Nonstationary Time Series Research Articles

A novel SARCIMA model based on central difference and its application in solar power generation of China

Due to the influence of exogenous factors such as weather, solar power generation shows the characteristics of instability and volatility, so the prediction of solar power generation has become especially important. In this paper, a new seasonal central difference autoregressive moving average model abbreviate as SARCIMA is proposed to accurately predict solar power generation. Firstly, the non-stationary time series is subjected to central difference until the difference series is stationary. Then, by analyzing autocorrelation images and partial autocorrelation images respectively, the number of the autoregressive term p and the moving average term q are obtained, so as to obtain the ARCIMA model. Next, the seasonal difference is combined with the ARCIMA model, thus establishing the SARCIMA model. Finally, the new model is applied to the prediction of solar power generation data exhibiting periodic characteristics and compared with five competing models. Mean absolute percentage error and residuals are separately calculated as indicators to evaluate the accuracy and stability of the models. In addition, the prediction of solar power generation in the next three months shows that the new model has better effectiveness, which provides a scientific basis for predicting solar power generation in the future.

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.

Evolution of temporal fluctuation scaling exponent in nonstationary time series using supersymmetric theory of stochastic dynamics.

Temporal fluctuation scaling (TFS) is an emergent property of complex systems that relates the variance (Ξ_{2}) and the mean (M_{1}) from an empirical data set in the form Ξ_{2}∼M_{1}^{α_{TFS}}, where the dispersion (fluctuation) of the data has been described in terms of Ξ_{2}. At present, it has been shown that this law of complex systems has different multidisciplinary applications such as characterizing the market rate based on its exponent, explaining the spatial spread of a pandemic or measuring dispersion in a counting process, among others, if it is known how the average value M_{1} of a representative quantity in a system changes. Then, using the path integral formalism and Parisi-Sourlas method, we propose an extension of path integral formalism to understand the origin of the temporal fluctuation scaling and the evolution of its exponent over time in nonstationary time series. To this end, we first show how the probability of transition between two states of a stochastic variable x(t) can be expressed once it is known its cumulant generating function. Also, we introduce a nonlinear term in a cumulant generating function of the form H^{(n)}(p,t;γ)∼p^{n} to obtain a model where the nth moment of the probability distribution evolves arbitrarily. Subsequently, in order to reproduce the temporal fluctuation scaling, a linear combination of H^{(n)}(p,t;γ) with n∈{1,2} is used. Therefore this allows describing how the mean M_{1}(t) and the variance Ξ_{2}(t) of empirical time series evolve. Thence, an analytical expression is deduced for the evolution of the temporal evolution of the temporal fluctuation scaling exponent α_{TFS}(t). Likewise, the validity of the expression found for α_{TFS}(t) is verified with a toy model based on white noise. Finally, this approach is verified in two stock indices (Dow Jones and Sao Paulo stock index) and two currencies (GBP-USD and EUR-USD) with daily data. It is found that this approach accurately captures the evolution of the mean and variance of these four financial derivatives after contrasting the results with a coefficient of determination that depends on H^{(n)}(p,t;γ). Also, it is shown that the temporal fluctuation scaling exponent is a measure of uncertainty or volatility in financial time series.

Ensemble Prediction Method Based on Decomposition–Reconstitution–Integration for COVID-19 Outbreak Prediction

Due to the non-linear and non-stationary nature of daily new 2019 coronavirus disease (COVID-19) case time series, existing prediction methods struggle to accurately forecast the number of daily new cases. To address this problem, a hybrid prediction framework is proposed in this study, which combines ensemble empirical mode decomposition (EEMD), fuzzy entropy (FE) reconstruction, and a CNN-LSTM-ATT hybrid network model. This new framework, named EEMD-FE-CNN-LSTM-ATT, is applied to predict the number of daily new COVID-19 cases. This study focuses on the daily new case dataset from the United States as the research subject to validate the feasibility of the proposed prediction framework. The results show that EEMD-FE-CNN-LSTM-ATT outperforms other baseline models in all evaluation metrics, demonstrating its efficacy in handling the non-linear and non-stationary epidemic time series. Furthermore, the generalizability of the proposed hybrid framework is validated on datasets from France and Russia. The proposed hybrid framework offers a new approach for predicting the COVID-19 pandemic, providing important technical support for future infectious disease forecasting.

Enhancing the predictive performance of non-stationary time series data through various transformations

Enhancing the predictive performance of non-stationary time series data through various transformations

Adaptive Wavelet Domain Principal Component Analysis for Nonstationary Time Series

High-dimensional multivariate nonstationary time series, that is, data whose second order properties vary over time, are common in many scientific and industrial applications. In this article we propose a novel wavelet domain dimension reduction technique for nonstationary time series. By constructing a time-scale adaptive principal component analysis of the data, our proposed method is able to capture the salient dynamic features of the multivariate time series. We also introduce a new time and scale dependent cross-coherence measure to quantify the extent of association between a multivariate nonstationary time series and its proposed wavelet domain principal component representation. Theoretical results establish that our associated estimation scheme enjoys good bias and consistency properties when determining wavelet domain principal components of input data. The proposed method is illustrated using extensive simulations and we demonstrate its applicability on a real-world dataset arising in a neuroscience study. Supplementary materials, with proofs of theoretical results, additional simulations and code, are available online.

Online Detection and Fuzzy Clustering of Anomalies in Non-Stationary Time Series

We consider the challenge of detecting and clustering point and collective anomalies in streaming data that exhibit significant nonlinearities and seasonal structures. The challenge is motivated by detecting problems in a communications network, where we can measure the throughput of nodes, and wish to rapidly detect anomalous traffic behaviour. Our approach is to train a neural network-based nonlinear autoregressive exogenous model on initial training data, then to use the sequential collective and point anomaly framework to identify anomalies in the residuals generated by comparing one-step-ahead predictions of the fitted model with the observations, and finally, we cluster the detected anomalies with fuzzy c-means clustering using empirical cumulative distribution functions. The autoregressive model is sufficiently general and robust such that it provides the nearly (locally) stationary residuals required by the anomaly detection procedure. The combined methods are successfully implemented to create an adaptive, robust, computational framework that can be used to cluster point and collective anomalies in streaming data. We validate the method on both data from the core of the UK’s national communications network and the multivariate Skoltech anomaly benchmark and find that the proposed method succeeds in dealing with different forms of anomalies within the nonlinear signals and outperforms conventional methods for anomaly detection and clustering.

A new spectral distance based on adaptive selection algorithm for non-stationary time series

We propose a distance metric to quantify the dissimilarity between time series from the perspective of the frequency domain, especially non-stationary time series, called the Wasserstein Hilbert marginal spectral distance (WHMS). It calculates the Wasserstein distance between the Hilbert marginal spectral (HMS) of the time series, where HMS is the integral of Hilbert spectrum. Hilbert spectrum is obtained by the Hilbert transform performed on the intrinsic mode functions (IMFs) generated after the Empirical Mode Decomposition (EMD) operation. On this basis, we propose an IMF adaptive selection algorithm to improve the feature extraction accuracy of the HMS, free from a large number of prior experiments to set the threshold. We demonstrate that WHMS distance can be used as a general measure of time series from both theoretical and practical aspects. Combined with multidimensional scaling (MDS), we comprehensively evaluate the proposed method through simulation experiments and empirical data. The results confirm the good applicability of the WHMS distance combined with the IMF adaptive selection algorithm, which can distinguish different types of complex systems, is robust to noise, and is superior to Wasserstein–Fourier (WF) distance, Wasserstein–Fourier distance with short time Fourier transform (STFT), Euclidean distance and Chebyshev distance.

TIME SERIES WITH MULTIPLE CHANGE POINTS AND CENSORED OBSERVATIONS

This article examines a Bayesian model for a nonstationary time series with an unknown number of change points and censored observations. Each segment is assumed to be an autoregressive process with order one. To estimate the number and locations of change points, we use the reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. The censored problem is solved by imputing the censored values from a multivariate normal distribution based on the observed part. A numerical example shows that the estimates of the number of change points and their localizations have little bias. Additionally, the estimates are robust to the censoring percentage.

Analisis Data Time Series Menggunakan Model Kernel: Pemodelan Data Harga Saham MDKA

<p>Classic time series data analysis techniques, such as autoregressive, model stationary data in which the values of prior observations influence the current observations through a process known as linear regression. There are several requirements for error assumptions in autoregressive, including independence, normal distribution with a zero mean and constant variance. It is frequently discovered that these assumptions are challenging to verify when modelling real data. Kernel time series regression is an alternative model that does not require error assumptions. Non-stationary time series data can be effectively modelled using the kernel time series method. Time series data that isn't yet stationary is made stationary first, then the data is modified by forming the current stationary time series data as the response variable and the previous period data as the predictor variable. Next, regression kernel modelling is carried out while applying kernel weight function and determining the optimal bandwidth. For development of science, the optimal bandwidth can be achieved by minimizing the MSE, CV, GCV, or UBR values. It is possible to use R<sup>2</sup> or MAPE as the kernel time series regression model's goodness metric. A strong model is generated while modelling MDKA stock price data using kernel regression utilizing the Gaussian kernel function and optimal bandwidth selection using GCV since R<sup>2</sup> is 0.9828372 more than 0.67 and MAPE is 1.985681% under 10%.</p><p><strong>Keywords</strong><strong>: </strong>3 time series; kernel regression; GCV; MDKA stock price.</p>

Oracle-efficient estimation and trend inference in non-stationary time series with trend and heteroscedastic ARMA error

The non-stationary time series often contain an unknown trend and unobserved error terms. The error terms in the proposed model consist of a smooth variance function and the latent stationary ARMA series, which allows heteroscedasticity at different time points. The theoretically justified two-step B-spline estimation method is proposed for the trend and variance function in the model, and then residuals are obtained by removing the trend and variance function estimators from the data. The maximum likelihood estimator (MLE) for the latent ARMA error coefficients based on the residuals is shown to be oracally efficient in the sense that it has the same asymptotic distribution as the infeasible MLE if the trend and variance function were known. In addition to the oracle efficiency, a kernel estimator is obtained for the trend function and shown to converge to the Gumbel distribution. It yields an asymptotically correct simultaneous confidence band (SCB) for the trend function, which can be used to test the specific form of trend. A simulation-based procedure is proposed to implement the SCB, and simulation and real data analysis illustrate the finite sample performance.

Distribution network planning method: Integration of a recurrent neural network model for the prediction of scenarios

The integration of new types of consumers and prosumers into distribution networks presents significant challenges for network planners, especially given the uncertainties of future expansions. This paper introduces a method for distribution network planning that integrates a Long-Short Term Memory (LSTM) model with a confidence interval threshold for long-term scenario predictions. The proposed forecasting model considers datasets collected from monitoring systems which include aggregated demand and self-consumption from photovoltaic sources. The use of the LSTM model enables a more accurate prediction of expansions, as it captures nonlinear patterns in time series data while taking into account the inherent characteristics of non-stationary time series data. The proposed method was tested on an existing radial medium voltage distribution network in Spain, taking into account the operational thermal limits for lines and substation time series measurements. The proposed planning approach also considers asset costs, active planning solutions, and time-series load flow analysis. The results obtained indicate that the use of time-based forecasts from aggregated generation and demand is a more realistic solution for flexible planning solutions when they are compared to traditional strategies.

MetaCAE: Causal autoencoder with meta-knowledge transfer for brain effective connectivity estimation

Using machine learning methods to estimate brain effective connectivity networks from functional magnetic resonance imaging (fMRI) data has gradually become one of the hot subjects in the fields of neuroscience. In particular, the encoder–decoder based methods can effectively extract the connections in fMRI time series, which have achieved promising performance. However, these methods generally use Granger causality model, which may identify false directions due to the non-stationary characteristic of fMRI data. Additionally, fMRI datasets have limited sample sizes, which significantly constrains the development of these methods. In this paper, we propose a novel brain effective connectivity estimation method based on causal autoencoder with meta-knowledge transfer, called MetaCAE. The proposed approach employs a causal autoencoder (CAE) to extract causal dependencies from non-stationary fMRI time series, and leverages meta-knowledge transfer to improve the estimation accuracy on small-sample data. More specifically, MetaCAE first employs a temporal convolutional encoder to extract non-stationary temporal information from fMRI time series. Then it uses a structural equation model-based decoder to decode causal relationships between brain regions. Finally, it utilizes a model-agnostic meta-learning method to learn the meta-knowledge of the shared brain effective connectivity among different subjects, and transfers the meta-knowledge to the CAE to enhance its estimation ability on small-sample fMRI data. Comprehensive experiments on both simulated and real-world data demonstrate the efficacy of MetaCAE in estimating brain effective connectivity.

Weighted Nonlinear Regression With Nonstationary Time Series

Weighted Nonlinear Regression With Nonstationary Time Series

ОБНАРУЖЕНИЕ РАЗЛАДОК В НЕСТАЦИОНАРНЫХ РЯДАХ НАБЛЮДЕНИЙ: ОБЗОР ТЕХНОЛОГИЙ И КРИТИЧЕСКИЙ АНАЛИЗ

A critical analysis of modern mathematical technologies used to solve the problem of nonstationary time series change point, taking into account the chaotic nature of the observed processes, is considered. The known methods of the quickest change point detection in nonstationary time series described by traditional models of random processes are considered as initial assumptions. The discrepancy between the known time series observation models and the real processes occurring in unstable immersion environments described by the concept of stochastic chaos is shown. Time series segmentation technologies based on a posteriori analysis of changes in the properties of observation sites of chaotic processes are presented. Options for building online algorithms for the quickest change point detection based on methods of multiregressive data analysis and metric machine learning technologies are considered.

Deep Learning LSTM-based approaches for 10.7 cm solar radio flux forecasting up to 45-days

Accurate forecasting of F10.7 index is important on short, medium and long-term timescales since F10.7 is an excellent proxy of solar activity and it plays an important role within the Space Weather framework. The analysis of the signatures of transient solar radio emission and its prediction are a challenging task as the underpinning physical processes are typically nonlinear, non-stationary and chaotic. In this paper we want to present three Deep Learning approaches for the daily forecasting of the adjusted F10.7 solar radio flux up to 45 days, using a family of Long Short Term Memory (LSTM) based models. We investigated two novel hybrid architectures: the LSTM model used in combination with Fast Iterative Filtering as decomposition algorithm (FIF-LSTM) and a method based on Multi-Head-Attention architecture (FIF-LSTM-MHA). FIF is a robust decomposition signal method very suitable for analyzing non-linear and non-stationary time series and it is used to separate the original time series into different oscillation components according to frequency, derived without leaving the time domain before to be fed into the neural network. The Attention mechanism is able to keep track of long-term dependencies in data sequences and improve the computational efficiency of the prediction model by reducing the effect of irrelevant information, mimicking human attention and selecting the most critical input. Our comparative analysis evaluated the models’ performance for different time lags and solar activity levels. The results indicated that the hybrid models achieve better performance than the LSTM model for mid-range F10.7 predictions while the LSTM achieves better performance within the first few time lags. FIF-LSTM-MHA gives more promising output for longer forecasts since it tends to smooth the prediction curve due to the peculiarity of the Attention module to discard less relevant features of the time series and highlight the global trend.

Long-term prediction of runoff of Heiher River in China based on EMD-LSTM networks

Due to the influence of global climate change and human activities, the time series data of river runoff become complex and non-stationary. These properties make sequence prediction difficult and with low accuracy. In order to improve the prediction accuracy, Empirical Mode Decomposition (EMD) was introduced to Long Short-Term Memory (LSTM) networks in this paper. The EMD decomposed a non-stationary time series into multiple components. We trained an LSTM network for each component, and added their predictions to obtain the final forecasting value of original sequence. At last, the EMD-LSTM model was applied to the annual runoff sequence in the upper Heihe River. By comparing with single LSTM, it shows that the EMD-LSTM has higher accuracy for the long-term prediction of river runoff.

A Phase-cum-Time Variant Fuzzy Time Series Model for Forecasting Non-stationary Time Series and its Application to the Stock Market

A Phase-cum-Time Variant Fuzzy Time Series Model for Forecasting Non-stationary Time Series and its Application to the Stock Market

Financial time series forecasting methods

The paper presents the development of time series forecasting algorithms based on the Integrated Autoregressive Moving Average Model (ARIMA) and the Fourier Expansion model. These models were applied to non-stationary time series of stock quotes after bringing these series to a stationary form. In the paper, ARIMA and Fourier Expansion model were constructed, using Python development environment. The developed algorithms were tested on Russian and American stock indices using the Mean Absolute Percentage Error metric.

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>