中国矿亚天整CHINA UNIVERSITYOF MININGANDTECHNOLOGY二、线性最小二乘拟合1．基本思想给定(x,,J,)(i=0,l,,m)，设x,v的关系为 y=S(x)其中S(x)来自函数类Φ如(1)中y(x)来自线性函数类设函数类Φ的基函数为(x)(i=0,1,,n）一般要求n≤mΦ = span(p,(x),P,(x),...,P,(x)S(x)= ap,(x)+ap(x)...+a,p,(x)e@Iol, =8? =(S(x,)- y,)仍然定义平方误差i=0=0
CHINA UNIVERSITY OF MINING AND TECHNOLOGY y = S x( ) 其中 S ( x )来自函数类 Φ 如 ( 1 ) 中 y ( x )来自线性函数类 ( , )( 0,1, , ) i i x yi m = " ( x)( i 0 , 1 , , n ) 设函数类 Φ的基函数为 ϕi = " 一般要求 n ≤ m 0 1 { ( ), ( ), , ( )} n Φ = span x x x ϕ ϕ ϕ " 2 2 2 0 m i i δ δ = = ∑ 2 0 (( ) ) m i i i Sx y = 仍然定义平方误差 = − ∑ 00 11 () () () () Sx a x a x a x = ϕ + + ϕ ϕ " n n ∈ Φ 二、线性最小二乘拟合 ⒈ 基本思想 给定 ，设x ,y的关系为
中国矿亚大医CHINAUNIVERSITY OF MININGANDTECHNOLOGY[8*2 =Z(S*(x)-y)21-0= minol2 = min Z(S(x,)- y,)22S(x)edS(x)eΦi=0称满足条件(2)的求函数 S*(x)=a;,(x)的方法为i=0线性最小二乘拟合S*(x)=a,p,(x)为最小二乘解=0S(x)=a,g,(x)为拟合函数,a,(j=0,1,…,n)为拟合系数j=0s*称为最小二乘解的最小偏差
CHINA UNIVERSITY OF MINING AND TECHNOLOGY 2 2 δ * ∑= = − m i i i S x y 0 2 ( * ( ) ) ∑= ∈ Φ = − m i i i S x S x y 0 2 ( ) min ( ( ) ) 2 2 ( ) min δ ∈ Φ = S x -(2) ∑ 为最小二乘解 = = n j j j S x a x 0 * * ( ) ϕ ( ) ( ) ( )为拟合函数 , ( 0 , 1 , , )为拟合系数 0 S x a x a j j n n j = ∑ j j = " = ϕ 2 δ * * 0 *( ) ( ) n j j j Sx a x ϕ = 称满足条件(2)的求函数 的方法为 = ∑ 线性最小二乘拟合 。 称为最小二乘解的最小偏差
中国矿亚大整CHINAUNIVERSITY OFMININGANDTECHNOLOGY2. 法方程组S(x)=a,p,(x)由j=0y=)2可知=S(x)a,p,(x)i=0i=01=0为拟合系数a(j=0,1,,n)的函数二次函数因此可假设y(ao,at,,an) =Z(Za,p,(x.)-y.)i=0j=0因此求最小二乘解转化为求y(a,a,.,a,)的最小值点a,a,.,a,的问题
CHINA UNIVERSITY OF MINING AND TECHNOLOGY ∑ ∑ = = = − m i i n j j j i a x y 0 2 0 ∑ ( ϕ ( ) ) = = − m i i i S x y 0 2 ( ( ) ) ⒉ 法方程组 2 2 δ ∑= = n j j j S x a x 0 由 ( ) ϕ ( ) 为拟合系数 a j ( j = 0 , 1 , " , n )的函数 可知 因此可假设 ( , , , ) 0 1 n ψ a a " a ∑ ∑ = = = − m i i n j j j i a x y 0 2 0 ( ϕ ( ) ) 因此求最小二乘解转化为 二次函数 01 01 (, , ) , , n n ψ aa a aa a 求 " " ∗ ∗ ∗ 的最小值点 的问题
中国矿亚大医CHINA UNIVERSITYOF MININGANDTECHNOLOGY由多元函数取极值的必要条件y(ao.aa,) - 0k=0,1...,nOakay=Z[2(Za,g,(x,)-y,)Pk(x,) = 0得aaki=0j=0Z(Za,p,(x,)g(x,)- y,px(x)] =0即i=0j=0mI2Zajp,(x,)p(x)-Zy,e(x)双i=0i=0 j=0
CHINA UNIVERSITY OF MINING AND TECHNOLOGY 由多元函数取极值的必要条件 0 ( , , , ) 0 1 = ∂ ∂ k n a ψ a a " a k = 0 , 1 , " , n [ 2 ( ( ) ) ( )] 0 0 k i m i i n j j j i ∑ ∑ a ϕ x y ϕ x = = = − a k ∂ ∂ψ 得 = 0 即 ∑∑ ∑ = = = = m i i k i m i k i n j j j i a x x y x 0 0 0 ϕ ( )ϕ ( ) ϕ ( ) [ ( ) ( ) ( )] 0 0 0 ∑ ∑ − = = = k i m i i n j j j i k i a ϕ x ϕ x y ϕ x
中国矿亚大鉴CHINA UNIVERSITY OF MININGAND TECHNOLOGY2Za,,(x):(x)=Zye:(x)i=0i=0j=0Z(g,(x,)p(x,)la, =Zy,pk(x,)j=0 1=0i=04k=0,1,.".,n即aZgo(x)p(x,)+aZo(x,)p(x,)+.+a,,(x,)p(x,)i=0i=0i=0mZyp(x)k =0,1,...,ni=0
CHINA UNIVERSITY OF MINING AND TECHNOLOGY ∑∑ ∑ = = = = m i i k i m i k i n j j j i a x x y x 0 0 0 ϕ ( )ϕ ( ) ϕ ( ) ∑ ∑ ∑ = = = = m i i k i n j k i j m i j i x x a y x 0 0 0 [ ϕ ( )ϕ ( )] ϕ ( ) k = 0 , 1 , " , n -(4) ∑ ∑ ∑ ∑ = = = = = + + + m i i k i k i m i k i n n i m i k i i m i i y x a x x a x x a x x 0 0 0 1 1 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ϕ ϕ ϕ ϕ ϕ " ϕ ϕ k = 0 , 1 , " , n 即