Errors in Variables and Ridge Regression for Structural Data from Synthesized Amorphous Calcium Phosphate
Adam L. Pintar
As vertebrates form mineralized tissues such as bone and teeth they use amorphous mineral precursors to store ions for use as needed. One such precursor is thought to be amorphous calcium phosphate (ACP), but this idea has lately come into question because of lower than expected compositional measurements of the calcium to phosphate ratio (Ca/P) in the precursor. The questions have been raised in part because the low Ca/P measurements cannot be reconciled with the prevailing model for ACP in which its structure does not depend on its composition (namely Ca/P). There were 81 ACP samples synthesized to nominally span the Ca/P range of 1.0 to 1.45. On each sample, duplicate synchrotron based X-ray total scattering measurements were made to obtain a one-dimensional representation of the atomic structure and five measurements of the chemical composition. The goal of the statistical analysis is to quantify the dependence between interesting properties of the structure and the Ca/P. Since the Ca/P is measured thus not known exactly, errors in variables regression is necessary. Further, a theoretical functional relationship between the interesting structural properties and the Ca/P is not known. To compensate, a polynomial of a sufficiently high degree is used as an approximation and the coefficients are regularized to avoid overfitting. The regularization constraints correspond to the familiar case of ridge regression.