Tests for periodic components in multiple time series

: This report describes some interpretations and uses of eigenvalues and eigenvectors of spectral and sample spectral density matrices of multiple stationary time series. The spectral density matrix of a zero-mean multiple stationary time series is defined. Eigenvalues and eigenvectors of the spectral density matrix are discussed and principal component theory is presented. Statistical distribution theory and related results are used to investigate the eigenvalues of a sample spectral density matrix. This investigation gives methods for obtaining simultaneous confidence bounds on the elements of the true spectral density matrix and its inverse, and also methods for obtaining confidence bounds on the eigenvalues of the true spectral density matrix.

Novel approaches in Extended Principal Component Analysis to compare spatio-temporal patterns among multiple image time series

The aim of this article is to introduce new resampling scheme for nonstationary time series, called generalized resampling scheme (GRS). The proposed procedure is a generalization of well known in the literature subsampling procedure and is simply related to existing block bootstrap techniques. To document the usefulness of GRS, we consider the example of model with almost periodic phenomena in mean and variance function, where the consistency of the proposed procedure was examined. Finally, we prove the consistency of GRS for the spectral density matrix for nonstationary, multivariate almost periodically correlated time series. We consider both zero mean and non‐zero mean case. The consistency holds under general assumptions concerning moment and α‐mixing conditions for multivariate almost periodically correlated time series. Proving the consistency in this case poses a difficulty since the estimator of the spectral density matrix can be interpreted as a sum of random matrixes whose dependence grow with the sample size.

Signal processing of flutter flight test data enables verification of aircraft flutter design, and the signal from a flutter flight test excited by atmospheric turbulence is a particularly important form of the flutter test. Owing to the randomness of atmospheric turbulence excitation, multi-channel analysis of turbulence responses at various positions in the same component can improve the analytical accuracy of flutter signal processing. The relationship between the maximum singular value of the multi-channel turbulence response power spectral density matrix and the system self-power spectral density function is elucidated herein using a frequency domain decomposition method. However, there is a contradiction in the power spectral density function between the spectral line density and the spectral line smoothing calculated based on the periodogram of the frequency domain decomposition. By applying an autoregressive spectral model, the power spectral density function of the turbulence response is calculated to achieve spectral line smoothing and sufficient spectral line density. Additionally, the power spectral density function is then used to construct the power spectral density function matrix of the multi-channel turbulence response, and the maximum singular value curve is calculated based on the singular value decomposition of each spectral pin. Finally, the modal parameters of the turbulence response signal are estimated via multi-modal frequency domain fitting. The developed approach is validated based on simulations and flutter flight test turbulence response signals.

Test procedures for assessing whether two stationary and independent time series with unequal lengths have the same spectral density (or same auto‐covariance function) are investigated. A new test statistic is proposed based on the wavelet transform. It relies on empirical wavelet coefficients of the logarithm of two spectral densities' ratio. Under the null hypothesis that two spectral densities are the same, the asymptotic normal distribution of the empirical wavelet coeffcients is derived. Furthermore, these empirical wavelet coefficients are asymptotically uncorrelated. A test statistic is proposed based on these results. The performance of the new test statistic is compared to several recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed test is very comparable to the current test statistics in most cases. The main advantage of our proposed test statistic is that it is constructed very simply and is easy to implement.

Gait variability is a sensitive metric for assessing functional deficits in individuals with mobility impairments. To correctly represent the temporal evolution of gait kinematics, nonlinear measures require extended and uninterrupted time series. In this study, we present and validate a novel algorithm for concatenating multiple time-series in order to allow the nonlinear analysis of gait data from standard and unrestricted overground walking protocols. The full-body gait patterns of twenty healthy subjects were captured during five walking trials (at least 5 minutes) on a treadmill under different weight perturbation conditions. The collected time series were cut into multiple shorter time series of varying lengths and subsequently concatenated using a novel algorithm that identifies similar poses in successive time series in order to determine an optimal concatenation time point. After alignment of the datasets, the approach then concatenated the data to provide a smooth transition. Nonlinear measures to assess stability (Largest Lyapunov Exponent, LyE) and regularity (Sample Entropy, SE) were calculated in order to quantify the efficacy of the concatenation approach using intra-class correlation coefficients, standard error of measurement and paired effect sizes. Our results indicate overall good agreement between the full uninterrupted and the concatenated time series for LyE. However, SE was more sensitive to the proposed concatenation algorithm and might lead to false interpretation of physiological gait signals. This approach opens perspectives for analysis of dynamic stability of gait data from physiological overground walking protocols, but also the re-processing and estimation of nonlinear metrics from previously collected datasets.

Coevolving multiple time series are ubiquitous and naturally appear in a variety of high-impact applications, ranging from environmental monitoring, computer network traffic monitoring, motion capture, to physiological signal in health care and many more. In many scenarios, the multiple time series data is often accompanied by some contextual information in the form of networks. In this paper, we refer to such multiple time series, together with its embedded network as a network of coevolving time series. In order to unveil the underlying patterns of a network of coevolving time series, we propose DCMF, a dynamic contextual matrix factorization algorithm. The key idea is to find the latent factor representation of the input time series and that of its embedded network simultaneously. Our experimental results on several real datasets demonstrate that our method (1) outperforms its competitors, especially when there are lots of missing values; and (2) enjoys a linear scalability w.r.t. the length of time series.

Measuring and testing Granger causality over the spectrum: An application to European production expectation surveys

This paper considers the inference of trends in multiple, nonstationary time series. To test whether trends are parallel to each other, we use a parallelism index based on the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -distances between nonparametric trend estimators and their average. A central limit theorem is obtained for the test statistic and the test's consistency is established. We propose a simulation-based approximation to the distribution of the test statistic, which significantly improves upon the normal approximation. The test is also applied to devise a clustering algorithm. Finally, the finite-sample properties of the test are assessed through simulations and the test methodology is illustrated by a cell phone download data collected in the United States.

The need for detection, identification and prediction of anomalies emerged just a short time ago. This task brings multiple challenges that are mainly associated with recording and detection of novel attacks and influences. It draws attention of experts in network security and diagnostics of information and industrial systems. Detection of anomalies in dynamic dataflows largely determines the efficiency of management of computer network information security within information and industrial systems. At the same time, the technology of dynamic dataflow prediction is also very important for building systems intended to detect anomalies in data protection within industrial control systems. Any automated systems are based on available computer capabilities and advances in management theory, mathematical modeling and optimizations methods. Processes that occur in industrial automated systems and that are reflected in dataflows being observed constitute complex multi-component processes, thus making it more complicated to predict such processes. In this case, complex multi-component time series data should be predicted at different (short-term, medium-term and long-term) time scales. A neural network architecture matching the structure of the multi-component time series being predicted should be built for generating a multi-component prediction. It is proposed to decompose the complex multi-component time series into several basic components using the digital signal processing technology, i. e. to perform a preliminary structural analysis of multi-component time series within the observed range of all time series that reflect operation of the industrial control system. Separate predictions with different time horizon are formed for each basic component of the multi-component time series using the available neural network architecture and machine learning taking into account dynamic characteristics of the above components. Anomalies in the observed range of multiple time series that reflect operation of the industrial control system are detected (identified) through component-wise comparison of each component (resulted from the above preliminary digital processing) of any time series within the observed range of all time series, with each prediction of the relevant component of the above time series within the observed range of all time series. This approach that implies component-wise comparison will allow to detect anomalies within the range of observed time series of the industrial control system separately by their different dynamic characteristics, and thus will improve the efficiency of management of information security within information and industrial systems.

Robust multivariate time series forecasting is crucial in many cyberphysical and Internet of Things applications. Existing state-of-the-art robust forecasting models decompose time series into independent functions covering trends and periodicities. However, these independent functions fail to capture correlations among multiple time series, thereby reducing prediction accuracy. Moreover, existing robust forecasting models treat certain abrupt but normal changes, e.g., caused by holidays, as outliers because they occur infrequently and have data distributions that resemble those of outliers. This exacerbates model bias and reduces prediction accuracy. This paper aims to capture correlations across multiple time series and abrupt but normal changes, thereby improving prediction accuracy. We employ weak labels to partition the dataset into source and target domains. Then, we propose the Domain Adversarial Robust Forecaster (DARF). This forecasting model is based on adversarial domain adaptation and includes two novel modules: Correlated Robust Forecaster (CORF) and Domain Critic. Specifically, CORF constitutes an encoder-decoder framework proficient at robust multivariate time series forecasting, and Domain Critic works to reduce data bias. Extensive experiments and discussions show that DARF is capable of state-of-the-art forecasting accuracy.

Relationships among the components of a multiple time series are shown to simplify in the frequency domain to elementary, linear time-lagged relationships. The result provides a time-lag interpretation for the partial group delay between two components of a multiple time series.

Testing nonparametric and semiparametric hypotheses in vector stationary processes

<title>Abstract</title> The MGM(1,n) model is a predictive framework developed for multivariate time series analysis. Its parameter estimation requires solving the covariance matrix \(\:{P}^{T}P\:\)derived from multiple time series. However, when these time series exhibit multicollinearity, the matrix \(\:{P}^{T}P\) may become singular, causing parameter calculation failures. Avoiding the occurrence of singular matrices in \(\:{P}^{T}P\) is crucial for applying the MGM(1, n) model. This study employs principal component analysis (PCA) to project high-dimensional variables into a lower-dimensional space. By constructing predictive models using the projected principal component time series and behavioral feature time series, the results demonstrate that this approach effectively resolves matrix singularity issues. The model achieves 3.13% and 2.13% fitting accuracy and prediction accuracy respectively, outperforming both the GM(1,1) and ARMA(1,1) models. When applied to electricity demand forecasting in Taiyuan Province, the model predicts that the province's electricity demand will reach 686.25 billion kWh by 2030.

In this chapter we consider problems of statistical inference for time series based on frequency-domain properties of the series. The fundamental tool used is the periodogram, which is defined in Section 10.1 for any time series {x 1,..., x n }. Section 10.2 deals with statistical tests for the presence of “hidden periodicities” in the data. Several tests are discussed, corresponding to various different models and hypotheses which we may wish to test. Spectral analysis for stationary time series, and in particular the estimation of the spectral density, depends very heavily on the asymptotic distribution as n → ∞ of the periodogram ordinates of the series {X 1,..., X n }. The essential results are contained in Theorem 10.3.2. Under rather general conditions, the periodogram ordinates I n (λ i ) at any set of frequencies λ 1,..., λ m , 0 < λ 1 < ⋯ < λ m < π, are asymptotically independent exponential random variables with means 2πf(λ i ), were f is the spectral density of {X t }. Consequently the periodogram I n is not a consistent estimator of 2πf. Consistent estimators can however be constructed by applying linear smoothing filters to the periodogram. The asymptotic behaviour of the resulting discrete spectral average estimators can be derived from the asymptotic behaviour of the periodogram as shown in Section 10.4. Lag-window estimators of the form \({\left( {2\pi } \right)^{ - 1}}\sum\nolimits_{\left| h \right| \leqslant r} {w\left( {h/r} \right)\hat \gamma \left( h \right)} {e^{ - ih\omega }}\), where w(x), −1 ≤ x ≤ 1, is a suitably chosen weight function, are also discussed in Section 10.4 and compared with discrete spectral average estimators. Approximate confidence intervals for the spectral density are given in Section 10.5. An alternative approach to spectral density estimation, based on fitting an AR MA model to the data and computing the spectral density of the fitted process, is discussed in Section 10.6. An important role in the development of spectral analysis has been played by the fast Fourier transform algorithm, which makes possible the rapid calculation of the periodogram for very large data sets. An introduction to the algorithm and its application to the computation of autocovariances is given in Section 10.7. The chapter concludes with a discussion of the asymptotic behaviour of the maximum likelihood estimators of the coefficients of an ARMA(p, q) process.