系统的描述与模型建立.doc

第三章 系统的描述与模型建立Chapter 3. Description of Systems and Modeling3.5.3 主元分析法/主成分分析法3.5.3 Principle Component Analysis (CPA)(先复习有关矩阵运算) (First review the knowledge about matrix manipulation)找出若干彼此不相关的综合因素来代表原来为数总多的变量, 将多个描述指标(参数)化为少数几个综合指标.Seek out some integrate factors to represent the numerous original variables, perform transformation of more parameters to less ones called integrate parameters. 例: 设某应变量W与两个自变量(描述指标) x1,x2 有关. 现有n组数据 x1j,x2j (j=1,n)Ex. Let variable W related to two self-variables, x1,x2, Suppose there exists n groups of data, x1j,x2j (j=1,n)x1,x2,y1,y2,即: Namely, 将n组数据”绘制在x1,x2 的座标中. 将座标旋转一个角度, 得到新座标系y1,y2 , 则各点在新座标系中的座标值为:To plot this n-group data point ” in the coordinate system x1,x2, then rotate the coordinate by an angle , the new coordinate system y1,y2 is thus established. The data points in the new coordinate system are ,其中U 为正交变换矩阵, UT=U-1, U UT =Iwhere U is orthogonal transform matrix, UT=U-1, U UT =I即that is 若将y1取为椭圆长轴方向, 则新座标值机具有下列性质:If take y1 as the major axis, the new coordinates possess the following properties 1) N 个点y1j, y2j (j=1,n)纵座标值相关几乎为零.1) For N points y1j, y2j (j=1,n), their correlation is almost zero. 2) N 个点的方差大部分是由于y1轴分量引起, 而沿y2轴引起的较小.2) For N points y1j, y2j, more significant contributions to the variances are from the weight of y1 axis compared with that of y2. y1和y2是x1和x2 的线性组合, 即称为综合变量. y1 and y2 are the linear combination of x1 and x2, namely integrate variable. 由于沿y1的方差较大, 所以差异性反映在方面较突出, 若仅以y1的座标值做为代表, 损失的信息量最小, 则y1被称为第一主元(主分量).Because the variance along y1 is relative large, so the variation of the data is mainly reflected in this orientation. If use y1 to represent original data points, the information loss will be minimized. Hence, y1 is called the 1st principle component. y2与y1 正交, 对应方差较小, 称为第二主元. y2 is orthogonal to y1, and the corresponding variance is relatively smaller, referred to as the 2nd principle component. 设原有p 个指标(或描述变量) x1, x2, xp , 有n个样本(集合), 经过主元变换, 则可合成p个综合指标y1,y2, yp .Suppose there are p original description variables x1, x2, xp, and n samples, after transformation,p integrate variables are formed 综合指标与原描述变量的关系为:The relation between the integrate variables and the original description variables isy1=c11x1 + c12x2 + + c1pxpy1=c21x1 + c22x2 + + c2pxpyp=cn1x1 + cn2x2 + + cnpxp即namely, 系数cij的选择由下列条件确定;The coefficients selection may be determined through the conditions as follows, 1) 1) 2) yi与yj () 相互独立 (或正交, 彼此轴上的投影为零)2) yi and yj are independent to each other. 3) y1是满足上列变换式中方差贡献最大的一个, y2是次大的一个, y33) y1 is the variable that gives the maximum contribution to the variance, the next is y2, and 求出的y1,y2, yp分别称为原指标(原描述变量)的第一, 第二, 第p个主元(主分量).y1,y2, yp obtained are the 1st, 2nd, 3rd,pth principle component of the original description variable, relatively. 求取主元:Computing for the principle components: 设由p个原描述变量,Suppose p original variables,X=x1, x2 xpT 变换后相应的p个综合指标, correspondingly, p integrate variables are obtained after transformation, Y=y1, y2 ypT进行主元变换perform the principle component transform, Y= CXC 为正交矩阵, 满足 C T= C -1, C C T =IC is the orthogonal matrix, and satisfying C T= C -1, C C T =I由于座标旋转, 则新座标轴仍然正交, 即仍为直角座标系. 变换后的n个点在y1方向的方差最大, y2 次之, , 而对不同的yi与yj 轴的协方差为零. As a result of coordinate rotation, the new coordinate axes remain orthogonal to each other. The n points after transformation have the greatest variance in the direction of y1, and next is in y2 , , and for different yi and yj, their co-variances are zero. 即要求Y的协方差为That is, it requires the co-variance of Y is Y YT=(CX) (CX)T= CXX T C T=其中,where现令 XX T=R, 则有Now let XX T=R , hence C R C T=, R C T= C -1= C 1 (注为对角线矩阵)( Note is the diagonal matrix)因为C T= C -1, 有 because C T= C -1, thenR- C T=0C T为矩阵R的特征向量, C T is the characteristic matrix of R,C=c1 c2 c3 cp=或 orC T=c1T c2 T c3 T cp T= R的特征根为The characteristic roots of R is | R-|=0由上式可得p个根 (i=1,p), 以及相应的各个cij.p roots can thus be acquired, denoted as (i=1,p), and the corresponding cij.注: 若有p个相异的特征根, 就有p个线性无关的特征向量. 例如 对应于c1, 即Note: if exist p different characteristic roots, then there are p independent characteristic vectors. That is, c1=yi 的方差为Var(ciX)= ciXXTciT= ciRciT=Variance of yi is 所以y1 对应于最大方差, y2其次, , 即变换后, y1 corresponds to the greatest variance , y2 and to the next, this means after transformation:y1, y2, y3, yp, 彼此不相关, 方差依次为 , . , 它们是第一, 第二,第p个主分量/主元.y1, y2, y3, yp, are independent to each other, and their variances are , . , respectively, the they correspond to the 1st, 2nd, 3rd, , pth principle component. 注意: 因为主元分析法得出的主元是原变量的线性组合, 不一定具有明确的技术/物理意义.Note: due to the fact that the principle components are obtained from the linear combination of the original variables, they may not have the concrete or physical meanings. 具体计算方法:The specific computation method: 对于p个自变量, n组样本,For p variables, and n batch samples, (自变量数目) (# of variables)(样本数目) (# of samples)标准化处理:Standardization, ( i=1,n, k=1,p ) 上式中,in the expression above, 即第k个自变量的n次观测的平均值. The mean of kth self-variable based on n observations. ( k=1,p ) 即第k个自变量的方差.The variance of kth self=-variance. 然后计算相关系数矩阵then compute for correlation matrix coefficients, R =XX T 求 rij. seek rij.再求出特征根 Then calculate for characteristic roots, | R - I |=0再求出对应于 的特征向量 ci , 即可得出变换方程的系数. Next is to calculate the characteristic vector ci corresponding to , thus the coefficients of the transformation equation can be obtained. 主元分析法的目的是减少变量数目, 最后综合变量的数目m应满足 ml2, and correspondingly, let e1 and e2 be the resultant eigenvectors. The direction of vector e1 indicates the orientation of the particles major eigen axis, and likewise, the direction of e2 coincides with the direction of its