Second-order statistics

So far, we have been discussing so called hrst-order statistics, since we have only been describing the results of measurements at a single point. If we wish to describe the relationship between two measurements (e.g., values of the random variable measured at two different points in space, or at two different times), then we must use second order statistics. The correlation of two measurements at points x and X2 is defined as... 

Applying a chromatographic method it is sometimes possible to separate copolymer molecules according to their size Z and composition [5]. The SCD found in such a way can be compared with that calculated within the framework of the chosen kinetic model. The first- and second-order statistical moments of SCD are of special importance. 

Each protein-ligand titration was done in duplicate. Values were determined by fitting the average data at similar conditions. A subsampling method was used to evaluate the second order statistics of the parameters. 

Let us consider a catalytic cycle with random rate constants. For a given sample constants k, ...,k the ith order statistics is equal its ith smallest value. We are interested in the first order (the minimal) and the second order statistics. 

It should however be emphasized that there is usually no choice but to keep the unmodified phase because of the lack of hypotheses concerning the unknown signal (recall that only the second order statistics of the signal and noise are supposed to be... 

The residual, being assumed stochastic, is characterized by second-order statistics i.e., the specific phase of the residual is of no importance. Nevertheless, this selection must be made carefully since spectral phase significantly influences the temporal properties of the signal. 

Fig. 7.32 The two textures (left field and right field) have the same first-order statistics (the same number of black dots), but they differ in second-order statistics. In the left field the dots fall at random, whereas in the right field there are at least 10 dot diameters between dots. [Reprinted by permission from B. Julesz, Experiments in the Visual Perception of Texture Sci. Am., 232, 34 (1975).]...

For measuring second-order statistics, Julesz (68) suggests dropping a dipole (e.g., a needle) on the two textures and observing the frequency with which both ends of the dipole land on black dots. Identical frequencies imply identical second-order statistics. His experiments indicate that our visual system can discriminate patterns solely by the perceptual process only if they differ in second-order statistics. He made a similar observation with musical textures, and found that random melodies could be perceived as being different only if they possessed different second-order statistics. 

Figure 7.34 shows a number of him samples extruded under different conditions. They have approximately the same carbon black concentration and are grossly uniform or have the same first-order statistics, but clearly exhibit different texture, and thus have different second-order statistics. 

The OSP algorithm is based on maximizing the SNR (signal-to-noise ratio) in the subspace orthogonal to the background subspace and only depends on the noise second-order statistics. It also does not provide any estimate of the abundance measure for the desired end member in the mixed pixel. A linear mixture model for pixel y consisting of p spectral bands is described by... 

The time dependence of the energy and the second-order statistical moments (variances) of the field mode quadrature components, Oab = (ab + ba) — (ia)(b), is governed by the equations following from Eq. (220) ... 

Recall that fi -) is the probability distribution over the type of agent i, and Fi -) the cumulative distribution. This priority level, sometimes called the virtual valuation, is less than an agent s type by the expectation of the second-order statistic of the distribution over its type. Economically, one can imagine that this represents the information rent of an agent, the expected payoff that an agent can extract from the private information that it has about its own type. 

The KLT depends on the second-order statistics of the noisy signal and is known to be computationally intensive. The DFT is computationally much less intensive and is a signal independent transformation. 

In this section we present the principles of the signal subspace approach and its relations to Wiener filtering and spectral subtraction. Our presentation follows [7] and [11]. This approach assumes that the signal and noise are noncorrelated, and that their second-order statistics are available. It makes no assumptions about the distributions of the two processes. 

We note that when estimates of the second-order statistics of the signal and noise replace the true second-order statistics in a given estimation scheme, the optimality of that scheme can no longer be guaranteed. The quality of the estimates of these second-order statistics is key for the overall performance of the speech enhancement system. Estimation of the second-order statistics can be performed in various ways usually outlined in the theory of spectral estimation [19]. Some of these approaches are reviewed in this section. 

On-line estimation of the second-order statistics of a speech signal from a sample function of the noisy signal has proven to be a better choice. Since the analysis frame length is usually relatively small, the covariance matrices of the speech signal maybe assumed Toeplitz. Thus, the autocorrelation function of the clean signal in each analysis frame must be estimated. The Fourier transform of a windowed autocorrelation function estimate provides estimates of the variances of the clean signal spectral components. 

Wide-sense stationarity A property of a random process whose second-order statistics do not change with time. 

Probabilistic response analysis consists of computing the probabilistic characterization of the response of a specific structure, given as input the probabilistic characterization of material, geometric and loading parameters. An approximate method of probabilistic response analysis is the mean-centred First-Order Second-Moment (FOSM) method, in which mean values (first-order statistical moments), variances and covariances (second-order statistical moments) of the response quantities of interest are estimated by using a mean-centred, first-order Taylor series expansion of the response quantities in terms of the random/uncertain model parameters. Thus, this method requires only the knowledge of the first- and second-order statistical moments of the random parameters. It is noteworthy that often statistical information about the random parameters is limited to first and second moments and therefore probabilistic response analysis methods more advanced than FOSM analysis cannot be fully exploited. 

The first- and second-order statistical moments of the response quantities r are approximated by the corresponding moments of the above linearized response quantities, i.e.. 

In this study, the Nataf model (Ditlevsen Madsen 1996) was used to generate realizations of the random parameters 0. It requires specification of the marginal PDFs of the random parameters 0 and their correlation coefficients. It is therefore able to reproduce the given first- and second-order statistical moments of random parameters 0. The same three-dimensional three-story reinforced concrete building presented in Section 2.4, but on rigid supports, is considered as application example. Table 1 provides the marginal distributions and their statistical parameters for the material parameters modelled as correlated random variables. Other details on the modelling of the structure and the statistical correlation of the random parameters can be found in Barbato et al. (2006). 

A method for accurately estimating the second-order statistical responses required for robust design optimization in case of nonlinear response. This is accomplished by a modified method of stochastic equivalent linearization, especially purported to take into account the actual non Gaussian behavior of hysteretic oscillators proposed by the author in (Hurtado and Barbat 1996 Hurtado and Barbat 2000). 

The difficulty in obtaining accurate values of the failure probabilities complicates the application of the reliability-based optimization approaches. A convenient alternative is the robust optimization approach, based on second order statistical information. 

The essential point is to measure the fluctuations of the current flowing between two identical electrodes kept at the same potential by means of a zero resistance ammeter (ZRA), and at the same time measure their voltage fluctuations with respect to a reference electrode (RE). The voltage measurement can be made between the two electrodes connected by the ZRA and a third electrode, which may be identical to the others or be a RE, (see configuration b in Fig. 7-21). The noise resistance is then calculated as the ratio of a second-order statistics of the voltage fluctuations divided by the same quantity relative to the current fluctuations. Often, the quantities chosen are the standard deviations measured over a fixed period of time. 

Many researchers have studied the problem of classification from ultrasound images mainly in the area of liver tissue [2]-[10]. Initial attempts to characterize diffuse diseases have utilized different signal processing techniques in order to obtain useful information from the raw radiofrequency signal [2], [3]. In a series of papers, Momenan et al. showed that second order statistical parameters from envelope-detected or intensity echo signals have discriminatory power [4], [5]. Some researchers have treated the task of tissue quantification from the point of description and classification with numerical texture... 

Since the filter bank performing the QT represents one special case of the local linear transform approach for the texture characterization, N iterations of the quincunx decomposition can be seen as a (N+l)-channel filter bank, whose outputs Ii,I2,...In+i serve for the estimation of texture quality in the corresponding frequency sub band. The texture is then, characterized by the set of N+1 first-order probability density functions estimated at the output of each channel. Another, psychophysical justification was offered by Pratt et al. [20], who showed that natural textures are visually indistinguishable if they possess the same first and second-order statistics. 

Kornev, N., Hassel, E. (2006). Synthesis of homogenous anisotropic divergence-free turbulent fields with prescribed second-order statistics by vortex dipoles. Physics of Fluids. 19, 068101. 

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