Maximal Covariance

All current taxometric procedures are based on a single statistical method termed Coherent Cut Kinetics (CCK). We decipher the meaning of this term in the next section in the example of the MAXCOV-HITMAX (MAXCOV stands for MAXimal COVariance the reason for this name will become clear in the next section) technique. However, we emphasize that it is not the shared statistical method that defines taxometrics. Adherence to a particular set of epistemological principles distinguishes taxometrics from other approaches. In other words, any analytic procedure that can identify taxa may... 

For a situation with one dependent variable, a score vector is sought which has maximal covariance with y. For the two-way case the loading vector is found directly as X y/HX yH. For the three-way case the weights can also be found directly as the first left and right singular vectors from a singular value decomposition of a matrix Z (J x K) which is obtained as the inner product of X and y (see proof in Appendix 6.B). Each element in Z is given by... 

Another popular method is called PLS regression. This method also creates a new coordinate system based on orthogonal linear transformations, but the optimization criterion is different. Whereas PCA maximizes the variance in X, PLS will maximize covariance between X and Y. 

As discussed in the previous paragraphs, one of the main characteristics of PLS is that the dependent variable(s) Y influence the data compression of the independent matrix X. Accordingly, when more than one response is to be modelled, the PLS algorithm described in Section 3.4.1 has to be slightly modified to take into account the requirement that the PLS factors should be optimal for the simultaneous prediction of all the dependent variables. Mathematically, this is accomplished through the introduction of a decomposition step also for the block of responses Y, which is compressed into a set of scores U, so that this second set of scores has, component-wise, the maximal covariance with the corresponding scores of the X-block T. Consequently, the criterion in Equation (30) is modified to ... 

A second PLS factor is extracted in a similar way maximizing the covariance of linear combinations of the residual matrices E, and F,. Subsequently, E, and F, are regressed on t2, yielding new residual matrices E2 and F2 from which a third PLS... 

The set of selected wavelengths (i.e. the experimental design) affects the variance-covariance matrix, and thus the precision of the results. For example, the set 22, 24 and 26 (Table 41.5) gives a less precise result than the set 22, 32 and 24 (Table 41.7). The best set of wavelengths can be derived in the same way as for multiple linear regression, i.e. the determinant of the dispersion matrix (h h) which contains the absorptivities, should be maximized. 

If the covariance matrices of the response variables are unknown, the maximum likelihood parameter estimates are obtained by maximizing the Loglikeli-hood function (Equation 2.20) over k and the unknown variances. Following the distributional assumptions of Box and Draper (1965), i.e., assuming that i= 2=...=En= , it can be shown that the ML parameter estimates can be obtained by minimizing the determinant (Bard, 1974)... 

PLS was originally proposed by Herman Wold (Wold, 1982 Wold et al., 1984) to address situations involving a modest number of observations, highly collinear variables, and data with noise in both the X- and Y-data sets. It is therefore designed to analyze the variations between two data sets, X, Y). Although PLS is similar to PCA in that they both model the A -data variance, the resulting X space model in PLS is a rotated version of the PCA model. The rotation is defined so that the scores of X data maximize the covariance of X to predict the Y-data. 

The score vector, zx, is found in a two-step operation to guarantee that the covariance of the scores is maximized. Once z, i, i, and qx have been found, the procedure is repeated for the residual matrices ExA and Ev i to find z2,ol2,u2, and q2. This continues until the residuals contain no... 

In literature, PLS is often introduced and explained as a numerical algorithm that maximizes an objective function under certain constraints. The objective function is the covariance between x- and y-scores, and the constraint is usually the orthogonality of the scores. Since different algorithms have been proposed so far, a natural question is whether they all maximize the same objective function and whether their results lead to comparable solutions. In this section, we try to answer such questions by making the mathematical concepts behind PLS and its main algorithms more transparent. The main properties of PLS have already been summarized in the previous section. 

The first PLS component is found as follows Since we deal with the sample covariance, the maximization problem (Equation 4.67) can be written as maximization of... 

It is not immediately visible why this algorithm solves the initial problem (Equation 4.67) of maximizing the covariance between x-scores and y-scores. This can be shown by relating the equations in the above pseudocode. For example, when starting with the equation in step 2 for iteration j+1, we obtain w 1 = XTu / ((h,1)tV1). Now we can plug in the formula for iix from step 7 can again replace Cj by step 6 and step 5, and so on. Finally we obtain... 

PLS regression as described in Section 4.7 allows finding (linear) relations between two data matrices X and Y that were measured on the same objects. This is also the goal of CCA, but the linear relations are determined by using a different objective function. While the objective in the related method PLS2 is to maximize the covar lance between the scores of the x- and y-data, the objective of CCA is to maximize their correlation. In CCA, it is usually assumed that the number n of objects is larger than the rank of X and of Y. The reason is that the inverse of the covariance matrices are needed which would otherwise not be computable applicability of CCA to typical chemistry data is therefore limited. 

Maximizing the posterior probabilities in case of multivariate normal densities will result in quadratic or linear discriminant rules. However, the mles are linear if we use the additional assumption that the covariance matrices of all groups are equal, i.e., X = = Xk=X- In this case, the classification rule is based on linear discriminant scores dj for groups j... 

M-step In this maximization step, the parameters /xr Xj, and pj are estimated. For fij and Xj weighted means and covariances are computed, with weights derived in the E-step, pj can be estimated by summing up all weights for group j and dividing by the sample size n. 

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