Principal component analysis derivation, algorithms

The field points must then be fitted to predict the activity. There are generally far more field points than known compound activities to be fitted. The least-squares algorithms used in QSAR studies do not function for such an underdetermined system. A partial least squares (PLS) algorithm is used for this type of fitting. This method starts with matrices of field data and activity data. These matrices are then used to derive two new matrices containing a description of the system and the residual noise in the data. Earlier studies used a similar technique, called principal component analysis (PCA). PLS is generally considered to be superior. 

The extent of homogeneous mixing of pharmaceutical components such as active drug and excipients has been studied by near-IR spectroscopy. In an application note from NIRSystems, Inc. [47], principal component analysis and spectral matching techniques were used to develop a near-IR technique/algorithm for determination of an optimal mixture based upon spectral comparison with a standard mixture. One advantage of this technique is the use of second-derivative spectroscopy techniques to remove any slight baseline differences due to particle size variations. 

Cluster analysis was considered in our discussion of conformational analysis (see Section 9.13) for compound selection one would typically want to select a representative molecule or molecules from each cluster. A practical consideration when deciding which cluster analysis method to use is that for large numbers of molecules some algorithms may not be feasible because they require an excessive amount of memory or may have a long execution time. Another consideration with cluster analysis (and with some of the other methods that we will discuss) is the need to calculate the distance between each pair of molecules from the vector of descriptors (or from their scaled derivatives or from a set of principal components, if these are being used). For binary descriptors such as molecular fingerprints this distance is often given by 1 — S, where S is the similarity coefficient (Table 12.3). 

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