Means covariates

Concentration-time curves for three individuals with different kidney functions CLCr are shown in Figure 1. The broken lines CLcr represent the time dependent CLcr as a measure of the kidney function. For the subject shown in the center panel, the CLCr decreases at 3.5 days causing a steep increase in the drug concentration. Dots are observations CONC (observed concentration), full lines PRED (model predictions for the population with p = 0) correspond to the model predictions for a typical individual with a specific set of mean covariates CLCr, WT and SEX (fixed effects). The broken lines IPRE (model predictions for the... 

A total of 10,000 patients were simulated. The simulated mean weight was 70 kg with a range of 45-90 kg. The simulated mean CrCL was 7.2 L/h with a range of 5.7-8.7 L/h. The correlation between weight and CrCL was 0.23. The means, covariance, and correlation for the simulated data were acceptable in simulating a patient with normal renal function. 

For the present model, the means, covariances, and correlation functions can be found from the expanded jump moments (see [2]). In terms of the coefficients given in Eqs. (10) and (11) they are... 

This argument obviously can be generalized to any number of variables. Equation (2-65) describes the propagation of mean square error, or the propagation of variances and covariances. 

Finally variance estimates of the parameters are calculated by means of the Cj/ elements, as in the following relationships (where var and cov represent variance and covariance, respectively). 

Finally, an infinite set of random vectors is defined to be statistically independent if all finite subfamilies are statistically independent. Given an infinite family of identically distributed, statistically independent random vectors having finite means and covariances, we define their normalized sum to be the vector sfn, , sj where... 

We shall conclude this section by investigating the very interesting behavior of the probability density functions of Y(t) for large values of the parameter n. First of all, we note that both the mean and the covariance of Y(t) increase linearly with n. Roughly speaking, this means that the center of any particular finite-order probability density function of Y(t) moves further and further away from the origin as n increases and that the area under the density function is less and less concentrated at the center. For this reason, it is more convenient to study the normalized function Y ... 

The relationship between the noise and atmospheric covariances is also evident in Eq. 17. If the noise on the measurements is large the N term dominates the inverse which means only the large eigenvalues of C contribute to the inverse. Consequently only the low order modes are compensated and a smooth reconstruction results. When the data is very noisy then 1 and hence a tend to zero. If the data is very noisy, then no estimate of the basis coefficients is made. 

It follows that for h close to bf the approximate marginal mean and covariance of yi is ... 

If the approximation (Lq. 3.5) is assumed to hold exactly we can derive the usual asymptotic results. The Plb estimator is asymptotically normal with mean p and covariance matrix... 

In eq. (33.3) and (33.4) x, and Xj are the sample mean vectors, that describe the location of the centroids in m-dimensional space and S is the pooled sample variance-covariance matrix of the training sets of the two classes. 

When all are considered equal, this means that they can be replaced by S, the pooled variance-covariance matrix, which is the case for linear discriminant analysis. The discrimination boundaries then are linear and is given by... 

As stated earlier, LDA requires that the variance-covariance matrices of the classes being considered can be pooled. This is only so when these matrices can be considered to be equal, in the same way that variances can only be pooled, when they are considered equal (see Section 2.1.4.4). Equal variance-covariance means that the 95% confidence ellipsoids have an equal volume (variance) and orientation in space (covariance). Figure 33.10 illustrates situations of unequal variance or covariance. Clearly, Fig. 33.1 displays unequal variance-covariance, so that one must expect that QDA gives better classification, as is indeed the case (Fig. 33.2). When the number of objects is smaller than the number of variables m, the variance-covariance matrix is singular. Clearly, this problem is more severe for QDA (which requires m < n ) than for LDA, where the variance-covariance matrix is pooled and therefore the number of objects N is the sum of all objects... 

Therefore, on statistical grounds, if the error terms (e,) are normally distributed with zero mean and with a known covariance matrix, then Q( should be the inverse of this covariance matrix, i.e.,... 

A valuable inference that can be made to infer the quality of the model predictions is the (l-a)I00% confidence interval of the predicted mean response at x0. It should be noted that the predicted mean response of the linear regression model at x0 is y0 = F(x0)k or simply y0 = X0k. Although the error term e0 is not included, there is some uncertainty in the predicted mean response due to the uncertainty in k. Under the usual assumptions of normality and independence, the covariance matrix of the predicted mean response is given by... 

The covariance matrix COV(k ) is obtained by Equation 3.30. Let us now concentrate on the expected mean response of a particular response variable. The (l-a)100% confidence interval of yl0 (i=l.,w). the i,h element of the response vector y0 at x0 is given below... 

The above expressions for the CO l (k ) and of are valid, if the statistically correct choice of the weighting matrix Q, (i=1,...,N) is used in the formulation of the problem. Namely, if the errors in the response variables (e, i=l,...,N) are normally distributed with zero mean and covariance matrix,... 

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... 

We also use a linearized covariance analysis [34, 36] to evaluate the accuracy of estimates and take the measurement errors to be normally distributed with a zero mean and covariance matrix Assuming that the mathematical model is correct and that our selected partitions can represent the true multiphase flow functions, the mean of the error in the estimates is zero and the parameter covariance matrix of the errors in the parameter estimates is ... 

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