Nonstationary Time Series Models Research Articles

A Hybrid Forecasting Model for Nonstationary and Nonlinear Time Series in the Stochastic Process of CO2 Emission Trading Price Fluctuation

Predicting CO2 emission prices is an important and challenging task for policy makers and market participants, as carbon prices follow a stochastic process of complex time series with nonstationary and nonlinear characteristics. Existing literature has focused on highly precise point forecasting, but it cannot correctly solve the uncertainties related to carbon price datasets in most cases. This study aims to develop a hybrid forecasting model to estimate in advance the maximum or minimum loss in the stochastic process of CO2 emission trading price fluctuation. This model can granulate raw data into fuzzy-information granular components with minimum (Low), average (R), and maximum (Up) values as changing space-description parameters. Furthermore, it can forecast carbon prices’ changing space with Low, R, and Up as inputs to support a vector regression. This method’s feasibility and effectiveness is examined using empirical experiments on European Union allowances’ spot and futures prices under the European Union’s Emissions Trading Scheme. The proposed FIG-SVM model exhibits fewer errors and superior performance than ARIMA, ARFIMA, and Markov-switching methods. This study provides several important implications for investors and risk managers involved in trading carbon financial products.

A Comparative Study of VMD-Based Hybrid Forecasting Model for Nonstationary Daily Streamflow Time Series

Data-driven methods are very useful for streamflow forecasting when the underlying physical relationships are not entirely clear. However, obtaining an accurate data-driven model that is sufficiently performant for streamflow forecasting remains often challenging. This study proposes a new data-driven model that combined the variational mode decomposition (VMD) and the prediction models for daily streamflow forecasting. The prediction models include the autoregressive moving average (ARMA), the gradient boosting regression tree (GBRT), the support vector regression (SVR), and the backpropagation neural network (BPNN). The latest decomposition model, the VMD algorithm, was first applied to extract the multiscale features from the entire time series and to decompose them into several subseries, which were predicted after that using forecast models. The ensemble forecast was finally reconstructed by summing. Historical daily streamflow series recorded at the Wushan and Weijiabao hydrologic stations from 1 January 2001 to 31 December 2014 in China were investigated using the proposed VMD-based models. Three quantitative evaluation indexes, including the Nash–Sutcliffe efficiency coefficient (NSE), the root mean square error (RMSE), and the mean absolute error (MAE), were used to evaluate and compare the predicted results of the proposed VMD-based models with two other models such as nondecomposition method (BPNN) and BPNN based on ensemble empirical mode decomposition (EEMD-BPNN). Furthermore, a comparative analysis of the performance of the VMD-BPNN model under different forecast periods (1, 3, 5, and 7 days) was performed. The results evidenced that the proposed VMD-based models could always achieve good performance in the testing stage and had relatively good stability and representativeness. Specifically, the VMD-BPNN model considered both the prediction accuracy and computation efficiency. The results show that the reliability of the forecasting decreased as the foresight period increased. The model performed satisfactorily up to 7-d lead time. The VMD-BPNN model could be applied as a promising, reliable, and robust prediction tool for short-term streamflow forecasting modelling.

Some notes on nonlinear cointegration: A partial review with some novel perspectives

Some recent work on the analysis of nonlinear and nonstationary time series models is reviewed. A couple of novel results are obtained in extending nonlinear cointegrating regression models to a time series situation. All through the paper focus is on aspects that could lead to a more well-defined concept of nonlinear cointegration.

Helsinki Stock Exchange Index during World War One: Statistical Test of Hypotheses on the Basis of Counter-Factual Modeling

The article proposes an improved model of St. Petersburg Stock Exchange index dynamics and constructs a similar model of Helsinki Stock Exchange index on the basis of published results of a counterfactual model predicting the hypothetical dynamics of St. Petersburg Stock Exchange index after July 1914 under the assumption that there is no war. The author hypothesizes that internal economic factors that determined the downward trend of St. Petersburg Stock Exchange index also influenced the dynamics of Helsinki Stock Exchange index under the assumption that there was no war. To test this hypothesis the author has constructed (in the R software environment) the ARIMA statistical model that is an integrated autoregressive-moving average model which extends the ARMA model for non-stationary time series. The constructed counterfactual models proved that while the influence of pre-war factors remained, the dynamics of both indices did not show similar trends thus suggesting that the Finnish stock market was developing without any noticeable look at St. Petersburg Stock Exchange and inner economic factors of the Russian Empire.

Chernoff Approximations for Nonstationary Random Walk Modeling

In this paper we discuss two aspects of kinetic approach for time series modeling in terms of dynamical system. One method is based on the interpretation of kinetic equation for empirical distribution function density as a reduced description of statistical mechanics for appropriate dynamical system. For example, if distribution function density is satisfied to Liouville equation with some velocity, then this velocity can be treated as an average velocity of particle in phase space. The second method is based on the so-called Chernoff theorem from the group theory. According to the consequence from this theorem some iteration procedure exists for construction of group or semigroup, which is equivalent in some sense to average shift generator over the trajectory of appropriate dynamical system. Connection between these two methods enables us to construct a strict approach to nonstationary time series modeling with non-parametric estimation of statistical properties of corresponding sample distribution function. Also the notion of Chernoff-equivalent semigroup can be used for the calculation optimization procedure.

Sparse time series modeling of the baseline vibration from a gearbox under time-varying speed condition

Abstract Time series model-based approach (TSMBA) is promising in processing vibration signals and assessing the health condition of gearboxes. Accurate time series modeling of the baseline vibration is critical to the TSMBA. Gearboxes often operate under time-varying speed condition, which makes the baseline vibration non-stationary. To accurately model such signals, non-stationary time series models are in demand. Conventional functional pooled autoregression (FP-AR) model is a good option. However, conventional FP-AR assumed 1) consecutive AR terms and 2) identical functional space that describes the dependency between AR parameters and rotating speed, which limited its modeling accuracy. To improve modeling accuracy, this paper proposes a sparse FP-AR model that uses sparse AR terms and non-identical functional spaces. To obtain such a sparse FP-AR model, a new model selection procedure is developed by adopting the least absolute shrinkage and selection operator. The sparse FP-AR model has been validated using simulation signals from a simulation model for a fixed-axis gearbox and experimental signals from two independent fixed-axis gearbox test-rigs. The modeling accuracy was measured by mean squared errors and randomness tests of the modeling residuals, goodness-of-fit between the one-step ahead prediction and real gear vibration, and time-frequency spectra. Results have shown that the proposed sparse FP-AR model has higher modeling accuracy than the conventional one. Meanwhile, TSMBA that uses the sparse FP-AR model was applied for detecting gear tooth crack faults under time-varying speed condition. Results have shown that the proposed method benefits the fixed-axis gearbox in early detection of faults and better assessment of fault progressions.

Stochastic models of joint non-stationary time-series of air temperature, relative humidity and atmospheric pressure

In this paper two numerical stochastic models of the joint non-stationary time-series of air temperature, relative humidity and atmospheric pressure are proposed. The first model is based on an assumption that real weather processes are periodically correlated random processes with a period equal to 1 day. This assumption takes into account the diurnal variation of real meteorological processes, determined by the day/night alternation. Within the framework of the second model, real weather processes are considered as non-stationary random processes. The input parameters of both models (one-dimensional distributions and correlation structure of the joint time-series) are determined from the data of long-term real observations at weather stations. The results of the models verification are presented.

A regime-switching recurrent neural network model applied to wind time series

This paper proposes a regime-switching recurrent network model (RS-RNN) for non-stationary time series. The RS-RNN model emits a mixture density with dynamic nonlinear regimes that fit flexibly data distributions with non-Gaussian shapes. The key novelties are: development of an original representation of the means of the component distributions by dynamic nonlinear recurrent networks, and derivation of a corresponding expectation maximization (EM) training algorithm for finding the model parameters. The results show that the RS-RNN applied to a real-world wind speed time series achieves standardized residuals similar to popular previous models, but it is more accurate distribution forecasting than other linear switching (MS-AR) and nonlinear neural network (MLP and RNN) models.

BINAR(1) negative binomial model for bivariate non-stationary time series with different over-dispersion indices

The existing stationary bivariate integer-valued autoregressive model of order 1 (BINAR(1)) with correlated Negative Binomial (NB) innovations is capable of modelling stationary count series where the innovation terms of both series have same over-dispersion index. Such BINAR(1) may not be useful to model real-life series that are affected by common time-dependent covariates whereby the two series may display non-stationarity as well as different over-dispersion indices. In this paper, we propose a novel BINAR(1) model with the pair of innovations following a joint NB distribution that accommodates different over-dispersion indices. The estimation of parameters is conducted using generalized quasi-likelihood (GQL) approach that operates in two phases. Monte Carlo simulations are implemented to assess the performance of the proposed GQL under the wide range of combinations of the model parameters. This BINAR(1) model is also applied to analyze the daily series of day and night accident data in some regions of Mauritius.

A non-stationary NDVI time series modelling using triplet Markov chain

Nowadays, vegetation monitoring using remotely sensed data is an important far-reaching real-world issue. The main purpose of this study is to build a triplet Markov chain (TMC) to model and analyse vegetation dynamics on large-scales using non-stationary normalised difference vegetation index (NDVI) time series. TMC is a generalisation of hidden Markov models (HMMs), which have been widely used to represent satellite time series images but which they proved to be inefficient for non-stationary data. The TMC model proposed in this paper overcomes this limit by adding an auxiliary process which allows modelling non-stationarity. In order to assess the performance of the proposed model, experimentation is carried out using moderate resolution imaging spectroradiometer (MODIS) NDVI time series of the north-western region of Tunisia. The TMC model is compared to standard HMM and seasonal auto regressive integrated moving average model (SARIMA) and proved to achieve the best performance with an overall accuracy prediction rate of 92.8% and a kappa coefficient of 0.885.

Stochastic modeling of nonstationary earthquake time series with long-term clustering effects

Earthquake time series are widely used to characterize the main features of regional seismicity and to provide useful insights into earthquake dynamics. Properties such as intermittency and nonstationary clustering are common in earthquake time series, highlighting the complex nature of the earthquake generation process. In the present work we introduce a stochastic model with memory effects that reproduces the temporal scaling behavior observed in regional seismicity. For nonstationary earthquake activity, where the average seismic rate fluctuates, the solution of the stochastic model is the $q$-generalized gamma function that presents two power-law regimes for short and long waiting times, respectively, while for stationary activity it reduces to the standard gamma function. To validate the derived model, we study nonstationary earthquake time series from Southern California and Japan. The analysis shows that for various threshold magnitudes and spatial areas and after rescaling with the mean waiting time, the normalized probability density functions fall onto a unique curve, which is characterized by two power-law regimes for short and long waiting times, respectively, a scaling behavior that can exactly be recovered by the derived $q$-generalized gamma function. The results show the validity of the stochastic model and the derived scaling function, further signifying both short- and long-term clustering effects and memory in the evolution of seismicity.

A non-stationary NDVI time series modelling using triplet Markov chain

Nowadays, vegetation monitoring using remotely sensed data is an important far-reaching real-world issue. The main purpose of this study is to build a triplet Markov chain (TMC) to model and analyse vegetation dynamics on large-scales using non-stationary normalised difference vegetation index (NDVI) time series. TMC is a generalisation of hidden Markov models (HMMs), which have been widely used to represent satellite time series images but which they proved to be inefficient for non-stationary data. The TMC model proposed in this paper overcomes this limit by adding an auxiliary process which allows modelling non-stationarity. In order to assess the performance of the proposed model, experimentation is carried out using moderate resolution imaging spectroradiometer (MODIS) NDVI time series of the north-western region of Tunisia. The TMC model is compared to standard HMM and seasonal auto regressive integrated moving average model (SARIMA) and proved to achieve the best performance with an overall accuracy prediction rate of 92.8% and a kappa coefficient of 0.885.

Nonstationary fuzzy forecasting of wind and wave climate in very long-term scales

Global climate change may have serious impact on human activities in coastal and other areas. Climate change may affect the degree of storminess and, hence, change the wind-driven ocean wave climate. This may affect the risks associated with maritime activities such as shipping and offshore oil and gas. So, there is a recognized need to understand better how climate change will affect such processes. Typically, such understanding comes from future projections of the wind and wave climate from numerical climate models and from the stochastic modelling of such projections. This work investigates the applicability of a recently proposed nonstationary fuzzy modelling to wind and wave climatic simulations. According to this, fuzzy inference models (FIS) are coupled with nonstationary time series modelling, providing us with less biased climatic estimates. Two long-term datasets for an area in the North Atlantic Ocean are used in the present study, namely NORA10 (57 years) and ExWaCli (30 years in the present and 30 years in the future). Two distinct experiments have been performed to simulate future values of the time series in a climatic scale. The assessment of the simulations by means of the actual values kept for comparison purposes gives very good results.

Gaussian Process Time-Series Models for Structures under Operational Variability

A wide range of vibrating structures are characterized by variable structural dynamics resulting from changes in environmental and operational conditions, posing challenges in their identification and associated condition assessment. To tackle this issue, the present contribution introduces a stochastic modeling methodology via Gaussian Process (GP) time-series models. In the presently introduced approach, the vibration response is represented by means of a random coefficient time-series model, whose coefficients comply with a GP regression on the environmental and operational parameters. The approach may be implemented in conjunction to any type of linear-in-the-parameters time-series model, ranging from simple AR models to more complex non-linear or non-stationary time-series models. The obtained GP time-series modeling approach provides an effective and compact global representation of the vibrational response of a structure under a wide span of environmental and operational conditions. The effectiveness of the postulated GP time-series models is demonstrated through two case studies: the first involves the identification of the vertical vibration response of the Humber bridge, evaluated over a period of three years; the second considers the long-term simulated vibration response of a wind turbine featuring non-stationary dynamics stemming from the rotor speed. In both cases, the variation of the average wind speed is the main driver of uncertainty, while, through application of the proposed GP time-series models, it is possible to track the resulting variation in modal quantities.

On the Identification and Estimation of Partially Nonstationary ARMAX Systems

This paper extends current theory on the identification and estimation of vector time series models to nonstationary processes. It examines the structure of dynamic simultaneous equations systems or ARMAX processes that start from a given set of initial conditions and evolve over a given, possibly infinite, future time horizon. The analysis proceeds by deriving the echelon canonical form for such processes. The results are obtained by amalgamating ideas from the theory of stochastic difference equations with adaptations of the Kronecker index theory of dynamic systems. An extension of these results to the analysis of unit-root, partially nonstationary (cointegrated) time series models is also presented, leading to straightforward identification conditions for the error correction, echelon canonical form. An innovations algorithm for the evaluation of the exact Gaussian likelihood is given and the asymptotic properties of the approximate Gaussian estimator and the exact maximum likelihood estimator based upon the algorithm are derived. Examples illustrating the theory are discussed and some experimental evidence is also presented.

Estimating the parameters of a BINMA Poisson model for a non-stationary bivariate time series

ABSTRACTThis article proposes a novel non-stationary BINMA time series model by extending two INMA processes where their innovation series follow the bivariate Poisson under time-varying moment assumptions. This article also demonstrates, through simulation studies, the use and superiority of the generalized quasi-likelihood (GQL) approach to estimate the regression effects, which is computationally less complicated as compared to conditional maximum likelihood estimation (CMLE) and the feasible generalized least squares (FGLS). The serial and bivariate dependence correlations are estimated by a robust method of moments.

Nonparametric Inference for Time-Varying Coefficient Quantile Regression

The article considers nonparametric inference for quantile regression models with time-varying coefficients. The errors and covariates of the regression are assumed to belong to a general class of locally stationary processes and are allowed to be cross-dependent. Simultaneous confidence tubes (SCTs) and integrated squared difference tests (ISDTs) are proposed for simultaneous nonparametric inference of the latter models with asymptotically correct coverage probabilities and Type I error rates. Our methodologies are shown to possess certain asymptotically optimal properties. Furthermore, we propose an information criterion that performs consistent model selection for nonparametric quantile regression models of nonstationary time series. For implementation, a wild bootstrap procedure is proposed, which is shown to be robust to the dependent and nonstationary data structure. Our method is applied to studying the asymmetric and time-varying dynamic structures of the U.S. unemployment rate since the 1940s. Supplementary materials for this article are available online.

SUSPEND: Determining software suspiciousness by non-stationary time series modeling of entropy signals

Commercial anti-virus software traditionally memorizes specific byte sequences (known as “signatures”) in the file contents of previously encountered malware. However, malware authors can evade signature-based detection in many ways; for instance, by using obfuscation techniques such as “packing” (encryption or compression) to hide snippets of malicious code; by writing metamorphic malware; or by tampering with existing malware. We hypothesize that certain evasion techniques can leave traces in the file’s entropy signal, revealing either similarities to known malware or the presence of tampering per se. To this end, we present SUSPEND (SUSPicious ENtropy signal Detector), an expert system which evaluates the suspiciousness of an executable file’s entropy signal in order to subserve malware classification. Whereas traditionally, entropy analysis has been used for the goal of packer detection (and therefore entropy-based features often merely comprise mean entropy or the entropy of a few file subcomponents), SUSPEND applies non-stationary time series modeling to aid in malware detection. In particular, SUSPEND (a) quantifies the “amount of structure” in the entropy signal (through detrended fluctuation analysis), (b) finds the location and size of sudden jumps in entropy (through mean change point modeling), and (c) computes the distribution of entropic variation across multiple spatial scales (through wavelet decomposition). In addition, SUSPEND (d) summarizes the entropy signal’s empirical probability distribution. Because SUSPEND’s run time can be made to scale linearly in file size, it is well-suited for large-scale malware analysis. We apply SUSPEND to a large-scale malware detection task with 500,000 heterogeneous real-world samples and over 1 million features. We find that SUSPEND boosts the predictive performance of traditional entropy analysis (as found in packer detectors) from 77.02% to 96.62%. Moreover, SUSPEND’s focus on entropy signals makes it a natural candidate for combining with other types of features; for instance, combining SUSPEND with a strings-based feature set boosts predictive accuracy from 97.18% to 98.62%. Thus, whereas traditionally, entropy analysis has focused on detecting that a file is packed, SUSPEND’s more comprehensive representation of the entropy signal helps to determine that a file is malicious. We illustrate the application of SUSPEND by studying 18 pieces of VirRansom, a family of viral ransomware which could cost millions to large organizations. SUSPEND is able to detect 100% of the studied files with over 99% confidence, whereas a more traditional strings-based model was very close to undecided and represents the entire family with a single string.

A Bayesian nonparametric Markovian model for non-stationary time series

Stationary time series models built from parametric distributions are, in general, limited in scope due to the assumptions imposed on the residual distribution and autoregression relationship. We present a modeling approach for univariate time series data, which makes no assumptions of stationarity, and can accommodate complex dynamics and capture non-standard distributions. The model for the transition density arises from the conditional distribution implied by a Bayesian nonparametric mixture of bivariate normals. This results in a flexible autoregressive form for the conditional transition density, defining a time-homogeneous, non-stationary Markovian model for real-valued data indexed in discrete time. To obtain a computationally tractable algorithm for posterior inference, we utilize a square-root-free Cholesky decomposition of the mixture kernel covariance matrix. Results from simulated data suggest that the model is able to recover challenging transition densities and non-linear dynamic relationships. We also illustrate the model on time intervals between eruptions of the Old Faithful geyser. Extensions to accommodate higher order structure and to develop a state-space model are also discussed.

Adaptive spectral estimation for nonstationary multivariate time series

Following the nonstationary univariate time series model of Rosen et al. (2012), we propose an adaptive estimation of time-varying spectra and cross-spectra for analyzing possibly nonstationary multivariate time series. Under the Bayesian framework, the estimation is implemented by smoothing stochastic approximation Monte Carlo (SSAMC) methods. We show by simulation study that the proposed method achieves good performance for time series whether changing abruptly or smoothly. The superiority to the other existing methods is also investigated. An application to longitudinal vibration data of the containership provides a wave-approach angle range, which should be recommended when sailing under a harsh sea condition.