Spectral statistical properties

Bonamy L., Nguyen Minh Hoang P. Far infrared absorption of diatomic polar molecules in simple liquids and statistical properties of the interactions. I. Spectral theory, J. Chem. Phys. 67, 4423-30 (1977) ... 

Moreover, in recent years broad band lasers have appeared which lack any frequency modal structure, at the same time retaining such common properties of lasers as directivity and spatial coherence of the light beam at sufficiently high spectral power density. The advantages of such a laser consist of fairly well defined statistical properties and a low noise level. In particular, the authors of [245] report on a tunable modeless direct current laser with a generation contour width of 12 GHz, and with a spectral power density of 50 /xW/MHz. The constructive interference which produces mode structure in a Fabry-Perot-type resonator is eliminated by phase shift, introduced by an acoustic modulator inserted into the resonator. 

Phillips OM (1985) Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J Fluid Mech 156 505-531 Plant WJ (1982) A relationship between wind stress and wave slope. J Geophys Res 87 1961-1967... 

Chapters 3 and 4 introduce two recently developed Bayesian methods for updating the mathematical models of dynamical systems. Chapter 3 presents the Bayesian spectral density approach. The spectral density estimator is defined to take into account of the aliasing and leakage effect. The statistical properties of the spectral density estimator are examined and... 

In this chapter, the Bayesian spectral density approach, which is a frequency-domain approach, for modal/model updating using wide-band response data is presented. It utilizes the statistical properties of the spectral density estimator to obtain not only the optimal values of model parameters but also their associated uncertainty by means of the updated probability distribution of the uncertain parameters. Uncertainty quantification is important for many applications, such as damage detection and reliability analysis. 

In this section, the statistical properties of the spectral density estimator are investigated in order to construct the likelihood function of the data. Firstly, it can be shown that the spectral density estimator is asymptotically unbiased ... 

Statistical Properties of the Spectral Density Matrix Estimator... 

Next, the statistical properties of the spectral density matrix estimator Sy j cok) are investigated. Denote by Vn((Ok) and licok) the real and imaginary part, respectively, of (cok) so ... 

The Bayesian fast Fourier transform approach uses the statistical properties of discrete Fourier transforms, instead of the spectral density estimators, to construct the likelihood function and the updated PDF of the model parameters [292]. It does not rely on the approximation of the Wishart distributed spectrum. Expressions of the covariance matrix of the real and imaginary parts of the discrete Fourier transform were given. The only approximation was made on the independency of the discrete Fourier transforms at different frequencies. Therefore, the Bayesian fast Fourier transform approach is more accurate than the spectral density approach in the sense that one of the two approximations in the latter is released. However, since the fast Fourier transform approach considers the real and imaginary parts of the discrete Fourier transform, the corresponding covariance matrices are 2No x 2Nq, instead of No x No in the spectral density approach. Therefore, the spectral density approach is computationally more efficient than the fast Fourier transform approach. 

A more general process known as least-squares filtering or Wiener filtering can be used when noise is present, provided the statistical properties of the noise are known. In this approach, g is deblurred by convolving it with a filter m, chosen to minimize the expected squared difference between / and m g. It can be shown that the Fourier transform M of m is of the form (1///)[1/(1 - - j], where S is related to the spectral density of the noise note that in the absence of noise this reduces to the inverse filter M = /H. A. number of other restoration criteria lead to similar filter designs. 

The statistical properties of a Gaussian random process is completely characterized by giving its two-point correlation function. We define a two-point correlation function ( ri — r2 ) = and the associated spectral function f k) by a Fourier transform relation... 

The basic difference between an atomic beam from a BEG (an atom laser) and an atomic beam from a thermal atomic oven is the same as the difference between a laser beam and a spectrally filtered and collimated light beam from a thermal light source. According to quantum theory (Glauber 1963), a laser field is a coherent state of light with minimal fluctuations in its amplitude and phase. In other words, the essence of an atomic beam from a BEG, as well as a laser beam, lies in its statistical properties. 

In general, several tools have been developed to match reference spectra with those measured by an imaging spectrometer. The most common approach is based on the use of standard supervised classification techniques, where known spectra are used to determine the statistical properties of each class based on spectral characteristics. Examples of supervised classification approaches applied to hyperspectral data are described in McKeown et al. (1999) and Roessner et al. (2001), where the maximum likelihood classifier (MLC) was applied to map urban land cover. Other techniques are based on the use of support vector machines (SVM) (Melgani and Bruzzone 2004) and neural networks (NNs) (Licciardi et al. 2009, 2012). Other approaches have been designed explicitly for the analysis of imaging spectrometry data, such as the Spectral Angle Mapper (SAM Kruse et al. 1993). 

The main problem of elementary chemical reaction dynamics is to find the rate constant of the transition in the reaction complex interacting with its environment. This problem, in principle, is close to the general problem of statistical mechanics of irreversible processes (see, e.g., Blum [1981], Kubo et al. [1985]) about the relaxation of initially nonequilibrium state of a particle in the presence of a reservoir (heat bath). If the particle is coupled to the reservoir weakly enough, then the properties of the latter are fully determined by the spectral characteristics of its susceptibility coefficients. 

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