Confidence ellipsoid

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... 

In summary, BA has been used in the space of principal eigenvectors in problems about oils and wines, and the plot of the first two eigenvectors has been used to display the confidence ellipsoids (class space) and their changes after outher deletion this is the procedure to obtain an improved class model. 

With regards to the analysis of the quality of the various parts of the model, one may use the same methods as are used for practical identifiability analysis. Since the same methods are used, albeit with different objectives, one sometimes refers to this model quality analysis as a posteriori identifiability (and the previous analysis as a priori identifiability). Now, however, one is also interested in how the parametric uncertainty translates to an uncertainty in the various model predictions. For instance, it might be so that even though two individual parameters have a high uncertainty, they are correlated in such a manner that their effect on a specific (non-measured) model output is always the same. Such a translation may be obtained by simulations of the model using parameters within the determined confidence ellipsoids. A global alternative to this is to consider the outputs for all parameters that correspond to a cost function that is below a certain threshold, for example 2% above the found minimum. 

The D-optimal design minimizes the volume of the confidence ellipsoid of the regression coefficients. This means that the regression coefficients obtained from D-optimal designs are determined with the highest possible precision. 

A criterion that is closely related to D-optimality is E-optimality. The D-optimality criterion minimizes the volume of the confidence ellipsoid of the regression coefficients. Hence, it minimizes the overall uncertainty in the estimation of the regression coefficients. The E-optimality criterion minimizes the length of the longest axis of the same confidence ellipsoid. It minimizes the uncertainty of the regression coefficient that has the worst estimate (highest variance). 

Originally limited to ellipsoids, the use of Mahalonobis distances allows the use of more variables as the confidence ellipsoid can be transformed to a confidence or tolerance hypersphere. These ideas were examined using the microecosystem test method developed by Kersting for the examination of multispecies systems. These three-compartment microecosystems are comprised of an autotrophic, herbivore, and decomposer subsystems that are connected by tubing and pumps. Although relatively simple and small, these systems are operable over a number of years. 

The smaller the volume of the confidence ellipsoid, the more precise will be the estimates of the coefficients. It may be shown that, for a significance level a and a given experimental variance, this volume is proportional to the determinant 1 (X X) I of the dispersion matrix, defined in chapter 4, section II.C.4. 

The confidence ellipsoid is characterised by its volume I (X X) I, but also by its shape - whether it is flattened or not and its direction with respect to the axes b,. The trace of the dispersion matrix (X X) is the sum of its diagonal elements Sc". For a given volume of ellipsoid (given by the determinant), the trace takes its minimum value when all the terms c" are equal, corresponding to a spherical confidence ellipsoid, as in figure 8.3a, where the estimations of the coefficients are independent (uncorrelated). 

Asanuma et al. (2001) proposed a modification to the collapsing method. In this modified method the movements of the locations are dependent on the shape of the distribution of the seismic events within the confidence ellipsoid, and not just the... 

Figure 4.3. Confidence ellipsoids for some ceramics. The grain size effect is shown for MgO where the MgO becomes softer with decreasing grain size, shown in brackets in /xm alongside the ellipses.

In the rare case where the parameters are uncorrelated, the matrix (Af. Y) is diagonal, the axes of the confidence ellipsoid would be parallel to the coordinates of the parameter space, and the individual parameter confidence intervals would hold for each parameter independently. However, since the parameters are usually correlated, the extent of the correlation can be measured from the correlation coefficient matrix, R. This is obtained by applying Eq. (7.34) to the variance-covariance matrix (7.135) ... 

The correlation between parameters causes the axes of the confidence ellipsoids of the linear model to be at an angle to the coordinates of the parameter space. Therefore, the individual parameter confidence limits will not represent the true interval within which a parameter may lie and still remain within the confidence ellipsoid. 

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