Orthogonal molecular descriptors

One of the central oversights and indifferences in use of MRA in QSAR relates not to using orthogonal molecular descriptors. Orthogonal molecular descriptors, which were introduced almost a quarter of a century ago [7], continue to be overlooked by a vast majority of QSAR researchers, who apparently do not realize that use of orthogonal molecular descriptors in multiple regression analysis is not an option but a must if they want to elaborate on the structure-property or structure-activity relationship ... 

The orthogonality of a set of molecular descriptors is a very desirable property. Classification methodologies such as CART (11) (or other decision-tree methods) are not invariant to rotations of the chemistry space. Such methods may encounter difficulties with correlated descriptors (e.g., production of larger decision trees). Often, correlated descriptors necessitate the use of principal components transforms that require a set of reference data for their estimation (at worst, the transforms depend only on the data at hand and, at best, they are trained once from some larger collection of compounds). In probabilistic methodologies, such as Binary QSAR (12), approximation of statistical independence is simplified when uncorrelated descriptors are used. In addition,... 

The table shows a number of representative descriptor types (there are many more) that can be used to define chemical spaces. Each descriptor adds a dimension (with discrete or continuous value ranges) to the chemical space representation (e.g., selection of 18 descriptors defines an 18-dimensional space). Axes of chemical space are orthogonal only if the applied molecular descriptors are uncorrelated (which is, in practice, hardly ever the case). 

A major practical issue affecting MP calculations is caused by use of correlated molecular descriptors. During subsequent MP steps, exact halves of values (and molecules) are only generated if the chosen descriptors are uncorrelated (orthogonal), as shown in Fig. 1A. By contrast, the presence of descriptor correlations (and departure from orthogonal reference space) leads to overpopulated and underpopulated, or even empty, partitions (see also Note 5), as illustrated in Fig. ID. For diversity analysis, compounds should be widely distributed over computed partitions and descriptor correlation effects should therefore be limited as much as possible. However, for other applications, the use of correlated descriptors that produce skewed compound distributions may not be problematic or even favorable (see Note 5). 

Orthogonalized descriptors are used in similarity/diversity analysis and quantitative -> structure/response correlations with the aim of eliminating the bias provided by the interdependence of common molecular descriptors. Moreover, the interpretation of regression models should be facilitated, as the information encoded in each descriptor is unique. 

The 2N molecular descriptors Sj are obtained by summing, over all the A atoms, the Lorentzian function values of the two orthogonal circles (N for each circle) ... 

Molecular descriptors calculated by applying the - Wiener operator to symmetric matrices derived from molecular graphs or the - Wiener orthogonal operator to unsymmetric matrices. 

Randic, M. (1991e). Orthogonal Molecular Descriptors. New J.Chem., 15,517-525. 

MLR Simple to use Models are easy to interpret Molecular descriptors should be orthogonal to one another Number of compounds in the training set should exceed the number of molecular descriptors by at least a factor of 5 Assumes a linear relationship between target property and molecular descriptors... 

These are obtained by applying an orthogonalization procedure to a selected set of molecular descriptors. A descriptor is made orthogonal to another descriptor Xi simply by regressing it against X and using as the new orthogonal descriptor the residual — Xj, where Xj is the value... 

The matrix of molecular descriptors frequently contains columns of values that are correlated with each other. This might arise from a poorly designed set in which there is a correlation between the MR and n of a substituent, or from descriptors that are intrinsically correlated, such as TPSA and the counts of nitrogen and oxygen atoms and the protons bonded to them. Principal component analysis considers the matrix of properties and extracts orthogonal components that contain a contribution (loading) from one or more of the... 

We repeat, for the third time, that the orthogonalization of molecular descriptors was introduced into structure-property-activity studies about 25 years ago [7,20-28] and has only occasionally been used since then [29-41]. It is still today not widely known, not recognized as indispensible in QSAR and QSPR modeling if people want to interpret their models in terms of descriptors used in the construction of the regression equation. Repeating the same message three times deliberately is... 

Here we have another subject, orthogonal molecular descriptors [30,31], the topic of MRA, which, despite some visibility (the two papers listed above have together about 450 citations), continue not to be implemented by most MRA users. This suggests... 

The origin of the instability of the regression equations can be traced to the interrelatedness of the descriptors used. Molecular descriptors, as a rule, are interrelated, often highly interrelated, the phenomenon referred to as colinearity. Only recently has the remedy for this hopeless situation been found. Instead of using the initial descriptors which are interrelated one can construct novel descriptors that will not be correlated at all. We refer to uncorrelated descriptors as orthogonal, and their construction as orthogonalization. 

Randi6,M. (1991b). Orthogonal molecular descriptors. A/ew, J. Chem. 15,517-525. 

Principle 5 a mechanistic interpretation, if possible hierarchy of minimum paths across a spectrum of computed/predicted end-points towards identifying the specific influence of molecular descriptors within the orthogonal space of chemical-biological complex interaction by molecular structural characters/indices and of their algebraic behavior and predicted statistics. 

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