Nonstationarity In Time Series Research Articles

U-Mixer: An Unet-Mixer Architecture with Stationarity Correction for Time Series Forecasting

Time series forecasting is a crucial task in various domains. Caused by factors such as trends, seasonality, or irregular fluctuations, time series often exhibits non-stationary. It obstructs stable feature propagation through deep layers, disrupts feature distributions, and complicates learning data distribution changes. As a result, many existing models struggle to capture the underlying patterns, leading to degraded forecasting performance. In this study, we tackle the challenge of non-stationarity in time series forecasting with our proposed framework called U-Mixer. By combining Unet and Mixer, U-Mixer effectively captures local temporal dependencies between different patches and channels separately to avoid the influence of distribution variations among channels, and merge low- and high-levels features to obtain comprehensive data representations. The key contribution is a novel stationarity correction method, explicitly restoring data distribution by constraining the difference in stationarity between the data before and after model processing to restore the non-stationarity information, while ensuring the temporal dependencies are preserved. Through extensive experiments on various real-world time series datasets, U-Mixer demonstrates its effectiveness and robustness, and achieves 14.5% and 7.7% improvements over state-of-the-art (SOTA) methods.

On Comparing and Assessing Robustness of Some Popular Non-Stationary BINAR(1) Models

Intra-day transactions of stocks from competing firms in the financial markets are known to exhibit significant volatility and over-dispersion. This paper proposes some bivariate integer-valued auto-regressive models of order 1 (BINAR(1)) that are useful to analyze such financial series. These models were constructed under both time-variant and time-invariant conditions to capture features such as over-dispersion and non-stationarity in time series of counts. However, the quest for the most robust BINAR(1) models is still on. This paper considers specifically the family of BINAR(1)s with a non-diagonal cross-correlation structure and with unpaired innovation series. These assumptions relax the number of parameters to be estimated. Simulation experiments are performed to assess both the consistency of the estimators and the robust behavior of the BINAR(1)s under mis-specified innovation distribution specifications. The proposed BINAR(1)s are applied to analyze the intra-day transaction series of AstraZeneca and Ericsson. Diagnostic measures such as the root mean square errors (RMSEs) and Akaike information criteria (AICs) are also considered. The paper concludes that the BINAR(1)s with negative binomial and COM–Poisson innovations are among the most suitable models to analyze over-dispersed intra-day transaction series of stocks.

Probing Intrinsic Neural Timescales in EEG with an Information-Theory Inspired Approach: Permutation Entropy Time Delay Estimation (PE-TD).

Time delays are a signature of many physical systems, including the brain, and considerably shape their dynamics; moreover, they play a key role in consciousness, as postulated by the temporo-spatial theory of consciousness (TTC). However, they are often not known a priori and need to be estimated from time series. In this study, we propose the use of permutation entropy (PE) to estimate time delays from neural time series as a more robust alternative to the widely used autocorrelation window (ACW). In the first part, we demonstrate the validity of this approach on synthetic neural data, and we show its resistance to regimes of nonstationarity in time series. Mirroring yet another example of comparable behavior between different nonlinear systems, permutation entropy-time delay estimation (PE-TD) is also able to measure intrinsic neural timescales (INTs) (temporal windows of neural activity at rest) from hd-EEG human data; additionally, this replication extends to the abnormal prolongation of INT values in disorders of consciousness (DoCs). Surprisingly, the correlation between ACW-0 and PE-TD decreases in a state-dependent manner when consciousness is lost, hinting at potential different regimes of nonstationarity and nonlinearity in conscious/unconscious states, consistent with many current theoretical frameworks on consciousness. In summary, we demonstrate the validity of PE-TD as a tool to extract relevant time scales from neural data; furthermore, given the divergence between ACW and PE-TD specific to DoC subjects, we hint at its potential use for the characterization of conscious states.

A systematic nonstationary causality analysis framework for root cause diagnosis of faults in manufacturing processes

Due to production planning adjustments and equipment faults, most variables in the manufacturing processes exhibit nonstationary characteristics. The fault may be covered by the nonstationary trends of variables, making it difficult to identify fault propagation paths and locate root causes by means of traditional fault diagnosis methods. Thus, a systematic causality analysis framework is designed for root cause diagnosis of faults, which fully considers the nonstationarity of time series after faults and the propagation characteristics of faults in manufacturing processes. To be specific, the Kwiatkowski–Phillips–Schmidt–Shin and augmented Dickey–Fuller tests are firstly combined for assessing whether the industrial time series after faults are stationary or not. Then, for the nonstationary variables, the Gonzalo–Granger decomposition is used for extracting stationary factors. Furthermore, the stationary variables and stationary factors are merged into a new data set, an attention based gated recurrent unit is proposed for nonlinear and dynamic causality analysis aiming at root cause diagnosis of faults. Finally, a typical manufacturing process, the hot rolling process is used for verification, and the simulation and comparison results show the practicability and feasibility of the proposed scheme.

Investigating the Impact of Globalization on Environmental Degradation in Kazakhstan

As a consequence of globalization, the world’s economies are merging; nonetheless, concerns about how globalization trends may harm the environment locally and worldwide are growing. The globalization-environment nexus has now become a contentious issue among policymakers. As a result, several empirical research studies on the globalization-environment nexus have been conducted. In the context of the environmental Kuznets curve theory, this study explores the influence of globalization on environmental deterioration in Kazakhstan (EKC). Furthermore, energy is a control variable in this research. Unit roots and structural break are used to solve the issue of non-stationarity in time series. The autoregressive distributed lagged (ARDL) model is used in this research for long run relationships between variables as well as to estimate long run and short run coefficients. Income, energy, and globalization are discovered to have a long-term link. Furthermore, the findings revealed that long-term environmental deterioration was exacerbated by economic, political, and social globalization, since these factors were shown to have a positive impact on carbon emissions in Kazakhstan. Energy is to blame for environmental damage both in the short and long term. Furthermore, this research reveals that the EKC theory exists in Kazakhstan. On the basis of the findings, policy suggestions are made.

How large are temporal representativeness errors in paleoclimatology?

Abstract. Ongoing work in paleoclimate reconstruction prioritizes understanding the origins and magnitudes of errors that arise when comparing models and data. One class of such errors arises from assumptions of proxy temporal representativeness (TR), i.e., how accurately proxy measurements represent climate variables at particular times and time intervals. Here we consider effects arising when (1) the time interval over which the data average and the climate interval of interest have different durations, (2) those intervals are offset from one another in time (including when those offsets are unknown due to chronological uncertainty), and (3) the paleoclimate archive has been smoothed in time prior to sampling. Because all proxy measurements are time averages of one sort or another and it is challenging to tailor proxy measurements to precise time intervals, such errors are expected to be common in model–data and data–data comparisons, but how large and prevalent they are is unclear. This work provides a 1st-order quantification of temporal representativity errors and studies the interacting effects of sampling procedures, archive smoothing, chronological offsets and errors (e.g., arising from radiocarbon dating), and the spectral character of the climate process being sampled. Experiments with paleoclimate observations and synthetic time series reveal that TR errors can be large relative to paleoclimate signals of interest, particularly when the time duration sampled by observations is very large or small relative to the target time duration. Archive smoothing can reduce sampling errors by acting as an anti-aliasing filter but destroys high-frequency climate information. The contribution from stochastic chronological errors is qualitatively similar to that when an observation has a fixed time offset from the target. An extension of the approach to paleoclimate time series, which are sequences of time-average values, shows that measurement intervals shorter than the spacing between samples lead to errors, absent compensating effects from archive smoothing. Nonstationarity in time series, sampling procedures, and archive smoothing can lead to changes in TR errors in time. Including these sources of uncertainty will improve accuracy in model–data comparisons and data comparisons and syntheses. Moreover, because sampling procedures emerge as important parameters in uncertainty quantification, reporting salient information about how records are processed and assessments of archive smoothing and chronological uncertainties alongside published data is important to be able to use records to their maximum potential in paleoclimate reconstruction and data assimilation.

Optimal scale Gaussian process regression model in Insulated Gate Bipolar Transistor remaining life prediction

The new energy industry has received extensive attention. The Insulated Gate Bipolar Transistor power module is the core device for new energy equipment, and accurately predicting its remaining useful life (RUL) is conducive to its timely maintenance and replacement, promoting the sustainable development of the society. When the power module fails, its collector–emitter saturation voltage drop (Vce) significantly increases. This feature is used to determine RUL of the power module. However, the time series have obvious random non-stationary features. Current methods are ineffective to such features. Therefore, the optimal scale Gaussian process (OSGP) model is proposed. The optimal parameters in the model are determined by Ant Lion Optimizer. The variable-scale function of OSGP model makes it able to handle the randomness and non-stationarity in time series. The results show that OSGP prediction accuracy is higher than that of support vector machine, BP neural network, Gaussian process and Gaussian process mixture. The proposed model has higher prediction accuracy and can adapt to less training samples.

Nonstationary time series transformation methods: An experimental review

Data preprocessing is a crucial step for mining and learning from data, and one of its primary activities is the transformation of data. This activity is very important in the context of time series prediction since most time series models assume the property of stationarity, i.e., statistical properties do not change over time, which in practice is the exception and not the rule in most real datasets. There are several transformation methods designed to treat nonstationarity in time series. However, the choice of a transformation that is appropriate to the adopted data model and to the problem at hand is not a simple task. This paper provides a review and experimental analysis of methods for transformation of nonstationary time series. The focus of this work is to provide a background on the subject and a discussion on their advantages and limitations to the problem of time series prediction. A subset of the reviewed transformation methods is compared through an experimental evaluation using benchmark datasets from time series prediction competitions and other real macroeconomic datasets. Suitable nonstationary time series transformation methods provided improvements of more than 30% in prediction accuracy for half of the evaluated time series and improved the prediction in more than 95% for 10% of the time series. Furthermore, the adoption of a validation phase during model training enables the selection of suitable transformation methods.

Do foreign exchange and equity markets co-move in Latin American region? Detrended cross-correlation approach

Abstract This paper investigates the dynamics of the relationship between foreign exchange markets and stock markets through time varying co-movements. In this sense, we analyzed the time series monthly of Latin American countries for the period from 1991 to 2015. Furthermore, we apply Granger causality to verify the direction of causality between foreign exchange and stock market and detrended cross-correlation approach ( ρ DCCA ) for any co-movements at different time scales. Our empirical results suggest a positive cross correlation between exchange rate and stock price for all Latin American countries. The findings reveal two clear patterns of correlation. First, Brazil and Argentina have positive correlation in both short and long time frames. Second, the remaining countries are negatively correlated in shorter time scale, gradually moving to positive. This paper contributes to the field in three ways. First, we verified the co-movements of exchange rate and stock prices that were rarely discussed in previous empirical studies. Second, ρ DCCA coefficient is a robust and powerful methodology to measure the cross correlation when dealing with non stationarity of time series. Third, most of the studies employed one or two time scales using co-integration and vector autoregressive approaches. Not much is known about the co-movements at varying time scales between foreign exchange and stock markets. ρ DCCA coefficient facilitates the understanding of its explanatory depth.

Comment on 'Nonstationarity in Time Series of State Densities' by Y. Chang, C. S. Kim and J. Y. Park

In this comment on Chang, Kim and Park [Journal of Econometrics, 192, 152-167 (2016)] I point out that the time series of densities which the authors purport to model as a nonstationary cointegrated process is in fact stationary under their assumptions, aside from a deterministic component.

Nonstationarity in time series of state densities

This paper proposes a new framework to analyze the nonstationarity in the time series of state densities, representing either cross-sectional or intra-period distributions of some underlying economic variables. We regard each state density as a realization of Hilbertian random variable, and use a functional time series model to fit a given time series of state densities. This allows us to explore various sources of the nonstationarity of such time series. The potential unit roots are identified through functional principal component analysis, and subsequently tested by the generalized eigenvalues of leading components of normalized estimated variance operator. The asymptotic null distribution of the test statistic is obtained and tabulated. We use the methodology developed in the paper to investigate the state densities given by the cross-sectional distributions of individual earnings and the intra-month distributions of stock returns. We find some clear evidence for the presence of strong persistency in their time series.

A frequency domain test for detecting nonstationary time series

We propose a frequency domain generalized likelihood ratio test for testing nonstationarity in time series. The test is constructed in the frequency domain by comparing the goodness of fit in the log-periodogram regression under the varying coefficient fractionally exponential models. Under such a locally stationary specification, the proposed test is capable of detecting dynamic changes of short-range and long-range dependences in a regression framework. The asymptotic distribution of the proposed test statistic is known under the null stationarity hypothesis, and its finite sample distribution can be approximated by bootstrap. Numerical results show that the proposed test has good power against a wide range of locally stationary alternatives.

A joint test for structural stability and a unit root in autoregressions

A test of the joint null hypothesis of parameter stability and a unit root within an ADF style autoregressive specification whose entire parameter structure is potentially subject to a structural break at an unknown time period is developed. The proposed test is a useful diagnostic tool for assessing the interactions of breaks and unit root type of nonstationarities in time series, in addition to offering a powerful device for detecting changes in persistence. As a byproduct the limiting behaviour of a related Wald statistic designed to test solely the null of parameter stability in an environment with a unit root is also obtained. These distributions are free of nuisance parameters and easily tabulated. The finite sample properties of the tests are assessed through a series of simulations, and an application to macroeconomic data illustrates their usefulness.

Determinants of interest rate swap spreads in the US: bounds testing approach to cointegration

This article empirically analyses the determinants of US interest rate swap spreads, and makes two key contributions. First, it considers the nonstationarity of time series, which previous studies have not done, and conducts a cointegration test using the bounds testing approach. The empirical results reveal that there exists a cointegration relationship between interest rate swap spreads and four determinants: the corporate bond spread, the slope of the yield curve, the T bill and Eurodollar (TED) spread and yield volatility. Second, it analyses the determinants of swap spreads using the Dynamic Ordinary Least Squares (DOLS). Considering the cointegration relationship, all explanatory variables were significant within the 5% level.

Assessment of apparent nonstationarity in time series of annual inflow, daily precipitation, and atmospheric circulation indices: A case study from southwest Western Australia

The southwest region of Western Australia has experienced a sustained sequence of low annual inflows to major water supply dams over the past 30 years. Until recently, the dominant interpretation of this phenomenon has been predicated on the existence of one or more sharp breaks (change or jump points), with inflows fluctuating around relatively constant levels between them. This paper revisits this interpretation. To understand the mechanisms behind the changes, we also analyze daily precipitation series at multiple sites in the vicinity and time series for several indices of regional atmospheric circulation that may be considered as drivers of regional precipitation. We focus on the winter half‐year for the region (May to October) as up to 80% of annual precipitation occurs during this “season”. We find that the decline in the annual inflow is in fact more consistent with a smooth declining trend than with a sequence of sharp breaks, the decline is associated with decreases both in the frequency of daily precipitation occurrence and in wet‐day amounts, and the decline in regional precipitation is strongly associated with a marked decrease in moisture content in the lower troposphere, an increase in regionally averaged sea level pressure in the first half of the season, and intraseasonal changes in the regional north‐south sea level pressure gradient. Overall, our approach provides an integrated understanding of the linkages between declining dam inflows, declining precipitation, and changes in regional atmospheric circulation that favor drier conditions.

A Scale-space Approach for Detecting Non-stationarities in Time Series

A Scale-space Approach for Detecting Non-stationarities in Time Series

Analysis of Local State Space Models for Feature Extraction in Structural Health Monitoring

Error metrics based on nonlinear dynamical predictive models has been implemented earlier to determine discrepancies or nonstationarity in time series caused by external sources, such as noise or filtering. The structural health monitoring (SHM) field has adapted various forms of such prediction error metrics for damage detection. This process usually requires randomly selecting a number of fiducial points on a reconstructed attractor, tracking their time evolutions, and then comparing them to a corresponding set (in a general sense) of points on another attractor. Such an approach has been successful, but it is a process that is globally averaged over the whole data set. However, changes to dynamics resulting from processes such as damage may only noticeably affect small regions within state space if the damage manifests itself locally in time and/or space. In such a case, a generalized selection of points covering the entire attractor may not be able to identify that a dynamical change has taken place. This work examines local dynamics and local dynamical stability and how these concepts relate to prediction error for similarly local regions in phase space. Further, this work considers whether computing features such as prediction error in these local regions can improve the ability to detect changes, including in the presence of additive noise on the output. This work considers a Lorenz-driven three-degree-of-freedom oscillator subject to both simulated linear and nonlinear damage scenarios in the presence of noise, as well as an experimental frame structure subject to bolt preload loss.

Flicker-Noise Spectroscopy for Analyzing Chaotic Time Series of Dynamic Variables: Problem of Signal-to-Noise Relation

The underlying principles of flicker-noise spectroscopy (FNS), which is a new method for obtaining information contained in chaotic time signals, are presented. The method's essence is the imparting of an information value to sequences of distinguishable irregularities (spikes, jumps, discontinuities of derivatives of various orders) in dynamic variables that characterize systems under test. The information distinguishability of the types of irregularities introduced here follows from the fact that the “passport data” that characterize the aggregate properties of the irregularities are distinguishably extracted when analyzing power spectra and difference momenta (“structural functions”) of different orders. Shown are potentialities of the FNS method associated with the introduction of quantitative criteria for the nonstationarity of time series produced by real systems. Revealing such criteria is of interest in connection with the all-pervading character of the nonstationarity phenomenon and with the search for “harbingers” of most dramatic changes in the state of evolving systems. The FNS method may be used for extracting phenomenological properties when analyzing time series that characterize evolution of complex systems. As an example, the dynamics of the electric potential fluctuation in an electromembrane system with a cation-exchange membrane in the region of the overlimiting current is considered. The introduced techniques of parametrization of a chaotic signal may effectively be used for the signal isolation against a “noisy” background even in the case where in fact one has to solve the problem of the isolation of two informatively-valuable signals if the information concerning parameters of the target signal is prespecified.

Dynamic support vector machines for non-stationary time series forecasting

This paper proposes a modified version of support vector machines (SVMs), called dynamic support vector machines (DSVMs), to model non-stationary time series. The DSVMs are obtained by incorporating the problem domain knowledge -- non-stationarity of time series into SVMs. Unlike the standard SVMs which use fixed values of the regularization constant and the tube size in all the training data points, the DSVMs use an exponentially increasing regularization constant and an exponentially decreasing tube size to deal with structural changes in the data. The dynamic regularization constant and tube size are based on the prior knowledge that in the non-stationary time series recent data points could provide more important information than distant data points. In the experiment, the DSVMs are evaluated using both simulated and real data sets. The simulation shows that the DSVMs generalize better than the standard SVMs in forecasting non-stationary time series. Another advantage of this modification is that the DSVMs use fewer support vectors, resulting in a sparser representation of the solution.

Determining the Order of Differencing in Autoregressive Processes

One way of handling nonstationarity in time series is to compute first differences and fit a model to the differenced series unless the differenced series also looks nonstationary. In that case, second- or higher-order differencing is done. To decide if the current degree of differencing is sufficient, one can look at the autocorrelation function for slow decay. A formal statistical test for the need to difference further is available if one is willing to assume that at most one more difference will render the series stationary. In this article, we present a proper sequence of statistical tests that allows the practitioner to handle cases in which a high order of differencing may be needed. The proper sequence is not the traditional sequence, which begins with a test for a single unit root.