Multichannel relative-entropy spectrum analysis

A new method is presented for estimating the power spectral density matrix for multichannel data, given correlation values for linear combinations of the channels and given an initial estimate of the spectral density matrix. A derivation of the method from the relative-entropy principle is given. The results differ significantly from the Multisignal Relative-Entropy (Minimum-Cross-Entropy) Spectrum Analysis of Johnson and Shore because the present method does not arbitrarily force the final distributions of the various channels to be independent. For the special case when correlation values are given only for the sum of the channels, Multichannel Relative-Entropy Spectrum Analysis is shown to reduce to a two-stage procedure: first a smooth power-spectrum model is fitted to the correlations of the sum; then final estimates of the spectra and cross spectra are obtained through linear filtering. Illustrative numerical examples are presented.

: 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.

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.

This paper presents a time series analysis of fluctuations in long‒term annual mean air temperature series of Turkey, which were tested for non‒randomness by the autocorrelation function and subjected to power spectrum methods. Power spectrum estimates are computed for the purpose of finding evidence of a cyclic component within the series. The analysis covered the period of 1930–1995 and included 71 stations in seven geographical regions. For most of the series, the correlograms revealed no apparent patterns of cyclicity. The arithmetic average of the positive correlation values is about 11 years, and that corresponds well with the sun-spot cycle effect on the temperature values. A further analysis of the series by power spectrum indicated little evidence for statistically significant differences from the white‒noise spectrum. In most cases the spectral densities above the white‒noise level tended to peak at frequencies corresponding to 2·4–3·1 and 3·3–4·3 years, being particularly pronounced in the east, Mediterranean, central, and south‒eastern Anatolia regions. The 3·3–4·3‒year peaks in the central and south‒eastern Anatolia regions were significantly different from white noise at the 95 per cent level. Roughly two groups of regions with different spectral shapes are identified. The coastal regions almost exhibited a white‒noise spectrum whereas the inland regions tended to approximate a grey‒noise spectrum. © 1998 Royal Meteorological Society.

It is shown that under certain conditions cross spectra between correlated inputs may be neglected when estimating frequency response of multiple-input, single-output systems. When the input time histories are similar except for specified time lags, the above conditions exist, providing appropriate frequency smoothing is used. An additional requirement is that the magnitude of the spectra of the inputs should be constant, or vary periodically over the smoothing bandwidth. When these conditions are fulfilled, the frequency-smoothed input cross spectra will be considerably less than the input auto spectra and can be neglected in the spectral matrix. Neglecting input cross spectra can save experimental and computer time, since for n inputs, n(n − 1) cross spectra need to be measured. The disadvantage is in a loss of frequency resolution due to the frequency smoothing which forms the basis of the technique. However, in most estimation procedures, some frequency smoothing is needed to reduce the random error. Two examples are presented to illustrate the effect for a two-input and a six-input system. This work has been directed at measuring the frequency responses and predicting the noise of a six-cylinder diesel engine.

A technique is presented for generating a modified cross- spectral density matrix (CSDM) that is sensitive to fluctuations in the frequency bin amplitudes and phases of sinusoidal signals (plus noise) and noise. The beamformed output of such a fluctuation-based CSDM provides signal-to- noise ratio (SNR) gains in excess of those achieved using a conventional CSDM. The additional gains are inversely proportional to the beamformed ratio of the signal bin fluctuations to the noise bin fluctuations. An unalerted auto-detection capability is another advantage of this new technique.

The fluctuating brightness of cosmic X-ray sources, particularly accreting black holes and neutron star systems, has enabled enormous progress in understanding the physics of turbulent accretion flows, the behaviour of matter on the surfaces of neutron stars and improving the evidence for black holes. Most of this progress has been made by analysing and modelling time series data in terms of their power and cross spectra, as will be discussed in other articles in this volume. Recently, attempts have been made to make use of other aspects of the data, by testing for non-linearity, non-Gaussianity, time asymmetry and by examination of higher order Fourier spectra. These projects, which have been made possible by the vast increase in data quality and quantity over the past decade, are the subject of this article.

The paper presents how to identify natural frequencies and mode shapes of structures by Operational Modal Analysis (OMA) technique, in which the Frequency Domain Decomposition (FDD) method is used. This method is an experimental method only base on the data of measuring the dynamic response of the structures under the excitation due to ambient forces and operational loads to determine the vibration characteristics. Measure vibration (acceleration) and determine spectral density matrix, using the singular values decomposition method of spectral density matrix to determine the natural frequencies and mode shapes of structures. The calculation results show that the natural frequencies, the mode shapes form determined by the OMA technique is consistent with the calculation results according to the theory and show the reliability of the method.KeywordsNatural frequencyMode shapeIdentifyEMAOMAFDD

SUMMARY The problem of testing for periodicities in multiple time series is considered. The test statistic proposed is based on the Euclidean norm of the matrix of periodograms. We first obtain the exact distribution of our test statistic under the hypothesis that observations are taken from an n-vector time series consisting of independent series of independent normal variables. We extend the theory to normal linear series with a specified spectral density matrix and then show that a suitable estimate can be used to replace this matrix in our test statistic. These results are asymptotic in nature. Extreme value theorems are stated for the case of normal variates and it is shown that the distributions for nonnormal series can be approximated adequately by those for normal series. An exact test for periodic components in an univariate time series was first proposed by Fisher (1929) who considered the null hypothesis of normal white noise. The test was based upon the maximum of a set of periodogram ordinates each divided by their sum, the division making it unnecessary to know in advance the variance of the components of the series. The use of the rth largest of these ordinates, r > 1, in testing for periodicities was also discussed by Fisher (1940). Whittle (1951, p. 101) showed how one could apply the Fisher tests to normal linear series provided the number of observations was large and one could specify the spectral density for the series. Hannan (1961) proposed a modification to this test that removed the necessity of knowing beforehand the spectral density function. Nicholls (1967) proposed a variation on Hannan's test statistic that improves its power under certain alternatives. Some extreme value results for periodogram ordinates were given by Walker (1965), who also demonstrated that dropping the normality condition has little effect on the large sample distributions, at least in the upper tails. In the sequel we consider the problem of testing for periodicities in multiple time series and pursue a course that parallels the historical development for univariate series as outlined in the paragraph above. Our tests are based on the Euclidean norm of the matrix of periodo- grams. In this connection Nicholls (1969) has discussed a test for discrete mass in a cross- spectrum. We consider real, discrete time parameter, n-vector, time series, Z(t) (t = 0, + 1, + 2, ...), where Z(t) = It+m(t) +X(t), with ,u+m(t) being a mean value function and X( ) being a weakly stationary, zero mean, time series possessing a spectral density matrix fxx(A) with tr{fxx(A)} <1 M < oo (JAI < 7r). We also assume the existence of a moving average representation 00 wo,t) , CO v - )

Frequency Domain Principal Components Estimation of Fractionally Cointegrated Processes

Testing nonparametric and semiparametric hypotheses in vector stationary processes

Two main topics form the content of this thesis. In the first part, non-linear cosmological structure formation is studied within the time renormalization group (TRG) formalism. The tree-level perturbative trispectrum was included in the time evolution of the bispectrum. Using Gaussian initial growing mode conditions we achieved an improvement in the predictions of the dark matter power spectrum in the mildly non-linear regime. We reached percent accuracy for wave numbers k 0:25 h/Mpc the perturbative description of the trispectrum breaks down. Our results emphasize the importance of higher order correlators for the non-linear power spectrum evolution, but the fast increase in numerical cost limits the applicability of the TRG method. Subject of the second part are signal-to-noise estimates for possible cross-correlation measurements between the integrated Sachs-Wolfe (iSW) effect and the tracer galaxy density field assuming data from the Planck mission and a Euclid-like galaxy survey. Orthogonalized polynomial line-of-sight weighting functions for the galaxy field are employed to resolve tomographical information of the cross-spectrum. For the equation-of-state parameter w = -0.9 our tomographic method provides a 15% increase in the signal-to-noise ratio Sigma of the cross-spectrum (10% for w = -1.0). Furthermore, cross-bispectra and cross-trispectra are studied with respect to a possible detection of the non-linear iSW effect. Finding values of Sigma = 0.83 for the mixed bispectrum and Sigma = 0.19 in case of the trispectrum , the effect has to be regarded as undetectable in correlations with future galaxy surveys.

Multichannel arma spectral estimation by the modified Yule-Walker method

In this paper, we employ the fact that all experimental vibration data, whether in the form of a set of FRFs or a set of output-only spectra, is a summation of resonance curves, each curve due to a mode of vibration. We also use this superposition property of modes to calculate a modal participation matrix, a measure of the participation of each mode in the experimental vibration data. First we show how this superposition property can be used to curve fit a set of FEA mode shapes to EMA mode shapes or ODS's. The modal participation matrix is calculated as a least-squared-error solution, so any number of FEA mode shapes can be curve fit to any number of EMA mode shapes or ODS's. Next we show how an expanded and enhanced set of FRFs, Cross spectra or ODS FRFs is obtained by curve fitting FEA mode shapes to experimental data. This approach in an alternative to FEA Model Updating, where an FEA model is modified so that its modes more closely correlate with experimental data. By curve fitting FEA shapes to experimental data, an extending and enhanced dynamic model is obtained which is more suitable for machinery & structural health monitoring, and for troubleshooting noise & vibration problems using SDM and MIMO methods.

Context. Timing analysis of accreting systems is key to probing the structure and dynamics around compact objects. In black-hole low-mass X-ray binaries (BH LMXBs), the compact object accretes matter from a low-mass companion star via Roche-Lobe overflow and forms an accretion disk that occasionally exhibits bright eruptions. The BH LMXB Swift J1727.8−1613 (hereafter J1727) underwent one of the brightest outbursts ever recorded in X-rays in August 2023. Aims. We study the timing properties of J1727 in the decaying phase of its outburst based on XMM–Newton data with a high-time resolution. Methods. We analyzed the power spectrum (PS) and cross spectrum (CS) of J1727, which we modeled with Lorentzians. The PS reveals the power distribution of the source across frequencies, and the real and imaginary parts of the CS compare the displacement of the light curves in different energy bands for the different observations. Finally, we simultaneously derived the phase lags and the coherence using a constant phase-lag model. Results. While the first (soft-state) observation shows no strong variability, the two harder observations exhibit quasi-periodic oscillations (QPOs). Because the QPO is more significantly detected in the imaginary part of the CS than in the PS, we refer to it as the “imaginary QPO”. The QPO is more prominent in the soft 0.3−2 keV band than in the hard 2−12 keV band. As the source evolves toward the hard state, the imaginary QPO shifts to lower frequencies, the broadband fractional rms amplitude in the 0.3−2 keV energy band increases, and the rms covariance of the imaginary QPO decreases. Simultaneously, the phase lags increase, and the coherence function drops at the imaginary QPO frequency. Conclusions. This analysis provides the first type-C QPO detection in a BH XB during the soft–to–hard transition using XMM–Newton data. The QPO is detected at particularly low energy (0.3−2 keV). Notably, the QPO is significantly detected in the imaginary part of the CS and the PS. Thus, we confirm the physical origin of the coherence drop and the phase-lag excess, which were only observed with NICER before.