Extending the trend vector: the trend matrix and sample-based partial least squares

Trend vector analysis [Carhart, R.E. et al., J. Chem. Inf. Comput. Sci., 25 (1985) 64], in combination with topological descriptors such as atom pairs, has proved useful in drug discovery for ranking large collections of chemical compounds in order of predicted biological activity. The compounds with the highest predicted activities, upon being tested, often show a several-fold increase in the fraction of active compounds relative to a randomly selected set. A trend vector is simply the one-dimensional array of correlations between the biological activity of interest and a set of properties or 'descriptors' of compounds in a training set. This paper examines two methods for generalizing the trend vector to improve the predicted rank order. The trend matrix method finds the correlations between the residuals and the simultaneous occurrence of descriptors, which are stored in a two-dimensional analog of the trend vector. The SAMPLS method derives a linear model by partial least squares (PLS), using the 'sample-based' formulation of PLS [Bush, B.L. and Nachbar, R.B., J. Comput.-Aided Mol. Design, 7 (1993) 587] for efficiency in treating the large number of descriptors. PLS accumulates a predictive model as a sum of linear components. Expressed as a vector of prediction coefficients on properties, the first PLS component is proportional to the trend vector. Subsequent components adjust the model toward full least squares. For both methods the residuals decrease, while the risk of overfitting the training set increases. We therefore also describe statistical checks to prevent overfitting. These methods are applied to two data sets, a small homologous series of disubstituted piperidines, tested on the dopamine receptor, and a large set of diverse chemical structures, some of which are active at the muscarinic receptor. Each data set is split into a training set and a test set, and the activities in the test set are predicted from a fit on the training set. Both the trend matrix and the SAMPLS approach improve the predictions over the simple trend vector. The SAMPLS approach is superior to the trend matrix in that it requires much less storage and CPU time. It also provides a useful set of axes for visualizing properties of the compounds. We describe a randomization method to determine the optimum number of PLS components that is very much faster for large training sets than leave-one-out cross-validation.