Data-driven shrinkage of the spectral density matrix of a high-dimensional time series

Four data-driven methods—random forest (RF), support vector machine (SVM), feed-forward neural network (FNN), and convolutional neural network (CNN)—are applied to discriminate surface and underwater vessels in the ocean using low-frequency acoustic pressure data. Acoustic data are modeled considering a vertical line array by a Monte Carlo simulation using the underwater acoustic propagation model, KRAKEN, in the ocean environment of East Sea in Korea. The raw data are preprocessed and reorganized into the phone-space cross-spectral density matrix (pCSDM) and mode-space cross-spectral density matrix (mCSDM). Two additional matrices are generated using the absolute values of matrix elements in each CSDM. Each of these four matrices is used as input data for supervised machine learning. Binary classification is performed by using RF, SVM, FNN, and CNN, and the obtained results are compared. All machine-learning algorithms show an accuracy of >95% for three types of input data—the pCSDM, mCSDM, and mCSDM with the absolute matrix elements. The CNN is the best in terms of low percent error. In particular, the result using the complex pCSDM is encouraging because these data-driven methods inherently do not require environmental information. This work demonstrates the potential of machine learning to discriminate between surface and underwater vessels in the ocean.

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Spectral Analysis for Bivariate Time Series with Long Memory

Econometric Theory ◽

10.1017/s0266466600007155 ◽

1996 ◽

Vol 12 (5) ◽

pp. 773-792 ◽

Cited By ~ 4

Author(s):

J. Hidalgo

Keyword(s):

Time Series ◽

Density Matrix ◽

Spectral Density ◽

Long Memory ◽

Limit Theorems ◽

Stationary Process ◽

Absolute Summability ◽

Spectral Density Matrix ◽

Mixing Coefficients ◽

Long Memory Processes

This paper provides limit theorems for spectral density matrix estimators and functionals of it for a bivariate covariance stationary process whose spectral density matrix has singularities not only at the origin but possibly at some other frequencies and, thus, applies to time series exhibiting long memory. In particular, we show that the consistency and asymptotic normality of the spectral density matrix estimator at a frequency, say λ, which hold for weakly dependent time series, continue to hold for long memory processes when λ lies outside any arbitrary neighborhood of the singularities. Specifically, we show that for the standard properties of spectral density matrix estimators to hold, only local smoothness of the spectral density matrix is required in a neighborhood of the frequency in which we are interested. Therefore, we are able to relax, among other conditions, the absolute summability of the autocovariance function and of the fourth-order cumulants or summability conditions on mixing coefficients, assumed in much of the literature, which imply that the spectral density matrix is globally smooth and bounded.

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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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Testing collinearity of vector time series

Econometrics Journal ◽

10.1093/ectj/uty002 ◽

2019 ◽

Vol 22 (2) ◽

pp. 97-116

Author(s):

Tucker S McElroy ◽

Agnieszka Jach

Keyword(s):

Time Series ◽

Density Matrix ◽

Spectral Density ◽

Time Series Data ◽

Series Data ◽

Test Statistics ◽

Rank Reduction ◽

Spectral Density Matrix ◽

Schur Complements ◽

Vector Time Series

Summary We investigate the collinearity of vector time series in the frequency domain, by examining the rank of the spectral density matrix at a given frequency of interest. Rank reduction corresponds to collinearity at the given frequency. When the time series is nonstationary and has been differenced to stationarity, collinearity corresponds to co-integration at a particular frequency. We examine rank through the Schur complements of the spectral density matrix, testing for rank reduction via assessing the positivity of these Schur complements, which are obtained from a nonparametric estimator of the spectral density. New asymptotic results for the test statistics are derived under the fixed bandwidth ratio paradigm; they diverge under the alternative, but under the null hypothesis of collinearity the test statistics converge to a non-standard limiting distribution. Subsampling is used to obtain the limiting null quantiles. A simulation study and an empirical illustration for 6-variate time series data are provided.

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Eigenvector Directions of Spectral Density Matrix

Underwater Acoustics and Signal Processing ◽

10.1007/978-94-009-8447-9_46 ◽

1981 ◽

pp. 571-575

Author(s):

J. P. Pignon

Keyword(s):

Density Matrix ◽

Spectral Density ◽

Spectral Density Matrix

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Modal Participation Estimated from the Response Correlation Matrix

Shock and Vibration ◽

10.1155/2019/9347075 ◽

2019 ◽

Vol 2019 ◽

pp. 1-10

Author(s):

Rune Brincker ◽

Sandro D. R. Amador ◽

Martin Juul ◽

Manuel Lopez-Aenelle

Keyword(s):

Density Matrix ◽

Spectral Density ◽

Correlation Matrix ◽

Analytical Form ◽

Mode Shapes ◽

Transformation Matrices ◽

Spectral Density Matrix ◽

Spectral Density Functions ◽

Input Load ◽

The Time Domain

In this paper, we are considering the case of estimating the modal participation vectors from the operating response of a structure. Normally, this is done using a fitting technique either in the time domain using the correlation function matrix or in the frequency domain using the spectral density matrix. In this paper, a more simple approach is proposed based on estimating the modal participation from the correlation matrix of the operating responses. For the case of general damping, it is shown how the response correlation matrix is formed by the mode shape matrix and two transformation matrices T1 and T1 that contain information about the modal parameters, the generalized modal masses, and the input load spectral density matrix Gx. For the case of real mode shapes, it is shown how the response correlation matrix can be given a simple analytical form where the corresponding real modal participation vectors can be obtained in a simple way. Finally, it is shown how the real version of the modal participation vectors can be used to synthesize empirical spectral density functions.

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Some Elements of Operational Modal Analysis

Shock and Vibration ◽

10.1155/2014/325839 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 15

Author(s):

Rune Brincker

Keyword(s):

Density Matrix ◽

Spectral Density ◽

Modal Analysis ◽

Mode Shape ◽

Fourier Integral ◽

Operational Modal Analysis ◽

Difficult Case ◽

Domain Identification ◽

Spectral Density Matrix ◽

Random Responses

This paper gives an overview of the main components of operational modal analysis (OMA) and can serve as a tutorial for research oriented OMA applications. The paper gives a short introduction to the modeling of random responses and to the transforms often used in OMA such as the Fourier series, the Fourier integral, the Laplace transform, and the Z-transform. Then the paper introduces the spectral density matrix of the random responses and presents the theoretical solutions for correlation function and spectral density matrix under white noise loading. Some important guidelines for testing are mentioned and the most common techniques for signal processing of the operating signals are presented. The algorithms of some of the commonly used time domain and frequency domain identification techniques are presented and finally some issues are discussed such as mode shape scaling, and mode shape expansion. The different techniques are illustrated on the difficult case of identifying the three first closely spaced modes of the Heritage Court Tower building.

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Cross-spectral density matrix of a random electromagnetic beam propagating through an apertured axially nonsymmetrical optical system

Physics Letters A ◽

10.1016/j.physleta.2008.03.004 ◽

2008 ◽

Vol 372 (22) ◽

pp. 4135-4140 ◽

Cited By ~ 4

Author(s):

Xinyue Du ◽

Daomu Zhao ◽

Olga Korotkova

Keyword(s):

Density Matrix ◽

Spectral Density ◽

Optical System ◽

Electromagnetic Beam ◽

Spectral Density Matrix ◽

Cross Spectral Density

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Bayesian recurrent neural network as a tool for reconstructing dynamical systems from multidimensional data

10.5194/egusphere-egu2020-8787 ◽

2020 ◽

Author(s):

Alexander Feigin ◽

Aleksei Seleznev ◽

Dmitry Mukhin ◽

Andrey Gavrilov ◽

Evgeny Loskutov

Keyword(s):

Neural Network ◽

Time Series ◽

Cost Function ◽

Recurrent Neural Network ◽

Bayesian Optimization ◽

Data Driven ◽

High Dimensional ◽

Model Learning ◽

Low Dimensional ◽

The Cost

We suggest a new method for construction of data-driven dynamical models from observed multidimensional time series. The method is based on a recurrent neural network (RNN) with specific structure, which allows for the joint reconstruction of both a low-dimensional embedding for dynamical components in the data and an operator describing the low-dimensional evolution of the system. The key link of the method is a Bayesian optimization of both model structure and the hypothesis about the data generating law, which is needed for constructing the cost function for model learning. &#160;The form of the model we propose allows us to construct a stochastic dynamical system of moderate dimension that copies dynamical properties of the original high-dimensional system. An advantage of the proposed method is the data-adaptive properties of the RNN model: it is based on the adjustable nonlinear elements and has easily scalable structure. The combination of the RNN with the Bayesian optimization procedure efficiently provides the model with statistically significant nonlinearity and dimension. The method developed for the model optimization aims to detect the long-term connections between system&#8217;s states &#8211; the memory of the system: the cost-function used for model learning is constructed taking into account this factor. In particular, in the case of absence of interaction between the dynamical component and noise, the method provides unbiased reconstruction of the hidden deterministic system. In the opposite case when the noise has strong impact on the dynamics, the method yield a model in the form of a nonlinear stochastic map determining the Markovian process with memory. Bayesian approach used for selecting both the optimal model&#8217;s structure and the appropriate cost function allows to obtain the statistically significant inferences about the dynamical signal in data as well as its interaction with the noise components. Data driven model derived from the relatively short time series of the QG3 model &#8211; the high dimensional nonlinear system producing chaotic behavior &#8211; is shown be able to serve as a good simulator for the QG3 LFV components. The statistically significant recurrent states of the QG3 model, i.e. the well-known teleconnections in NH, are all reproduced by the model obtained. Moreover, statistics of the residence times of the model near these states is very close to the corresponding statistics of the original QG3 model. These results demonstrate that the method can be useful in modeling the variability of the real atmosphere.The work was supported by the Russian Science Foundation (Grant No. 19-42-04121).

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