Vector graphics column

Since U and V express one and the same set of latent vectors, one can superimpose the score plot and the loading plot into a single display as shown in Fig. 31,2e. Such a display was called a biplot (Section 17.4), as it represents two entities (rows and columns of X) into a single plot [10]. The biplot plays an important role in the graphic display of the results of PCA. A fundamental property of PCA is that it obviates the need for two dual data spaces and that instead of these it produces a single space of latent variables. 

Each of these columns of this symmetrical matrix may be seen as representing a molecule in the subspace formed by the density functions of the N molecules that constitute the set. Such a vector may also be seen as a molecular descriptor, where the infinite dimensionality of the electron density has been reduced to just N scalars that are real and positive definite. Furthermore, once chosen a certain operator in the MQSM, the descriptor is unbiased. A different way of looking at Z is to consider it as an iV-dimensional representation of the operator within a set of density functions. Every molecule then corresponds to a point in this /V-dimensional space. For the collection of all points, one can construct the so-called point clouds, which allow one to graphically represent the similarity between molecules and to investigate possible relations between molecules and their properties [23-28]. 

Organizing J in a three-dimensional array is elegant, but it does not fit well into the standard routines of MATLAB for matrix manipulation. There is no command for the calculation of the pseudoinverse J+ of such a three-dimensional array. There are several ways around this problem one of them is discussed in the following. The matrices R(k) and R(k + 5k) as well as each matrix < RIdk, are vectorized, i.e., unfolded into long column vectors r(k) and r(k + 5k). The nk vectorized partial derivatives then form the columns of the matricized Jacobian J. The structure of the resulting analogue to Equation 7.13 can be represented graphically in Equation 7.17. 

When it comes to the behavior of a CS, the DPE dictates that profiles are influenced by both S and M, and the vector addition of these two results in the movement of the column profiles. Figure 3.13 graphically displays nature of these vectors. Notice how the vector sum is tangent to the column profile. 

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