Nonlinear Time Series Research Articles

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.

Unveiling an asymmetric relationship between global crude oil and local food prices in an oil-importing economy

AbstractRecent swift comovements of local food and global crude oil prices have attracted the attention of policymakers and researchers. To evaluate this relationship, many studies have used time series models to explore global crude oil and local food prices. However, robust research based on advanced nonlinear time series models that incorporate control variables for their formation is lacking. In this paper, nonlinear techniques are applied to assess the asymmetric nexus between Brent oil prices and local retail food prices in Slovakia. To estimate this value, we extend the single-threshold NARDL approach to the MTNARDL model. The nominal exchange rate and industrial production index are used as the control variables. Compared with conventional NARDL models, the MTNARDL model provides a more detailed representation of global oil‒local food price linkages and detects the asymmetric effect of global oil prices on food prices from both long- and short-term perspectives. Interestingly, with respect to long- and short-term food price volatility, changes in response to oil price fluctuations are greatest under a regime with rather a small number of positive and moderate changes.

Boosting Financial Market Prediction Accuracy With Deep Learning and Big Data

The accuracy of financial market trend prediction directly impacts investors' decisions and financial institutions' risk management, making it a focal point of research in the financial domain. While traditional statistical methods lay the foundation for market forecasting, they still face limitations in handling complex, nonlinear, and non-stationary financial time series data with accuracy and adaptability. Time series analysis plays a pivotal role in financial market prediction, revealing historical patterns and future trends within the data. Addressing the limitations of existing methods, this study proposes a deep learning-based CEEMDAN-CNN-LSTM model. Leveraging the CEEMDAN decomposition method to capture the multi-frequency components of time series, CNN extracts spatial features, and LSTM learns temporal dependencies. Experimental results demonstrate that the performance of the CEEMDAN-CNN-LSTM model surpasses traditional models on standardized evaluation metrics such as RMSE, MAE, and MAPE across four major financial datasets.

Modal volatility function

We in this article propose a novel non‐parametric estimator for the volatility function within a broad context that encompasses nonlinear time series models as a special case. The new estimator, built on the mode value, is designed to complement existing mean volatility measures to reveal distinct data features. We demonstrate that the suggested modal volatility estimator can be obtained asymptotically as well as if the conditional mean regression function were known, assuming observations are from a strictly stationary and absolutely regular process. Under mild regularity conditions, we establish that the asymptotic distributions of the resulting estimator align with those derived from independent observations, albeit with a slower convergence rate compared to non‐parametric mean regression. The theory and practice of bandwidth selection are discussed. Moreover, we put forward a variance reduction technique for the modal volatility estimator to attain asymptotic relative efficiency while maintaining the asymptotic bias unchanged. We numerically solve the modal regression model with the use of a modified modal‐expectation‐maximization algorithm. Monte Carlo simulations are conducted to assess the finite sample performance of the developed estimation procedure. Two real data analyses are presented to further illustrate the newly proposed model in practical applications. To potentially enhance the accuracy of the bias term, we in the end discuss the extension of the method to local exponential modal estimation. We showcase that the suggested exponential modal volatility estimator shares the same asymptotic variance as the non‐parametric modal volatility estimator but may exhibit a smaller bias.

An entropy measure-based study on flow pattern of gas–liquid two-phase flow in a U-Tube

U-tubes are widely applied in gas–liquid two-phase transportation in chemical engineering. The diverse flow patterns within these tubes significantly affect the pressure loss, heat transfer efficiency, and even the fluid-induced vibration amplitude of the tubes. This study explores the complex flow pattern features in a U-tube in a vertical plane and focuses on recognizing them. For the acquisition and classification of flow patterns, a Computational Fluid Dynamics (CFD) model for gas–liquid two-phase flow is first established, and its quantitative calculation error is ensured to be less than 5%. Then, the spatiotemporal evolution characteristics of flow patterns is analyzed. The real-time pressure drop response is chosen as the representation signal, and its nonlinear features in the time and frequency domain under different flow patterns are explored. A nonlinear time series is constructed by extracting a segment from the real-time pressure drop data, and six entropy measures are applied to analyze and identify them. Finally, the sensitivity of entropy measures to both the time series lengths and the tested sections are evaluated. Results show that there are six typical flow patterns in a U-tube. According to most entropy measures, the bubble flow has the highest complexity; however, the plug flow presents the lowest complexity. In the U-bend, pressure drop signals for the bubble and annular flows show random fluctuations within a specific range, in contrast to the marked periodicity in plug flow signals, while wavy and slug flows exhibit intermittent peak values. Including the upstream and downstream straight pipes in the analysis, rather than focusing solely on the U-bend, significantly increases the complexity of the stratified, plug, and slug flows. Fuzzy entropy is an effective tool for identifying the six flow patterns, demonstrating good resilience to variations in the length of the data series. This characteristic makes it highly useful for real-time identification of flow patterns in the U-bend sections of non-transparent U-tubes, offering considerable potential in chemical equipment.

Optimised Deep Learning for Time-Critical Load Forecasting Using LSTM and Modified Particle Swarm Optimisation

Short-term electric load forecasting is critical for power system planning and operations due to demand fluctuations driven by variable energy resources. While deep learning-based forecasting models have shown strong performance, time-sensitive applications require improvements in both accuracy and convergence speed. To address this, we propose a hybrid model that combines long short-term memory (LSTM) with a modified particle swarm optimisation (mPSO) algorithm. Although LSTM is effective for nonlinear time-series predictions, its computational complexity increases with parameter variations. To overcome this, mPSO is used for parameter tuning, ensuring accurate forecasting while avoiding local optima. Additionally, XGBoost and decision tree filtering algorithms are incorporated to reduce dimensionality and prevent overfitting. Unlike existing models that focus mainly on accuracy, our framework optimises accuracy, stability, and convergence rate simultaneously. The model was tested on real hourly load data from New South Wales and Victoria, significantly outperforming benchmark models such as ENN, LSTM, GA-LSTM, and PSO-LSTM. For NSW, the proposed model reduced MSE by 91.91%, RMSE by 94.89%, and MAPE by 74.29%. In VIC, MSE decreased by 91.33%, RMSE by 95.73%, and MAPE by 72.06%, showcasing superior performance across all metrics.

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.

Predicting e-commerce product prices through the integration of variational mode decomposition and deep neural networks

Product prices frequently manifest nonlinear and nonstationary time-series attributes, indicating potential variations in their behavioral patterns over time. Conventional linear models may fall short in adequately capturing these intricate properties. In addressing this, the present study leverages the adaptive and non-recursive attributes of the Variational Mode Decomposition (VMD) methodology. It employs VMD to dissect the intricate time series into multiple Intrinsic Mode Functions (IMF). Subsequently, a method rooted in the minimum fuzzy entropy criterion is introduced for determining the optimal modal number (K) in the VMD decomposition process. This method effectively mitigates issues related to modal confusion and endpoint effects, thereby enhancing the decomposition efficacy of VMD. In the subsequent phase, deep neural networks (DNN) are harnessed to forecast the identified modes, with the cumulative modal predictions yielding the ultimate e-commerce product price prognostications. The predictive efficacy of the proposed Variational Mode Decomposition-deep neural network (VMD-DNN) decomposition model is assessed on three public datasets, wherein the mean absolute percentage error (MAPE) on the E-commerce Price Prediction Dataset and Online Retail Dataset is notably low at 0.6578 and 0.5414, respectively. This corresponds to a remarkable error reduction rate of 66.5% and 70.4%. Moreover, the VMD-DNN decomposition model excels in predicting e-commerce product prices through DNN, thereby amplifying the VMD decomposition capability by 4%. The VMD-DNN model attains superior results in terms of directional symmetry, boasting the highest Directional Symmetry (DS) score of 86.25. Notably, the forecasted trends across diverse price ranges closely mirror the actual trends.

Drone motion prediction from flight data: a nonlinear time series approach

In this paper, we explore the application of data-driven predictive systems in enhancing unmanned aerial vehicle (UAV) control capabilities. We introduce a new model for predicting the motion of individual drones by utilizing fundamental flight control data. The model aims to improve the autonomy of individual drones and circumvent the complexity of traditional flight control systems, thus eliminating intricate nested controls. The proposed model lays the foundation for studying collective behaviours within a cluster of drones, thereby advancing the research into swarm behaviour exhibited by drones. The research findings demonstrate the potential of data-driven methods in the construction of UAV control systems. In particular, we here show a comparison of the prediction performances between two neural network architectures using real drone flight data involved in various kinds of motions. We explore the utility of using long short term memory (LSTM) and nonlinear autoregressive with exogenous inputs (NARX) family of nonlinear time series models in developing a virtual drone model using real experimental data.

Real-Time Prediction of Multi-Degree-of-Freedom Ship Motion and Resting Periods Using LSTM Networks

This study presents a novel real-time prediction technique for multi-degree-of-freedom ship motion and resting periods utilizing Long Short-Term Memory (LSTM) networks. The primary objective is to enhance the safety and efficiency of shipborne helicopter landings by accurately predicting heave, pitch, and roll data over an 8 s forecast horizon. The proposed method utilizes the LSTM network’s capability to model complex nonlinear time series while employing the User Datagram Protocol (UDP) to ensure efficient data transmission. The model’s performance was validated using real-world ship motion data collected across various sea states, achieving a maximum prediction error of less than 15%. The findings indicate that the LSTM-based model provides reliable predictions of ship resting periods, which are crucial for safe helicopter operations in adverse sea conditions. This method’s capability to provide real-time predictions with minimal computational overhead highlights its potential for broader applications in marine engineering. Future research should explore integrating multi-model fusion techniques to enhance the model’s adaptability to rapidly changing sea conditions and improve the prediction accuracy.

Short-term air quality prediction based on EMD-transformer-BiLSTM

Actual acquired air quality time series data are highly volatile and nonstationary, and accurately predicting nonlinear time series data containing complex noise is an ongoing challenge. This paper proposes an air quality prediction method based on empirical mode decomposition (EMD), a transformer and a bidirectional long short-term memory neural network (BiLSTM), which is good at addressing the ultrashort-term prediction of nonlinear time-series data and shows good performance for application to the air quality dataset of Patna, India (6:00 am on October 3, 2015–0:00 pm on July 1, 2020). The AQI sequence is first decomposed into intrinsic mode functions (IMFs) via EMD and subsequently predicted separately via the improved transformer algorithm based on BiLSTM, where linear prediction is performed for IMFs with simple trends. Finally, the predicted values of each IMF are integrated using BiLSTM to obtain the predicted AQI values. This paper predicts the AQI in Patna with a time window of 5 h, and the RMSE, MAE and MAPE are as low as 5.6853, 2.8230 and 2.23%, respectively. Moreover, the scalability of the proposed model is validated on air quality datasets from several other cities, and the results prove that the proposed hybrid model has high performance and broad application prospects in real-time air quality prediction.

Physical reservoir computing with visible-light signals using dye-sensitized solar cells

Physical reservoir computing (PRC) with visible-light signals was demonstrated using dye-sensitized solar cells. The short-term memory required for PRC was confirmed using light pulse inputs. Waveform learning was demonstrated for nonlinear autoregressive moving-average time series level 2 (NARMA2) signals with normalized mean square error of 0.027. The relatively slow (milliseconds to seconds) and complex charge transfer dynamics in the TiO2 porous layer with redox reactions in the solution phase provided the characteristics required for PRC.

Optimal segmentation of non-linear and non-stationary time series based on fractal dimension and Poincare section and its application in solving EEG-signal

Non-linearity and non-stationarity characteristics in physiological and biological time series limit the reliability of conventional analysis. The main challenge in pattern recognition problems is determining the Optimal-Window-Duration (OWD) for segmenting these time series. Our study presents an innovative algorithm, based on Individual-Optimal-Segmentation (IOS), which integrates chaos theory using fractal dimension, and Poincare section into the estimation of the OWD for segmenting time series with non-linear and non-stationary characteristics. In the logistic map case study, the IOS algorithm proved to be effective in detecting cycles in a range of numerical examples, from period-2 to chaotic behavior. So that its performance was superior to previous approaches, especially when it comes to detecting chaotic behaviors. Furthermore, the combination of the IOS algorithm with the Welch signal processing method shows promise in identifying abnormal Power-Spectral-Density (PSD) curves related to Beta and Gamma rhythms in depression diagnosis scenarios. The generated depression diagnosis system attained an accuracy (%) of 99.07±0.35 (with 2-channel) using the Monte Carlo simulation method. These findings suggest a potentially valuable tool for addressing segmentation challenges in time series analysis, particularly in contexts involving complex, dynamic data. While, it is suggested that future research investigate the computational efficiency and robustness of the IOS algorithm when handling large-scale or real-time data and increase its application in various fields in data science and signal processing. Furthermore, expanding the application range of the IOS algorithm for other pattern recognition problems with physiological or biological data can strengthen the generalizability and practical application of the algorithm.

Construction of battery charge state prediction model for new energy electric vehicles

New energy vehicles have made considerable achievements after nearly a decade of development, but the range and safety of automobile battery packs are still a difficult problem for the industry to overcome. The charge state of the battery can reflect the charging and discharging situation and battery health, so it is significant to construct an accurate prediction model of the battery charge state. In view of this, an RBF neural network is utilized to predict the battery state of charge. Considering that the battery charge state is a nonlinear time series, the study introduces an integrated learning algorithm to further optimize the prediction model. Experimentally, the loss rate of the raised model is almost stable around 150 steps and the loss rate tends to be close to 0. The accuracy of the raised model also reached 81.60%, which is 2.84% and 1.80% higher than the control model, respectively; The recall rate of the raised model reached 77.92%, which was 4.73% and 25.31% higher than the control model, respectively. From the effectiveness test results, the raised model has higher accuracy and faster convergence rate than the control model, and the recall rate of the raised model is also higher, so the raised model has a certain progressiveness.

Complex Function-Based Fault Detection: A New Method to Measure the Complexity of Nonlinear Time Series

In this paper, we propose a new complexity computation based on the complex function. This measure exploits the dispersion Lempel–Ziv complexity (DLZC) and the dispersion statistical complexity measure based on Jensen–Shannon divergence (DCJS) of the analytic signal and constructs the binary complex function, called complex number complexity (CNC). The statistical measure depicts the complex system from different angle and mines more potential information. Through simulation experiments, we prove that the proposed method is able to detect data fluctuations more sensitively and accurately. For an application, the CNC is applied to fault detection which has different fault diameters and rotational speeds. The results deliver that the CNC measure can well represent the complexity of faulty bearings and has significant differences in any two fault types. The technique is effective to characterize different fault.

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.

Parsimonious streamflow forecasting system based on a dynamical systems approach

Operational hydrologic forecasting often relies on the use of physically-based computer models at the catchment scale. However, our understanding of streamflow generation is still evolving and incomplete, and oftentimes linearization assumptions must be made to make physically-based hydrologic and hydraulic models tractable. In addition to uncertainty in the physical runoff-generation mechanisms, many physically-based approaches do not formally assimilate observed streamflow data. Streamflow is their output, deterministically simulated from observed inputs (such as precipitation, temperature, etc.). Divergences between observed and simulated streamflow may be due to input uncertainty, calibration issues, or problems with the inherent physical process representation. Thus, data-driven approaches that are responsive to observed streamflow by including assimilation as an essential feature are promising in operational hydrology. In this study, we use a novel forcings-weighted k-Nearest Neighbor (kNN) nonlinear timeseries forecasting method developed in the context of the modeling of nonlinear dynamical systems. The rainfall-runoff transform is assumed to be a low-dimensional system of (unknown) nonlinear differential equations; the deterministic state space is reconstructed based on the streamflow timeseries; and a forecast is generated based on how “similar” the current hydrograph is to historical events encoded within the reconstructed state space, weighted by the similarity of forcings. We tested the model in a long-term test period with no real data assimilation, and in forecast mode with daily data assimilation. When used for forecasting, we also illustrate two different uncertainty quantification approaches: forecast postprocessing and internal-to-the-model ensemble generation. The method is conceptually and computationally simple, but offers promising results. We tested the model in three different scenarios. First, investigating a long-term test period with no real data assimilation in a rainfall-dominated research catchment at the Coweeta Hydrologic Lab in North Carolina, USA. The simulation exhibited an NSE of 0.73. Next, we included a conceptual snowmelt model to drive the long term simulation and studied boreal-temperate transition zone catchments in northern Minnesota, USA. The study area included a small (<0.1 mi2) reference research catchment at the Marcell Experimental Forest, and a large river basin (>19 k mi2) with incomplete forcings information. The kNN method yielded NSEs from 0.28 to 0.50. And finally, we conducted a pseudo-operational forecasting test for a large river basin in northern Minnesota, USA with a focus on performance forecasting a flood-of-record on the Rainy River. The test involved reforecasting, data assimilation, and implementation of ensemble and postprocessing bias correction uncertainty-quantification schemes. The technique demonstrated good performance capturing a flood-of-record on a large catchment (−5.7 to −3.3 % crest error with 1-day of lead time). For the entire test period in forecast mode, the 1-day lead time NSE was 0.99, going down to 0.79 for the 7-day lead time, showing its general utility as an operational forecasting tool.

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.

Time-Series Feature Extraction by Return Map Analysis and Its Application to Bearing-Fault Detection

Developing new features for time-series characterization is a current challenge in data science and machine learning. In this paper, we propose a new metric based on a simple and efficient algorithm, namely, the return map. The return map analysis is well established in the field of non-linear dynamics, in particular, for fitting the parameters of a chaotic system from a waveform, or to attack a chaotic communication channel. We show that our metrics work well for both non-linear dynamics and time-series feature extraction problems in the field of machine learning. In an experiment aiming to classify vibration signals of normal and damaged bearings, we compare our method with two other methods that reported to have excellent accuracy, based on entropy and statistical feature distribution, respectively. We show that our method achieves higher accuracy with almost the lowest time costs, which was confirmed in experiments with two different datasets containing three main classes of bearings: normal, with inner race faults, and with outer race faults, having different damage origins and recorded in various conditions. In particular, for the dataset supplied by Case Western Reserve University, our method reached an accuracy of 100% at signals of 5000 sample points length, with a total time of 0.4 s required for feature estimation, while the entropy-based method reached an accuracy of 95% with a time of 100 s, and a statistical feature distribution method reached an accuracy of 93% with a total time of 1.9 s. Results show that the developed method is better suited to real-time bearing condition monitoring applications than most of the methods reported to date.

Modeling Exchange Rate of Naira to Euro with the APLSTAR-GARCH model

Application of the asymmetric power logistic smooth transition autoregressive (APLSTAR) model proposed by [1] to naira/Euro exchange rate spanning from January, 2006 to April, 2021, which is a nonlinear macroeconomic time series was considered. The APLSTAR model was justifiably fitted to the series and the fit of the APLSTAR model compared with the fits of the competing models revealed that the APLSTAR model fits the data exchange rate of naira to Euro better than the other asymmetric STAR models. Lagrange Multiplier tests for autoregressive conditional heteroscedastic (ARCH) effects were carried out and there was no substantial evidence to reject the presence of ARCH effects in the set of residuals used. Hence, we compared hybrid smooth transition autoregressive-generalized ARCH (STAR-GARCH) models using model evaluation criteria. On balance, the APLSTAR-GARCH (0, 1) model outperforms the other models under consideration.