Deflated matrix

We continue with deriving the next set of components by maximizing the initial problem (Equation 4.67). This maximum is searched in a direction orthogonal to tx, and searching in the orthogonal complement is conveniently done by deflation of X. The deflated matrix X is... 

Just as a known root of an algebraic equation can be divided out, and the equation reduced to one of lower order, so a known root and the vector belonging to it can be used to reduce the matrix to one of lower order whose roots are the yet unknown roots. In principle this can be continued until the matrix reduces to a scalar, which is the last remaining root. The process is known as deflation. Quite generally, in fact, let P be a matrix of, say, p linearly independent columns such that each column of AP is a linear combination of columns of P itself. In particular, this will be true if the columns of P are characteristic vectors. Then... 

A key operation in the power algorithm is the calculation of the deflated cross-product matrix which is independent of the contribution by the first eigenvector. This is achieved by means of the instmction ... 

After a PC has been calculated the information of this component is peeled off from the currently used X-matrix (Figure 3.12b). This process is called deflation, and is a projection of the object points on to a subspace which is orthogonal to p, the previously calculated loading vector. The obtained X-residual matrix Xres is then used as a new X-matrix for the calculation of the next PC. The process is stopped after the desired number of PCs is calculated or no further PCA components can be calculated because the elements in Xres are very small. 

Next, the information (variance) of this component is removed from the x-data. This process is called peeling or deflation actually it is a projection of the x-space on to a (hyper-)plane that is orthogonal to the direction of the found component. The resulting residual matrk Xres has the same number of variables as the original X-matrix but the intrinsic dimensionality is reduced by one. 

Further PLS components (t2, Pi. and so on) are obtained by the same algorithm as the first components using the deflated X matrix obtained after calculation of the previous component. The procedure is continued until a components have been extracted. 

The kernel algorithm also works for univariate t-data (PLS1). Like for PLS2, the deflation is carried out only for the matrix X. Now there exists only one positive eigenvalue for Equation 4.69, and the corresponding eigenvector is the vector w1. In this case, the eigenvectors for Equation 4.70 are not needed. 

Steps 11-13 are the OLS estimates using the regression models (Equations 4.62 through 4.64). Step 14 performs a deflation of the X and of the Y matrix. The residual matrices Xt and Yi are then used to derive the next PLS components, following the scheme of steps 1-10. Finally, the regression coefficients B from Equation 4.61 linking the y-data with the x-data are obtained by B = Y(P Y) C ... 

The resulting weights Wj and scores tj are stored as columns in the matrices W and T, respectively. Note that the matrix W differs now from the previous algorithms because the weights are directly related to X and not to the deflated matrices. Step 2 accounts for the orthogonality constraint of the scores tj to all previous... 

The cross-covariance matrix I., is then estimated by, and the PLS weight vectors ra are computed as in the SIMPLS algorithm, but now starting with instead of S. In analogy with Equation 6.32, the x-loadings py are defined as p = / (vf X/ ) Then the deflation of the scatter matrix is performed as... 

Subsequent eigenvalues can be found by transforming the matrix, A, such that the remaining eigensystem is retained, but the influence of z is removed. This process is known as deflation and may be achieved in a number of ways. Again, see Gourlay and Watson (1973) for further details. 

Salt. Salt adds flavor but also controls the yeast by slowing the fermentation process, keepir the bread matrix from overstretchir and subsequently deflating. For a proper rise, you have to get the salt-to-yeast tario r ht. Plain table salt will do, though some prefer the flavor of sea salt. 

Once the first component is computed on the basis of the Equations (30)-(33), a deflation step is necessary, to eliminate from both X and y the portion of variation aheady accounted for. Since the weights Wi describe the covariance between the X- and Y-blocks, the deflation step relies, for the matrix X, on the computation of a second set of coefficients, p, which resembles the PC A loadings and is calculated as ... 

The loadings pi, together with the corresponding scores, are then used to deflate the independent matrix according to ... 

As in the single-y case, the PLS algorithm is sequential, so that, once the first latent variable is extracted, it is necessary to deflate both the X- and Y-blocks to proceed with the calculation of the second component. Accordingly, it is necessary to compute the X-loadings pi, as already described in Equation (34), to obtain the residual matrix as ... 

Then, extraction of the second PLS component is carried out by repeating all the steps in Equations (48)-(51) with X and Y being substituted by their deflated versions. Once the second PLS factor also is computed a further deflation step takes place and new residual matrices are created, so that the process continues until the desired number of component F is calculated. In this respect, while the Equatirms (38)-(42), summarizing the model for the X-block, remain the same also in the multiple-y case, the formulas accounting for the description and the prediction of the responses are different. Indeed, gathering all the Y-loadings into the matrix Q... 

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