Variance matrix

We have assumed that the prior information can be described by the multivariate normal distribution, i.e., k is normally distributed with mean kB and co-variance matrix VB. 

The covariances between the parameters are the off-diagonal elements of the covariance matrix. The covariance indicates how closely two parameters are correlated. A large value for the covariance between two parameter estimates indicates a very close correlation. Practically, this means that these two parameters may not be possible to be estimated separately. This is shown better through the correlation matrix. The correlation matrix, R, is obtained by transforming the co-variance matrix as follows... 

The proposed NDDR algorithm was applied with a history horizon of five time steps. The variance matrix was defined as... 

Of the six desirable properties, we note that (i) will be satisfied if Sg(0)Cg(0) becomes independent of 0 when the variance matrix S,P approaches zero. Since they are expected to be smooth functions near 0 = (0), property (i) will be satisfied. 

Prove the result that the restricted least squares estimator never has a larger variance matrix than the unrestricted least squares estimator. 

A method that is Bayesian in nature is that proposed by Racine-Poon (38). The method uses the estimates of the individual parameters j and asymptotic variance matrix Vj obtained from the individual fits, with very weak assumptions about the prior distribution of the population parameters to calculate a posterior density function from which and Q can be obtained. In an iterative method suggested by Dempster et al. (39) the EM algorithm is used to calculate the posterior density function. Simulation studies in which several varying and realistic conditions were... 

The variance matrix of each of a collection of estimates was calculated for a variety of distributions using the Monte Carlo techniques of Andrews and co-workers1 and Relles.s The distributions were all expressed in terms of mixtures of Gaussian distributions. G(0, 1) denotes a standard Gaussian distribution with mean 0 and variance 1. The distribution formed by contaminating this with 10 percent of a Gaussian distribution with mean 0 and variance 9 is denoted by 10% G(0, 9). 

Under all these models, the generic residuals are assumed to be independent, have zero mean, and constant variance residual variance components, a2, is referred to as the residual variance matrix (X), the elements of which are not necessarily independent, i.e., the residual variance components can be correlated, which is referred to as autocorrelation. It is generally assumed that and r are independent, but this condition may be relaxed for residual error models with proportional terms (referred to as an -q — s interaction, which will be discussed later). 

Suppose Y = f(x, 0, t ) + g(z, e) where nr] — (0, il), (0, ), x is the set of subject-specific covariates x, z, O is the variance-covariance matrix for the random effects in the model (t ), and X is the residual variance matrix. NONMEM (version 5 and higher) offers two general approaches towards parameter estimation with nonlinear mixed effects models first-order approximation (FO) and first-order conditional estimation (FOCE), with FOCE being more accurate and computationally difficult than FO. First-order (FO) approximation, which was the first algorithm derived to estimate parameters in a nonlinear mixed effects models, was originally developed by Sheiner and Beal (1980 1981 1983). FO-approximation expands the nonlinear mixed effects model as a first-order Taylor series approximation about t) = 0 and then estimates the model parameters based on the linear approximation to the nonlinear model. Consider the model... 

A final point regarding the probabilistic modelling of observations eoncems issues with the co-variance matrix. If we use say 13 acoustic coefficients (often a standard number made from 12 cepstral coefficients and one energy coefficient), and then use delta and acceleration coefficients, we have a total of 39 coefficients in each observation frame and so the mean vector has 39 values. 

A D-optimal design experiment (D like determinant) with N experiments minimizes the determinant of the variance matrix. 

The quantity J is identical to J of Eq. (52) in 3DVAR, except that the analysis field at r = 0 is denoted by Xo. Because all observations during r = 0 to t are used, the constraint is expressed as the integral form of J2 of Eq. (53) with respect to time. The observational error co-variance matrix O depends on time t. The quantity J3 can be interpreted in the same way as J, exeept that the prediction model errors are involved instead of the observational errors. The quantity P in Eq. (58) denotes a covariance matrix that deseribes predietion error statistics. 

Element on the y row and the A column of the inverse of the variance matrix of the responses Space time s Weisz modulus... 

For the second order moments, let us consider the phase-space variance matrix,... 

The variance matrix for the parameters consists of the variance for each parameter in the diagonal and co-variances in the other positions. Equation (7.59) contains an example of a model with four parameters (bo, b, b2 and b ) shown to illustrate this relationship ... 

The correlation matrix, C, can be retrieved from the variance matrix according to the following equation ... 

Fig. 3 Delay time tomography from the inversion of calculated from the model variance matrix, w/ 7c wvc/tcrf...

If the xk)t are known by solving (3.77), Eq. (3.80) is a system of non-homogeneous linear differential equations with given time dependent coefficients for the variance matrix Oik(t). 

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