\uf0d8 离散数据的线性拟合 \uf0d8 数据拟合的线性模型 \uf0d8 超定方程组的最小二乘解 \uf0d8 离散数据的多项式拟合 第四章 数据拟合与函数逼近 已知数据表 x x 1 x 2 ·········· x m f(x) y 1 y 2 ·········· y m 离散数据的线性拟合 求拟合函数: \uf06a \uf06a (x) = a + b x a + b x 1 = y 1 a + b x 2 = y 2 ·················· a + b x m = y m r 1 =(a + b x 1 ) - y 1 r 2 =(a + b x 2 ) - y 2 ·················· r m =(a + b x m ) - y m 2/18 残差的范数表示: | ) ( | max || ||1k km ky bx a r \uf02d \uf02d \uf02b \uf02b \uf03d \uf03d\uf0a3 \uf0a3 \uf0a3 \uf0a3\uf0a5 \uf0a5\uf0e5 \uf0e5\uf03d \uf03d\uf02d \uf02d \uf02b \uf02b \uf03d \uf03dmkk ky bx a r11| ) ( | || ||\uf0e5 \uf0e5\uf03d \uf03d\uf02d \uf02d \uf02b \uf02b \uf03d \uf03dmkk ky bx a r122] ) [( || ||最小二乘法: 确定拟合函数使残差2- 范数最小 残差: r(x k )= (a + bx k ) – y k ( k = 1,2,···,m) 3/18 \uf0e5 \uf0e5\uf03d \uf03d\uf02d \uf02d \uf02b \uf02b \uf03d \uf03d \uf03d \uf03dmkk ky bx a r b a S12 22] ) [( || || ) , (0 21\uf03d \uf02d \uf02b\uf0e5\uf03dmkk ky bx a ) (\uf0e5\uf03d\uf03d \uf02d \uf02bmkk k kx y bx a10 2 ) (\uf0ef\uf0ef\uf0ee\uf0ef\uf0ef\uf0ed\uf0ec\uf03d \uf02b\uf03d \uf02b\uf0e5 \uf0e5 \uf0e5\uf0e5 \uf0e5\uf03d \uf03d \uf03d\uf03d \uf03dmkk kmkkmkkmkkmkky x b x a xy b x ma1 1211 10 \uf03d\uf0b6\uf0b6aS0 \uf03d\uf0b6\uf0b6bS求 求 a, b 使 S(a, b)= min 4/18 \uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0e5 \uf0e5 \uf0e5 \uf0e5\uf0e5 \uf0e5\uf03d \uf03d \uf03d \uf03d\uf03d \uf03dmmmkkmkkmkkxxxx x xx xx m1111 1 1212 11211\uf04d \uf04d \uf04c\uf04c\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0e5 \uf0e5\uf0e5 \uf0e5\uf03d \uf03d\uf03d \uf03dmmmkk kmkkyyyx x xy xy\uf04d \uf04c\uf04c212 1111 1 1方程组 系数矩阵 方程组 右端项 5/18 例 例1. 已知实验数据如下, 求拟合曲线。 解 解: 设拟合曲线方程为: \uf06a \uf06a (x)= a + b x x 1 2 3 4 5 f(x) 4 4.5 6 8 9 5a + 15b = 31.5 15a +55b =108 a =2.25, b= 1.35 \uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e99865 . 445 14 13 12 11 1ba6/18 1 2 3 4 5246810||r || 2 = 0.7583 残差向量: \uf06a \uf06a (1) -4= - -0.40 \uf06a \uf06a (2) -4.5= 0.45 \uf06a \uf06a (3) -6= 0.30 \uf06a \uf06a (4) -8=- - 0.35 \uf06a \uf06a (5) -9= 0 \uf06a \uf06a (x)= 2.25+1.35 x 7/18 多项式拟合函数: \uf06a \uf06a(x)=a 0 + a 1 x + a 2 x 2 + ······ + a n x n 数据拟合的线性模型 给定数据表 x x 1 x 2 ·········· x m f(x) y 1 y 2 ·········· y m 数据拟合的线性模型 \uf06a \uf06a(x)=a 0 \uf06a \uf06a 0 (x) + a 1 \uf06a \uf06a 1 (x) + ······ + a n \uf06a \uf06a n (x) 例如: [\uf06a \uf06a 0 (x) , \uf06a \uf06a 1 (x) , ··· , \uf06a \uf06a n (x)]=[1, x, ···, x n ] [\uf06a \uf06a 0 (x) , \uf06a \uf06a 1 (x) , ··· , \uf06a \uf06a n (x)]=[1, cos x, ···, cos nx] 8/18 ) ( ) ( x a xjnjj \uf06a \uf06a\uf06a \uf06a\uf0e5\uf03d\uf03d0拟合函数: 拟合数据: f(x j )= y j , ( j = 1,2,3,·····, m ) 超定方程组: 1 1 )( y x \uf03d \uf03d \uf06a \uf06a2 2 )( y x \uf03d \uf03d \uf06a \uf06a3 3 )( y x \uf03d \uf03d \uf06a \uf06am my x \uf03d \uf03d ) ( \uf06a \uf06a·············· \uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9m n m n m mnnyyyaaax x xx x xx x x\uf04d \uf04d\uf04c\uf04c \uf04c \uf04c \uf04c\uf04c\uf04c21212 12 2 2 2 11 1 2 1 1) ( ) ( ) () ( ) ( ) () ( ) ( ) (\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a9/18 min ] ) ( [ ] ) ( [120 12\uf03d \uf03d \uf02d \uf02d \uf03d \uf03d \uf02d \uf02d\uf0e5 \uf0e5 \uf0e5 \uf0e5 \uf0e5 \uf0e5\uf03d \uf03d \uf03d \uf03d \uf03d \uf03dmii injjmii iy x a y x \uf06a \uf06a \uf06a \uf06a使得 \uf0e5 \uf0e5\uf03d \uf03d\uf02d \uf03dminji i j j ny x a a a a a S1 022 1 0] ) ( [ ) , , , , ( \uf06a \uf06a \uf04c 记 记 求 求S(a 0 , a 1 ,…, a n ) 最小值 , 得方程组 ) ( ) ( x a xjnjj \uf06a \uf06a\uf06a \uf06a\uf0e5\uf03d\uf03d0求 求 ( n < m ) 令 令 0 \uf03d\uf0b6\uf0b6kaS\uf0e5 \uf0e5\uf03d \uf03d\uf03d \uf02dnji i j jmii ky x a x0 10 ] ) ( [ ) ( \uf06a \uf06a \uf06a \uf06a( k = 0,1,2,···, n) 10/18 , ) ( )] ( ) ( [\uf0e5 \uf0e5 \uf0e5\uf03d \uf03d \uf03d\uf03dmii i knjmij i j i ky x a x x1 0 1\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a记 记 Tm j j j jx x x )] ( ) ( ) ( [ \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf04c\uf0722 1\uf03dTmy y y y ] [ \uf04c\uf0722 1\uf03djTk j k\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf072 \uf072 \uf072 \uf072\uf03d ) , (y yTk k\uf072 \uf072 \uf072 \uf072\uf06a \uf06a \uf06a \uf06a \uf03d ) , (称为正规方程组( 或法方程) 则方程组可写为 ) , ( ) , ( y aknjj j k\uf072 \uf072 \uf072 \uf072\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf03d\uf0e5\uf03d0 k = 0, 1, ······,n 11/18 \uf05b \uf05b \uf05d \uf05dnTnTTn n n nnn\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf072\uf04c\uf072 \uf072\uf072\uf04d\uf072\uf072\uf072 \uf072\uf04c\uf072 \uf072 \uf072 \uf072\uf04c \uf04c \uf04c \uf04c\uf072 \uf072\uf04c\uf072 \uf072 \uf072 \uf072\uf072 \uf072\uf04c\uf072 \uf072 \uf072 \uf0721 0101 01 1 1 0 10 1 0 0 0) , ( ) , ( ) , () , ( ) , ( ) , () , ( ) , ( ) , (\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9) , () , () , () , ( ) , ( ) , () , ( ) , ( ) , () , ( ) , ( ) , (10101 01 1 1 0 10 1 0 0 0yyyaaan n n n n nnn\uf072 \uf072\uf04d\uf072 \uf072\uf072 \uf072\uf04d\uf072 \uf072\uf04c\uf072 \uf072 \uf072 \uf072\uf04c \uf04c \uf04c \uf04c\uf072 \uf072\uf04c\uf072 \uf072 \uf072 \uf072\uf072 \uf072\uf04c\uf072 \uf072 \uf072 \uf072\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06ayyyyTnTTn\uf072\uf072\uf04d\uf072\uf072\uf072 \uf072\uf04d\uf072 \uf072\uf072 \uf072\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a1010) , () , () , (系数矩阵 方程组右端项 12/18 已知数据如下, 求二次拟合P(x)=a 0 +a 1 x+a 2 x 2 x 1 2 3 4 5 f(x) 4 4.5 6 8 9 解 解: 将数据点代入, 得 \uf0fa\uf0fa\uf0fa\uf0fa\uf0fa\uf0fa\uf0fb\uf0f9\uf0ea\uf0ea\uf0ea\uf0ea\uf0ea\uf0ea\uf0eb\uf0e9\uf03d\uf0fa\uf0fa\uf0fa\uf0fb\uf0f9\uf0ea\uf0ea\uf0ea\uf0eb\uf0e9\uf0fa\uf0fa\uf0fa\uf0fa\uf0fa\uf0fa\uf0fb\uf0f9\uf0ea\uf0ea\uf0ea\uf0ea\uf0ea\uf0ea\uf0eb\uf0e99865 4425 5 116 4 19 3 14 2 11 1 1210.aaaa 0 + a 1 + a 2 = 4 a 0 + 2a 1 + 4a 2 = 4.5 ·················· a 0 + 5a 1 + 25a 2 = 9 13/18 得 得 P(x)=3+0.7071x + 0.1071x 2 1 2 3 4 5345678910二次拟合误差: || r || 2 = 0.6437 \uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e94291085 . 31979 225 55225 55 1555 15 5210aaaa 0 =3, a 1 =0.7071, a 2 =0.1071 线性拟合误差: ||r || 2 = 0.7583 14/18 \uf0e5\uf03d\uf02d \uf03d \uf03dmjjxmx x x11 011 ) ( , ) ( \uf06a \uf06a \uf06a \uf06a 取 取 求拟合函数: \uf06a \uf06a(x)=a 0 \uf06a \uf06a 0 (x) + a 1 \uf06a \uf06a 1 (x) 给定数表 x x 1 x 2 ·········· x m f(x) y 1 y 2 ·········· y m \uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9mm myyyaax xx xx x\uf04d\uf04d\uf04d\uf04d\uf04d21101 02 1 2 01 1 1 0) ( ) () ( ) () ( ) (\uf06a \uf06a \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a\uf05b \uf05b \uf05d \uf05d y a\uf072 \uf072 \uf072 \uf072\uf03d \uf03d1 0\uf06a \uf06a \uf06a \uf06a15/18 \uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9) , () , () , ( ) , () , ( ) , (10101 1 0 11 0 0 0yyaa\uf072 \uf072\uf072 \uf072\uf072 \uf072 \uf072 \uf072\uf072 \uf072 \uf072 \uf072\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a \uf06a\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf03d \uf03d\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9\uf0fa \uf0fa\uf0fb \uf0fb\uf0f9 \uf0f9\uf0ea \uf0ea\uf0eb \uf0eb\uf0e9 \uf0e9) , () , () , () , (10101 10 0yyaa\uf072 \uf072\uf072 \uf072\uf072 \uf072\uf072 \uf072\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a) , () , (0 000\uf06a \uf06a \uf06a \uf06a\uf06a \uf06a\uf072 \uf072\uf072 \uf072ya \uf03d \uf03d) , () , (1 111\uf06a \uf06a \uf06a \uf06a\uf06a \uf06a\uf072 \uf072\uf072 \uf072ya \uf03d \uf03d) () , () , () () , () , () (11 1100 00xyxyx \uf06a \uf06a\uf06a \uf06a \uf06a \uf06a\uf06a \uf06a\uf06a \uf06a\uf06a \uf06a \uf06a \uf06a\uf06a \uf06a\uf06a \uf06a\uf072 \uf072\uf072 \uf072\uf072 \uf072\uf072 \uf072\uf02b \uf02b \uf03d \uf03d16/18 MATLAB 中的多项式拟合命令 P= polyfit (X,Y,n) 求出( 最小二乘意义下)n 次拟合多项式 P(x)=a 0 x n + a 1 x n-1 + ······ + a n-1 x + a n 计算结果为系数: P=[ a 0 , a 1 , ······ , a n-1 , a n ] 多项式求值命令: y1=polyval(P, x) 其中,P 是n 次多项式的系数,x 是自变量的值,y1是多项式在x 处的值 17/18 x=-5:5;y=1./(1+x.^2); t=-5:.05:5;y0=1./(1+t.^2); p=polyfit(x,y,10);y1=polyval(p,t); plot(t,y0,x,y,'o',t,y1,'.') -5 -4 -3 -2 -1 0 1 2 3 4 5-0.500.511.52多项式拟合命令实现插值 18/18