最大离差值极小响应面及按Kreisselmerier-Steinhauser函数的拟合  

Min-Max Distance Response Surface and Kreisselmerier-Steinhauser Function

作　　者：宇慧平[1] 隋允康[1] 丁力[1] 叶红玲[1] 

机构地区：[1]北京工业大学机械工程与应用电子技术学院,北京100124

出　　处：《科技导报》2010年第1期63-68,共6页Science & Technology Review

基　　金：北京市教委基金项目(KM200710005032);国家自然科学基金项目(10872012);北京市自然科学基金项目(3093019);高等学校博士学科点专项科研基金项目(20060005010);工业装备结构分析国家重点实验室基于网格应用的专项研究基金(GZ0819)

摘　　要：现有的响应曲面是按最小二乘法实现的。提出一种响应面拟合方法:使响应面函数同样本值之间的最大距离极小。该方法运用Kreisselmerier-Steinhauser函数的特性,建立数学模型,采用求导、泰勒展开、数值迭代等方法确定响应面函数的系数。在数值迭代过程中以最小二乘法获得的结果作为迭代初值。通过一系列算例得出,Kreisselmerier-Steinhauser函数法获得的响应面函数能够达到最大距离最小的目的;该方法获得的响应面函数与最小二乘法拟合的响应面函数相比,在与样本值差的均方根相对增加不大的情况下,能显著减少最大距离;采用变熵参数方法能获得更好的解。本途径不仅丰富了响应面的构造方法,而且可满足实际工程问题中希望响应函数与样本值最大距离极小化的需求。The current Response Surface Methodology （RSM） is based on the least squares method. This paper proposes a new response surface fitting method, which let the maximum distance among the response hypersurface function and sample values be minimized. In this method, a mathematical model is built based on the characteristics of Kreisselmerier-Steinhauser function. The coefficients of the response hyper-surface function can be determined by taking derivatives, carrying out Taylor expansions and numerical iterations. The result of the least squares method is used as the initial value in the process of numerical iteration. According to a series of calculated cases, three conclusions are reached. First, the response hyper-surface function obtained from the Kreisselmerier-Steinhauser function method satisfies the condition of minimizing the maximum distance. Second, compared with the counterpart obtained from the least squares method, the response hyper-surface function obtained from this method could noticeably reduce the maximum distance in case that the RMS of sample values increases in a small scale. Third, the adoption of a method of changing stretching factor could lead to a better solution. The method proposed in this paper not only provides a response surface method but also can ensure the maximum distance among the response hyper-surface function and sample values minimized in practical engineering problems.

关 键 词：响应面方法 最大值极小 Kreisselmerier—Steinhauser函数 最小二乘法 

分 类 号：O241.5[理学—计算数学]