Wishart distribution

The Wishart distribution is a multivariate gamma distribution, which itself is a general case of a chi-squared distribution. The Wishart also has the desirable property that random samples of any matrix from this distribution will always be positive definite. This is useful for simulating variance-covariance matrices, which have a positive determinant, and ensuring correlations that lie between -1 and 1. 

Similar to the prior of the mean population parameter values, the prior for the between-subject variance can also be selected to have a more plausible range for PK/PD systems. If we consider the coefficient of variation of between-subject variability for most PK/PD parameters as being approximately <100%, then a choice of p for the Wishart distribution that provided a 97.5th percentile value of around this level would be biologically plausible. This is not quite as straightforward as for the precision of the population mean parameter values, since the minimum size of p is indexed to the minimum dimension of the variance-covariance matrix of between-subject effects, and p affects all variance parameters equally. The value of p required to provide a similar level of weak informativeness will vary with the dimension of the matrix. A series of simulations have been performed from the Wishart distribution, where the mean value of the variance of between-subject effects was set at 0.2. The value of p required to provide a similar level of weak informativeness will vary with the dimension of the matrix (Table 5.1). 

An empirical method to estimate p may be gained by simulation, where candidate matrices of the inverse of Q are simulated from a Wishart distribution and the empirical distribution of each variance component is compared to the empirical... 

One problem with the use of the inverse Wishart distribution is that the distribution does account for correlation among the random effects nor can it express any correlation among 0 and ft. To account for this Gisleskog, Karlsson, and Beal propose using a normal normal prior where now correlation can be accounted... 

Keywords ambient vibration correlation function Duffing oscillator hydraulic jump information entropy modal identification optimal sensor placement spectral density structural health monitoring Wishart distribution... 

The spectral density estimators at different frequencies possess tractable properties so that they follow independent Wishart distributions in a certain frequency band, regardless of the distribution of the signal in the time domain. The method is efficient in the sense that most of the information from the data for identification of the model parameters, especially those related to the frequency stmcture, concentrates in a very limited bandwidth around the peaks in the spectrum. Therefore, the number of frequencies involved in the computation of the posterior PDF is significantly smaller than the total number of frequencies in the spectmm, i.e., INT(A /2)+l. However, computation of the inverse and determinant of the matrices [Sy,iv(r A)] 6 kefC, is required for each frequency included in establishing the... 

The Bayesian spectral density approach approximates the spectral density matrix estimators as Wishart distributed random matrices. This is the consequence of the special structure of the covariance matrix of the real and imaginary parts of the discrete Fourier transforms in Equation (3.53) [295]. Another approximation is made on the independency of the spectral density matrix estimators at different frequencies. These two approximations were verified to be accurate at the frequencies around the peaks of the spectmm. The spectral density estimators in the frequency range with small spectral values will become dependent since aliasing and leakage effects have a greater impact on their values. Therefore, the likelihood function is constructed to include the spectral density estimators in a limited bandwidth only. In particular, the loss of information due to the exclusion of some of the frequencies affects the estimation of the prediction-error variance but not the parameters that govern the time-frequency structure of the response, e.g., the modal frequencies or stiffness of a structure. 

The spectral density estimator is Wishart distributed regardless of the distribution of the orig-inai time-domain signal. Therefore, the expressions for the likelihood function in Chapter 3 are valid even for nonlinear systems. The only challenge is on the computation of the mean spectrum but this may be accommodated by simulations. However, the Bayesian spectral density approach is not applicable for nonstationary response measurements. 

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. 

The riverbed levels upstream and downstream, Z and Zv respectively, are imcertain and their uncertainty will be quantified by a bivariate normal distribution N(/m., E). Indeed, as upstream and downstream section are quite close it seems more reasonable to model them as possibly dependent variables. 29 couples of data (zm z ) are available to perform Bayesian inference and setting the posterior distribution of fi and E. The prior distributions for both components of vector /i are normal with means equal to 56 and 50 m for ix and fj,2 respectively and standard deviations equal to 1 m. This prior translates a vague knowledge around two reference values. Concerning the prior of E, a classical choice is the inverse-Wishart distribution ... 

For this posterior density, of the form of a Wishart distribution, the marginal postenor density for 0 can be computed analytically ... 

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