Central Composite Designs for Reduced Models: I_V-Optimality, Heredity Rules, and High-Dimensional Heuristics
Keywords:
Central Composite Design, Iv-Optimality, Strong Heredity, Reduced Models, High-Dimensional Heuristics, Real-World Case StudyAbstract
Evaluations of Central Composite Designs (CCDs) concerning reduced second-order response surface models have historically emphasized $D$-, $A$-, and $G$-optimality under weak heredity for limited factor dimensions ($k \le 5$). Yet, practical experimentation often eliminates non-significant terms, necessitating thorough assessments of combined prediction variance, strict enforcement of model hierarchy, sampling in higher dimensions ($k \ge 6$), and validation through empirical means. This study expands the assessment framework for Central Composite Circumscribed (CCCD), Inscribed (CCID), and Face-Centered (CCFD) designs within reduced model spaces by: (1) incorporating $I_V$-optimality through the integration of exact moment matrices; (2) analyzing efficiency sensitivity when transitioning from weak to strong heredity constraints; (3) creating an Adaptive Genetic Algorithm Sampling Scheme to investigate combinatorial model spaces for $k \ge 6$; (4) performing Monte Carlo simulations to evaluate the Mean Integrated Prediction Error (MIPE); and (5) confirming results through a practical bio-chemical extraction yield case study. Precise algebraic computations demonstrate that CCCD consistently provides better average prediction outcomes ($I_V$-efficiency of $26.45\%$ compared to $18.27\%$ for CCFD and $7.76\%$ for CCID at $k=3$), while maintaining robust heredity stabilizes candidate spaces without changing relative design efficiency rankings. Additionally, high-dimensional heuristic sampling ($k=6,7$) shows that the efficiency benefit of CCCD increases exponentially with factor dimensions, and Monte Carlo simulations demonstrate that algebraic efficiencies correspond directly to reduced empirical prediction errors ($\text{MIPE}=0.1484$). Researchers are highly encouraged to utilize CCCD when axial extensions are possible.
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References
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