不确定约束混料域上的D-最优试验设计
|
摘要
使用混料试验设计研究中草药配方的过程中,需要考虑一类特殊的约束-加法取小不等式组成的不确定约束系统。该系统虽可简洁表达中药配方复杂的分量间关系,但却给试验设计过程带来了困难。本文旨在构造该约束系统形成约束域上的D-最优设计,首先给出一个将该系统化为线性约束不等式组的方法,然后使用改进的CONSIM算法求出极端顶点点集,并使用改进的MDRS算法求得D-最优的设计点集和该点集设计达D-最优时的测度,最后给出一个例子来展示算法,并提出有待进一步研究的问题。
In the process of finding the best recipe of herbal medicine, a kind of special constrained system formed by addition-min uncertain inequalities was introduced. The system can clearly express the relationship of components, which is complex according to herbal medicine recipe. The property of uncertain constraints make the confirming of experimental design region difficult. In this paper, we propose a method to convert uncertain constraints into combination of linear inequalities, which can be handled by the CONSIM algorithm. Here, we use modified CONSIM algorithm to speed up the process of getting extreme points of the region. The extreme point set can be used to improve the MDRS algorithm in the process of finding the D-optimal design point set and associated measure. At last, we use an example to show the efficiency of our algorithms.
In the process of finding the best recipe of herbal medicine, a kind of special constrained system formed by addition-min uncertain inequalities was introduced. The system can clearly express the relationship of components, which is complex according to herbal medicine recipe. The property of uncertain constraints make the confirming of experimental design region difficult. In this paper, we propose a method to convert uncertain constraints into combination of linear inequalities, which can be handled by the CONSIM algorithm. Here, we use modified CONSIM algorithm to speed up the process of getting extreme points of the region. The extreme point set can be used to improve the MDRS algorithm in the process of finding the D-optimal design point set and associated measure. At last, we use an example to show the efficiency of our algorithms.
引文
[1]陈博照,张崇岐.混料模型的混合最优设计[J].数理统计与管理,2015, 34(3):401-408.
[2]燕飞,张崇岐.组合投资的试验设计方法研究[J].数理统计与管理,2016, 35(1):142-153.
[3]张崇岐.混料试验设计研究[J].广州大学学报(自然科学版),2005, 4(5):381-385.
[4] SchefféH. Experiments with mixtures[J]. J R Statist Soc B, 1958, 20(2):344-360.
[5]闫湛,张崇岐.混料试验设计的变量选择[J].数理统计与管理,2016, 35(5):786-793.
[6] Cornell J A. Experiments with Mixtures[M]. New York:Wiley, 2002.
[7]关颖男.混料试验设计[M].上海:上海科技出版社,1990.
[8]朱志彬,张崇岐.非线性分式可加混料模型及其最优设计[J].广州大学学报(自然科学版),2012, 11(3):24-26,
[9]黄秀秀,张崇岐.混料模型的假设检验[J].数理统计与管理,2014, 33(4):620-627.
[10]李俊鹏,张小峰,张崇岐.二阶可加混料模型稳健D-最优设计[J].数理统计与管理,2017, 36(5):843-852.
[11] Fedorov V V, Hackl P. Model-oriented Design of Experiments[M]. New York:Springer-Verlag, 1997.
[12] Kiefer J. General equivalence theory for optimum designs(approximate theory)[J]. Ann Statist,1974, 2(5):849-879.
[13] Nobuhara H, Pedrycz W, Hirota K. Fast solving method of fuzzy relational equation and its application to lossy image compression/reconstruction[J]. IEEE Trans. Fuzzy Syst, 2000, 8(3):325-334.
[14] Sanchez E. Truth-qualification and fuzzy relations in natural languages, application to medical diagnosis[J]. Fuzzy Sets Syst, 1996, 84(2):155-167.
[15] Yang S J. An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition-min composition[J]. Fuzzy Sets Syst, 2014, 255:41-51.
[16] Yang X P, Zhou X G, Cao B Y. Min-max programming problem subject to addition-min fuzzy relation inequalities[J]. IEEE Trans. Fuzzy Syst. 2016, 24(1):111-119.
[17] Snee R D. Experimental design for mixture systems with multicomponent constraints[J]. Commun.Stat.:Theory Methods, 1979, 8(4):303-326.
[18]李光辉,张崇岐.具有复杂约束混料试验的渐近D-最优设计[J].应用概率统计,2017, 34(2):203-220.
[19] Wang Y, Fang K T. Uniform design of experiments with mixtures[J]. Sci. China(Ser A), 1996,44(3):345-350.
[2]燕飞,张崇岐.组合投资的试验设计方法研究[J].数理统计与管理,2016, 35(1):142-153.
[3]张崇岐.混料试验设计研究[J].广州大学学报(自然科学版),2005, 4(5):381-385.
[4] SchefféH. Experiments with mixtures[J]. J R Statist Soc B, 1958, 20(2):344-360.
[5]闫湛,张崇岐.混料试验设计的变量选择[J].数理统计与管理,2016, 35(5):786-793.
[6] Cornell J A. Experiments with Mixtures[M]. New York:Wiley, 2002.
[7]关颖男.混料试验设计[M].上海:上海科技出版社,1990.
[8]朱志彬,张崇岐.非线性分式可加混料模型及其最优设计[J].广州大学学报(自然科学版),2012, 11(3):24-26,
[9]黄秀秀,张崇岐.混料模型的假设检验[J].数理统计与管理,2014, 33(4):620-627.
[10]李俊鹏,张小峰,张崇岐.二阶可加混料模型稳健D-最优设计[J].数理统计与管理,2017, 36(5):843-852.
[11] Fedorov V V, Hackl P. Model-oriented Design of Experiments[M]. New York:Springer-Verlag, 1997.
[12] Kiefer J. General equivalence theory for optimum designs(approximate theory)[J]. Ann Statist,1974, 2(5):849-879.
[13] Nobuhara H, Pedrycz W, Hirota K. Fast solving method of fuzzy relational equation and its application to lossy image compression/reconstruction[J]. IEEE Trans. Fuzzy Syst, 2000, 8(3):325-334.
[14] Sanchez E. Truth-qualification and fuzzy relations in natural languages, application to medical diagnosis[J]. Fuzzy Sets Syst, 1996, 84(2):155-167.
[15] Yang S J. An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition-min composition[J]. Fuzzy Sets Syst, 2014, 255:41-51.
[16] Yang X P, Zhou X G, Cao B Y. Min-max programming problem subject to addition-min fuzzy relation inequalities[J]. IEEE Trans. Fuzzy Syst. 2016, 24(1):111-119.
[17] Snee R D. Experimental design for mixture systems with multicomponent constraints[J]. Commun.Stat.:Theory Methods, 1979, 8(4):303-326.
[18]李光辉,张崇岐.具有复杂约束混料试验的渐近D-最优设计[J].应用概率统计,2017, 34(2):203-220.
[19] Wang Y, Fang K T. Uniform design of experiments with mixtures[J]. Sci. China(Ser A), 1996,44(3):345-350.