Central composite designs

Fix $k\ge 1$, coded factor coordinates $x\in {\mathbb{R}}^k$, an axial distance $\alpha >0$, and an integer $m_0\ge 1$. A factorial core $\mathcal{F}$ is either the full $2^k$ run set in Two-level full factorial designs and coded effects or the run set of a regular fractional factorial as defined by Regular two-level fractions, signed defining relations, and resolution. In the standard second-order fractional construction the core has resolution V or higher; its generators and resolution are declared with the core.

Let $e_j$ be the $j$th standard coordinate vector. The axial runs are ${\mathcal{A}}_{\alpha }=\{\alpha e_j,-\alpha e_j:1\le j\le k\}$, and the center runs are $m_0$ copies of $0\in {\mathbb{R}}^k$. A central composite design (CCD) is the multiset union $\mathcal{D}=\mathcal{F}\uplus {\mathcal{A}}_{\alpha }\uplus \{0{\}}^{(m_0)}\mathrm{.}$

The definition specifies runs only. It does not imply that a particular fitted polynomial is estimable, that the design is rotatable, or that blocking is orthogonal; those properties require separate checks for the chosen runs and model matrix.