方开泰、刘民千、周永道《试验设计与建模》课件-8.pptVIP

不同设计变换后的比较 原来的均匀性好 原来的均匀性差 * 试验结果 筛选后的模型： ? = 0.317 + 1.823x1 ? 2.568x12 + 1.257x1x2 ?0.947x22, 其 R2 = 0.995，σ2 = 0.0002436 最优解：x1 = 0.4238，x2 =0.2813 时，y 达到最大值 0.7033，此时 x3 = 0.2949 。 * 在Ts 上的均匀性测度 F-偏差：单纯形上偏差与变换前的偏差一样 均方距离 均方根偏差 最大距离偏差 DM2偏差：直接定义在 Ts 上 * 8.4 有限制的混料均匀设计 上下界限制条件： 0 ? ai ? Xi ? bi ? 1, i = 1,?,s 组合限制条件： 保序限制条件： 0 ≤ xi1≤ xi2≤ · · · ≤ xik≤ 1, * Chapter four Uniform Design of experiments with mixtures. Introduction Many products are formed by mixing two or more ingredients together. Suppose we have the coffee bread. The ingredients include: (the ingredients comes out as the ingredients are named) flour, water, sugar, vegetable shortening, flaked coconut, salt, yeast, emulsifiers, calcium propionate, coffee powder, liquid flavor, and food coloring. How to decide the proportions of the ingredients? What we needed are the experience and experiments. Experiments for such a purpose is called “Experiments with Mixtures”. (Jackson: Pls draw a picture of coffee bread, and put all the ingredients into the bread.) In general, there are s factors, x-one to x-s, such that x-i is greater than or equal to zero, for all i from 1 to s, and the summation of all x-i is one. The experimental domain is the simplex, T-s. Many designs of experiments with mixtures have been suggested, such as simplex-lattice designs; simplex-centroid designs; cox designs and Axial designs. The above transformation method is based on the following theory: Let X be uniformly distributed on T-s. Let x-i be the product of c-i-square and the product of all s-j-square, for j from 1 to i minus one, and x-s be the product of all s-j-square for j equals one to s minus one where s-i and c-i have the definition of (middle equations show) then we have (a) phi-i are mutually independent, for i from one to s minus one, (b) the cumulative distribution function of phi-i is (the last equation shows) with the inverse transformation method formula 4.1 follows. Suppose that we