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数学模型实验课（二） 最小二乘法与直线拟合
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康尔乃奶粉 32.4元 400g; 元 900g. 4 k1 + 42/3 k2 = 32.4 9 k1 + 92/3 k2 = 67.1 解得: k1 = , k2 = 模型: C(W)= W W2/3. 预测: W=1800, C(W) = W=2500, C(W) = 实际: W=1800, C(W)=115.9 W=2500, C(W)=146.85
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>>x=[4,9,18,25];y=[32.4,67.1,115.9,146.85]; >>x1=[x(1),x(2)]’,x2=x1.^(2/3); >>y1=[y(1),y(2)]’;X=[x1,x2]; >>k=X\y1 >>x1=[x(1),x(2),x(3)];x2=x1.^(2/3); >>y1=[y(1),y(2),y(3)];X=[x1,x2];
![>>x=[4,9,18,25];y=[32.4,67.1,115.9,146.85]; >>x1=[x(1),x(2)]’,x2=x1.^(2/3); >>y1=[y(1),y(2)]’;X=[x1,x2];](http://slidesplayer.com/slide/17317536/100/images/3/%3E%3Ex%3D%5B4%2C9%2C18%2C25%5D%3By%3D%5B32.4%2C67.1%2C115.9%2C146.85%5D%3B+%3E%3Ex1%3D%5Bx%281%29%2Cx%282%29%5D%E2%80%99%2Cx2%3Dx1.%5E%282%2F3%29%3B+%3E%3Ey1%3D%5By%281%29%2Cy%282%29%5D%E2%80%99%3BX%3D%5Bx1%2Cx2%5D%3B.jpg ">>k=X\y1. >>x1=[x(1),x(2),x(3)];x2=x1.^(2/3); >>y1=[y(1),y(2),y(3)];X=[x1,x2];")
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设：Ax+ε= y，求 x 使得误差 ||ε|| 最小。
如果 . 设：Ax+ε= y，求 x 使得误差 ||ε|| 最小。
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A’A x = A’y x = (A’A)-1A’y X b = y 二. 直线拟合 1. 矩阵左除
数据（xi, yi）,拟合直线 y = a + bx, 误差最小 模型：yi = a + bxi +ε X b = y
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>>y=[17,16,17,23,26,27,41,49]; >>x=[1953,2015,2015,2821,3049,3049,5133,5592]; >>X=[ones(8,1),x’];b=X\y’; >>y1=b(1)+b(2).*x; >>plot(x,y1,’r’,x,y,’*r’)
![>>y=[17,16,17,23,26,27,41,49]; >>x=[1953,2015,2015,2821,3049,3049,5133,5592]; >>X=[ones(8,1),x’];b=X\y’;](http://slidesplayer.com/slide/17317536/100/images/6/%3E%3Ey%3D%5B17%2C16%2C17%2C23%2C26%2C27%2C41%2C49%5D%3B+%3E%3Ex%3D%5B1953%2C2015%2C2015%2C2821%2C3049%2C3049%2C5133%2C5592%5D%3B+%3E%3EX%3D%5Bones%288%2C1%29%2Cx%E2%80%99%5D%3Bb%3DX%5Cy%E2%80%99%3B.jpg ">>y1=b(1)+b(2).*x; >>plot(x,y1,’r’,x,y,’*r’)")
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2. 直线拟合的效果（相关系数）
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>>X1=[x’,y’];m=size(X1);n=m(1);p=m(2);
>>X=[ones(n,1),x’];B=X\y’;a=B(1);b=B(2:p); >>y1=a+b*x;plot(x,y1,’r’,x,y,’*r’); >>Cov=X1’*(eye(n)-ones(n)./n)*X1; >>Lyy=Cov(p,p);Lxx=Cov(1:(p-1),1:(p-1)); >>R=(b’*Lxx*b)/Lyy;r=sqrt(R);
![>>X1=[x’,y’];m=size(X1);n=m(1);p=m(2);](http://slidesplayer.com/slide/17317536/100/images/9/%3E%3EX1%3D%5Bx%E2%80%99%2Cy%E2%80%99%5D%3Bm%3Dsize%28X1%29%3Bn%3Dm%281%29%3Bp%3Dm%282%29%3B.jpg ">>X=[ones(n,1),x’];B=X\y’;a=B(1);b=B(2:p); >>y1=a+b*x;plot(x,y1,’r’,x,y,’*r’); >>Cov=X1’*(eye(n)-ones(n)./n)*X1; >>Lyy=Cov(p,p);Lxx=Cov(1:(p-1),1:(p-1)); >>R=(b’*Lxx*b)/Lyy;r=sqrt(R);")
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3. 多项式拟合 b=polyfit(x,y,1); 4.线性回归 [b,r,j,k,l]=regress(y’,X);
![3. 多项式拟合 b=polyfit(x,y,1); 4.线性回归 [b,r,j,k,l]=regress(y’,X);](http://slidesplayer.com/slide/17317536/100/images/10/3.+%E5%A4%9A%E9%A1%B9%E5%BC%8F%E6%8B%9F%E5%90%88+b%3Dpolyfit%28x%2Cy%2C1%29%3B+4.%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92+%5Bb%2Cr%2Cj%2Ck%2Cl%5D%3Dregress%28y%E2%80%99%2CX%29%3B.jpg "3. 多项式拟合 b=polyfit(x,y,1); 4.线性回归 [b,r,j,k,l]=regress(y’,X);")
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