Cross-dimensional stability and spectral characteristics of central composite designs under alphabetic optimality criteria

The behavior of Central Composite Designs (CCDs) as the dimension increases remains poorly understood, particularly with respect to structural stability. In this study, a cross-dimensional evaluation approach is proposed to investigate the evolution of CCD properties as dimensionality increases from \(k = 3\) to \(k = 10\) . Three types of CCDs are considered: Rotatable (RCCD), Spherical (SCCD), and Face-Centered (FCCD). The evaluation is based on the spectral properties of the information matrix. The results reveal a clear divergence between the growth of the determinant and the stability of the design. While the determinant of the information matrix increases monotonically with dimension, the eigenvalue distribution becomes increasingly dispersed, leading to a rapid increase in the condition number. These findings indicate that spectral properties play a critical role in determining the behavior of CCDs. Among the designs considered, RCCD exhibits more favorable properties, particularly in terms of maintaining larger minimum eigenvalues, compared to SCCD and FCCD. Using the Cross-Dimensional Stability (CDS) framework, it is observed that the optimality criteria for RCCD exhibit positive scaling behavior, with scaling exponents ranging from 0.052 to 0.229. In contrast, FCCD demonstrates a negative scaling trend, with exponents ranging from \(-1.087\) to \(-1.543\) , indicating a deterioration in performance as dimensionality increases.