标题: ATLAB 脚本进行最小二乘法拟合 [打印本页] 作者: 2744557306 时间: 2023-12-31 16:12 标题: ATLAB 脚本进行最小二乘法拟合 这是一个 MATLAB 脚本，用于进行最小二乘法拟合。脚本首先要求用户输入已知点的 x 和 y 坐标，然后输入拟合的多项式次数 n。脚本使用最小二乘法拟合数据，并绘制了原始数据点和拟合曲线的图表。以下是对代码的主要部分的解释：, v( x+ i C) R
function fp = fitpt()- Z* y( A1 Z! X, O. A( `2 n+ G
% 最小二乘2 c' o3 |' s7 J! s1 [' F
% 基取 {1, x, ...}2 O4 Z w% Y7 I/ g* O7 x( v/ v( I
% fitpt.m+ T3 K2 S, ?/ L
`+ m2 Y6 t9 f0 f- y) y" M % 默认算例为课本：P65，例3.2% o/ F$ [: w) Z+ h0 U( H! P% n1 v
% x = [0,1,2,3,4,5,6,7]& J5 v! Q7 K8 h8 j8 x: z i
% y = [3.95,6.82,9.78,12.91,15.74,19.26,21.73,24.07] {" Y+ U. h6 v! I7 A % 结果：P(x) = 4.005 + 2.936x 平方误差=0.6162 ) r1 x; j# z( {$ w . p$ H3 w7 H, {9 A4 E % MatLab函数：polyfit(x, y, n)) D q+ c3 a; h) E
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s = input('<最小二乘>\n输入已知点的x坐标：（回车表示[0,1,2,3,4,5,6,7]）\n', 's');+ n4 c7 E5 J5 Y! l; f
if isempty(s) - T1 M, Q/ T ?' U( {3 D3 Q$ q s = '[0,1,2,3,4,5,6,7]'; 6 m& o9 ^1 D; u9 ~7 E7 Q; m' s else 9 X8 R" z9 P' A e& E if (s(1) ~= '[') - Q4 z: u6 _4 W2 { s = strcat('[', s);( t: p! s& _7 ~# V; h5 R
s = strcat(s, ']');& u' a2 I& l c! w9 w, f8 | m$ X
end6 u. ?% Q A7 N% C" u6 N
end$ A6 E) l. e. X; i& G) o+ l
x = sym(s); / y" l. X, @3 ` & c* J) ?5 E9 A$ ~6 I s = input('输入已知点的y坐标：（回车表示[3.95,6.82,9.78,12.91,15.74,19.26,21.73,24.07]）\n', 's');$ D/ w: m2 v; q) i* o; Y
if isempty(s) 2 s! E7 d. {6 B* O9 s- e s = '[3.95,6.82,9.78,12.91,15.74,19.26,21.73,24.07]'; - C' w6 J3 ]' f% `' Q else / T0 g" @, ^0 c9 k: J if (s(1) ~= '[') ) j: T- n6 ~1 H. R s = strcat('[', s); 1 g/ N4 V8 n6 F( b$ X4 | s = strcat(s, ']'); " ]7 H, c; B& Z0 K end2 I% E! C) A) H/ S4 m3 I ?
end1 ?1 n h; T5 L% s9 m" e
y = sym(s);0 h: D' p% i2 r$ ^6 O, @
sz = size(x);; e9 b w: z1 t6 ]% Z3 X
sz = sz(2); 9 Z( E- u0 d! A2 C. X n = input('输入多项式次数n：'); # P; C7 U9 H/ Q if (n + 1 > sz) ) d( `* K$ R+ P3 j- V/ \ n = input('多项式次数需要小于已知点个数，请重新输入n：'); 1 q9 j/ ~7 K% H# ^, s5 s7 j/ l# } end - g6 E7 x9 h' f! f4 G/ T! O9 X if (n + 1 > sz) ; v: m8 w% E: K: @+ z error('多项式次数不能小于已知点个数！');& X( E f* x' f
end * l$ |0 v! _1 M0 \+ O6 K; @; i fp = s_fitpt_p(x, y, n); # C* B. n; B+ W/ w& M9 s) D+ r$ G i+ Q) S% m2 ]) h
% 绘制原始数据点和拟合曲线/ I0 O' a: `9 P: [ i/ _1 {9 p q+ l
plot(double(x), double(y), 'r*')9 W: U/ \" U& ]2 N5 X& D
hold on6 M0 T6 A+ s# B" Z
a = double(x(1));7 V" \/ x" O8 L" ^
b = double(x(sz));9 V0 f* a" J% {# p4 [7 S* ]# b
x = a:abs(b - 1)/100:b;( H4 }1 Y- D$ W4 x( h7 |, N
y = subs(fp, x);$ S/ n0 \8 @2 Y; l- S5 R N
plot(x, y)# i6 b' |! L6 A- d" v' } B
end6 N$ [4 H) Z0 T4 S! ~1 l+ z
; u# G1 ?9 o0 [. {6 n- p. G%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ( q/ I! V* g3 s: c$ P 9 E+ Y* U1 p' l) D, k+ ~function f = s_fitpt_p(x, y, n) ) r# |1 L: ]( f) s4 ] % 用 n 次多项式实现的最小二乘法" j0 i M) k7 q. P: n: z( M, ]6 z
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sz = size(x);& l) b. J) G- Y2 Z
sz = sz(2);# `) @/ j# r! S. |$ W% x
A = zeros(sz, n + 1); & ^: K5 }7 n' X0 c8 H v = vh(n); U# v& Z8 u1 b for i = 1:sz0 C7 L; w0 v) e9 q9 E
A(i, = subs(v, double(x(i))); . G( I U) m& A& c. G end G! t2 X- j3 }3 V$ @+ G5 ~0 \
f = linsolve(A' * A, A' * y');4 d, Z% A* o; J
f = vpa(f, 4);# m+ K% g7 b( F! Y7 T0 h+ e
f = v * f; ' z+ m; q& |% Dend ' k- H1 O/ J% E$ K 5 ?3 M% G: G% S; }8 V. Z3 k%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%7 n6 x1 m4 m( D
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function v = vh(n) % J* Y f- N5 u7 z. M % Create vector in horizontal style, such as 5 c+ \9 r0 \2 e6 Q9 K4 g: D % v = [1, x, x^2, ..., x^n]" j9 m6 D8 P2 A. H
1 `( T1 H Q4 y if (n < 0 || n > 9)0 V& X i$ F+ j( P# W& S
error('Make sure ''n'' is in range of [0, 9]') * A0 k- J* m& T4 Y. N0 a end , ?6 @: S/ `) W& g; u s = '';+ W) L( e* Y4 ]4 B9 S; o' o
for i = 0:n6 ?" V6 `) u- n7 j) Y# T
s = strcat(s, ',x^');2 K2 D5 C6 _) Y1 o
s = strcat(s, num2str(i)); . S" ?" w4 C5 b/ l' Q# X+ F4 D end 7 o- Y+ r! o4 D4 @8 {; B/ b& V: z' k s(1) = '[';3 n# i j6 {) n- H# F; Y
sz = size(s);5 a6 K- \0 R; T3 Q, @
s(sz(2) + 1) = ']';7 z9 g4 C" x- G' ?$ s& c% W
5 `& ] q0 e+ j9 c7 G0 Z v = simplify(sym(s)); - V4 t1 \ y5 ^! {5 R' J# Aend # @1 T/ F4 z. A% Q- l; S5 G7 ^: Q3 ~. I7 j. Y/ {3 n
这个脚本首先获取用户输入的已知点的 x 和 y 坐标，然后使用最小二乘法进行拟合。最后，脚本绘制了原始数据点和拟合曲线的图表。 9 e1 n+ j& X3 H/ _- Q) q( |" L y& T% z; ~% Z: \" g
% I* ?9 s8 p3 ] v: i2 E* I
= subs(v, double(x(i)));