency domain using wavelets. Subsequently, a topological feature-based time series clustering is performed by reconstructing a higher-dimensional phase space through a delay embedding method. Our findings reveal a prominent 7-day periodicity in case numbers from mid-2021 to the end of 2022. The state-wise daily cases are moderately correlated in both amplitudes and phases during the Delta and Omicron waves. Biweekly averaged data significantly enhances the detection of topological loops associated with these waves. Selangor demonstrates unique case trajectories, while Pahang shows the highest similarity with other states. This methodological framework provides a more detailed understanding of epidemiological time series data, offering valuable insights for preparing for future public health crises.

Comparative Performance of ARIMA and ARCH Models On Time Series of Monthly Libyan Brent Oil Price

Many economic and financial time series exhibit a phenomenon known as heteroscedastic, where the variance of the series changes over time. This research study focuses on financial time series modelling, with special application to modelling the price of Libyan Brent Oil. In particular, the theory of univariate nonlinear time series analysis is explored and applied to the price Libyan of Brent Oil, spanning from January 2000 to December 2010. The data was obtained from Bullent of the Platts Market Wire of Statistics. This study aims to evaluate the performance of ARIMA as a linear model and ARCH as a nonlinear modelling data. Multiple time series models were considered for fitting the data, and the best ARIMA models were selected based on the Akaike Information Criteria (AIC). ARIMA (0, 1, 1), and ARIMA (1, 1, 0) were identified as the best models. After estimating the parameters of the selected models, model checking revealed that these models were not suitable for modeling the data, as they lacked validity according to the test of squared residuals. The goodness of fit was assessed using the AIC, and based on minimum AIC values, the best fit ARCH models were found to be ARIMA (0, 1, 1) - ARCH (1). After estimating the parameters of the selected model, a series of diagnostic and forecast accuracy tests were performed. Based on this model, a twelve-month forecast of the price of Libyan Brent crude was made.

Spray swirling flame dynamic under combined modulation of air and fuel

Combustion instability is still a main challenge in developing advanced gas turbine combustors. It is essential to conduct instability control once it occurs. In this paper, liquid fuel modulation by a high-speed on/off valve is used to control flame oscillation generated by loudspeakers. Spray swirling flame dynamic under the combined modulation of air and fuel is experimentally studied. Flame dynamical characteristics and coherent structures at different excitation phase delays are discussed with nonlinear time-series analysis and proper orthogonal decomposition method. Experimental results show flame heat release rate fluctuation is decreased by 86% at 40° while flame instability is intensively enhanced at 220°. Phase portraits and recurrence plots show the flame is in a chaotic oscillation with low amplitude aperiodic oscillations at 40° while it suffers from the largest fluctuation at 220°. Difference in flame dynamic is caused by the destructive (out of phase) and constructive (in phase) interference of heat release rate fluctuations induced by air excitation and fuel modulation. Flame returns back to its nature dynamical behavior under out-of-phase modulation. Longitudinal modes of the spray flame are enhanced while the spiral modes are inhibited under in-phase modulation.

Functional index coefficient models for locally stationary time series

In the analysis of nonlinear time series, we propose a novel functional index coefficient model for the locally stationary data. The proposed model can effectively capture the dynamic interaction effects between variables and the nature of data evolution. Drawing the idea from the spline-backfitted method, we propose a three-step estimation procedure and establish the asymptotic properties of the resulting nonparametric estimators. We further construct simultaneous confidence bands for the time-varying functions to explore the global variation of the original data. We also develop a test statistic to check the time-varying properties based on a bootstrap procedure. Simulation studies have been conducted to investigate the finite sample performance of the proposed methods. Two real applications in the finance market and the Hong Kong respiratory and circulatory disease data are analysed for illustration.

Strain variation and anomalous climate synergistically influence cholera pandemics.

Explanations for the genesis and propagation of cholera pandemics since 1817 have remained elusive. Evolutionary pathogen change is presumed to have been a dominant factor behind the 7th "El Tor" pandemic, but little is known to support this hypothesis for preceding pandemics. The role of anomalous climate in facilitating strain replacements has never been assessed. The question is of relevance to guide the understanding of infectious disease emergence today and in the context of climate change. We investigate the roles of climate and putative strain variation for the 6th cholera pandemic (1899-1923) using newly assembled historical records for climate variables and cholera deaths in provinces of former British India. We compare this historical pandemic with the 7th (El Tor) one and with the temporary emergence of the O139 strain in Bangladesh and globally. With statistical methods for nonlinear time series analysis, we examine the regional synchrony of outbreaks and associations of the disease with regional temperature and rainfall, and with the El Niño Southern Oscillation (ENSO). To establish future expectations and evaluate climate anomalies accompanying historical strain replacements, climate projections are generated with multi-model climate simulations for different 50-year periods. The 6th cholera pandemic featured the striking synchronisation of cholera outbreaks over Bengal during the El Niño event of 1904-07, following the invasion of the Bombay Presidency with a delay of a few years. Accompanying anomalous weather conditions are similar to those related to ENSO during strain replacements and pandemic expansions into Africa and South America in the late 20th century. Rainfall anomalies of 1904-05 at the beginning of the large cholera anomaly fall in the 99th percentile of simulated changes for the regional climate. Evolutionary pathogen change can act synergistically with climatic conditions in the emergence and propagation of cholera strains. Increased climate variability and extremes under global warming provide windows of opportunity for emerging pathogens.

Machine learning analysis of pressure fluctuations in a gas-solid fluidized bed

Pressure fluctuations in a gas-solid fluidized bed involve much information of the dynamic system. To uncover the value and significance of the pressure fluctuations time series, two meaningful tasks, i.e., the fluidization regime classification and the future state prediction, are investigated using Machine Learning (ML) algorithms, including Back Propagation neural network (BP), Convolutional Neural Network (CNN), Long Short-Term Memory network (LSTM), Radial Basis Function network (RBF), Random Forest (RF) and Support Vector Machine (SVM). Findings indicate that the six ML methods rank their performance for the classification task from high to low as: BP ∼ RF > SVM > CNN ∼ LSTM ∼ RBF. For the one-step-ahead future state prediction, the accuracy goes from high to low as: BP ∼ RBF > CNN ∼ LSTM > SVM ∼ RF, whilst for the multistep-ahead prediction the LSTM approach shows the best performance. Additionally, the sampling frequency of pressure time series has significant effects on both tasks. Overall, this study demonstrates the capability of machine learning methods for the value analysis of nonlinear time series and the better operation of gas-solid fluidization systems.

Mathematical modeling and seasonal solar radiation variability in Nigeria’s geo-political zones: A recurrence and multifractal analysis

This article delves into the seasonal variation of solar radiation patterns across Nigeria’s four geo-political zones, exploring their complex, scale-dependent behaviors. By employing chaotic quantifiers, the study characterizes irregular and self-similar patterns, enhancing our understanding of solar radiation variability and heterogeneity. The research uniquely fits meteorological parameters onto a global solar radiation model and focuses on the underexplored nonlinear aspects within tropical regions, specifically in the Nigerian context. Utilizing daily data from ERA INTERIM satellite archives for representative stations, the study employs nonlinear time series analysis methods like recurrence plots, recurrence qualitative analysis, and multifractal spectral analysis to comprehensively explore the unpredictable behaviors observed. The quantifier spectrums play a key role in revealing intricate scaling behaviors and correlation structures among environmental variables, shedding light on patterns often concealed by conventional statistical methods. For instance, we found that solar radiation variability in the northern region increases more significantly during the dry season compared to the wet season, unlike in the southern region. Additionally, the multifractal spectral analysis revealed a higher degree of complexity in solar radiation patterns during transition periods between seasons. The findings reveal a low recurrence quantitative analysis, long right tail, and truncations at both ends of the spectrum. This suggests instability in solar radiation across different seasons and locations. Nonetheless, the results also demonstrate that solar radiation is consistently available throughout the year, which is typical of tropical regions.

Local-manifold-distance-based regression: an estimation method for quantifying dynamic biological interactions with empirical time series.

Quantifying species interactions based on empirical observations is crucial for ecological studies. Advancements in nonlinear time-series analyses, particularly S-maps, are promising for high-dimensional and non-equilibrium ecosystems. S-maps sequentially perform a local linear model fitting to the time evolution of neighbouring points on the reconstructed attractor manifold, and the coefficients can approximate the Jacobian elements corresponding to interaction effects. However, despite that the advantages in nonlinear forecasting with noise-contaminated data, these methodologies have a limitation in the Jacobian estimation accuracy owing to non-equidistantly stretched local manifolds in the state space. Herein, we therefore introduced a local manifold distance (LMD) concept, a non-equidistant measure based on the multi-faceted state dependency. By integrating LMD with advanced computation techniques, we presented a robust and efficient analytical method, LMD-based regression (LMDr). To validate its advantages in prediction and Jacobian estimation, we analysed synthetic time series of model ecosystems with different noise levels and applied it to an experimental protozoan predator-prey system with established biological information. The robustness to noise was the highest for LMDr, which also showed a better correspondence to expected predator-prey interactions in the protozoan system. Thus, LMDr can be applied to study complex ecological networks under dynamic conditions.

Identification of climatic tipping points and transitions in Chinese loess grain-size records utilizing nonlinear time series analysis

As one of the most important terrestrial sediments, Chinese loess provides valuable information on regional and global climatic and environmental changes and holds great potential for studying on nonlinear behaviors of the East Asian monsoon system. Utilizing objective and quantitative methods to identify tipping points and climate transitions in paleoclimatic records can help us understand the climatic change in the Chinese Loess Plateau (CLP). This study explores critical tipping points and nonlinear climate transitions within the CLP using the Chiloparts record, a comprehensive 2600-ka paleoclimate dataset. We pinpointed potential tipping points using recurrence quantification analysis and the augmented Kolmogorov-Smirnov test, ultimately leading to 15 critical tipping points. We argued that these 15 tipping points represent some of the most significant climatic changes recorded in the Chinese loess paleoclimate record. Employing recurrence quantification analysis, recurrence networks, and visibility graphs, we also identified several climate transitions and provided some nonlinear information, including the Mid-Pleistocene Transition (MPT) as well as the Mid-Brunhes Transition (MBT). We particularly highlight a significant climatic regime transition around 500 ka that may reflect a nonlinear response to variations in the Atlantic Meridional Overturning Circulation (AMOC). Our research also contributes to the understanding of the complex interplay between loess deposition, environmental change, and tectonic activity, emphasizing the need for further investigations to elucidate the mechanisms driving these transitions.

Advanced multiscale modal and frequency analysis of swirling spray flame near to lean blowout

This study provides an in-depth characterization of Jet-A1 swirled flames' dynamics using advanced decomposition techniques on high-speed OH* chemiluminescence images. Research conducted in a 300-kW combustor with global fuel-to-air ratios Φ = 0.36, 0.24, and 0.18 reveals dominant low-frequency components under ultra-lean conditions through statistical and wavelet analyses. Integrating Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD), and Multiscale Proper Orthogonal Decomposition (mPOD) enhances understanding by detailing the energy and frequency of flame modal structures. Observations show that as Φ decreases, flame size and intensity reduce, while fluctuations increase, indicating a transition towards unstable combustion dynamics. Furthermore, the frequency and intensity of burst phenomena increase, particularly in higher modes, indicating the system's approach to lean blowout (LBO). The application of nonlinear time series analysis, including autocorrelation (AFC) and phase space reconstruction, to mPOD eigenvalues, detects early warning signs of LBO. At lower Φ, the ACF reveals significant and frequent bursts, indicating high instability and intermittent behavior. Phase space reconstruction shows distorted orbits with a spiral-like structure, characteristic of substantial damping and irregular behavior near blowout conditions.

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.