现代检测技术课程实验编程：最小二乘法应用编程
一、最小二乘法编程题目描述
在对量程为10MPa的压力传感器进行标定时，传感器输出电压值与压力值之间的关系如下表所示，请简述最小二乘法准则的意义，并分析下列电压-压力直线中哪一条最符合最小二乘法准则？（使用计算机辅助进行计算）
二、最小二乘法编程题目要求
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使用计算机软件（VB、VC、JAVA、LabVIEW、Matlab、Python均可）编程完成本次编程题目；
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所编程序要有较为美观的GUI界面，可以通过人机界面输入校准数据xi/yi，和备选直线方程的参数。
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所编程序，要能够直接显示哪条直线为最佳直线，不能人为进行判断。
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对所编程序的原理和运行结果进行介绍和分析。
三、什么是最小二乘法
四、最小二乘法编程步骤
4.1、 界面的设计
- 1 界面可以进行测量数据的压力和电压的输入、五条拟合直线的回归参数a、b。
- 2 点击计算按钮，可以完成最小二乘法直线拟合的回归参数a、b。五条直线的残差平方和。自动判断哪条直线最符合最小二乘法准则。
- 3 页面最右边的轴可以显示最小二乘法直线拟合的图像。
4.2、 程序的编写
4.2.1、程序在计算按钮如下的回调函数中编写
function pushbutton1_Callback(hObject, eventdata, handles)
4.2.2、编辑的文本框输入的数据转换成数字类型的数据
Xi1 = get(handles.edit_Xi1, 'String'); Yi1 = get(handles.edit_Yi1, 'String');
Xi2 = get(handles.edit_Xi2, 'String'); Yi2 = get(handles.edit_Yi2, 'String');
Xi3 = get(handles.edit_Xi3, 'String'); Yi3 = get(handles.edit_Yi3, 'String');
Xi4 = get(handles.edit_Xi4, 'String'); Yi4 = get(handles.edit_Yi4, 'String');
Xi5 = get(handles.edit_Xi5, 'String'); Yi5 = get(handles.edit_Yi5, 'String');
Xi1 = str2num(Xi1); Yi1 = str2num(Yi1);
Xi2 = str2num(Xi2); Yi2 = str2num(Yi2);
Xi3 = str2num(Xi3); Yi3 = str2num(Yi3);
Xi4 = str2num(Xi4); Yi4 = str2num(Yi4);
Xi5 = str2num(Xi5); Yi5 = str2num(Yi5);
a1 = get(handles.edit_a1, 'String'); b1 = get(handles.edit_b1, 'String');
a2 = get(handles.edit_a2, 'String'); b2 = get(handles.edit_b2, 'String');
a3 = get(handles.edit_a3, 'String'); b3 = get(handles.edit_b3, 'String');
a4 = get(handles.edit_a4, 'String'); b4 = get(handles.edit_b4, 'String');
a5 = get(handles.edit_a5, 'String'); b5 = get(handles.edit_b5, 'String');
a1 = str2num(a1); b1 = str2num(b1);
a2 = str2num(a2); b2 = str2num(b2);
a3 = str2num(a3); b3 = str2num(b3);
a4 = str2num(a4); b4 = str2num(b4);
a5 = str2num(a5); b5 = str2num(b5);
4.2.3、将Xi、Yi数据存放与数组中
Xi = [Xi1 Xi2 Xi3 Xi4 Xi5];
Yi = [Yi1 Yi2 Yi3 Yi4 Yi5];
4.2.4、计算最小二乘法直线拟合的回归参数a、b
squareXi = Xi .* Xi;
squareYi = Yi .* Yi;
mulXiYi = Xi .* Yi;
sumXi = sum(Xi);
sumYi = sum(Yi);
sumSquareXi = sum(squareXi);
sumSquareYi = sum(squareYi);
sumMulXiYi = sum(mulXiYi);
Lxx = sumSquareXi - sumXi * sumXi / 5;
Lxy = sumMulXiYi - sumXi * sumYi / 5;
b = Lxy / Lxx;
a = sumYi / 5 - b * sumXi / 5;
4.2.5、计算五条直线的残差平方和
SubYiXi1 = Yi - (a1 + b1 * Xi);
squareSubYiXi1 = SubYiXi1 .* SubYiXi1;
sumSub1 = sum(squareSubYiXi1);
SubYiXi2 = Yi - (a2 + b2 * Xi);
squareSubYiXi2 = SubYiXi2 .* SubYiXi2;
sumSub2 = sum(squareSubYiXi2);
SubYiXi3 = Yi - (a3 + b3 * Xi);
squareSubYiXi3 = SubYiXi3 .* SubYiXi3;
sumSub3 = sum(squareSubYiXi3);
SubYiXi4 = Yi - (a4 + b4 * Xi);
squareSubYiXi4 = SubYiXi4 .* SubYiXi4;
sumSub4 = sum(squareSubYiXi4);
SubYiXi5 = Yi - (a5 + b5 * Xi);
squareSubYiXi5 = SubYiXi5 .* SubYiXi5;
sumSub5 = sum(squareSubYiXi5);
4.2.6、判断最佳的最小二乘法直线的拟合
subArrays = [sumSub1 sumSub2 sumSub3 sumSub4 sumSub5];
minSub = subArrays(1);
subJudge = 1;
for i = 2: 5
if minSub > subArrays(i)
minSub = subArrays(i);
subJudge = i;
end
end
strJudge = '最符合最小二乘法准侧的直线是: 第';
subJudge = num2str(subJudge);
strJudge1 = '条直线';
strJudge = strcat(strJudge, subJudge, strJudge1);
4.2.7、数据和图像的显示
set(handles.text_judge, 'String', num2str(strJudge));
set(handles.edit1, 'String', num2str(sumSub1));
set(handles.edit2, 'String', num2str(sumSub2));
set(handles.edit3, 'String', num2str(sumSub3));
set(handles.edit4, 'String', num2str(sumSub4));
set(handles.edit5, 'String', num2str(sumSub5));
set(handles.edit_result_a, 'String', num2str(a));
set(handles.edit_result_b, 'String', num2str(b));
plot(Xi, Yi, '*');
hold on
y1 = a1 + b1 * Xi;
axes(handles.axes1);
plot(Xi, y1, 'm');
hold on
y2 = a2 + b2 * Xi;
axes(handles.axes1);
plot(Xi, y2, 'r');
hold on
y3 = a3 + b3 * Xi;
axes(handles.axes1);
plot(Xi, y3, 'y');
hold on
y4 = a4 + b4 * Xi;
axes(handles.axes1);
plot(Xi, y4, 'k');
hold on
y5 = a5 + b5 * Xi;
axes(handles.axes1);
plot(Xi, y5, 'g');
hold on
三、 程序的运行结果
- 程序可以自动进行最佳拟合直线的判断。从运行的结果可以确定是第五条直线是最佳最小二乘法直线的拟合。
五、最小二乘法编程总结
- 用MATLAB所编写的GUI页面程序实现了计算最小二乘法直线拟合的回归参数的计算。
- 自动判断哪条直线最符合最小二乘法直线拟合的准侧。
- 最小二乘法直线拟合的图像显示。