Quantile cross-spectral density: A novel and effective tool for clustering multivariate time series

We derive uniform convergence results of lag-window spectral density estimates for a general class of multivariate stationary processes represented by an arbitrary measurable function of iid innovations. Optimal rates of convergence, that hold as both the time series and the cross section dimensions diverge, are obtained under mild and easily verifiable conditions. Our theory complements earlier results, most of which are univariate, which primarily concern in-probability, weak or distributional convergence, yet under a much stronger set of regularity conditions, such as linearity in iid innovations. Based on cross spectral density functions, we then propose a new test for independence between two stationary time series. We also explain the extent to which our results provide the foundation to derive the double asymptotic results for estimation of generalized dynamic factor models.

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ON TESTING FOR SERIAL CORRELATION WITH A WAVELET-BASED SPECTRAL DENSITY ESTIMATOR IN MULTIVARIATE TIME SERIES

Econometric Theory ◽

10.1017/s0266466606060312 ◽

2006 ◽

Vol 22 (04) ◽

Cited By ~ 8

Author(s):

Pierre Duchesne

Keyword(s):

Time Series ◽

Spectral Density ◽

Serial Correlation ◽

Multivariate Time Series ◽

Density Estimator

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Asymptotic Expansions of the Distributions of Statistics Related to the Spectral Density Matrix in Multivariate Time Series and Their Applications

Econometric Theory ◽

10.1017/s0266466600004928 ◽

1990 ◽

Vol 6 (1) ◽

pp. 75-96 ◽

Cited By ~ 2

Author(s):

Masanobu Taniguchi ◽

Koichi Maekawa

Keyword(s):

Time Series ◽

Density Matrix ◽

Spectral Density ◽

Asymptotic Expansions ◽

Multivariate Time Series ◽

Principal Component ◽

Autoregressive Process ◽

Reduced Form ◽

Gaussian Stationary Process ◽

Spectral Density Matrix

Let {X(t)} be a multivariate Gaussian stationary process with the spectral density matrix f0(ω), where θ is an unknown parameter vector. Using a quasi-maximum likelihood estimator θ̂ of θ, we estimate the spectral density matrix f0(ω) by fθ̂(ω). Then we derive asymptotic expansions of the distributions of functions of fθ̂(ω). Also asymptotic expansions for the distributions of functions of the eigenvalues of fθ̂(ω) are given. These results can be applied to many fundamental statistics in multivariate time series analysis. As an example, we take the reduced form of the cobweb model which is expressed as a two-dimensional vector autoregressive process of order 1 (AR(1) process) and show the asymptotic distribution of θ̂, the estimated coherency, and contribution ratio in the principal component analysis based on θ̂ in the model, up to the second-order terms. Although our general formulas seem very involved, we can show that they are tractable by using REDUCE 3.

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F4: An All-Purpose Tool for Multivariate Time Series Classification

Mathematics ◽

10.3390/math9233051 ◽

2021 ◽

Vol 9 (23) ◽

pp. 3051

Author(s):

Ángel López-Oriona ◽

José A. Vilar

Keyword(s):

Time Series ◽

Multivariate Time Series ◽

Discrete Wavelet ◽

Time Series Classification ◽

East Anglia ◽

Ecg Signals ◽

Novel Approach ◽

Cross Spectral Density ◽

The University ◽

Electrocardiogram Ecg

We propose Fast Forest of Flexible Features (F4), a novel approach for classifying multivariate time series, which is aimed to discriminate between underlying generating processes. This goal has barely been addressed in the literature. F4 consists of two steps. First, a set of features based on the quantile cross-spectral density and the maximum overlap discrete wavelet transform are extracted from each series. Second, a random forest is fed with the extracted features. An extensive simulation study shows that F4 outperforms some powerful classifiers in a wide variety of situations, including stationary and nonstationary series. The proposed method is also capable of successfully discriminating between electrocardiogram (ECG) signals of healthy subjects and those with myocardial infarction condition. Additionally, despite lacking shape-based information, F4 attains state-of-the-art results in some datasets of the University of East Anglia (UEA) multivariate time series classification archive.

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Robust Methods for Soft Clustering of Multidimensional Time Series

Engineering Proceedings ◽

10.3390/engproc2021007060 ◽

2021 ◽

Vol 7 (1) ◽

pp. 60

Author(s):

Ángel López-Oriona ◽

Pierpaolo D’Urso ◽

José A. Vilar ◽

Borja Lafuente-Rego

Keyword(s):

Time Series ◽

Spectral Density ◽

Simulation Study ◽

Robust Methods ◽

Robust Algorithms ◽

Fuzzy C Means ◽

Multidimensional Time Series ◽

Cross Spectral Density ◽

Alternative Procedures ◽

Underlying Processes

Three robust algorithms for clustering multidimensional time series from the perspective of underlying processes are proposed. The methods are robust extensions of a fuzzy C-means model based on estimates of the quantile cross-spectral density. Robustness to the presence of anomalous elements is achieved by using the so-called metric, noise and trimmed approaches. Analyses from a wide simulation study indicate that the algorithms are substantially effective in coping with the presence of outlying series, clearly outperforming alternative procedures. The usefulness of the suggested methods is also highlighted by means of a specific application.

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