Covariance partitioning

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 ... 

The first two terms on the RHS of equation 70-20 are the variances of X and Y. The third term, the numerator of which is known as the cross-product term, is called the covariance between X and Y. We also note (almost parenthetically) here that multiplying both sides of equation 70-20 by (re - 1) gives the corresponding sums of squares, hence equation 70-20 essentially demonstrates the partitioning of sums of squares for the multivariate case. 

There are several critical facts that come out of the partitioning of sums of squares and its consequences, as shown in equations 70-20 and 70-22. One is the fact that in the multivariate case, variances add only as long as the variables are uncorrelated, that is, the correlation coefficient (or the covariance) is zero. 

Introducing this partitioning into the equations, we have for the covariance matrix of the residuum in the balances... 

In order to be able to proceed with differentiation, let us partition the covariance matrix of the ith measurement as... 

If, as might be expected, the unfilterable solid concentration covaries with the total solid concentration, then at high solid concentrations (KcMc> 1) a decrease in the apparent partition coefficient with increasing Me will result (see Morel and Gschwend, 1987). 

The variance of the selectivity is a function of the individual variances of the partition coefficients which are due to small variations in the extraction liquid composition and of the covariance Op pj between these partition coefficients (equation (18)). The variances for the individual partition coefficients has been derived above. 

The covariance of the partition coefficients can be estimated by the correlation between the tangent planes of the response surfaces in a given mixture composition. This is explained in the next part of this paragraph. 

The asymptotic covariance matrix for the joint estimator is the inverse of this matrix. To compare this to the asymptotic variance for the marginal estimator of a, we need the upper left element of this matrix. Using the formula for the partitioned inverse, we find that this upper left element in the inverse is... 

As discussed below, knowledge of NEE and S Cnee are prerequisites for the partitioning of NEE into its photosynthetic and respiratory components. Preliminary results of using this approach gave close agreement with surface-based, eddy covariance and isotopic measurements. 

The dynamics of canopy-scale net fluxes of water and CO2 exchanged between vegetation and the atmosphere are routinely measured today with micrometeorological methods (e.g., with eddy covariance www.daac.ornl.gov/FLUXNET/ fluxnet.html). Combining these methods with isotopic measurements allows to partition a net flux into its gross flux components. The approach here is similar to that used on the global scale to... 

Thus far all mixtures considered have been static, in the sense that the probability model did not change as a function of covariate values. Recall that in the introduction, isoniazid acetylator polymorphism was used as an example to introduce the concept of mixture modeling utility. In that example it was stated that race was associated with how patients were partitioned between slow and fast acetylator status. So, given an isoniazid PK data set without acetylator genotype, but with race as a covariate, one might want to introduce race as a covariate in our mix block to help model the patients as either fast or slow acetylators. 

Performing principal component analysis on the ranks can help to assess the dimensionality of the ordering context. Since the marginal distributions of the ranks are the same except for ties, the difference between covariance matrix and correlation matrix is not critical. If there are subsets of indicators that segregate strongly in their loadings, then complexity is confirmed and it may be prudent to consider partitioning of the prioritization process. 

In their second simulation, they examined the Type I error rate when a covariate influenced residual variability, i.e., residual variability was partitioned into two groups. FOCE-I and FO-approximation had a Type I error rate of about 0.075 and 0.11, respectively, with two observations per subject (collected at 1.75 and 7 h after administration) and a residual error of 10%. When residual error increased to 31%, the Type I error rate for FOCE-I decreased to the nominal value, but increased to 0.16 with FO-approximation. With 19 observations per subject, the Type I error rate for both FO-approximation and FOCE-I was unchanged as residual error increased from 10 to 31%, but FOCE-I remained near 0.05. 

Structural model adequacy Does the structural model explain the data Application of structural model for all subjects on all occasions Are there subjects/occasions for which the model is inadequate Covariate modeling strategy Is the process adequate to identify important relationships or the magnitude of variance partitions ... 

Data splitting is fairly straightforward and covered in detail in the next section on validation. It simply implies that data to be modeled are partitioned based on differences in sampling (i.e., windows where suspect 0 are believed to be constant). The most common data splits to explore pharmacokinetic time dependencies would be single-dose, chronic non-steady-state, and steady-state conditions. Data subsets are modeled individually with all parameters and variability estimates along with any relevant covariate expressions compared in a manner similar to a validation procedure (see next section). Data can be combined in a leave-one-out strategy (see cross-validation description) to examine the uniformity of data windows. ... 

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