最小二乘法拟合曲线的VB程序源代码
vba 最小二乘法
Function AxA T(A() As Double, m, n) Dim AA T() As DoubleDim i, j, k As IntegerReDim AA T(1 To m, 1 To m)For i = 1 To mFor k = 1 To mAA T(i, k) = 0For j = 1 To nAA T(i, k) = A(j, i) * A(j, k) + AA T(i, k)NextNextNextFor i = 1 To mFor k = 1 To mA(i, k) = AA T(i, k)NextNextEnd FunctionFunction HLS(A() As Double, m)Dim i, j, k As IntegerDim rt As DoubleFor i = 1 To mFor j = 2 To mA(1, i) = A(j, i) + A(1, i)NextNextFor k = 1 To m - 1For i = k + 1 To mFor j = m To k Step -1A(i, j) = A(i, j) - A(k, j) * A(i, k) / A(k, k) NextNextNextrt = 1For i = 1 To mrt = rt * A(i, i)NextHLS = rtEnd FunctionFunction QiuBanSuiZhen(A() As Double, m)Dim bsz() As Double '伴随阵Dim yzs() As Double '代数余子式ReDim bsz(1 To m, 1 To m) As Double '伴随阵ReDim yzs(1 To m - 1, 1 To m - 1) As Double '代数余子式Dim i, j, r, l As IntegerFor i = 1 To mFor j = 1 To mFor r = 1 To mFor l = 1 To mIf r < i ThenIf l < j Thenyzs(r, l) = A(r, l)End IfIf l > j Thenyzs(r, l - 1) = A(r, l)End IfEnd IfIf r > i ThenIf l < j Thenyzs(r - 1, l) = A(r, l)End IfIf l > j Thenyzs(r - 1, l - 1) = A(r, l)End IfEnd IfNextNextbsz(j, i) = 1For r = 1 To i + jbsz(j, i) = bsz(j, i) * (-1)Nextbsz(j, i) = bsz(j, i) * HLS(yzs, m - 1)Cells(j + 8, i) = bsz(j, i)NextNextFor i = 1 To mFor j = 1 To mA(i, j) = bsz(i, j)NextNextEnd FunctionSub testAxA T()Dim r, l, mm, nn As IntegerDim A() As Double '原始参数阵Dim x() As Double '多项式的系数Dim AA T() As DoubleDim AA T_ni() As DoubleDim TempA() As DoubleDim y() As DoubleDim A Ty() As DoubleDim HlsZhi As Double'一次多项式X=a1*x+b1*y+c1'一次多项式Y=a2*x+b2*y+c2'mm=3'nn>=3'//////////////////////////////////////'二次多项式Y=a1*x*x+b1*x*y+c1*y*y+d1*x+e1*y+f1 '二次多项式X=a2*x*x+b2*x*y+c2*y*y+d2*x+e2*y+f2 'm=6'nn>=6mm = 3nn = 4ReDim A(1 To nn, 1 To mm)ReDim AA T(1 To nn, 1 To mm)ReDim AA T_ni(1 To nn, 1 To mm)ReDim TempA(1 To nn, 1 To mm)ReDim y(1 To nn)ReDim A Ty(1 To mm)ReDim x(1 To mm)'//////////////////////////////////////////'参数AFor r = 1 To nnFor l = 1 To mmA(r, l) = Cells(r, l)NextNext'///////////////////////////'y赋值For r = 1 To nny(r) = Cells(r, 4)Next'/////////////////////////////////////////'计算A乘A的转置For r = 1 To nnFor l = 1 To mmAA T(r, l) = A(r, l)NextNextCall AxA T(AA T, mm, nn)'///////////////////////'/////////////////////////'计算A乘A的转置的行列式For r = 1 To nnFor l = 1 To mmTempA(r, l) = AA T(r, l)NextNextHlsZhi = HLS(TempA, mm)'//////////////////////////////////'计算A乘A的转置的逆For r = 1 To nnFor l = 1 To mmAA T_ni(r, l) = AA T(r, l)NextNextCall QiuBanSuiZhen(AA T_ni, mm) For r = 1 To mmFor l = 1 To mmAA T_ni(r, l) = AA T_ni(r, l) / HlsZhi NextNext'/////////////////////'计算A T 乘yFor r = 1 To mmA Ty(r) = 0For l = 1 To nnA Ty(r) = A Ty(r) + A(l, r) * y(l) NextNext'////////////////////////////////////'//////////////////////////////////////'计算AA T的逆乘A TY结果就是x() For r = 1 To mmx(r) = 0For l = 1 To mmx(r) = x(r) + AA T_ni(r, l) * A Ty(l)NextNextEnd Sub。
最小二乘法代码
package com.song.badmath;public class LeastSquares {/*** 由最小二乘法拟合最逼近二维曲线* @param x x散点坐标* @param y y散点坐标* @return拟合的曲线系数，曲线次数为 length-1*/public static double[] leastSquares(double x[],double y[]){ if(x.length!=y.length)return null;int m = 1;double c[] = leastSquares(x,y,m);double backError = leastSquaresError(x,y,c);System.out.print("m:"+m+"->ERROR:"+backError);while(true){m++;double bc[] = leastSquares(x,y,m);double error = leastSquaresError(x,y,bc);if(error>=backError){System.out.println("=>final");break;}else{c = bc;backError = error;System.out.println("=>next");}System.out.print("m:"+m+"->ERROR:"+backError);}return c;}public static double[] leastSquares(double x[],double y[],double max_error){if(x.length!=y.length)return null;int m = 1;double c[] = leastSquares(x,y,m);double backError = leastSquaresError(x,y,c);System.out.print("m:"+m+"->ERROR:"+backError);while(true){m++;double bc[] = leastSquares(x,y,m);double error = leastSquaresError(x,y,bc);if(error>=backError||backError<max_error){System.out.println("=>final");break;}else{c = bc;backError = error;System.out.println("=>next");}System.out.print("m:"+m+"->ERROR:"+backError);}return c;}* 由最小二乘法拟合m次二维曲线* @param x x散点坐标* @param y y散点坐标* @param m 曲线次数* @return拟合的曲线系数*/public static double[] leastSquares(double x[],double y[],int m){ if(x.length!=y.length)return null;double A[][] = new double[m+1][m+1];double B[] = new double[m+1];double sx[] = new double[2*m+1];//sx[i] = Sx^idouble sy[] = new double[m+1];//sy[i] = Sx^iydouble px[][] = new double[x.length][2*m+1];//px[i][j] = x[i]^j double pxy[][] = new double[y.length][m+1];//pxy[i][j] = x[i]^j*y[i] //计算并存储x[i]^jfor(int i=0;i<px.length;i++){px[i][0] = 1;for(int j=1;j<px[i].length;j++)px[i][j] = px[i][j-1]*x[i];}//计算并存储x[i]^j*y[i]for(int i=0;i<pxy.length;i++){for(int j=0;j<pxy[i].length;j++)pxy[i][j] = px[i][j]*y[i];}//计算Sx^ifor(int i=1;i<sx.length;i++){sx[i] = 0;for(int j=0;j<px.length;j++){sx[i] += px[j][i];}}sx[0] = x.length;//计算Sx^iyfor(int i=0;i<sy.length;i++){sy[i] = 0;for(int j=0;j<pxy.length;j++){sy[i] += pxy[j][i];}}//计算矩阵Afor(int i=0;i<A.length;i++){for(int j=0;j<A[i].length;j++){A[i][j] = sx[i+j];}}//计算矩阵Bfor(int i=0;i<B.length;i++){B[i] = sy[i];return Matrix.linearEquations(A,B);}public static String polynomialToString(double c[]){String strPolynomial = "y=";for(int i=0;i<c.length;i++){if(i>0){if(c[i]==1){strPolynomial += "+x^"+i;}else{strPolynomial += "+"+c[i]+"x^"+i;}}else{strPolynomial += c[i];}}return strPolynomial;}public static String polynomialToStringNoZero(double c[]){ String strPolynomial = "y=";int j = 0;for(int i=0;i<c.length;i++){if(c[i]==0)continue;if(j>0){if(c[i]==1){strPolynomial += "+x^"+i;}else{strPolynomial += "+"+c[i]+"x^"+i;}}else if(i==0){strPolynomial += c[i];}else{if(c[i]==1){strPolynomial += "x^"+i;}else{strPolynomial += c[i]+"x^"+i;}}j++;}return strPolynomial;}public static double leastSquaresError(double x[],double y[],double a[]){double px[][] = new double[x.length][a.length];//px[i][j] = x[i]^j for(int i=0;i<px.length;i++){px[i][0] = 1;for(int j=1;j<px[i].length;j++)px[i][j] = px[i][j-1]*x[i];}double error = 0;for(int i=0;i<x.length;i++){double f = 0;for(int j=0;j<a.length;j++){f += px[i][j]*a[j];}double dy = f-y[i];error += dy*dy;}return error;}public static void test(){double x[] = {1,2,3};double y[] = {2.2f,6.2f,12.2f};double c[] = LeastSquares.leastSquares(x, y);System.out.println(LeastSquares.polynomialToString(c));System.out.println(LeastSquares.polynomialToStringNoZero(c));}}。
Python实现曲线拟合的最小二乘法
Python实现曲线拟合的最⼩⼆乘法本⽂实例为⼤家分享了Python曲线拟合的最⼩⼆乘法，供⼤家参考，具体内容如下模块导⼊import numpy as npimport gaosi as gs代码"""本函数通过创建增⼴矩阵，并调⽤⾼斯列主元消去法模块进⾏求解。
"""import numpy as npimport gaosi as gsshape = int(input('请输⼊拟合函数的次数:'))x = np.array([0.6,1.3,1.64,1.8,2.1,2.3,2.44])y = np.array([7.05,12.2,14.4,15.2,17.4,19.6,20.2])data = []for i in range(shape*2+1):if i != 0:data.append(np.sum(x**i))else:data.append(len(x))b = []for i in range(shape+1):if i != 0:b.append(np.sum(y*x**i))else:b.append(np.sum(y))b = np.array(b).reshape(shape+1,1)n = np.zeros([shape+1,shape+1])for i in range(shape+1):for j in range(shape+1):n[i][j] = data[i+j]result = gs.Handle(n,b)if not result:print('增⼴矩阵求解失败！')exit()fun='f(x) = 'for i in range(len(result)):if type(result[i]) == type(''):print('存在⾃由变量！')fun = fun + str(result[i])elif i == 0:fun = fun + '{:.3f}'.format(result[i])else:fun = fun + '+{0:.3f}*x^{1}'.format(result[i],i)print('求得{0}次拟合函数为：'.format(shape))print(fun)⾼斯模块# 导⼊ numpy 模块import numpy as np# ⾏交换def swap_row(matrix, i, j):m, n = matrix.shapeif i >= m or j >= m:print('错误! : ⾏交换超出范围 ...')else:matrix[i],matrix[j] = matrix[j].copy(),matrix[i].copy()return matrix# 变成阶梯矩阵def matrix_change(matrix):m, n = matrix.shapemain_factor = []main_col = main_row = 0while main_row < m and main_col < n:# 选择进⾏下⼀次主元查找的列main_row = len(main_factor)# 寻找列中⾮零的元素not_zeros = np.where(abs(matrix[main_row:,main_col]) > 0)[0]# 如果该列向下全部数据为零，则直接跳过列if len(not_zeros) == 0:main_col += 1continueelse:# 将主元列号保存在列表中main_factor.append(main_col)# 将第⼀个⾮零⾏交换⾄最前if not_zeros[0] != [0]:matrix = swap_row(matrix,main_row,main_row+not_zeros[0])# 将该列主元下⽅所有元素变为零if main_row < m-1:for k in range(main_row+1,m):a = float(matrix[k, main_col] / matrix[main_row, main_col])matrix[k] = matrix[k] - matrix[main_row] * matrix[k, main_col] / matrix[main_row, main_col] main_col += 1return matrix,main_factor# 回代求解def back_solve(matrix, main_factor):# 判断是否有解if len(main_factor) == 0:print('主元错误,⽆主元！ ...')return Nonem, n = matrix.shapeif main_factor[-1] == n - 1:print('⽆解！ ...')return None# 把所有的主元元素上⽅的元素变成0for i in range(len(main_factor) - 1, -1, -1):factor = matrix[i, main_factor[i]]matrix[i] = matrix[i] / float(factor)for j in range(i):times = matrix[j, main_factor[i]]matrix[j] = matrix[j] - float(times) * matrix[i]# 先看看结果对不对return matrix# 结果打印def print_result(matrix, main_factor):if matrix is None:print('阶梯矩阵为空！ ...')return Nonem, n = matrix.shaperesult = [''] * (n - 1)main_factor = list(main_factor)for i in range(n - 1):# 如果不是主元列，则为⾃由变量if i not in main_factor:result[i] = '(free var)'# 否则是主元变量，从对应的⾏，将主元变量表⽰成⾮主元变量的线性组合else:# row_of_main表⽰该主元所在的⾏row_of_main = main_factor.index(i)result[i] = matrix[row_of_main, -1]return result# 得到简化的阶梯矩阵和主元列def Handle(matrix_a, matrix_b):# 拼接成增⼴矩阵matrix_01 = np.hstack([matrix_a, matrix_b])matrix_01, main_factor = matrix_change(matrix_01)matrix_01 = back_solve(matrix_01, main_factor)result = print_result(matrix_01, main_factor)return resultif __name__ == '__main__':a = np.array([[2, 1, 1], [3, 1, 2], [1, 2, 2]],dtype=float)b = np.array([[4],[6],[5]],dtype=float)a = Handle(a, b)以上就是本⽂的全部内容，希望对⼤家的学习有所帮助，也希望⼤家多多⽀持。
VB写的最小二乘法曲线拟合
Option ExplicitDim x() As Double, y() As DoubleDim A(20, 20) As Double, M As Double, B() As Double '最多取20次的拟合Dim N As Double, I As Double, j As DoubleDim xiaoA() As DoubleDim Xmin As Double, Xmax As DoubleDim Ymin As Double, Ymax As DoubleDim X0pos As Double, Y0pos As DoubleDim xmaxpos As Double, ymaxpos As DoubleDim xstep As Double, ystep As DoubleDim xl As Double, yl As DoubleDim xbc As Double, ybc As DoubleDim bc As DoubleDim Xh As DoublePrivate Sub HuaZuoBiao(x() As Double, y() As Double)ReDim xpos(I) As DoubleReDim ypos(I) As DoubleReDim x(I), y(I)X0pos = Width * 0.25 '坐标原点最左点Y0pos = Height * 0.75 '坐标原点最低点xmaxpos = Width * 0.85 '坐标最右点ymaxpos = Height * 0.15 '坐标最高点xstep = (xmaxpos - X0pos) / (Xmax - Xmin) '对应X轴上单位长度代表的屏幕宽度值ystep = (ymaxpos - Y0pos) / (Ymax - Ymin) '对应Y轴上单位长度代表的屏幕高度值'在屏幕上画直角坐标系ForeColor = vbBlueLine (Width * 0.1, Y0pos)-(Width * 0.9, Y0pos) '画X坐标轴,从左10%，到右的90%处Line (X0pos, Height * 0.1)-(X0pos, Height * 0.9) '画y坐标轴,从上10%，到下的90%处Font.Size = 20 '指定X轴，Y轴标志的字体大小CurrentX = Width * 0.9CurrentY = Y0pos + 100Print "X" '在横线上画X轴标志'在横线上画X轴箭头标志CurrentX = Width * 0.9CurrentY = Y0posLine (CurrentX - 200, CurrentY - 50)-(CurrentX, CurrentY)Line (CurrentX, CurrentY)-(CurrentX - 200, CurrentY + 50)CurrentX = X0pos - 500CurrentY = Height * 0.1Print "y" '在纵线上画Y轴标志'在纵线上画Y轴箭头标志CurrentX = X0posCurrentY = Height * 0.1Line (CurrentX - 50, CurrentY + 200)-(CurrentX, CurrentY)Line (CurrentX, CurrentY)-(CurrentX + 50, CurrentY + 200)CurrentX = X0pos + 200 '此为Y轴左边500绝对坐标处CurrentY = Y0pos + 400 '取当前Y轴上的相对坐标值Print "f=f(x)" '在Y轴左边500绝对坐标处对应显示Y轴相对坐标刻度值xl = Xmax - Xminyl = Ymax - YminIf xl < 0.01 Thenxbc = 0.001ElseIf xl <= 0.1 Thenxbc = 0.01ElseIf xl <= 2 Thenxbc = 0.1ElseIf xl <= 20 Thenxbc = 1ElseIf xl <= 120 Thenxbc = 10ElseIf xl <= 1000 Thenxbc = 100ElseIf xl <= 10000 Thenxbc = 1000Elsexbc = 10000End IfIf yl < 0.01 Thenybc = 0.001ElseIf yl <= 0.1 Thenybc = 0.01ElseIf yl <= 2 Thenybc = 0.1ElseIf yl <= 20 Thenybc = 1ElseIf yl <= 120 Thenybc = 10ElseIf yl <= 1000 Thenybc = 100ElseIf yl <= 10000 Thenybc = 1000Elseybc = 10000End IfFor bc = Xmin To Xmax Step xbcIf bc <= Xmax Thenx(j) = bc 'X轴上的相对坐标值xpos(j) = X0pos + (x(j) - Xmin) * xstepLine (xpos(j), Y0pos)-(xpos(j), ymaxpos), vbRed ' 画垂直于X轴的刻度线，只画了100个绝对尺寸ElseEnd IfFont.Size = 10 '指定X轴，Y轴坐标刻度值的字体大小CurrentX = xpos(j) - 200 '取当前X轴上的相对坐标值CurrentY = Y0pos + 100 '此为X轴下方100绝对坐标处Print x(j) '在X轴下方100绝对坐标处对应显示X轴相对坐标刻度值Next bcFor bc = Ymin To Ymax Step ybcIf bc <= Ymax Theny(j) = bc 'X轴上的相对坐标值ypos(j) = Y0pos + (y(j) - Ymin) * ystepLine (X0pos, ypos(j))-(xmaxpos, ypos(j)), vbRed ' 画垂直于X轴的刻度线，只画了100个绝对尺寸ElseEnd IfFont.Size = 10 '指定X轴，Y轴坐标刻度值的字体大小CurrentX = X0pos - 500 '取当前X轴上的相对坐标值CurrentY = ypos(j) - 100 '此为X轴下方100绝对坐标处Print y(j) '在X轴下方100绝对坐标处对应显示X轴相对坐标刻度值Next bcEnd SubPrivate Sub ZuoDian(x() As Double, y() As Double)ReDim xpos(I) As DoubleReDim ypos(I) As DoubleFor I = 0 To Nxpos(I) = X0pos + (x(I) - Xmin) * xstepypos(I) = Y0pos + (y(I) - Ymin) * ystepIf y(I) <= Ymax ThenDrawWidth = 4PSet (xpos(I), ypos(I)), vbRedElseEnd IfNext IDrawWidth = 1End SubPrivate Sub HuaQuXian(xiaoA() As Double)ReDim xpos(I) As DoubleReDim ypos(I) As DoubleDim Ysum As Double, Ii As DoubleFor Ii = Xmin To Xmax Step 1 / (Xmax - Xmin)Ysum = 0For j = 1 To MYsum = Ysum + xiaoA(j) * Ii ^ (j - 1)Next jxpos(I) = X0pos + (Ii - Xmin) * xstepypos(I) = Y0pos + (Ysum - Ymin) * ystepDrawWidth = 2If Ii = Xmin Thenxpos(0) = X0pos + (Ii - Xmin) * xstepypos(0) = Y0pos + (Ysum - Ymin) * ystepPSet (xpos(0), ypos(0))ElseEnd IfIf Ysum <= Ymax ThenDrawWidth = 2Line -(xpos(I), ypos(I)), vbBlueElseEnd IfNext IiDrawWidth = 1End SubPrivate Sub JieFangCheng(A() As Double, B() As Double, x() As Double) Dim nn As Doublenn = UBound(B)Dim TempA As Double, L As Double, K As Double, Kk As DoubleDim Ii As Double, ChuShu As Double, Sum As DoubleFor I = 1 To nnL = 0: Kk = 0For j = I To nnIf A(j, I) = 0 Then L = L + 1Next jFor j = I To nn - LIf A(j, I) = 0 ThenKk = Kk + 1For K = I To nnTempA = A(j, K)A(j, K) = A(nn - Kk + 1, K)A(nn - Kk + 1, K) = TempANext KTempA = B(j): B(j) = B(nn - Kk + 1): B(nn - Kk + 1) = TempAEnd IfNext jFor Ii = I To nn - LChuShu = A(Ii, I)For j = I To nnA(Ii, j) = A(Ii, j) / ChuShuNext jB(Ii) = B(Ii) / ChuShuNext IiFor Ii = I + 1 To nn - LFor j = I To nnA(Ii, j) = A(Ii, j) - A(I, j)Next jB(Ii) = B(Ii) - B(I)Next IiNext IFor I = 1 To nnFor j = 1 To I - 1A(I, j) = 0Next jNext Ix(nn) = B(nn) / A(nn, nn)For I = nn - 1 To 1 Step -1Sum = 0For j = I + 1 To nnSum = Sum + A(I, j) * x(j)Next jx(I) = (B(I) - Sum) / A(I, I)Next IEnd SubPrivate Sub Command1_Click()ClsXmin = 0 ' InputBox("请输入x坐标下限值", "x坐标下限值", 0) Ymin = 0 'InputBox("请输入y坐标下限值", "y坐标下限值", 0) Xmax = 10 ' InputBox("请输入x坐标上限值", "x坐标上限值度", 10) Ymax = 10 'InputBox("请输入y坐标上限值", "y坐标上限值度", 10) N = 20For I = 0 To NReDim Preserve x(I)ReDim Preserve y(I)Next ICall HuaZuoBiao(x, y)Private Sub Command2_Click()For I = 0 To Nx(I) = Xmin + I * (Xmax - Xmin) / N 'InputBox("请输入X坐标测量值", "X坐标值", "0") ' y(I) = Sin(x(I)) + 5 ' InputBox("请输入Y坐标测量值", "Y坐标值", "0") 'Next ICall ZuoDian(x, y)End SubPrivate Sub Command3_Click()M = 20 'InputBox("请输入拟合曲线次数M", "拟合曲线", 3)Erase B: Erase xiaoA: Erase A '必不可少***********ReDim B(M): ReDim xiaoA(1 To M)'形成方程组的各元素A(1, 1) = NFor I = 1 To NB(1) = B(1) + y(I)Next IFor j = 2 To MFor I = 1 To NA(1, j) = A(1, j) + x(I) ^ (j - 1)Next INext jFor I = 2 To MFor j = 1 To MFor Xh = 1 To NA(I, j) = A(I, j) + x(Xh) ^ (I + j - 2)If j = 1 ThenB(I) = B(I) + x(Xh) ^ (I - 1) * y(Xh)End IfNext XhNext jNext ICall JieFangCheng(A, B, xiaoA)ForeColor = vbBlackFor I = 1 To M'Print Tab(6); "a"; I - 1; Tab(12); "="; xiaoA(I); Next IDim Str As String: Str = "y="For I = 1 To M '写方程If I < M ThenStr = Str & xiaoA(I) & "*x^" & I - 1 & "+"ElseStr = Str & xiaoA(I) & "*x^" & I - 1 End IfNext IPrint vbCrLf; "曲线方程:"; vbCrLf & StrCall HuaQuXian(xiaoA)End SubPrivate Sub Command4_Click()EndEnd SubPrivate Sub Form_Load()Width = Screen.Width * 1 '取屏幕宽度的一半'Height = Screen.Height * 0.5 '取屏幕高度的一半Height = Screen.Width * 1 '取屏幕宽度的一半Left = (Screen.Width - Width) / 2 '使窗体居屏幕中心Top = (Screen.Height - Height) / 2 '使窗体居屏幕中心End Sub。
python最小二乘法拟合曲线程序
Python最小二乘法拟合曲线程序1. 简介在数据分析和机器学习领域，拟合曲线是一种常见的技术，用于找到最佳曲线来描述数据的关系。
其中，最小二乘法是一种常用的拟合曲线方法之一。
Python作为一种流行的编程语言，在科学计算和数据分析方面具有广泛的应用。
本文将介绍如何使用Python实现最小二乘法来拟合曲线。
2. 最小二乘法最小二乘法是一种数学优化方法，用于找到与给定数据点最能匹配的曲线或函数。
它通过最小化残差平方和来实现这一目标。
残差是指观测值与拟合值之间的差异。
假设我们有一个包含n个数据点的样本集合：{(x1, y1), (x2, y2), …, (xn, yn)}。
我们希望找到一个函数f(x)来近似描述这些数据点。
最小二乘法通过寻找使得残差平方和最小化的函数参数来实现这一目标。
3. Python实现在Python中，我们可以使用scipy库提供的curve_fit()函数来执行最小二乘法拟合曲线。
首先，我们需要导入必要的库：import numpy as npimport matplotlib.pyplot as pltfrom scipy.optimize import curve_fit然后，我们定义一个用于拟合的函数。
这个函数的参数将在最小二乘法过程中进行优化调整。
例如，我们可以使用一个多项式函数来拟合数据：def polynomial_func(x, *coefficients):y = 0for i, c in enumerate(coefficients):y += c * x**ireturn y接下来，我们准备好我们的数据。
在这个例子中，我们使用一个简单的正弦曲线作为示例：x = np.linspace(0, 2*np.pi, 100)y = np.sin(x) + np.random.normal(0, 0.1, 100)现在，我们可以使用curve_fit()函数来执行最小二乘法拟合曲线：popt, pcov = curve_fit(polynomial_func, x, y)popt是一个包含了最佳拟合参数的数组，pcov是协方差矩阵。
VBA最小二乘法多模型拟合问题
VBA最小二乘法多模型拟合问题n = 1 '表头列'数据行数获取----------------------------------------------------------------------------------------------------m = 0For i = n To 10000If Cells(i, 1) <> "" Thenm = m + 1End IfNext i'获取数据-------------------------------------------------------------------------------------------------------Dim x() As DoubleDim y() As DoubleReDim x(1 To m) '动态数组定义方法ReDim y(1 To m) '动态数组定义方法t = n - 1For i = 1 To mx(i) = Cells(i + t, 1)y(i) = Cells(i + t, 2)Next i'算法部分---------------------------------------- 模型：Y = aX + b -------------------------------------Dim a(1 To 3) As DoubleDim b(1 To 4) As DoubleDim r(1 To 4) As Double ' 拟合度 R2Dim w(1 To 4) As Double'线性拟合-------------------------------------------------------------------------------------------------------Dim x1() As DoubleDim y1() As DoubleReDim x1(1 To m)ReDim y1(1 To m)For i = 1 To mx1(i) = x(i)y1(i) = y(i)Next ia(1) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y1(), x1(), True, True), 1, 1)b(1) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y1(), x1(), True, True), 1, 2)r(1) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y1(), x1(), True, True), 3, 1)'指数拟合-------------------------------------------------------------------------------------------------------Dim x2() As DoubleDim y2() As DoubleReDim x2(1 To m)ReDim y2(1 To m)For i = 1 To mx2(i) = x(i)y2(i) = Application.WorksheetFunction.Ln(y(i))Next ia(2) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y2(), x2(), True, True), 1, 1)b(2) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y2(), x2(), True, True), 1, 2)r(2) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y2(), x2(), True, True), 3, 1)b(2) = Exp(b(2))'幂指数拟合---------------------------------------------------------------------------------------------------Dim x3() As DoubleDim y3() As DoubleReDim x3(1 To m)ReDim y3(1 To m)For i = 1 To mx3(i) = Application.WorksheetFunction.Ln(x(i))y3(i) = Application.WorksheetFunction.Ln(y(i))Next ia(3) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y3(), x3(), True, True), 1, 1)Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y3(), x3(), True, True), 1, 2)r(3) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y3(), x3(), True, True), 3, 1)b(3) = Exp(b(3))'多项式拟合-----------------------------------四次多项式------------------------------------------------------Dim x4() As DoubleDim y4()As DoubleReDim x4(1 To m, 1 To 4)ReDim y4(1 To m, 1 To 1)'----------------数组要进行转置不然对应不上---------------For i = 1 To my4(i, 1) = y(i)x4(i, 1) = x(i)x4(i, 2) = x(i) ^ 2x4(i, 3) = x(i) ^ 3x4(i, 4) = x(i) ^ 4Next iw(1) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y4(), x4(), True, True), 1, 1)Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y4(), x4(), True, True), 1, 2)w(3) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y4(), x4(), True, True), 1, 3)w(4) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y4(), x4(), True, True), 1, 4)b(4) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y4(), x4(), True, True), 1, 5)r(4) = Application.WorksheetFunction.Index(Application.WorksheetFu nction.LinEst(y4(), x4(), True, True), 3, 1)'---------------------------------------------模-型-优-先-----------------------------------------------------rmax = Application.WorksheetFunction.Max(r())For i = 1 To 4If r(i) = rmax Thenrn = iEnd IfNext i'填写X、Y所在行-------------------------------------------------------------'rn = 3yn = 4 'Y输出行xn = 3 'X输入行'---------------------------------------------------------------------------mn = 0For i = n To 10000If Cells(i, xn) <> "" Thenmn = mn + 1End IfNext i'---------------------------------------------------------------------------For i = 1 To 4If rn = 1 ThenFor j = 1 To mnxa = Cells(j, xn).ValueCells(j, yn) = a(1) * xa + b(1) '线性模型Next jEnd If'------------------------If rn = 2 ThenFor j = 1 To mnxa = Cells(j, xn).ValueCells(j, yn) = b(2) * Exp(a(2) * xa) '指数模型Next jEnd If'------------------------If rn = 3 ThenFor j = 1 To mnxa = Cells(j, xn).ValueCells(j, yn) = b(3) * (xa ^ a(3)) '幂指数模型Next jEnd If'------------------------If rn = 4 ThenFor j = 1 To mnxa = Cells(j, xn).ValueCells(j, yn) = w(1) * xa ^ 4 + w(2) * xa ^ 3 + w(3) * xa ^ 2 + w(4) * xa ^ 1 + b(4) '4次多项式模型Next jEnd IfNext i'-----------------------------------------------模型优选--------------------------------------------------For i = 1 To 4Cells(i + 8, 8) = r(i)Next i。
最小二乘法曲线拟合_原理及matlab实现
曲线拟合（curve-fitting ）：工程实践中，用测量到的一些离散的数据},...2,1,0),,{(m i y x i i =求一个近似的函数)(x ϕ来拟合这组数据，要求所得的拟合曲线能最好的反映数据的基本趋势（即使)(x ϕ最好地逼近()x f ，而不必满足插值原则。
因此没必要取)(i x ϕ=i y ，只要使i i i y x -=)(ϕδ尽可能地小）。
原理：给定数据点},...2,1,0),,{(m i y x i i =。
求近似曲线)(x ϕ。
并且使得近似曲线与()x f 的偏差最小。
近似曲线在该点处的偏差i i i y x -=)(ϕδ，i=1,2,...,m 。
常见的曲线拟合方法:1.使偏差绝对值之和最小2.使偏差绝对值最大的最小3.使偏差平方和最小最小二乘法：按偏差平方和最小的原则选取拟合曲线，并且采取二项式方程为拟合曲线的方法,称为最小二乘法。
推导过程：1. 设拟合多项式为：k k x a x a a x +++=...)(10ϕ2. 各点到这条曲线的距离之和，即偏差平方和如下：3. 问题转化为求待定系数0a ...k a 对等式右边求i a 偏导数，因而我们得到了：.......4、把这些等式化简并表示成矩阵的形式，就可以得到下面的矩阵：5. 将这个德蒙得矩阵化简后可得到:6. 也就是说X*A=Y，那么A = (X'*X)-1*X'*Y，便得到了系数矩阵A，同时，我们也就得到了拟合曲线。
MATLAB实现：MATLAB提供了polyfit（）函数命令进行最小二乘曲线拟合。
调用格式：p=polyfit(x,y,n)[p,s]= polyfit(x,y,n)[p,s,mu]=polyfit(x,y,n)x,y为数据点，n为多项式阶数，返回p为幂次从高到低的多项式系数向量p。
x 必须是单调的。
矩阵s包括R（对x进行QR分解的三角元素）、df(自由度）、normr(残差）用于生成预测值的误差估计。
最小二乘法拟合曲线的VB程序源代码
最小二乘法拟合曲线的VB程序源代码'按教材定义LSS方法求系数的函数（P123），输入G，y()矩阵，不求误差（另外定义函数）Private Function answerX(G() As Single, y() As Single) As Single()Dim i As Integer, j As Integer, k As IntegerDim xx() As Single, w() As SingleDim sigma As Single, t As Single, bat As SingleDim m As Integer, n As IntegerDim all As Singlem = UBound(G, 1)n = UBound(G, 2)Dim GG() As SingleReDim GG(m, n + 1)ReDim xx(n)ReDim w(m)'将y放入G最后一列，定义为GG矩阵For i = 1 To mFor j = 1 To nGG(i, j) = G(i, j)Next jGG(i, n + 1) = y(i)Next i'开始LSS方法For k = 1 To n'形成矩阵Qkall = 0For i = k To mall = all + GG(i, k) ^ 2Next iall = Sqr(all)sigma = -Sgn(GG(k, k)) * allw(k) = GG(k, k) - sigmaFor j = k + 1 To mw(j) = GG(j, k)Next jbat = sigma * w(k)'变换GGk-1到GGkGG(k, k) = sigmaFor j = k + 1 To n + 1all = 0For i = k To mall = all + w(i) * GG(i, j)Next it = all / batFor i = k To mGG(i, j) = GG(i, j) + t * w(i)Next iNext j'解三角方程Rx=h1xx(n) = GG(n, n + 1) / GG(n, n)For i = n - 1 To 1 Step -1all = 0For j = i + 1 To nall = all + GG(i, j) * xx(j)Next jxx(i) = (GG(i, n + 1) - all) / GG(i, i)Next ianswerX = xx' Dim shuchu As String' Open fullpath & "xishua.txt" For Output As 1' shuchu = ""' For i = 1 To n' shuchu = shuchu & xx(i) & Space(5)' Next i' Print #1, shuchu' Close 1Next kEnd Function'定义n次多项式最小二乘法拟合时求系数矩阵函数'系数形势为p(x)=a(1)+a(2)x+a(3)x^2+...+a(n)x^(n-1)+a(n+1)x^n'因此，g1(x)=x^0,g2(x)=x,g3(x)=x^2...gn(x)=x^(n-1),g(n+1)=x^n'需给出x(1-m)和y(1-m)矩阵Private Function answerA(x() As Single, y() As Single, n As Integer) As Single() Dim m As IntegerDim i As Integer, j As IntegerDim aa() As Single, G() As Singlem = UBound(x)ReDim aa(n + 1)ReDim G(m, n + 1)'求出G矩阵For j = 1 To n + 1For i = 1 To mG(i, j) = x(i) ^ (j - 1)Next iNext j' Dim shuchu As String' Open fullpath & "MATG.txt" For Output As 1' For i = 1 To m' shuchu = ""' For j = 1 To n + 1' shuchu = shuchu & G(i, j) & Space(5)' Next j' Print #1, shuchu' Next i' Close 1'求出法方程的系数矩阵和右端矩阵,并求解系数矩阵Dim a() As Single, d() As SingleReDim a(n + 1, n + 1)ReDim d(n + 1)'a() = mattime(matT(G()), G(), False)'d() = mattime(matT(G()), y(), True)'aa() = answerX(a(), d())aa() = answerX(G(), y())answerA = aaEnd Function'定义n次多项式求值函数,a()为系数矩阵，xx为带入时刻（整数）Private Function Pxn(a() As Single, xx As Single) As SingleDim n As SingleDim i As Integern = UBound(a)Pxn = 0For i = 1 To nPxn = Pxn + a(i) * xx ^ (i - 1)Next iEnd Function'定义题目要求指数函数求值函数，按2次多项式求值，ae()为系数矩阵，x为带入时刻（整数）Private Function Pxe(a() As Single, xx As Single) As SinglePxe = a(1) * Exp(-a(2) * (xx - a(3)) ^ 2)End Function'定义n次多项式拟合求误差函数，n为次数Private Function EN(n As Integer) As SingleDim m As IntegerDim aa() As Singleaa = answerA(x(), y(), n)m = UBound(x())EN = 0Dim z As IntegerFor z = 1 To mEN = EN + (Pxn(aa, x(z)) - y(z)) ^ 2Next zEN = Sqr(EN)End Function'定义多项式函数形式生成函数，n为次数Private Function Pnstr(n As Integer) As StringDim m As IntegerDim aa() As Singleaa = answerA(x(), y(), n)m = UBound(x())Pnstr = ""Dim z As IntegerFor z = 1 To n + 1Pnstr = Pnstr & IIf(aa(z) > 0, "+" & aa(z), aa(z)) & IIf((z - 1) = 0, "", "t^" & CStr(z - 1)) Next zEnd Function'定义题目要求指数函数拟合误差函数Private Function EE() As SingleDim m As IntegerDim aa() As Singleaa = answerA(x(), ye(), 2)ae(1) = Exp(aa(1) - aa(2) ^ 2 / (4 * aa(3)))ae(2) = -aa(3)ae(3) = -aa(2) / (2 * aa(3))m = UBound(x)EE = 0Dim z As IntegerFor z = 1 To mEE = EE + (Pxe(ae(), x(z)) - ye(z)) ^ 2Next zEE = Sqr(EE)End FunctionPrivate Sub cmd1_Click()txte2 = EN(2)txte3 = EN(3)txte4 = EN(4)txtee = EEtxtpe = ae(1) & "Exp(" & -ae(2) & "(t-" & ae(3) & ")^2)"txtp2 = Pnstr(2)txtp3 = Pnstr(3)txtp4 = Pnstr(4)Command2.Enabled = TrueEnd SubPrivate Sub Command1_Click()EndEnd SubPrivate Sub Command2_Click()Dim i As SingleDim a2() As Single, a3() As Single, a4() As Singlea2() = answerA(x, y, 2)a3() = answerA(x, y, 3)a4() = answerA(x, y, 4)pic1.DrawWidth = 2For i = -2 To 26 Step 0.01pic1.Line (i - 0.01, Pxn(a2, (i - 0.01)))-(i, Pxn(a2, i)), vbMagenta Next iFor i = -2 To 26 Step 0.01pic1.Line (i - 0.01, Pxn(a3, (i - 0.01)))-(i, Pxn(a3, i)), vbBlue Next iFor i = -2 To 26 Step 0.01pic1.Line (i - 0.01, Pxn(a4, (i - 0.01)))-(i, Pxn(a4, i)), vbCyan Next iFor i = -2 To 26 Step 0.01pic1.Line (i - 0.01, Pxe(ae(), (i - 0.01)))-(i,Pxe(ae(), i)), vbGreenNext iPrivate Sub Form_Load()'窗口加载定义数组x,y,yeDim i As IntegerReDim x(25)ReDim y(25)ReDim ye(25)For i = 1 To 25x(i) = i - 1Next iy(1) = 15y(2) = 14y(3) = 14y(4) = 14y(5) = 14y(6) = 15y(7) = 16y(8) = 18y(9) = 20y(10) = 20y(11) = 23y(12) = 25y(13) = 28y(14) = 31y(15) = 32y(16) = 41y(17) = 29y(18) = 27y(19) = 25y(20) = 24y(21) = 22y(22) = 20y(23) = 18y(24) = 17y(25) = 16For i = 1 To 25ye(i) = Log(y(i))Next ifram2.Top = fram1.Topfram2.Left = fram1.Leftfram2.Visible = FalseCommand2.Enabled = False' If Right(App.Path, 1) = "\" Then fullpath = App.Path Else fullpath = App.Path & "\"Private Sub Form_activate()'绘制坐标线和各数据点Dim i As Integerpic1.Scale (-2, 45)-(26, 10)pic1.DrawWidth = 1For i = -2 To 26pic1.Line (i, 10)-(i, 45), vbBlackNext iFor i = 10 To 45 Step 5pic1.Line (-2, i)-(26, i), vbBlackNext ipic1.DrawWidth = 8For i = 1 To 25pic1.PSet (x(i), y(i)), vbRedNext iEnd SubPrivate Sub TabStrip1_Click()Select Case TabStrip1.SelectedItem.Index Case 1fram1.Visible = Truefram2.Visible = FalseCase Elsefram1.Visible = Falsefram2.Visible = TrueEnd SelectEnd Sub。
最小二乘法拟合原理及代码的实现
上式是关于 (a0 , a1 ,...., an ) 的线性方程组，用矩阵表示为：
m 1 m xi i 0 m x n i i 0
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m
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2 i
x
i 0
m
n 1 i
m n x y i i i 0 a0 i 0 m m xy a xin 1 i i 1 i 0 i 0 an m m n 2n x y xi i i i 0 i 0
最小二乘法拟合原理及代码实现
1. 最小二乘法定义 已知一组实验数据（xi, yi ）(i = 0,1,2,….,m),且观测数据有误差 求自变量 x 与因变量 y 之间的函数关系 y = F(x),不要求 y = F(x)经过 所有点，而只要求在给定点的误差 δi = F(xi) – yi (i = 0,1,2,…,m) 按某种标准最小。 度量的标准不同，将导致不同的结果，常用的准则有如下三 种： a． 残差的最大绝对值为最小
解方程，得
a0
a1
x y x x y m x x m x y x y m x x
2 i i i i i 2 i 2 i i i i i 2 i 2 i
C#程序代码的实现：
} if(Math.Abs(fmX * fmX - fmXX * fn) < 0.000001) { return false; }
fk = (fmY * fmX - fmXY * fn)/(fmX * fmX - fmXX * fn); fb = (fmY - fmX * fk)/fn; return true; }
曲线拟合的最小二乘法matlab举例
曲线拟合的最小二乘法学院：光电信息学院 姓名：赵海峰 学号：200820501001一、曲线拟合的最小二乘法原理：由已知的离散数据点选择与实验点误差最小的曲线)(...)()()(1100x a x a x a x S n n ϕϕϕ+++=称为曲线拟合的最小二乘法。
若记),()()(),(0i k i j mi i k j x x x ϕϕωϕϕ∑==k i k i mi i k d x x f x f ≡=∑=)()()(),(0ϕωϕ上式可改写为),...,1,0(;),(n k d a k j noj j k -=∑=ϕϕ这个方程成为法方程，可写成距阵形式d Ga =其中,),...,,(,),...,,(1010T n T n d d d d a a a a ==⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=),(),(),()(),(),(),(),(),(101110101000n n n n n n G ϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕ 。
它的平方误差为：.)]()([)(||||2022i i mi i x f x S x -=∑=ωδ二、数值实例：下面给定的是乌鲁木齐最近1个月早晨7:00左右(新疆时间)的天气预报所得到的温度数据表，按照数据找出任意次曲线拟合方程和它的图像。
下面应用Matlab编程对上述数据进行最小二乘拟合三、Matlab程序代码：x=[1:1:30];y=[9,10,11,12,13,14,13,12,11,9,10,11,12,13,14,12,11,10,9,8,7,8,9,11,9,7,6,5,3,1]; a1=polyfit(x,y,3) %三次多项式拟合%a2= polyfit(x,y,9) %九次多项式拟合%a3= polyfit(x,y,15) %十五次多项式拟合%b1=polyval(a1,x)b2=polyval(a2,x)b3=polyval(a3,x)r1= sum((y-b1).^2) %三次多项式误差平方和%r2= sum((y-b2).^2) %九次次多项式误差平方和%r3= sum((y-b3).^2) %十五次多项式误差平方和%plot(x,y,'*') %用*画出x,y图像%hold onplot(x,b1, 'r') %用红色线画出x,b1图像%hold onplot(x,b2, 'g') %用绿色线画出x,b2图像%hold onplot(x,b3, 'b:o') %用蓝色o线画出x,b3图像%四、数值结果：不同次数多项式拟和误差平方和为：r1 = 67.6659r2 = 20.1060r3 = 3.7952r1、r2、r3分别表示三次、九次、十五次多项式误差平方和。
曲线拟合的最小二乘法的Matlab程序实现
《曲线拟合的最小二乘法》目录1 实验要求 (1)1.1 问题提出 (1)1.2 要求 (1)1.3 目的和意义 (1)2 问题求解 (2)2.1 基本原理 (2)2.1.1 最小二乘法简介 (2)2.1.2 求解思路 (2)2.1.3 常用的多项式拟合 (4)2.2 求解步骤 (5)2.2.1 绘制散点图 (5)2.1.2 进行三次拟合 (6)2.1.3 三次拟合的误差分析 (7)2.1.4 进行五次拟合 (8)2.1.5 五次拟合的误差分析 (10)2.1.6 拟合效果比较 (11)3 参考文献 (13)1 实验要求1.1 问题提出从随机的数据中找出其规律性，给出其近似表达式的问题，在生产实践和科学实验中大量存在，通常利用数据的最小二乘法求得拟合曲线。
在某冶炼过程中，根据统计数据的含碳量与时间关系，试求含碳量 y 与时间t 的拟合曲线。
1.2 要求（1）用最小二乘法进行曲线拟合；（2）近似解析表达式为φ(t)=a1t+a2t2+a3t3；（3）打印出拟合函数φ(t)，并打印出φ(t j) 与y(t j) 的误差，j=1,2,⋯,12 ；（4）另外选取一个近似表达式，尝试拟合效果的比较；（5） * 绘制出曲线拟合图﹡。
1.3 目的和意义（1）掌握曲线拟合的最小二乘法；（2）最小二乘法亦可用于解超定线代数方程组；（3）探索拟合函数的选择与拟合精度间的关系。
2 问题求解2.1 基本原理2.1.1 最小二乘法简介对原始数据做适当变换或其他处理，发现其中隐藏的数学规律往往是数学建模中非常重要的一步。
通过对实验数据的分析，求出自变量、因变量之间近似的函数关系，进而对其进行进一步分析，而最小二乘法是对数据进行拟合常用的方法。
最小二乘法是研究观测数据的主要方法之一，若已知两变量满足线性关系y =ax +b ，对这两个变量进行了 n 次观测，从而得到了 n 对观测数据（x 1, y 1）,（x 2, y 2）......（x n , y n ），最小二乘法提供了一个方法，基本思想是找到离这 n 组观测数据最近的直线。
VB计算程序课程设计--用最小二乘法求拟合曲线
VB计算程序课程设计--用最小二乘法求拟合曲线测试与光电工程学院课程设计任务书测控技术与仪器系100813班学号10081329 姓名吴辉课程名称：用最小二乘法求拟合曲线课题要求：利用VB语言编程实现对给定离散点的拟合（不小于10个）的拟合用最小二乘法求数据的拟合曲线。
要求有良好的输入、输出界面，输出应包含直线方程并图形显示拟合效果。
完成软件的整体设计。
课题进程：1）熟悉VB编程语言、最小二乘法算法分析3天2）编写程序实现以上功能3天3）软件调试、测试2天4）撰写课程设计报告2天指导老师：杨琳瑜目录摘要 ------------------------------------------------------------------------------------ 2 第一章最小二乘法 ---------------------------------------------------------------- 21) 理论依据 ----------------------------------------- 错误!未定义书签。
2) 线性拟合分析 ----------------------------------- 错误!未定义书签。
3) 非线性拟合分析 ------------------------------------------------------- 5第二章系统设计-------------------------------------------------------------------- 51) 采用的软件及开发平台 ---------------------------------------------- 52) 项目的总体方案 ------------------------------------------------------- 53) 项目的详细设计 ------------------------------------------------------- 6第三章设计实现------------------------------------------------------------------- 101) 主要功能模块的具体实现 ------------------------------------------ 102) 主要技术问题或难题的解决方法 --------------------------------- 103) 亮点或创新点的实现 ------------------------------------------------ 11第四章结束语---------------------------------------------------------------------- 11 参考文献 ----------------------------------------------------------------------------- 12 附录 ----------------------------------------------------------------------------------- 14摘要最小二乘法最早是由高斯提出的，这是数据处理的一种很有效的统计方法。
matlab最小二乘法拟合曲线代码
在Matlab中使用最小二乘法进行曲线拟合是一项非常常见的任务。
最小二乘法是一种数学优化技术，用于对一组数据进行曲线拟合，以便找到最能代表数据趋势的曲线。
在本文中，我将深入探讨Matlab中最小二乘法拟合曲线的代码实现，并共享我对这一主题的个人理解。
让我们来了解一下什么是最小二乘法。
最小二乘法是一种数学优化技术，用于寻找一组数据的最佳拟合曲线。
在Matlab中，可以使用内置的polyfit函数来实现最小二乘法曲线拟合。
这个函数的基本语法是：```matlabp = polyfit(x, y, n)```其中，x和y分别是数据点的横纵坐标，n是要拟合的多项式次数。
这个函数将返回多项式系数向量p，使得拟合多项式最小化了实际数据点与拟合曲线之间的误差平方和。
举个例子，假设我们有一组数据点(x, y)，我们可以使用polyfit函数来进行二次多项式拟合：```matlabx = [1, 2, 3, 4, 5];y = [2, 3, 4, 3, 5];p = polyfit(x, y, 2);```在这个例子中，p将会是一个包含三个元素的向量，分别代表二次多项式的系数a、b和c。
通过这些系数，我们就可以得到拟合的二次多项式方程。
除了使用polyfit函数，我们还可以使用polyval函数来计算拟合曲线上的点。
其基本语法形式是：```matlaby_fit = polyval(p, x)```在这个例子中，p是通过polyfit得到的多项式系数向量，x是我们要计算拟合曲线上的点的横坐标，y_fit将是这些点的纵坐标。
另外，Matlab还提供了许多其他的拟合函数和工具箱，用于不同类型的数据和曲线拟合需求。
通过调用这些函数和工具箱，我们可以实现更复杂的曲线拟合任务，满足不同数据类型和拟合目标的需求。
总结来说，Matlab提供了丰富的工具和函数，用于实现最小二乘法曲线拟合。
通过调用polyfit函数和其他拟合工具箱，我们可以轻松地对一组数据进行曲线拟合，从而得到最能代表数据趋势的曲线。
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txte4 = EN(4)
txtee = EE
txtpe = ae(1) & "Exp(" & -ae(2) & "(t-" & ae(3) & ")^2)"
txtp2 = Pnstr(2)
txtp3 = Pnstr(3)
' shuchu = ""
' For i = 1 To n
' shuchu = shuchu & xx(i) & Space(5)
' Next i
' Print #1, shuchu
' Close 1
aa = answerA(x(), y(), n)
m = UBound(x())
EN = 0
Dim z As Integer
For z = 1 To m
EN = EN + (Pxn(aa, x(z)) - y(z)) ^ 2
Next z
Pxe = a(1) * Exp(-a(2) * (xx - a(3)) ^ 2)
End Function
'定义n次多项式拟合求误差函数，n为次数
Private Function EN(n As Integer) As Single
Dim m As Integer
Dim aa() As Single
For i = k To m
GG(i, j) = GG(i, j) + t * w(i)
Next i
Next j
'解三角方程Rx=h1
xx(n) = GG(n, n + 1) / GG(n, n)
For i = -2 To 26 Step 0.01
pic1.Line (i - 0.01, Pxn(a2, (i - 0.01)))-(i, Pxn(a2, i)), vbMagenta
Next i
For i = -2 To 26 Step 0.01
GG(i, j) = G(i, j)
Next j
GG(i, n + 1) = y(i)
Next i
'开始LSS方法
For k = 1 To n
'形成矩阵Qk
all = 0
ae(1) = Exp(aa(1) - aa(2) ^ 2 / (4 * aa(3)))
ae(2) = -aa(3)
ae(3) = -aa(2) / (2 * aa(3))
m = UBound(x)
EE = 0
Dim z As Integer
' shuchu = ""
' For j = 1 To n + 1
' shuchu = shuchu & G(i, j) & Space(5)
' Next j
' Print #1, shuchu
For i = k To m
all = all + GG(i, k) ^ 2
Next i
all = Sqr(all)
sigma = -Sgn(GG(k, k)) * all
w(k) = GG(k, k) - sigma
' Next i
' Close 1
'求出法方程的系数矩阵和右端矩阵,并求解系数矩阵
Dim a() As Single, d() As Single
ReDim a(n + 1, n + 1)
ReDim d(n + 1)
'a() = mattime(matT(G()), G(), False)
txtp4 = Pnstr(4)
Command2.Enabled = True
End Sub
Private Sub Command1_Click()
End
End Sub
Private Sub Command2_Click()
Dim i As Single
Pxn = Pxn + a(i) * xx ^ (i - 1)
Next i
End Function
'定义题目要求指数函数求值函数，按2次多项式求值，ae()为系数矩阵，x为带入时刻（整数）
Private Function Pxe(a() As Single, xx As Single) As Single
For i = n - 1 To 1 Step -1
all = 0
For j = i + 1 To n
all = all + GG(i, j) * xx(j)
Next j
Private Function Pxn(a() As Single, xx As Single) As Single
Dim n As Single
Dim i As Integer
n = UBound(a)
Pxn = 0
For i = 1 To n
'需给出x(1-m)和y(1-m)矩阵
Private Function answerA(x() As Single, y() As Single, n As Integer) As Single()
Dim m As Integer
Dim i As Integer, j As Integer
For j = k + 1 To n + 1
all = 0
For i = k To m
all = all + w(i) * GG(i, j)
Next i
t = all / bat
Next z
End Function
'定义题目要求指数函数拟合误差函数
Private Function EE() As Single
Dim m As Integer
Dim aa() As Single
aa = answerA(x(), ye(), 2)
n = UBound(G, 2)
Dim GG() As Single
ReDim GG(m, n + 1)
ReDim xx(n)
ReDim w(m)
'将y放入G最后一列，定义为GG矩阵
For i = 1 To m
For j = 1 To n
xx(i) = (GG(i, n + 1) - all) / GG(i, i)
Next i
answerX = xx
' Dim shuchu As String
' Open fullpath & "xishua.txt" For Output As 1
G(i, j) = x(i) ^ (j - 1)
Next i
Next j
' Dim shuchu As String
' Open fullpath & "MATG.txt" For Output As 1
' For i = 1 To m
'd() = mattime(matT(G()), y(), True)
'aa() = answerX(a(), d())
aa() = answerX(G(), y())
answerA = aa
End Function
'定义n次多项式求值函数,a()为系数矩阵，xx为带入时刻（整数）
Next k
End Function
'定义n次多项式最小二乘法拟合时求系数矩阵函数
'系数形势为p(x)=a(1)+a(2)x+a(3)x^2+...+a(n)x^(n-1)+a(n+1)x^n
'因此，g1(x)=x^0,g2(x)=x,g3(x)=x^2...gn(x)=x^(n-1),g(n+1)=x^n
EN = Sqr(EN)
End Function
'定义多项式函数形式生成函数，n为次数
Private Function Pnstr(n As Integer) As String
Dim m As Integer
Dim aa() As Single
aa = answerA(x(), y(), n)
Dim xx() As Single, w() As Single
Dim sigma As Single, t As Single, bat As Single
Dim m As Integer, n As Integer
Dim all As Single
m = UBound(G, 1)
Dim a2() As Single, a3() As Single, a4() As Single
a2() = answerA(x, y, 2)
a3() = answБайду номын сангаасrA(x, y, 3)
a4() = answerA(x, y, 4)
pic1.DrawWidth = 2
'按教材定义LSS方法求系数的函数（P123），输入G，y()矩阵，不求误差（另外定义函数）