Three-dimensional rank analysis

The number of linearly independent columns (or rows) in a matrix is called the rank of that matrix. The rank can be seen as the dimension of the space that is spanned by the columns (rows). In the example of Figure 4-15, there are three vectors but they only span a 2-dimensional plane and thus the rank is only 2. The rank of a matrix is a veiy important property and we will study rank analysis and its interpretation in chemical terms in great detail in Chapter 5, Model-Free Analyses. 

Finally, we considered the use of three dimensional analytical techniques such as the GC-MS-MS and video fluorometric monitoring of BPLC effluents. As with two dimensional techniques, we can use Rank Analysis, Rank Annihilation and Factor Analysis. However, in this case the data can be shown to be unambiguously decomposible, even when there is severe overlap among conq>onents. 

If the origin ( 0 ) is chosen at the centroid of the atoms, then it can be shown that distances from this point can be computed from the interatomic distances alone. A fundamental theorem of distance geometry states that a set of distances can correspond to a three-dimensional object only if the metric matrix g is rank three, i.e., if it has toee positive and N — 3 zero eigenvalues. This is not a trivial theorem, but it may be made plausible by thinking of the eigenanalysis as a principal component analysis all of the distance properties of the molecule should be describable in terms of three components , which would be the x, y and z coordinates. If we denote the eigenvector matrix as w and the eigenvalues A., the metric matrix can be written in two ways ... 

It is necessary to decide the appropriate number of components to use in a model. The appropriate dimensionality of a model may even change depending on what the specified purpose of the model is. Hence, appropriate dimensionality of, e.g., a PARAFAC model, is not necessarily identical to the three-way pseudo-rank of the data array. Appropriate model dimensionality is not only a function of the data but also a function of the context and aim of the analysis. Hence, a suitable PARAFAC model for exploring a data set may have a rank different from a PARAFAC model where the scores are used for a subsequent regression model. 

---