Nonstationary Time Series Models Research Articles

Interpretable predictive modeling of non-stationary long time series

Highly accurate and interpretable forecasting of non-stationary long-time series is a challenge. It is often difficult for existing models to capture the trends of non-stationary time series and achieve interpretable forecasts. To this end, by integrating causal learning and high-order fuzzy cognitive graphs (HFCM), this paper proposes the Causal-HFCM framework. Firstly, the Peter-Clark (PC) algorithm is used to obtain the causal relationships of data, so that the relationships between data can be interpretable and causal learning breaks the limitation that the fuzzy cognitive graph only considers the correlation of data. Secondly, to capture the non-stationary property of time series and improve the prediction ability of fuzzy cognitive maps, a method combining the learning weight matrix algorithm of Huber regression with high-order fuzzy cognitive maps is proposed. Finally, the effectiveness of the proposed method is verified by conducting experiments with three different non-stationary time series datasets of traffic, air, and bearings.

A Bayesian non-stationary heteroskedastic time series model for multivariate critical care data.

We propose a multivariate GARCH model for non-stationary health time series by modifying the observation-level variance of the standard state space model. The proposed model provides an intuitive and novel way of dealing with heteroskedastic data using the conditional nature of state-space models. We follow the Bayesian paradigm to perform the inference procedure. In particular, we use Markov chain Monte Carlo methods to obtain samples from the resultant posterior distribution. We use the forward filtering backward sampling algorithm to efficiently obtain samples from the posterior distribution of the latent state. The proposed model also handles missing data in a fully Bayesian fashion. We validate our model on synthetic data and analyze a data set obtained from an intensive care unit in a Montreal hospital and the MIMIC dataset. We further show that our proposed models offer better performance, in terms of WAIC than standard state space models. The proposed model provides a new way to model multivariate heteroskedastic non-stationary time series data. Model comparison can then be easily performed using the WAIC.

Predictive analysis of stochastic stock pattern utilizing fractional order dynamics and heteroscedastic with a radial neural network framework

Modelling of high-dimension chaotic, noisy, and non-stationary time series of complex fractal dynamics is a big challenge. In this research work, a novel approach of Leverage Convolution LC ARFIMA– GARCH model is presented for sequential learning of irregular, fractal dynamic patterns of stochastic time series dynamics. Different classical methods including the auto-regressive approach indecently are unable to capture fragile patterns and consequently convert some weak signals into random errors by creating white noise patterns. The Convolution Leverage provides an additional degree of freedom for critical noisy points and asymmetrical distribution dynamics to regulate the frequency of imbalance and skewed observation. The designed transformation in the ARFIMA model preserves the loss of information by shifting the skewed data distribution toward a normal pattern. The pattern of population parameters in the proposed leverage paradigm provides out of box approach to track and extract additional information on population variance patterns in the form of an additional degree of freedom to stabilize imbalance signals. The model can provide reliable performance for long-range dependence, particularly for mean reversion chaotic phase variation at critical non-differentiable points. The performance of the dynamic model is verified on chaotic real data of the Ireland Stock Market (ISEQ). The result statistics confirmed the optimized outcome with the addition of the GARCH heteroscedastic multimodal and radial basis neural network. The novel technique can help to address inherited challenges in the imbalance learning and extreme events of the time series modeling by monitoring the chaotic trajectory of fractal physical phenomena, particularly in, finance healthcare, climate, and intelligent sustainability.

Causal structure learning for high-dimensional non-stationary time series

Learning the causal structure of high-dimensional non-stationary time series can help in understanding the data generation mechanism, which is a crucial task in machine learning. However, current causal discovery methods for high-dimensional non-stationary time series face several challenges, including the inability to effectively capture non-stationarity, failure to ensure acyclicity of causal graphs, and reliance on subjective threshold definitions, leading to suboptimal algorithm performance. To address these challenges, we introduce a novel Causal Structure Learning model for High-dimensional Non-stationary Time Series (CSL-HNTS). Firstly, this model presents a graph neural network to model the non-stationarity of time series. Secondly, it introduces a novel Directed Acyclic Graph (DAG) sampling method that transforms the space of DAGs into a continuous space, enabling the search for causal graphs within this continuous space to ensure acyclicity. Finally, the model proposes an automatic threshold definition method, without prior knowledge, to convert the weighted adjacency matrix into the Boolean adjacency matrix of the causal graph, thereby avoiding time-consuming postprocessing steps. The proposed approach is validated using simulation datasets and two real datasets, and is benchmarked against current state-of-the-art methods and ablation experiments. The results demonstrate a significant improvement over existing methods, highlighting the efficacy of the proposed model.

The Circumstance-Driven Bivariate Integer-Valued Autoregressive Model.

The novel circumstance-driven bivariate integer-valued autoregressive (CuBINAR) model for non-stationary count time series is proposed. The non-stationarity of the bivariate count process is defined by a joint categorical sequence, which expresses the current state of the process. Additional cross-dependence can be generated via cross-dependent innovations. The model can also be equipped with a marginal bivariate Poisson distribution to make it suitable for low-count time series. Important stochastic properties of the new model are derived. The Yule-Walker and conditional maximum likelihood method are adopted to estimate the unknown parameters. The consistency of these estimators is established, and their finite-sample performance is investigated by a simulation study. The scope and application of the model are illustrated by a real-world data example on sales counts, where a soap product in different stores with a common circumstance factor is investigated.

A fuzzy time series forecasting model with both accuracy and interpretability is used to forecast wind power

Considering the current research focus on the interpretability and efficiency of wind speed prediction models, this research presents a novel prediction model for dynamic non-stationary fuzzy time series. The proposed model aims to effectively improve prediction accuracy and address the issues of low interpretability and excessive data preprocessing. Unlike existing mainstream hybrid systems for wind speed prediction, this model provides detailed explanations for almost every prediction step and eliminates the need for cumbersome data preprocessing steps. To improve the prediction accuracy of the proposed model, this research incorporates non-stationary sets to overcome the limitations of fuzzy time series in adapting to long-term changes. The developed algorithm, SFTSM, dynamically adjusts the fuzzy time series prediction to effectively address long-term prediction challenges. Furthermore, this study introduces an enhanced version of the artificial hummingbird algorithm, called SLG-AHA, to further improve the accuracy and stability of fuzzy time series prediction. Experimental results using data from the Shandong Penglai wind farm in China validate the effectiveness of the proposed model by showcasing its superior prediction accuracy and stability.

Using the Arima Method with Minitab Applications for Forecasting Work Order Projects of Casting Construction (Case Study: PT. Bumi Sarana Beton)

The ARIMA method is a non-stationary homogeneous time series model that uses the procedure for applying the Autoregressive model or scheme and the Moving Average in preparing its forecasts. The purpose of this study was to determine the application of the Autoregressive Integrated Moving Average (ARIMA) method and minitab application at PT. Bumi Sarana Beton in planning the estimated number of work orders. It can be concluded that for K-225 the best model is the ARIMA model (2,0,0) because it has the lowest MSE value and for K-400 the best model to use is the ARIMA model (2,0,2 ) because it has the smallest value. From the results of the research that has been done, it can be concluded that the production of K-225 in 2022 is 3573.50 m3, while using the ARIMA method in 2023 the total production is 3920.61 m3 and in 2024 the total production is 3573.50 m3. to 3824.36 m3. Production of K-400 in 2022 was 2015.00 m3, while using the ARIMA method in 2023 a total production of 1857.07 m3 was obtained and in 2024 a total production of 2045.89 m3 was obtained.

Trends, Shifting, or Oscillations? Stochastic Modeling of Nonstationary Time Series for Future Water‐Related Risk Management

AbstractHydrological time series often present nonstationarities such as trends, shifts, or oscillations due to anthropogenic effects and hydroclimatological variations, including global climate change. For water managers, it is crucial to recognize and define the nonstationarities in hydrological records. The nonstationarities must be appropriately modeled and stochastically simulated according to the characteristics of observed records to evaluate the adequacy of flood risk mitigation measures and future water resources management strategies. Therefore, in the current study, three approaches were suggested to address stochastically nonstationary behaviors, especially in the long‐term variability of hydrological variables: as an overall trend, shifting mean, or as a long‐term oscillation. To represent these options for hydrological variables, the autoregressive model with an overall trend, shifting mean level (SML), and empirical mode decomposition with nonstationary oscillation resampling (EMD‐NSOR) were employed in the hydrological series of the net basin supply in the Lake Champlain‐River Richelieu basin, where the International Joint Committee recently managed and significant flood damage from long consistent high flows occurred. The detailed results indicate that the EMD‐NSOR model can be an appropriate option by reproducing long‐term dependence statistics and generating manageable scenarios, while the SML model does not properly reproduce the observed long‐term dependence, that are critical to simulate sustainable flood events. The trend model produces too many risks for floods in the future but no risk for droughts. The overall results conclude that the nonstationarities in hydrological series should be carefully handled in stochastic simulation models to appropriately manage future water‐related risks.

Nonstationary time series forecasting using optimized-EVDHM-ARIMA for COVID-19.

The Coronavirus (COVID-19) outbreak swept the world, infected millions of people, and caused many deaths. Multiple COVID-19 variations have been discovered since the initial case in December 2019, indicating that COVID-19 is highly mutable. COVID-19 variation "XE" is the most current of all COVID-19 variants found in January 2022. It is vital to detect the virus transmission rate and forecast instances of infection to be prepared for all scenarios, prepare healthcare services, and avoid deaths. Time-series forecasting helps predict future infected cases and determine the virus transmission rate to make timely decisions. A forecasting model for nonstationary time series has been created in this paper. The model comprises an optimized EigenValue Decomposition of Hankel Matrix (EVDHM) and an optimized AutoRegressive Integrated Moving Average (ARIMA). The Phillips Perron Test (PPT) has been used to determine whether a time series is nonstationary. A time series has been decomposed into components using EVDHM, and each component has been forecasted using ARIMA. The final forecasts have been formed by combining the predicted values of each component. A Genetic Algorithm (GA) to select ARIMA parameters resulting in the lowest Akaike Information Criterion (AIC) values has been used to discover the best ARIMA parameters. Another genetic algorithm has been used to optimize the decomposition results of EVDHM that ensures the minimum nonstationarity and maximal utilization of eigenvalues for each decomposed component.

PROBABILISTIC MODELS AND METHODS OF REGRESSION ANALYSIS OF VOLATILE FINANCIAL TIME SERIES

The article considers financial and economic time series in production, banking, and investment branches. Empirical research in the field of economics increasingly uses data at the individual or household level obtained from surveys. Some variables are difficult enough to measure that such problems arise even when estimating simple bivariate regressions; when panel data are used in ways that effectively distinguish much of the true change while adding to the noise. Results of analysis of real information give the reasons to suppose that the most adequate mathematic models of non-stationary financial time series are homoscedastic and heteroscedastic probabilistic models with partially unknown impact factors. We propose the auto regression and moving average (ARMA) models for analysis of homoscedastic series and auto regression and integrated moving average (ARIMA) models for analysis of heteroscedastic series. These models cover rather wide class of random processes, which are non-stationary in wide and narrow sense. Correct choice of model order allows getting the results with acceptable errors (discrepancy) using rather simple models. We showed the principal useless of tendency to non-critical enlarging of order of moving average and regression equations. Moreover, the model gets much more complicated, and the errors of extrapolation, corresponding with forecasting, grow very quickly. The article attempts a preliminary survey and analysis of time series data for the specification of a model of the interrelationship of variables. It should be recognized that the practical implementation of the above rules is not trivial. In particular, it is obvious that it is possible to obtain satisfactory estimates of the spectrum of financial and economic time series, but at the moment it is not clear how to quantitatively estimate volatility values, cooperation processes in conflict conditions, etc. Only further analysis, both theoretical and empirical, can provide answers to these questions.

Bayesian circular lattice filters for computationally efficient estimation of multivariate time-varying autoregressive models

Nonstationary time series data exist in various scientific disciplines, including environmental science, biology, signal processing, econometrics, among others. Many Bayesian models have been developed to handle nonstationary time series. The time-varying vector autoregressive (TV-VAR) model is a well-established model for multivariate nonstationary time series. Nevertheless, in most cases, the large number of parameters presented by the model results in a high computational burden, ultimately limiting its usage. To address this issue, a computationally efficient multivariate Bayesian Circular Lattice Filter is developed, extending the usage of the TV-VAR model to a broader class of high-dimensional problems. The fully Bayesian framework allows both the autoregressive (AR) coefficients and innovation covariance to vary over time. The proposed estimation method is based on the Bayesian lattice filter (BLF), which is extremely computationally efficient and stable in univariate cases. To illustrate the effectiveness of the proposed approach, a comprehensive comparison with other competing methods is conducted through simulation studies and finds that, in most cases, the proposed approach performs superior in terms of the average squared error between the estimated and true time-varying spectral density. Finally, the methodology is demonstrated through applications to quarterly Gross Domestic Product (GDP) data and Northern California wind data.

Modelling customer demand for mobile value-added services: non-stationary time series models or neural networks time series analysis

Modelling customer demand for mobile value-added services: non-stationary time series models or neural networks time series analysis

Least absolute deviations estimation for nonstationary vector autoregressive time series models with pure unit roots

Least absolute deviations estimation for nonstationary vector autoregressive time series models with pure unit roots

Non-Stationary Time Series Model for Station-Based Subway Ridership During COVID-19 Pandemic: Case Study of New York City.

The COVID-19 pandemic in 2020 has caused sudden shocks in transportation systems, specifically the subway ridership patterns in New York City (NYC), U.S. Understanding the temporal pattern of subway ridership through statistical models is crucial during such shocks. However, many existing statistical frameworks may not be a good fit to analyze the ridership data sets during the pandemic, since some of the modeling assumptions might be violated during this time. In this paper, utilizing change point detection procedures, a piecewise stationary time series model is proposed to capture the nonstationary structure of subway ridership. Specifically, the model consists of several independent station based autoregressive integrated moving average (ARIMA) models concatenated together at certain time points. Further, data-driven algorithms are utilized to detect the changes of ridership patterns as well as to estimate the model parameters before and during the COVID-19 pandemic. The data sets of focus are daily ridership of subway stations in NYC for randomly selected stations. Fitting the proposed model to these data sets enhances understanding of ridership changes during external shocks, both in relation to mean (average) changes and the temporal correlations.

Research on the Restoration and Development of Tourism According to the Characteristics of Rural and Urban Groups --- Based on Cointegration and Error Correction Model

With the further establishment of the scientific development concept of "green water and green mountains are golden mountains and silver mountains", tourism has become an important part of the tertiary industry and an important way for urban development, poverty alleviation and prosperity. It is known as the "sunrise industry". During the epidemic period, even if the tourism industry and its related catering and accommodation fields are greatly impacted, the tertiary industry is still the driving force for maintaining local macroeconomic stability in all regions of China and the key to the victory of the "battle against poverty". Based on the modeling of non-stationary time series, this paper establishes cointegration and error correction models for the relationship between urban and rural residents' domestic tourism consumption, residents' income and social security, in order to put forward reasonable suggestions and measures on how to restore tourism in different regions after the epidemic. The innovation of this paper is to compare rural areas with urban areas, and explore and analyze the different effects and paths of influencing factors. The conclusion of this paper is that for urban residents, the policy of increasing residents' income or reducing personal tax to increase disposable income can better stimulate urban residents' tourism consumption, while residents in rural areas pay more attention to social welfare and security, Therefore, the economic recovery of tourism in the later stage of the epidemic should be developed according to local conditions and different consumer groups.

Activation Function Modulation in Generative Triangular Recurrent Neural Networks

Autonomous generation of time series is challenging because the network must capture short-term features while tracking long-term time dependencies. This paper introduces the modulation of the activation function slopes of the upper-lower triangular recurrent neural networks (ULTRNNs) for dynamic variation of memory through a secondary recurrent network with its own independent states. A zigzag propagation algorithm for weight updates is proposed that accounts for the dynamic interaction of the states between the ULTRNN and the secondary network. A novel training method is proposed that distributes the eigenvalues of the closed-loop system around the unit circle in the complex z-plane to ensure that the network behaves as a nonlinear oscillator with an output that neither collapses nor saturates but continues to emulate the target. Examples encompassing the Lorenz series, Santa Fe laser data, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">kolam</i> patterns, electrocardiogram (ECG) signals, stock pricing data, and smart grid data are presented to demonstrate that the proposed approach is highly effective in the generative modeling of complex periodic, chaotic, and nonstationary time series. The qualitative and quantitative performance of the ULTRNN obtained with the proposed activation-function modulation technique is comparable to that of state-of-the-art techniques including feedforward networks and generative adversarial networks, but with far fewer trainable parameters and shorter computation times.

Stochastic Identification of Guided Wave Propagation under Ambient Temperature via Non-Stationary Time Series Models.

In the context of active-sensing guided-wave-based acousto-ultrasound structural health monitoring, environmental and operational variability poses a considerable challenge in the damage diagnosis process as they may mask the presence of damage. In this work, the stochastic nature of guided wave propagation due to the small temperature variation, naturally occurring in the ambient or environment, is rigorously investigated and modeled with the help of stochastic time-varying time series models, for the first time, with a system identification point of view. More specifically, the output-only recursive maximum likelihood time-varying auto-regressive model (RML-TAR) is employed to investigate the uncertainty in guided wave propagation by analyzing the time-varying model parameters. The steps and facets of the identification procedure are presented, and the obtained model is used for modeling the uncertainty of the time-varying model parameters that capture the underlying dynamics of the guided waves. The stochasticity inherent in the modal properties of the system, such as natural frequencies and damping ratios, is also analyzed with the help of the identified RML-TAR model. It is stressed that the narrow-band high-frequency actuation for guided wave propagation excites more than one frequency in the system. The values and the time evolution of those frequencies are analyzed, and the associated uncertainties are also investigated. In addition, a high-fidelity finite element (FE) model was established and Monte Carlo simulations on that FE model were carried out to understand the effect of small temperature perturbation on guided wave signals.

Application of ARIMA model on prediction of China’s corn market

ARIMA is a common model for non-stationary time series modeling, the model is essentially the combination of ARMA model and differentiate operation. This study indicated that most of the non-stationary time series can be differentially stationary by the difference of the appropriate order, and can be fitted by the ARMA model. As a relatively mature statistical method, ARIMA has been widely used in the price prediction of energy, agricultural products and stocks due to its advantages such as solid theoretical foundation and good short-term forecasting effect. In this paper, ARIMA was used to take the monthly corn price data of 23 months from April 2019 to February 2021 as the sample, processed and analyzed the data, and made a forecast analysis of the corn price in March 2021 in China. By comparing the experimental results with the actual price, it can be seen that the model established has a good fitting degree and can predict the corn price in China more accurately.

Forecasting stock prices using hybrid non-stationary time series model with ERNN

In recent years, financial market dynamics forecasting has been a focus of economic research. To predict the price indices of stock markets, we developed a hybrid non-stationary model with Elman’s Recurrent Neural Networks (ERNN). The proposed model is non-stationary in trend component with lagged variable, average of all past observations and ERNN. This model can capture both linear and non-linear structures in time series. The non-linear structure is capture by ERNN. We derive the expression for the h-step ahead minimum mean square error (MMSE) forecast for the proposed model. Real data sets of stock prices were used to examine the forecasting accuracy of the proposed model and it is found that the proposed approach has the best forecasting accuracy.

A Model for Non-Stationary Time Series and its Applications in Filtering and Anomaly Detection

The information contained in a time series is more than what the values themselves are. In this paper, the Time-variant Local Autocorrelated Polynomial model with Kalman filter is proposed to model the underlying dynamics of a time series (or signal) and mine the deep pattern of it, except estimating the instantaneous mean function (also known as trend function), including: (1) identifying and predicting the peak and valley values of a time series; (2) reporting and forecasting the current changing pattern (increasing or decreasing pattern of the trend, and how fast it changes). We will show that it is this deep pattern that allows us to make higher-accuracy estimation and forecasting for a time series, to easily detect the anomalies (faults) of a sensor, and to track a highly-maneuvering target.