Nonlinear Time Series Analysis Research Articles

Package AdvEMDpy: Algorithmic variations of empirical mode decomposition in Python

AbstractThis work presents a $\textsf{Python}$ EMD package named AdvEMDpy that is both more flexible and generalises existing empirical mode decomposition (EMD) packages in $\textsf{Python}$ , $\textsf{R}$ , and $\textsf{MATLAB}$ . It is aimed specifically for use by the insurance and financial risk communities, for applications such as return modelling, claims modelling, and life insurance applications with a particular focus on mortality modelling. AdvEMDpy both expands upon the EMD options and methods available, and improves their statistical robustness and efficiency, providing a robust, usable, and reliable toolbox. Unlike many EMD packages, AdvEMDpy allows customisation by the user, to ensure that a broader class of linear, non-linear, and non-stationary time series analyses can be performed. The intrinsic mode functions (IMFs) extracted using EMD contain complex multi-frequency structures which warrant maximum algorithmic customisation for effective analysis. A major contribution of this package is the intensive treatment of the EMD edge effect which is the most ubiquitous problem in EMD and time series analysis. Various EMD techniques, of varying intricacy from numerous works, have been developed, refined, and, for the first time, compiled in AdvEMDpy. In addition to the EMD edge effect, numerous pre-processing, post-processing, detrended fluctuation analysis (localised trend estimation) techniques, stopping criteria, spline methods, discrete-time Hilbert transforms (DTHT), knot point optimisations, and other algorithmic variations have been incorporated and presented to the users of AdvEMDpy. This paper and the supplementary materials provide several real-world actuarial applications of this package for the user’s benefit.

Temporary disruption in language processing reflected as multiscale temporal discoordination in a recurrent network

By juxtaposing time series analyses of activity measured from a fully recurrent network undergoing disrupted processing and of activity measured from a continuous meta-cognitive report of disruption in real-time language comprehension, we present an opportunity to compare the temporal statistics of the state-space trajectories inherent to both systems. Both the recurrent network and the human language comprehension process appear to exhibit long-range temporal correlations and low entropy when processing is undisrupted and coordinated. However, when processing is disrupted and discoordinated, they both exhibit more short-range temporal correlations and higher entropy. We conclude that by measuring human language comprehension in a dense-sampling manner similar to how we analyze the networks, and analyzing the resulting data stream with nonlinear time series analysis techniques, we can obtain more insight into the temporal character of these discoordination phases than by simply marking the points in time at which they peak.

Temporary disruption in language processing reflected as multiscale temporal discoordination in a recurrent network

By juxtaposing time series analyses of activity measured from a fully recurrent network undergoing disrupted processing and of activity measured from a continuous meta-cognitive report of disruption in real-time language comprehension, we present an opportunity to compare the temporal statistics of the state-space trajectories inherent to both systems. Both the recurrent network and the human language comprehension process appear to exhibit long-range temporal correlations and low entropy when processing is undisrupted and coordinated. However, when processing is disrupted and discoordinated, they both exhibit more short-range temporal correlations and higher entropy. We conclude that by measuring human language comprehension in a dense-sampling manner similar to how we analyze the networks, and analyzing the resulting data stream with nonlinear time series analysis techniques, we can obtain more insight into the temporal character of these discoordination phases than by simply marking the points in time at which they peak.

Lyapunov exponents for temporal networks.

By interpreting a temporal network as a trajectory of a latent graph dynamical system, we introduce the concept of dynamical instability of a temporal network and construct a measure to estimate the network maximum Lyapunov exponent (nMLE) of a temporal network trajectory. Extending conventional algorithmic methods from nonlinear time-series analysis to networks, we show how to quantify sensitive dependence on initial conditions and estimate the nMLE directly from a single network trajectory. We validate our method for a range of synthetic generative network models displaying low- and high-dimensional chaos and finally discuss potential applications.

Nonlinear features in whistles produced by the short-beaked common dolphin (Delphinus delphis) off southeastern Brazil.

Animal vocalizations have nonlinear characteristics responsible for features such as subharmonics, frequency jumps, biphonation, and deterministic chaos. This study describes the whistle repertoire of a short-beaked common dolphin (Delphinus delphis) group at Brazilian coast and quantifies the nonlinear features of these whistles. Dolphins were recorded for a total of 67 min around Cabo Frio, Brazil. We identify 10 basic categories of whistle, with 75 different types, classified according to their contour shape. Most (45) of these 75 types had not been reported previously for the species. The duration of the whistles ranged from 0.04 to 3.67 s, with frequencies of 3.05-29.75 kHz. Overall, the whistle repertoire presented here has one of the widest frequency ranges and greatest level of frequency modulation recorded in any study of D. delphis. All the nonlinear features sought during the study were confirmed, with at least one feature occurring in 38.4% of the whistles. The frequency jump was the most common feature (29.75% of the whistles) and the nonlinear time series analyses confirmed the deterministic chaos in the chaotic-like segments. These results indicate that nonlinearities are a relevant characteristic of these whistles, and that are important in acoustic communication.

Characterizing the statistical complexity of nonlinear time series via ordinal pattern transition networks

Complex network approaches have lately emerged as innovative and complementary notions of nonlinear time series analysis, capable of revealing numerous aspects that more classic analytic methods fail to reveal. In this paper, we concentrate on ordinal pattern transition networks (OPTN) to characterize the statistical complexity of time series by considering each pattern as a network node and the probability of transition between patterns as a directed weighted edge between nodes. The traditional complexity measures of time series, permutation entropy (PE) and transition entropy (TE), have been first explained via OPTN, and a new statistical complexity technique termed permutation weighted statistical transition entropy (PWSTE) is suggested to address some drawbacks of some existing complexity measures. In the context of OPTN, the proposed PWSTE enhances PE and TE by taking into account both static information (i.e., network node probability distributions) and dynamic information (i.e., time series transition probabilities represented by nodes). It combines with the imbalance of OPTN, which enables a more accurate measurement of the statistical complexity of nonlinear time series. The suggested new technique for assessing statistical complexity is thoroughly described based on the simulated time series, and the findings demonstrate that the proposed algorithm is more sensitive and effective in identifying dynamic changes. When the proposed algorithm is used to medical data, the EEG time series of epileptic patients are examined to define the changes in EEG data over various stages. The findings demonstrate that the proposed PWSTE algorithm outperforms conventional methods in identifying the stage in which a patient’s EEG data is located.

Challenges and perspectives in recurrence analyses of event time series

The analysis of event time series is in general challenging. Most time series analysis tools are limited for the analysis of this kind of data. Recurrence analysis, a powerful concept from nonlinear time series analysis, provides several opportunities to work with event data and even for the most challenging task of comparing event time series with continuous time series. Here, the basic concept is introduced, the challenges are discussed, and the future perspectives are summarized.

A Mixture Autoregressive Model Based on an Asymmetric Exponential Power Distribution

In nonlinear time series analysis, the mixture autoregressive model (MAR) is an effective statistical tool to capture the multimodality of data. However, the traditional methods usually need to assume that the error follows a specific distribution that is not adaptive to the dataset. This paper proposes a mixture autoregressive model via an asymmetric exponential power distribution, which includes normal distribution, skew-normal distribution, generalized error distribution, Laplace distribution, asymmetric Laplace distribution, and uniform distribution as special cases. Therefore, the proposed method can be seen as a generalization of some existing model, which can adapt to unknown error structures to improve prediction accuracy, even in the case of fat tail and asymmetry. In addition, an expectation-maximization algorithm is applied to implement the proposed optimization problem. The finite sample performance of the proposed approach is illustrated via some numerical simulations. Finally, we apply the proposed methodology to analyze the daily return series of the Hong Kong Hang Seng Index. The results indicate that the proposed method is more robust and adaptive to the error distributions than other existing methods.

Nonlinear Analysis of Financial Time Series

Nonlinear Analysis of Financial Time Series

Effects of ambient temperature on mortality among elderly residents of Chengdu city in Southwest China, 2016–2020: a distributed-lag non-linear time series analysis

BackgroundWith complex changes in the global climate, it is critical to understand how ambient temperature affects health, especially in China. We aimed to assess the effects of temperature on daily mortality, including total non-accidental, cardiovascular disease (CVD), respiratory disease, cerebrovascular disease, and ischemic heart disease (IHD) mortality between 2016 and 2020 in Chengdu, China.MethodsWe obtained daily temperature and mortality data for the period 2016–2020. A Poisson regression model combined with a distributed-lag nonlinear model was used to examine the association between temperature and daily mortality. We investigated the effects of individual characteristics by sex, age, education level, and marital status.ResultsWe found significant non-linear effects of temperature on total non-accidental, CVD, respiratory, cerebrovascular, and IHD mortality. Heat effects were immediate and lasted for 0–3 days, whereas cold effects persisted for 7–10 days. The relative risks associated with extreme high temperatures (99th percentile of temperature, 28 °C) over lags of 0–3 days were 1.22 (95% confidence interval [CI]: 1.17, 1.28) for total non-accidental mortality, 1.40 (95% CI: 1.30, 1.50) for CVD morality, 1.34 (95% CI: 1.24, 1.46) for respiratory morality, 1.33 (95% CI: 1.20, 1.47) for cerebrovascular mortality, and 1.38 (95% CI: 1.20, 1.58) for IHD mortality. The relative risks associated with extreme cold temperature (1st percentile of temperature, 3.0 °C) over lags of 0–14 days were 1.32 (95% CI: 1.19, 1.46) for total mortality, 1.45 (95% CI: 1.24, 1.68) for CVD morality, 1.28 (95% CI: 1.09, 1.50) for respiratory morality, 1.36 (95% CI: 1.09, 1.70) for cerebrovascular mortality, and 1.26 (95% CI: 0.95, 1.68) for IHD morality. We found that hot and cold affects were greater in those over 85 years of age, and that women, individuals with low education levels, and those who were widowed, divorced, or never married, were more vulnerable.ConclusionsThis study showed that exposure to hot and cold temperatures in Chengdu was associated with increased mortality, with people over 85 years old, women, those with low education levels, and unmarried individuals being more affected by hot and cold temperatures.

Forecasting Carbon Price Using Double Shrinkage Methods.

It is commonly recognized that setting a reasonable carbon price can promote the healthy development of a carbon trading market, so it is especially important to improve the accuracy of carbon price forecasting. In this paper, we propose and evaluate a hybrid carbon price prediction model based on so-called double shrinkage methods, which combines factor screening, dimensionality reduction, and model prediction. In order to verify the effectiveness and superiority of the proposed model, this paper takes data from the Guangdong carbon trading market for empirical analysis. The sample interval is from 5 August 2013 to 25 March 2022. Based on the results of the empirical analysis, several main findings can be summarized. First, the double shrinkage methods proposed in this paper yield more accurate prediction results than various alternative models based on the direct application of factor screening methods or dimensionality reduction methods, when comparing R2, root-mean-square error (RMSE), and root absolute error (RAE). Second, LSTM-based double shrinkage methods have superior prediction performance compared to LR-based double shrinkage methods. Third, these findings are robust with the use of normalized data, different data frequencies, different carbon trading markets, and different dataset divisions. This study provides new ideas for carbon price prediction, which might have a theoretical and practical contributions to complex and non-linear time series analysis.

Commodity Price Pass-Through and Inflation in Japan: a Nonlinear Time Series Analysis

Commodity Price Pass-Through and Inflation in Japan: a Nonlinear Time Series Analysis

Nonlinear time-series analysis on the causality response of phytoplankton in a tributary of the Three Gorges Reservoir

水华的频发已成为当前三峡水库最为突出的生态环境问题之一。尽管水动力调控叠加上温度变暖和营养负荷增加会诱导水华暴发强度和频率增加,但仍缺乏有效的方法框架去利用野外观测数据评估环境因子与浮游植物间的因果关联。本研究以三峡水库澎溪河监测数据为例,采用非线性时序分析的建模框架来量化浮游植物的因果响应规律。数据来自于2007年6月至2018年9月澎溪河流域的高阳平湖和汉丰湖两个观测点,其中包含了水文、气象和水质及叶绿素a等11种变量。首先,利用奇异谱分析(SSA)分离了叶绿素a和环境因子的低维确定性动力学信号;其次,采用收敛交叉映射(CCM)方法检验了叶绿素a与环境因子间的因果关联。结果显示:① 气象因子、支流流量、水温、三峡大坝水位和上游调节坝水位是影响高阳平湖叶绿素a时序变化的重要因素;② 总氮、总磷为代表的营养盐只在汉丰湖观测点中表现出与叶绿素a的因果关系,且总氮较之于总磷对叶绿素a变化影响更为显著;③ CCM结果与传统的皮尔森相关性分析及格兰杰因果检验比较,证实非线性时序分析方法在分析浮游植物的因果响应上更具优势。本研究为水生态系统的因果建模提供了研究范例,也为推动利用长期观测数据评估三峡水库水华驱动因子提供了新的研究角度。;Harmful algae blooms in Three Gorges Reservoir (TGR) have become the most major environmental and ecological problem. Despite broad recognition that hydrological manipulation, warming climate and nutrient loading promote the intensity and frequency of bloom events, the main problem is still no science-based framework for evaluating the causal relationships between environmental factors and phytoplankton biomass by using field observation data. Thus, as a case study, a nonlinear time-series causality analysis framework was used to reveal the phytoplankton response in the TGR, China. Two sampling stations, named Lake Gaoyangping and Lake Hanfeng, were used to reconstruct the dynamics of real-world environmental systems with eleven parameters from June 2007 to September 2018. Singular spectrum analysis isolated low-dimensional, deterministic signals for chlorophyll-a(Chl.a) dynamics and other candidate drivers. Causal analysis with convergent cross-mapping (CCM) provided strong evidence that temperature, sunshine hours, precipitation, discharge in the tributary, and water level of TGR systematically influenced phytoplankton biomass dynamics in Lake Gaoyangping. In contrast, total nitrogen (TN) and total phosphorus (TP) showed a causal relationship with Chl.a only in Lake Hanfeng, and the effect of TN on Chl.a was more significant than that of TP. Finally, the research compares the nonlinear CCM analysis with the linear Pearson correlation analysis and Granger causality test. It confirms that the nonlinear causality method has more advantage in causal discovery based on long time-series monitoring data. Overall, this study provides a case for causal modelling of water ecosystems and a new research perspective for evaluating the environmental drivers of harmful algae blooms in TGR by using long-term observational data.

Non-linear time series analysis of Parkinsonian tremor signals

Nonlinear time series analysis of Parkinsonian tremor signals involves exploring the complex dynamics and nonlinear interactions within the tremor signals to gain insights into the underlying physiological processes. By applying advanced analytical techniques, researchers aim to uncover hidden patterns, chaotic behavior, and self-organization within the signals, which can provide valuable information for the diagnosis and monitoring of Parkinson's disease. The vibrational phenomena studied in this work regards the arm and forearm vibration with the purpose to detect and recognize the dynamic properties and correlations of onset of pathological tremor in patients affected by Parkinson disease. Experimental data measured by patients will be analyzed using multiscale recursive analysis methodologies through the TISEAN package (1). Recurrence plot analysis is a method that visually represents the recurrent patterns and dependencies within a time series and helps to identify important features such as deterministic dynamics, periodicity, and phase relationships (2,3).

A fast approach of implementing the Fourier decomposition method for nonlinear and non-stationary time series analysis

The Fourier Decomposition Method (FDM) is an advanced tool to gather information about signals from nonlinear and/or non-stationary systems. It decomposes a signal into a finite set of zero-mean band-limited oscillation modes, so-called analytic Fourier intrinsic band functions (AFIBFs). Owing to its amplitude and frequency modulation properties, each AFIBF enables local analysis of signals. Thus, the determination of AFIBFs is of the key point in performing the FDM. In the traditional case, AFIBFs are obtained iteratively by evaluating numerous inverse discrete Fourier transforms (IDFTs). Also, phase calculations and unwrapping operations must be performed on IDFTs to examine the positivity of instantaneous frequencies. Hence, the classical FDM suffers from heavy computational burden, making it challenging to analyze large-size signals. This paper proposes a new approach to implement the FDM faster than its traditional one, without the need for phase calculation, unwrapping and derivation operations, and also exploits the computational efficiency of the inverse fast Fourier transform. Despite its low computational cost and thus improved computation speed, the new approach decomposes the signal into the same AFIBFs as the conventional one. Simulation results show that the proposed approach outperforms its traditional counterpart in terms of both computational complexity and computation time.

Experimental evidence that rill-bed morphology is governed by emergent nonlinear spatial dynamics

Past experimental work found that rill erosion occurs mainly during rill formation in response to feedback between rill-flow hydraulics and rill-bed roughness, and that this feedback mechanism shapes rill beds into a succession of step-pool units that self-regulates sediment transport capacity of established rills. The search for clear regularities in the spatial distribution of step-pool units has been stymied by experimental rill-bed profiles exhibiting irregular fluctuating patterns of qualitative behavior. We hypothesized that the succession of step-pool units is governed by nonlinear-deterministic dynamics, which would explain observed irregular fluctuations. We tested this hypothesis with nonlinear time series analysis to reverse-engineer (reconstruct) state-space dynamics from fifteen experimental rill-bed profiles analyzed in previous work. Our results support this hypothesis for rill-bed profiles generated both in a controlled lab (flume) setting and in an in-situ hillside setting. The results provide experimental evidence that rill morphology is shaped endogenously by internal nonlinear hydrologic and soil processes rather than stochastically forced; and set a benchmark guiding specification and testing of new theoretical framings of rill-bed roughness in soil-erosion modeling. Finally, we applied echo state neural network machine learning to simulate reconstructed rill-bed dynamics so that morphological development could be forecasted out-of-sample.

Nonlinear time series analysis of limestone rock failure process

Nonlinear analysis of the rock mass failure process is a powerful tool for the study of instability and prevention of disaster in rock mass engineering. Nonlinear analysis has rapidly developed in recent decades. We propose a new method to determine the fractal characteristics of the failure process using time series analysis based on an iterative cycle division of the box dimension of fractal theory. We use AE technology to carry out time series analysis of fracture sources in the failure process of rock mass. We show that the disordered failure of rock mass exhibits notable fractal characteristics in the time series. The advantage of this method is that only the time and frequency of AE monitoring are used, resulting in a better identification and classification of damage, compared with traditional natural sequence frequency distribution. In addition, MATLAB numerical simulation analysis shows that these time series fractal dimensions can accurately characterize the failure characteristics of rock mass. In brief, we show the potential for future orderly time analyses of the complex and disorderly process of rock mass failure.

An Acceleration-Based Nonlinear Time-Series Analysis of Effects of Robotic Walkers on Gait Dynamics During Assisted Walking

This article aimed to identify the effects of a robotic walker on gait dynamics during assisted walking via the analysis of acceleration data. Walking experiments were conducted with 11 healthy young subjects under different conditions, including normal, rollator-assisted, and robotic-walker-assisted walking. We collected acceleration data from the trunk and pelvis to extract measures for evaluating the properties of gait dynamics, including periodicity, complexity, and determinism. These measures include maximum Lyapunov exponent (MLE), multiscale entropy (MSE), multiscale permutation entropy (MPE), correlation dimension (CD), and percentage of determinism (%DET, from recurrence plot). Significant differences in the complex measures, MSE and MPE between normal and walker-assisted walking for trunk acceleration data in anteroposterior (AP) and mediolateral (ML) directions were observed. The determinism measures, CD and %DET, demonstrated significant differences between normal and robotic-walker-assisted walking for trunk acceleration data in the AP direction, while significant differences in the periodicity measures and MLE were not observed between the three conditions. These results imply mostly unchanged local dynamic stability, reduced deterministic patterns, and increased complexity in the gait dynamics of healthy young subjects, especially for the upper body. The robotic walker’s assistance is considered appropriate because of no significant difference between rollator-assisted and robotic-walker-assisted walking. This study provides new insights into the biomechanical effects of the robotic walker on gait. It also suggests that the used methods can be applied to the assistance evaluation of walking assistive devices like lower-limb exoskeletons and assess the ability to resist perturbations with a robotic walker for rehabilitation applications such as perturbation-based gait training.

Validation of a non-linear index of heart rate variability to determine aerobic and anaerobic thresholds during incremental cycling exercise in women

Studies highlight the usage of non-linear time series analysis of heart rate variability (HRV) using the short-term scaling exponent alpha1 of Detrended Fluctuation Analysis (DFA-alpha1) during exercise to determine aerobic and anaerobic thresholds. The present study aims to further verify this approach in women. Gas exchange and HRV data were collected from 26 female participants with different activity levels. Oxygen uptake (VO2) and heart rate (HR) at first (VT1) and second ventilatory thresholds (VT2) were compared with DFA-alpha1-based thresholds 0.75 (HRVT1) and 0.50 (HRVT2). Results: VO2 at VT1 and VT2 were 25.2 ml/kg/min (± 2.8) and 31.5 ml/kg/min (± 3.6) compared with 26.5 ml/kg/min (± 4.0) and 31.9 ml/kg/min (± 4.5) for HRVT1 and HRVT2, respectively (ICC3,1 = 0.77, 0.84; r = 0.81, 0.86, p < 0.001). The mean HR at VT1 was 147 bpm (± 15.6) and 167 bpm (± 12.7) for VT2, compared with 152 bpm (± 15.5) and 166 bpm (± 13.2) for HRVT1 and HRVT2, respectively (ICC3,1 = 0.87, 0.90; r = 0.87, 0.90, p < 0.001). Bland–Altman analysis for VT1 vs. HRVT1 showed a mean difference of − 1.3 ml/kg/min (± 2.4; LoA: 3.3, − 6.0 ml/kg/min) for VO2 and of − 4.7 bpm (± 7.8; LoA: 10.6, − 20.0 bpm) for HR. VT2 vs. HRVT2 showed a mean difference of − 0.4 ml/kg/min (± 2.3; LoA: 4.1, − 4.9 ml/kg/min) for VO2 and 0.5 bpm (± 5.7; LoA: 11.8, − 10.8 bpm) for HR. DFA-alpha1-based thresholds showed good agreement with traditionally used thresholds and could be used as an alternative approach for marking organismic transition zones for intensity distribution in women.

Recurrence flow measure of nonlinear dependence

Couplings in complex real-world systems are often nonlinear and scale dependent. In many cases, it is crucial to consider a multitude of interlinked variables and the strengths of their correlations to adequately fathom the dynamics of a high-dimensional nonlinear system. We propose a recurrence-based dependence measure that quantifies the relationship between multiple time series based on the predictability of their joint evolution. The statistical analysis of recurrence plots (RPs) is a powerful framework in nonlinear time series analysis that has proven to be effective in addressing many fundamental problems, e.g., regime shift detection and identification of couplings. The recurrence flow through an RP exploits artifacts in the formation of diagonal lines, a structure in RPs that reflects periods of predictable dynamics. Using time-delayed variables of a deterministic uni-/multivariate system, lagged dependencies with potentially many time scales can be captured by the recurrence flow measure. Given an RP, no parameters are required for its computation. We showcase the scope of the method for quantifying lagged nonlinear correlations and put a focus on the delay selection problem in time-delay embedding which is often used for attractor reconstruction. The recurrence flow measure of dependence helps to identify non-uniform delays and appears as a promising foundation for a recurrence-based state space reconstruction algorithm.