Analysis Of Nonstationary Time Series Research Articles

Weak convergence of the conditional single index $ U $-statistics for locally stationary functional time series

<abstract><p>In recent years, there has been a notable shift in focus towards the analysis of non-stationary time series, driven largely by the complexities associated with delineating significant asymptotic behaviors inherent to such processes. The genesis of the theory of locally stationary processes arises from the quest for asymptotic inference grounded in nonparametric statistics. This paper endeavors to formulate a comprehensive framework for conducting inference within the realm of locally stationary functional time series by harnessing the conditional $ U $-statistics methodology as propounded by W. Stute in 1991. The proposed methodology extends the Nadaraya-Watson regression function estimations. Within this context, a novel estimator was introduced for the single index conditional $ U $-statistics operator, adept at accommodating the non-stationary attributes inherent to the data-generating process. The primary objective of this paper was to establish the weak convergence of conditional $ U $-processes within the domain of locally stationary functional mixing data. Specifically, the investigation delved into scenarios of weak convergence involving functional explanatory variables, considering both bounded and unbounded sets of functions while adhering to specific moment requirements. The derived findings emanate from broad structural specifications applicable to the class of functions and models under scrutiny. The theoretical insights expounded in this study constitute pivotal tools for advancing the domain of functional data analysis.</p></abstract>

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.

Application of singular spectrum analysis to nonstationary time series in flow-induced vibrations of a circular cylinder

In numerical simulations of flow-induced vibrations (FIV) of a circular cylinder, abundant time series data are available, including cylinder displacement and acting forces. Singular spectrum analysis (SSA) is employed to deal with nonstationary multi-component time series produced in two FIV cases with proper interpretation in physics. In the first case, the cylinder displacement time series is decomposed into two oscillatory components using SSA. The instantaneous frequencies of these two components are obtained by Hilbert transform (HT) and found to be in agreement with the wavelet transform of the cylinder displacement. In the second case, three oscillatory components are extracted from the cylinder displacement time series by SSA. The dominant component is characterized by steady oscillations at the vortex shedding frequency, which suggests a relatively steady vortex shedding process behind the rear cylinder. In contrast, the second component, which is closely associated with the alternate boundary layer separations from the front cylinder, features in the increasing amplitude with time. This implies that the unsteady flow field in the gap might be attributed to the nonstationary cylinder oscillations. This work demonstrates that SSA, in conjunction with HT, enables a comprehensive time-frequency analysis of nonstationary time series obtained in FIV.

Markov-Switching Models with Unknown Error Distributions: Identification and Inference Within the Bayesian Framework

Abstract The basic Markov-switching model has been extended in various ways ever since the seminal work of Hamilton (1989. “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle.” Econometrica 57: 357–84). However, the estimation of Markov-switching models in the literature has relied upon parametric assumptions on the distribution of the error term. In this paper, we present a Bayesian approach for estimating Markov-switching models with unknown and potentially non-normal error distributions. We approximate the unknown distribution of the error term by the Dirichlet process mixture of normals, in which the number of mixtures is treated as a parameter to estimate. In doing so, we pay special attention to the identification of the model. We then apply the proposed model and MCMC procedure to the growth of the postwar U.S. industrial production index. Our model can effectively control for irregular components that are not related to business conditions. This leads to sharp and accurate inferences on recession probabilities.

Reliable Detection Method of Variable Series Arc Fault in Building Integrated Photovoltaic Systems Based on Nonstationary Time Series Analysis

Arc fault characteristics would be influenced by diverse system elements and operation levels in building integrated photovoltaic (BIPV) systems, which would challenge the arc fault detection. This paper aims at improving the detection reliability and speed by applying the time series analysis at feature and classification layers to reflect the random evolution process of arc fault current. According to the arc ignition method in UL1699B, arc fault signals are firstly sampled with a residential storage element and ten DC load elements at three voltage levels in BIPV systems. After evaluating time-frequency information with established indexes, variable and early arc fault are discovered to have no obvious current and harmonic indications due to strong system element noises, sensor precision and low arc energy. Then, sample entropy (SE) feature and gated recurrent unit (GRU) classifier are proposed to improve the detection performance through involving the nonstationary time series correlation of the arc fault current. An arc fault simulation platform is built to generate arc current with low signal-to-noise ratio (SNR), which proves the detection algorithm integrates multi-source information to grasp the appearance of new signal pattern. Next, constrained optimization problems are designed for both feature and classifier layers to achieve the most distinguishable arc fault and fault-like performance. With matched optimization methods, the modified SE and GRU method could improve the detection accuracy by an average of 21.08% after comparing with existing features and classifiers. Finally, hardware implementation results indicate a higher detection accuracy and less runtime of the proposed detection method.

Characterization of infrasonic wind noise using the Discrete Orthonormal Stockwell Transform

The Discrete Orthonormal Stockwell Transform (DOST) (Stockwell, 2007) projects a finite-length, periodically sampled signal onto a set of orthonormal basis vectors that are localized in time and scale such that all basis functions at each scale are represented by a finite number of Discrete Fourier Transform (DFT) basis functions. It is based on the Stockwell Transform (Stockwell et al., 1996), sometimes referred to as the S-transform, which is applicable to continuously defined waveforms. Furthermore, the DFT basis functions corresponding to each DOST scale are non-overlapping. The shorter DOST scales correspond to higher-frequency DFT basis functions and greater bandwidths than the longer DOST scales. Thus, the DOST is a type of time-frequency transform and is suitable for non-stationary time-series analysis. Additionally, the DOST of an N-point record can be calculated efficiently in order N log N operations (Wang and Orchard, 2009). Several applications of the S-transform and the DOST have been documented including beamforming with array data (Collar and Frazier, 2022; Frazier, 2022). In this presentation, the DOST used is to examine the statistics of data measured with infrasound sensors as a function of the DOST scales, adaptively characterize the amplitude distributions of the DOST coefficients at each scale and perform near-optimal detection of transient signals. Results corresponding to infrasound data recorded under various conditions are presented.

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.

Optimal selection of wavelet transform parameters for spatio-temporal analysis based on non-stationary NDVI MODIS time series in Mediterranean region

The multi-resolution analysis based on wavelet transform (WT) has widely proved its effectiveness for the analysis of non-stationary time series (TS) in the remote sensing field. However, there are three essential parameters affecting wavelet-aided data in vegetation dynamics detection. This study proposes a new methodology that serves for the proper selection of the most critical parameters to carry out an optimal analysis for vegetation seasonal and long-term dynamics, specifically adapted to NDVI TS. The specific objectives are: (1) to simulate a TS dataset to understand the best WT parameter performance for trend and seasonal changes detection, and (2) to assess the impact of those WT parameters using NDVI MODIS TS in the Mediterranean area along the period 2001–2019. Different types of trends and WT were considered (i.e., Discrete WT (DWT) and Maximal Overlap DWT (MODWT)) in the simulated dataset. The best WT parameters were calculated based on an iterative algorithm that computes the trend and seasonal components using the optimal level of decomposition for each TS and mother wavelet (MW). Then, the findings obtained from the simulated data were exploited using Earth observation data. In the step, the Theil–Sen’s slope (Q) derived for the best MWs was calculated to provide the trend magnitude and direction. The z-score and energy to Shannon entropy ratio (R(S)) were computed to select the most adequate MW based on pixel characterization. On one hand, the simulated results proved the Multi-resolution analysis DWT (MRA-DWT) as the most optimal wavelet type. Level 5 of decomposition was obtained in 99% and 72% of simulated TS according to the trend and seasonal analysis, respectively. Symlet family gave the optimal results in 92% (trend) and 64% (seasonal) of simulated TS. On the other hand, based on R(S), sym12 MW from Symlet family showed the most dominant wavelet for forest, savanna and mosaic land covers, followed by db14 MW from Daubechies family which was dominated in cropland. From more practical adaptation, sym12 was revealed as an effective solution for all the pixels in the Mediterranean area with a relative error lower than 15%. The vegetation change analysis in the Mediterranean area showed a relationship between QNDVI and precipitation in almost 37% of the area, being Turkey, Greece and North Africa the regions with a high correspondence with precipitation. Around 5% of the study area showed to be possibly affected by land degradation processes.

Econometric analysis and modeling of the dynamics of the balance of payments’ development in Azerbaijan

Purpose of the study. The study is devoted to econometric analysis and modeling of the dynamics of the balance of payments’ development of Azerbaijan, the formation of a mathematical and statistical trend that can give a perspective assessment of the development of the balance of payments. In accordance with the goal, the tasks of choosing the best composition of explanatory factors for the model were set, using the characteristics and criteria of correlation and regression analysis, econometric tests, calculating estimates of the nature and closeness of the relationship between the explanatory factors, dependent and independent factors, testing the stationarity of the series.Materials and methods. The official statistical data of the State Statistics Committee and the Central Bank of Azerbaijan, scientific works and studies of scientists, specialists, both Azerbaijani and foreign, in the fields of economics, mathematical and economic modeling were used. For the empirical analysis of non-stationary time series, statistical methods of information processing are used inthe work; to check the adequacy and test the multivariate model, the appropriate criteria and modern econometric procedures are used, taking into account the impact of exogenous factors. For calculations, application packages such as Excel and Eviews 8 were used.Results. A multivariate regression model has been created that makes it possible to conduct an economic and statistical analysis of the dynamics of the current account of the balance of payments; the form and directions of the functional relationship between dependent and independent variables were determined, variability of variables was estimated, the results of multivariate regression analysis using econometric methods were analyzed; the quantitative characteristics of the mechanisms of influence of explanatory factors on the balance of payments were measured and interpreted; correlation dependencies for causal dependencies were investigated in the model, the Granger test was performed and factors were identified that reliably explain the outcome with high probabilities based on the Fisher criterion; the stationarity of the model was measured based on the Dickey-Fuller test. With differences of the first and second degree, the stationarity of the autoregressive model was determined based on the Student’s criterion by changing the lag value. In the process of modeling, the initially constructed model, covering the years 1995-2017 with five factors such as foreign investment, exports, imports, manat exchange rate, general investments, showed insufficient adequacy, that is, non-stationarity of the current account series of the balance of payments. The exchange rate of the national currency, which is involved in the model as an explanatory factor, subjected the values of the dependent series to large fluctuations, an increase in the variance in the residue, which created non-stationarity and which can be explained by the denomination of the national currency in 2006. In the next step, the period covering 2006-2017 was examined. In addition, in the process of research, independent factors were added to the model, such as state budget deficit and foreign exchange reserves. As a result, a multifactorial econometric model was created. Conclusion. The constructed autoregressive model is quite adequate, demonstrates stationarity for the time series of the dependent variable and can be considered suitable for predictive values of the current account of the balance of payments. To develop specific recommendations for the long-term development of the balance of payments, the results of the study, substantiated by the analysis of the dynamics of the development of the balance of payments, make it possible to identify real trends in the balance of payments of Azerbaijan on the current account and determine its interdependence with other macroeconomic variables.

Package AdvEMDpy: Algorithmic Variations of Empirical Mode Decomposition in Python

This paper introduces a new Empirical Mode Decomposition (EMD) Python package called AdvEMDpy that is demonstrably more flexible and which generalises in numerous important ways the existing EMD packages available in Python, R, and MATLAB. The extensions introduced by this AdvEMDpy package both significantly improve the options available to the modeller when applying variations of the EMD methodology as well as also improving the statistical robustness and efficiency of methods otherwise available in existing packages. Unlike many of the functions available for the EMD procedure in existing packages, this EMD package, AdvEMDpy, was developed for customisation by the user, to ensure that a broader class of linear, non-linear, and non-stationary time series analysis could be performed. The intrinsic mode functions (IMFs) extracted from various time series contain multi-frequency implicit structures which warrant further study and as such maximum customisability and intricate knowledge of the algorithm and algorithmic variations is required for users to exploit the full power of the EMD approach to time series decomposition. The edge effect is the most familiar and troublesome problem present in EMD. Various techniques, of varying intricacy from numerous works, have been developed, refined, and offered in AdvEMDpy. In addition to edge effects, numerous preprocessing, post-processing, detrended fluctuation analysis techniques, stopping criteria, spline methods, discrete-time Hilbert transforms (DTHT), knot point optimisations, and algorithmic variations have been incorporated and exposed to the user of AdvEMDpy for fine-tuning of applications of EMD.

Detrended Correlogram Method for Non-Stationary Time-Series Analysis

A correlogram is a statistical tool that is used to check time-series memory by computing the auto-correlation coefficient as a function of the time lag. If the time-series has no memory, then the auto-correlation must be close to zero for any time lag, otherwise if there is a memory, then the auto-correlations must be significantly different from zero. Therefore, based on the robust detrended cross-correlation coefficient, [Formula: see text], we propose the detrended correlogram method in this paper, which will be tested for some time-series (simulated and empirical). This new statistical tool is able to visualize a complete map of the auto-correlation for many time lags and time-scales, and can therefore analyze the memory effect for any time-series.

Empirical Fourier decomposition: An accurate signal decomposition method for nonlinear and non-stationary time series analysis

Signal decomposition is an effective tool to assist identification of modal information in time-domain signals. Two signal decomposition methods, including the empirical wavelet transform (EWT) and Fourier decomposition method (FDM), have been developed based on Fourier theory. However, the EWT can suffer from a mode mixing problem for signals with closely-spaced modes and from a trivial component problem resulting in a trivial residual in the first decomposed component. Decomposition results by FDM can suffer from an inconsistency problem. In this work, an accurate adaptive signal decomposition method, called the empirical Fourier decomposition (EFD), is proposed to solve the aforementioned problems. The proposed EFD combines the uses of an improved Fourier spectrum segmentation technique and a zero-phase filter bank. The segmentation technique solves the trivial component problem by an adaptive sorting process and the inconsistency problem by predefining the number of components in a signal to be decomposed. The zero-phase filter bank has no transition phases, which exist in the EWT, in its each filter function, and it can solve the mode mixing problem. Numerical investigations are conducted to study the effectiveness and accuracy of the EFD. It is shown that the EFD can yield accurate and consistent decomposition results for signals with multiple non-stationary modes and those with closely-spaced modes, compared with decomposition results by the EWT, FDM, variational mode decomposition and empirical mode decomposition. It is also shown that the EFD can yield accurate time–frequency representation results and it has the highest computational efficiency among the compared decomposition methods. An experimental validation is also conducted to study the effectiveness of the EFD for experimentally measured signals with closely-spaced modes. It is shown that the EFD can decompose a signal with closely-spaced modes with higher accuracy, compared with the other decomposition methods.

Forecast of the Surface Tilt Based on the Monitoring Data of Settlement of a Group of Buildings

This paper presents the results of long-term monitoring of ground subsidence above flooded mining. To assess the change in the shape of ground surface, the data obtained by monitoring the settlement of the foundation of a large group of buildings were used. The control of the slope was carried out using automated hydrostatic level systems mounted on the foundations of 40 buildings located in this area. The accumulated data on the long-term evolution of the tilt were processed using the methods of statistical analysis of non-stationary time series. Choice of forecast model was made based on a comparison of forecast and real monitoring data. A forecast based on 5-year monitoring data predicts that the observed deformation processes in the rock massif will not change over the next 2 years.

Investigation of Oscillatory States Involving Acoustic Mode Shifts in a Turbulent Syngas Combustion using Non-stationary Time-series Analysis

The transition of thermo-acoustic oscillations in a turbulent bluff-body syngas combustor is analyzed experimentally in the present work. The analysis was carried out for three syngas compositions using simultaneous, unsteady pressure measurement and OH* chemiluminescence across the variation of Reynolds number, Re, over 2289–8009 range. It is observed that across the variation of controlled parameters, the system undergoes series of dynamical states having different nonlinear oscillations. In this paper, we investigate the seldom observed transitions from low-frequency instability to high-frequency instability through two-scale oscillations, and finally, it attains the combustion noise state in response to the variation in Re. To identify the nature of these thermo-acoustic oscillations, time-series analysis based on wavelet transformation, phase portrait, and a novel wavelet-based measure is performed. Based on the results of time-resolved OH* chemiluminescence, the distinct flame behavior is observed in response to the change in Re. It is observed that syngas combustion instability is driven by small-scale structures due to flame stabilization and modulation in the shear layer. Further, the wavelet-based analysis also tracks the evolution of the dynamical state by quantifying the stable state and other non-stable states to be composed of continuously varying phase shifts resulting in stable/quasi-stable combustion.

EMD: Empirical Mode Decomposition and Hilbert-Huang Spectral Analyses in Python.

The Empirical Mode Decomposition (EMD) package contains Python (>=3.5) functions for analysis of non-linear and non-stationary oscillatory time series. EMD implements a family of sifting algorithms, instantaneous frequency transformations, power spectrum construction and single-cycle feature analysis. These implementations are supported by online documentation containing a range of practical tutorials.

Adaptive Bayesian Spectral Analysis of High-Dimensional Nonstationary Time Series

This article introduces a nonparametric approach to spectral analysis of a high-dimensional multivariate nonstationary time series. The procedure is based on a novel frequency-domain factor model that provides a flexible yet parsimonious representation of spectral matrices from a large number of simultaneously observed time series. Real and imaginary parts of the factor loading matrices are modeled independently using a prior that is formulated from the tensor product of penalized splines and multiplicative gamma process shrinkage priors, allowing for infinitely many factors with loadings increasingly shrunk toward zero as the column index increases. Formulated in a fully Bayesian framework, the time series is adaptively partitioned into approximately stationary segments, where both the number and locations of partition points are assumed unknown. Stochastic approximation Monte Carlo techniques are used to accommodate the unknown number of segments, and a conditional Whittle likelihood-based Gibbs sampler is developed for efficient sampling within segments. By averaging over the distribution of partitions, the proposed method can approximate both abrupt and slowly varying changes in spectral matrices. Performance of the proposed model is evaluated by extensive simulations and demonstrated through the analysis of high-density electroencephalography. Supplementary materials for this article are available online.

On one approach to the detection of infrasonic signals of irregular natural phenomena in the instrumental observations time series at the student interdisciplinary testing ground

The work is devoted to the analysis of time series and the problem of processing signals obtained as a result of the design approach implementation during the organization of instrumental observations of irregular natural phenomena at the student interdisciplinary testing ground. The objective of the work is to study the methods of processing noisy signals obtained as a result of monitoring the infrasonic environment, which make it possible to automate the search for fragments of the time series generated by irregular natural phenomena. At the beginning of the work, a brief explanation of the essence of the measuring scientific experiment carried out within the framework of the project approach used in the additional education of students and schoolchildren shall be given. The following is a review of publications describing various approaches to the analysis of nonstationary time series obtained in the process of instrumental observations. As the main method of time series analysis, it is proposed to use the algorithm for calculating the fractal dimension of the time series, proposed by T. Higuchi [1]. During studying of the time series of infrasonic signals, a number of regularities were discovered that contribute to the development of an original procedure for processing and transforming the signal under study, which makes it possible to determine the time intervals of fragments of the time series corresponding to the signals of the desired natural phenomena. The essence of the proposed approach lies in the preliminary preparation of the time series by processing the data with a simple normalized difference filter, previously smoothed by performing the coenvolution (convolution) operation with a Gaussian kernel; determining the step of segmenting the normalized time series, calculating fractal dimensions and averaged amplitudes for each of the segments of the time series and obtaining on their basis vectors of changes in dimensions and amplitudes with their subsequent element-wise multiplication. It is shown that the maximum values of the components of the resulting vector are indicators of timestamps for the location of the desired signals.

Wave-shape oscillatory model for nonstationary periodic time series analysis

The oscillations observed in many time series, particularly in biomedicine, exhibit morphological variations over time. These morphological variations are caused by intrinsic or extrinsic changes to the state of the generating system, henceforth referred to as dynamics. To model these time series (including and specifically pathophysiological ones) and estimate the underlying dynamics, we provide a novel wave-shape oscillatory model. In this model, time-dependent variations in cycle shape occur along a manifold called the wave-shape manifold. To estimate the wave-shape manifold associated with an oscillatory time series, study the dynamics, and visualize the time-dependent changes along the wave-shape manifold, we propose a novel algorithm coined Dynamic Diffusion map (DDmap) by applying the well-established diffusion maps (DM) algorithm to the set of all observed oscillations. We provide a theoretical guarantee on the dynamical information recovered by the DDmap algorithm under the proposed model. Applying the proposed model and algorithm to arterial blood pressure (ABP) signals recorded during general anesthesia leads to the extraction of nociception information. Applying the wave-shape oscillatory model and the DDmap algorithm to cardiac cycles in the electrocardiogram (ECG) leads to ectopy detection and a new ECG-derived respiratory signal, even when the subject has atrial fibrillation.

Modeling Customer Demand for Mobile Value-Added Services: Non-Stationary Time Series Models or Neural Networks Time Series Analysis

Modeling Customer Demand for Mobile Value-Added Services: Non-Stationary Time Series Models or Neural Networks Time Series Analysis

A similarity measure for second order properties of non-stationary functional time series with applications to clustering and testing

Due to the surge of data storage techniques, the need for the development of appropriate techniques to identify patterns and to extract knowledge from the resulting enormous data sets, which can be viewed as collections of dependent functional data, is of increasing interest in many scientific areas. We develop a similarity measure for spectral density operators of a collection of functional time series, which is based on the aggregation of Hilbert–Schmidt differences of the individual time-varying spectral density operators. Under fairly general conditions, the asymptotic properties of the corresponding estimator are derived and asymptotic normality is established. The introduced statistic lends itself naturally to quantify (dis)-similarity between functional time series, which we subsequently exploit in order to build a spectral clustering algorithm. Our algorithm is the first of its kind in the analysis of non-stationary (functional) time series and enables to discover particular patterns by grouping together ‘similar’ series into clusters, thereby reducing the complexity of the analysis considerably. The algorithm is simple to implement and computationally feasible. As a further application, we provide a simple test for the hypothesis that the second order properties of two non-stationary functional time series coincide.