Rank-deficient problem

An alternative to the method of Lagrange multipliers for imposing the necessary constraints is sketched below. It derives a lower dimensional unconstrained problem from the original constrained problem by using an orthogonal basis for the null space of the constraint matrix. This method is well suited to the potentially rank-deficient problem at hand, where steps may be taken to... 

The literature of the past three decades has witnessed a tremendous explosion in the use of computed descriptors in QSAR. But it is noteworthy that this has exacerbated another problem rank deficiency. This occurs when the number of independent variables is larger than the number of observations. Stepwise regression and other similar approaches, which are popularly used when there is a rank deficiency, often result in overly optimistic and statistically incorrect predictive models. Such models would fail in predicting the properties of future, untested cases similar to those used to develop the model. It is essential that subset selection, if performed, be done within the model validation step as opposed to outside of the model validation step, thus providing an honest measure of the predictive ability of the model, i.e., the true q2 [39,40,68,69]. Unfortunately, many published QSAR studies involve subset selection followed by model validation, thus yielding a naive q2, which inflates the predictive ability of the model. The following steps outline the proper sequence of events for descriptor thinning and LOO cross-validation, e.g.,... 

Hansen, P.C., Rank-Deficient and Discrete Ill-Posed Problems Numerical Aspects of Linear Inversion, SIAM, Philadelphia, 1998. 

Hansen, C., 1998, Rank-deficient and discrete ill-posed problems. Numerical aspects of linear inversion Department of mathematical modeling. Technical University of Denmark, Lyngby, 247 pp. 

The problem of rank deficiency is particularly acute for larger molecules, as one might expect considering classical electrostatic models. Classically, all... 

Sigfridsson and Ryde have recently shown that the method chosen for solving the least-squares problem as well as the method used to impose the constraints influence the rank deficiency. Those authors point out that the LLS solution for CHELP and CHELPG is found by starting with Aq = f, where q is the vector of charges and f is the data to be fit. Then left multiplying leads to... 

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