q-1阶塌落的单纯型-中心设计及A-最优配置

Design of Collapsed Simplex-Centroid of q-1 Degree and Its A-Optimation

对比例分量 q>1的混料进行了实验设计 ,响应 y依赖于各相关比例分量x1,x2 ,… ,xq,记x′ =(x1,x2 ,… ,xq)′,x′∈Sq -1={ (x1,x2 ,… ,xq)′|x1+x2 +… +xq=1,1≤xi≤q} ,Sq -1的几何解释是 q - 1维欧氏空间内的正规单纯形。在一般的 q - 1维正规单纯形利益区域和多重线性多项式模型为E(y) =∑qi=1βixi+∑qi<jβijxixj+… +∑qi1 <i2 <… <inβi1 i2 …inxi1 xi2 …xin,KieferandUranist分别在 q =2和q =3的情况下证明了上述单纯形中心设计是D -最优的 ;若A -最优的混料设计的柱点是所有正规单纯的各类的中心 ,令ri(i =1,2 ,… ,q)表示每个第i类中心点的设计测度 ,TONGYi(佟毅 )、GUANYing -nan(关颖男 )等研究了广义单纯形中心设计是A -最优的 ;对具有“冲淡剂”分量的混料系统 ,研究了 q-分量n阶塌落的多重线性多项式模型 ,并且证明了n阶塌落的单纯形中心设计是D -最优的。在一般的q - 1维正规单纯形利益区域和多重线性多项式模型的问题中 ,对于具有“冲淡剂”分量的混料系统 ,研究了相应模型的A -最优的混料设计 ,其测度比为 :rjr0 =(∑q- 1k =jk2 j2k-2 Ck-jq-j-1) 1/2 (j=1,2 ,… ,q - 1)的 q - 1阶塌落的单纯形 -中心设计是A

The mixture of q > 1 ingredients has been experimentally designed. The response y depends on the relative proportions x1, x2,...,xq of the ingredients. Let x prime = (x1, x2,...,xq) prime so that x belongs to the (q - 1) - dimensional simplex Sq-1={(x1, x2,...,xq) prime |x1 + x2 + ... + xq = 1, xi greater than or equal 0,1 [less-than or equal to] xi [less-than or equal to] q}. For general regular simplex interest region of (q - 1) - dimension and multilinear polynomial model is : E(y) = summation i=1qβixi + summation i less than or equal jqβijxi xj + ... + summation i(1) less than or equal i(2) less than or equal ... less than or equal i(n)qβi(1)i(2)...i(n )xi(1)xi(2)... xi(n), Kiefer and Uranist each has proved that Scheffe(1963) simplex-centroid designs are the D - optimal design when q = 2 and q = 3. If the support points of A - optimal mixture design are the centroids of the reqular simplex, if let ri(i = 1,2,...,q) denote design measure in every i - th class centroid, Tong yi and Guan ying - nan showed that the generalized simplex - centroid design rj/r0 = [ summation k=jq-1k2j2k-2Cq-j-1 k-j]1/2 (j = 1,2,...,q - 1) is A - optimal. For the mixture system with 'diluent component', Guan Ying - nan and Tong Yi suggested the collapsed multilinear polynomial models and proved that the collapsed simplex - centroid designs of n - degree of q - component are D - optimal. In this paper for general regular simplex interest region of (q - 1) - dimension and multilinear polynomial model, with 'diluent component', when n = q - 1, that the collapsed mixture design of q - 1 degree of q - component with measure rate is A - optimal has been proved.