Significant PLS components

The number of significant PLS components is established by testing the significance of each additional dimension (PLS component). This is done to avoid overfitted QSARs, which may exhibit lesser, or no, validity. The optimal number of PLS components to be used in conventional analyses is typically chosen from the analysis with the highest cross-validated value, and for component models with identical values, the model having the smallest standard error of prediction, PRESS (see also the following section). Unlike spectroscopic data, where a PLS model typically has more than 10 components, models in 3D-QSAR tend to exhibit less complexity. As a rule of thumb, two to four components should suffice when CoMFA standard fields are used." ... 

If the value of cvd sd, is below 1.0, this indicates that the PLS component is significant for the y th Y variable. Note that PRESS and RSS only involve Y. This is so because the PLS model is formulated as prediction of Yfrom X. In a recent simulation study, it has been shown how the 5 % -level varies with the number of objects and the percentage variance in X explained by the first PLS dimension [24],... 

The numbers of partial least squares (PLS) components were higher in CoMSIA than in CoMFA. This difference probably resulted from the significantly higher number of lattice points showing steadily varying field values (e.g., inside the molecules). The optimal numbers of components were selected on the basis of lowest Spress. 

Note that eq 5 has a pKa term even though there is no electronic component to the absorption process itself. This is needed to "correct" the log P for ionization. The fact that eq 4 can t accept a significant pl term in our view indicates absorption depends only on the amount of compound partitioning into lipid phases. 

After fitting the first PLS component, there may still be systematic variation left in the Y space which can be described by a second PLS component. As in principal components anafysis, the PLS components can be peeled oft one dimension at a time, until the systematic variation in the Y space has been described. The model can be determined by cross validation to ensure valid predictions. The number of significant PLS dimensions is usually denoted by A. 

A first PLS model was established from 124 reaction systems. To ensure that this set of reaction systems was not selected in such a way that the descriptor variables were correlated, a principal component analysis was made of the variation of the eight descriptors over the set. This analysis afforded eight significant principal components according to cross validation. This showed that the variance-covariance matrix of the descriptors was a full rank matrix and that there were no severe colinearities among the descriptors. 

Cross-validation estimates model robustness and predictivity to avoid overfitting in QSAR [27]. In 3D-QSAR models, PLS and NN model complexity are established by testing the significance of adding a new dimension to the current QSAR, i.e., a PLS component or a hidden neuron, respectively. The optimal number of PLS components or hidden neurons is usually chosen from the analysis with the highest q2 (cross-validated r2) value, Eq. (3). The most popular cross-validation technique is leave-one-out (LOO), where each compound is left out of the model once and only once, yielding reproducible results. An extremely fast LOO method, SAMPLS [42], which evaluates the covariance matrix only, allows the end user to rapidly estimate the robustness of 3D-QSAR models. Randomly repeated cross-validation rounds using leave 20% out (L5G), or leave 50% out (L2G), are routinely used to check internal... 

SDEP (the standard deviation of the error of predictions) corresponds to jpress but the number of degrees of freedom is not considered in the calculation of the SDEP value. The smallest -spress or SDEP value should be taken as the criterion for the optimum number of components. Alternatively, an increase of the value by a certain percentage, e.g., 5%, may be defined as the criterion to accept a further PLS component. As long as only significant components are extracted in the PLS analysis, PRESS, SDEP and Jpress will decrease if too many components are derived, overprediction results and PRESS, SDEP and spress increase. 

In any empirical modeling it is es.sential to determine the correct complexity of the model. With many and correlated X variables there is a substantial risk of overfitting , i.e.. obtaining a well-fitting model with little or no predictive power. Hence a strict test of the significance of each consecutive PLS component is necessary, this process being stopped when the components start to become non-significant. 

Thus, we see that CCA forms a canonical analysis, namely a decomposition of each data set into a set of mutually orthogonal components. A similar type of decomposition is at the heart of many types of multivariate analysis, e.g. PCA and PLS. Under the assumption of multivariate normality for both populations the canonical correlations can be tested for significance [6]. Retaining only the significant canonical correlations may allow for a considerable dimension reduction. 

Haaland and coworkers (5) discussed other problems with classical least-squares (CLS) and its performance relative to partial least-squares (PLS) and factor analysis (in the form of principal component regression). One of the disadvantages of CLS is that interferences from overlapping spectra are not handled well, and all the components in a sample must be included for a good analysis. For a material such as coal LTA, this is a significant limitation. 

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