Bivariate Time Series Analysis

All processes created by nature are random. Every time series is a sample record of some random process and it should be studied with methods based upon theory of random processes. This book offers easy-to-use tools for analysis of scalar, bivariate and trivariate time series to users not closely familiar with the theory. It contains programs AVESTA1 and AVESTA3 which produce information about statistical properties of time series in both time and frequency domains. AVESTA1 can forecast time series in agreement with the mathematically strict theory of extrapolation created by A. Kolmogorov and N. Wiener and unknown in natural sciences. The time domain characteristics of scalar and multivariate time series include covariance and correlation functions, autoregressive models and predictability characteristics while the frequency domain functions include spectra and coherent spectra, ordinary, multiple, and partial coherences, gain and phase factors, and frequency dependent time lags. All results are produced in one run of each program in accordance with user’s instructions. The book contains 26 examples which are supposed to be reproduced by the user to get experience in mathematically proper time series analysis. Hopefully, the reader agrees that methods of analysis should correspond to the probabilistic nature of the time series.

Spectral analysis of multivariate geochemical time series

PurposeThis study aims to examine the relationship between green finance, economic growth, renewable energy consumption (energy efficiency), energy import and CO2 emission in Vietnam using multivariate time series analysis.Design/methodology/approachThe data were collected from 1986 to 2018 since Vietnam initiated the economic reforms, namely “Doi Moi” in 1986. The concept and methods of cointegration, Granger causality and error correction model (ECM) were employed to establish the relationship between the variables of interest.FindingsOur results confirmed the existence of cointegration among the variables. The Granger causality test revealed unidirectional causality running from renewable energy consumption to CO2 emission and green investment to CO2 emission.Originality/valueThis study results confirm the existence of cointegration among the variables. The results of the study imply that policies on economic development impose a significant impact on pollution in Vietnam. This study has described Vietnam, its economic development, green manufacturing practices, its environmental health and level of carbon dioxide emission which was enhanced due to COVID-19.

Results obtained with interindividual techniques in a representative sample of a population are not necessarily generalizable to the individual members of this population. In this article the specific condition is presented that must be satisfied to generalize from the interindividual level to the intraindividual level. A way to investigate whether this condition is satisfied is by means of multivariate time series analysis. More generally, time series analysis can be used to investigate psychological processes situated within individuals. In this article we consider a well established class of multivariate stationary time series models that may be used to study the intraindividual covariance structure. We demonstrate the application of some of these models with an empirical example consisting of state measurements of behavior associated with the Five Factor Model of Personality. We illustrate how one can investigate whether individuals are similar with respect to their intraindividual structure of variation, and whether this structure is similar to the structure of interindividual variation.

A fast LSH-based similarity search method for multivariate time series

Time Series: State Space Methods

Many complex systems in physics, biology and engineering are modeled as dynamical networks and described using multivariate time series analysis. Recent developments have shown that the emergent dynamics of a network system are significantly affected by interactions involving multiple network nodes which cannot be described using pairwise links. While these higher-order interactions can be probed using information-theoretic measures, a rigorous framework to describe them in the frequency domain is still lacking. This work presents an approach for the spectral decomposition of multivariate information measures, capable of identifying higher-order synergistic and redundant interactions between oscillatory processes. We show theoretically that synergy and redundancy can coexist at different frequencies among the output signals of a network system and can be detected only using the proposed spectral method. To demonstrate the broad applicability of the framework, we provide parametric and non-parametric data-efficient estimators for the spectral information measures, and employ them to describe multivariate interactions in three complex systems producing rich oscillatory dynamics, namely the human brain, a ring of electronic oscillators, and the global climate system. In these systems, we show that the use of our framework for the spectral decomposition of information measures reveals multivariate and higher-order interactions not detectable in the time domain. Our results are exemplary of how the frequency-specific analysis of multivariate dynamics can aid the implementation of assessment and control strategies in realworld network systems.

SUMMARY We propose a method to specify an appropriate yet parsimonious vector autoregressive moving average (ARMA) model for a given multivariate time series. By considering contemporaneous linear transformations of the vector process, we introduce the concept of scalar component models within the vector ARMA framework (a) to reveal possibly hidden simplifying structures of the process, (b) to achieve parsimony in parameterization and (c) to identify the exchangeable models. The simplifying structures are of particular importance in the analysis of multivariate time series because they are often not obvious from the observed data but can be used to gain insights to the process under study. The analytical tool used to search for scalar component models is a canonical correlation analysis of vector processes and the proposed procedures are illustrated via two real examples.

Nowadays, great quantities of data are produced by a large and diverse family of sensors (e.g., remote sensors, biochemical sensors, wearable devices), which typically measure multiple variables over time, resulting in data streams that can be profitably organized as multivariate time-series. In practical scenarios, the speed at which such information is collected often makes the data labeling task uneasy and too expensive, so that limit the use of supervised approaches. For this reason, unsupervised and exploratory methods represent a fundamental tool to deal with the analysis of multivariate time series. In this paper we propose a deep-learning based framework for clustering multivariate time series data with varying lengths. Our framework, namely DeTSEC (Deep Time Series Embedding Clustering), includes two stages: firstly a recurrent autoencoder exploits attention and gating mechanisms to produce a preliminary embedding representation; then, a clustering refinement stage is introduced to stretch the embedding manifold towards the corresponding clusters. Experimental assessment on six real-world benchmarks coming from different domains has highlighted the effectiveness of our proposal.

Remote sea-surface temperature variations (2001–2019) in Kuwait Bay: Time series analysis in frequency and time domains

By extending Takens' embedding theorem [Dynamical Systems and Turbulence, Warwick 1980, edited by D. Rand and L.-S. Young (Springer, Berlin, 1981), pp. 366-381], Deyle and Sugihara [PLoS One 6, 1-8 (2011)] provided a theoretical justification for using parallel measurement time series to reconstruct a system's attractor. Building on Takens' framework, Brunton et al. [Nat. Commun. 8, 19 (2017)] introduced the Hankel alternative view of Koopman (HAVOK) algorithm, a data-driven approach capable of linearizing chaotic systems through delay embeddings. In this work, a modified version of the original algorithm (mHAVOK) is presented, a practical realization of Deyle and Sugihara's generalized embedding theory. mHAVOK extends the original algorithm from one to multiple input time series and introduces a systematic approach to separating linear and nonlinear terms. An R2-informed quality score is introduced and shown to be a reliable guide for the selection of the reduced rank. The algorithm is tested on the familiar Lorenz system, as well as the more sophisticated Sprott system, which features different behaviors depending on the initial conditions. The quality of the reconstructions is assessed with the Chamfer distance, validating how mHAVOK allows for a more accurate reconstruction of the system dynamics. The new methodology generalizes HAVOK by allowing the analysis of multivariate time series, fundamental in real-life data-driven applications.

Inhomogeneous Dependency Modelling with Time Varying Copulae

We propose an extension of a previously proposed method for lead-lag analysis of multivariate time series to include the analysis of spatial correlations. We applied the extended spatial and temporal method to CIRIGHTS, a large global human rights dataset, in order to determine the most influential and most influenced indicators of human rights, freedoms, and atrocities over time. We consider four target countries, each from a different continent. The previously proposed method used a weighted directed network with several lags of each variable as nodes and with edges weighted by transfer entropy. In this study, that method is extended to the spatial multivariate time series by investigating geographical correlations between each target country and its surrounding region to determine which variables best represent activity between the target country and the surrounding region. The nodes in this undirected network are variables in the target country and variables in the surrounding region, with edges weighted by mutual information. Eigenvector centrality is used to determine which variables best represent activity between the target country and the surrounding region. Key findings indicate that occupational and worker rights are the most influential and most influenced in the target countries over time and by region, and laws tend to influence future activity.

To solve the problem of the flashover forecasting of contaminated or polluted insulator, a flashover forecasting model of contaminated insulators based on multivariate nonlinear time series analysis is proposed in the paper. The equivalent salt deposit density (ESDD) is the key of flashover on polluted insulator. The ESDD value of insulator can be forecasted by the method of nonlinear time series analysis of the ESDD time series and a forecasting model of polluted insulator flashover is proposed in the paper. The forecasting model consists of two artificial neural networks that reflect relationship of environment, ESDD and flashover probability. The first is used to estimate the ESDD time series of insulator and the second is employed to calculate the probability of the flashover. A series of artificial pollution tests show that the results of the forecasting model is acceptable.

The purpose of this study is a retrospective analysis of long-term trends in the dynamics of the number of people employed in agriculture and the development of a scenario forecast of its further changes in the medium term. The information base was the official data of the Federal State Statistics Service, covering an unprecedentedly long period from 1939 to 2023 on key indicators: the number of people employed in the industry, the rural population and the labor productivity index. Using the tools of econometric analysis of multivariate time series, including the Dickie-Fuller test and the Angle-Granger procedure, the initial hypotheses were verified. It has been established that the time series of the number of workers and the rural population are non-stationary processes of the first order of integration (I(1)), while the dynamics of labor productivity is a stationary process (I(0)). A critically important result was the identification of a co-integration relationship between the number of employed and the rural population, which confirms the existence of a stable long-term equilibrium between them and avoids the construction of false regressions. Based on the identified dependencies, a statistically correct distributed lag autoregression (ADL) model was specified and evaluated, adequately describing the relationships between all three variables. The scenario forecast until 2030 carried out on its basis demonstrates that the trend towards a reduction in the personnel potential of the industry is likely to continue. According to the pessimistic scenario, the number of employed people may decrease to 3.4 million people, while even the most optimistic scenario assumes only the stabilization of the indicator at the current level of about 4.4 million people. The study proves that, unlike the post-war period, the modern growth of labor productivity does not compensate for the outflow of the population, and demographic and migration processes in rural areas remain the dominant factor determining the crisis of staffing in the agricultural sector of Russia.