Global sum of squares

From the numerical results in the previous example we find that the global sum of squares of the concentrations of all trace elements in all wind directions amounts to ... 

Each eigenvector or contributes an amount X to the global sum of squares c of X. Hence, eigenvectors can be ranked according to their contributions to c. From now on we assume that the columns in U and V are arranged in decreasing order of their contributions. 

An important aspect of latent vectors analysis is the number of latent vectors that are retained. So far, we have assumed that all latent vectors are involved in the reconstruction of the data table (eq. (31.1)) and the matrices of cross-products (eq. (31.3)). In practical situations, however, we only retain the most significant latent vectors, i.e. those that contribute a significant part to the global sum of squares c (eq.(31.8)). 

A measure for the goodness of the reconstruction is provided by the relative contribution y of the retained latent vectors to the global sum of squares c (eq. (31.8)) ... 

The global weighted sum of squares c of the transformed data Z can be shown to be equal to the global interaction 52 between rows and columns ... 

The trace of A is equal to the traces of the weighted cross-product matrices, which in turn are equal to the global weighted sum of squares c or global interaction 5 (eq. (32.27)) ... 

Since GSA requires an estimate of the magnitude of cost function response at the global optimum, an estimate of the residual sum of squares for the best fit was needed. Based on work with similar data and some trial and error, a value of le-4 was found to give good results. 

Figure 3.5 Example sum of squares contour plots, (a) contour plot of a linear model illustrating that the contours for such a model are perfecdy elliptical with the global minimum at the center, (b) example of a well defined nonlinear model having a single global minimum and nearly elliptical contours, (c) contour plot of a nonlinear model showing banana -shaped contours indicative of ill conditioning, (d) contour plot of a nonlinear model with multiple solutions. All nonlinear models were parameterized in terms of a and (3. Upper right and bottom left examples were suggested by Seber and Wild (1989). The symbol ( ) indicates global minimum. M indicates local minimum.

Contours of tlx sum of squares surface for Model 1. For both figures the A and E ranges are centered at the global minimum. Figure 4. lb has an E-range 1/5 that of Figure 4.1a and shows that the problem of identifying the minimum persists as we approach the minimum. 

In the case of the test on the global significance of the regression, the null hypothesis is to be rejected, meaning that the residual sum of squares should be as small as possible. Whereas tabulated values are typically of the order of a few units, the calculated values should amount up to a few hundreds if not thousands, before the model can be considered as a potentially good model. 

Finally, the logarithmic standard deviations of the individual safety factors are combined using square root of sum of squares (SRSS), in order to obtain the corresponding variabihty parameter of the global safety factor, Pr and jSy, respectively. 

There are three ways by which the global distance of chi-square can be meaningfully rewritten as a weighted sum. These correspond with the three different ways of closing the data in the original contingency table X, such as has been described above in Section 32.3. The metric matrices W and have to be defined differently for each of the three cases. 

In the case of the contingency Table 32.4 we obtained a chi-square of 15.3. Taking into account that the global sum equals 30, this produces a global interaction of 15.3/30 = 0.510. The square root of this value is the global distance of chi-square which is equal to 0.714. 

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