多项式拟合和最小二乘问题

背景知识

The method of least squares is a standard approach in regression analysis to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation.

多项式拟合

The first clear and concise exposition of the method of least squares was published by Legendre in 1805.[5] The technique is described as an algebraic procedure for fitting linear equations to data and Legendre demonstrates the new method by analyzing the same data as Laplace for the shape of the earth. The value of Legendre's method of least squares was immediately recognized by leading astronomers and geodesists of the time.

实验一

%%给定数据对，(x0,y0)
x0=0:0.1:1;
y0=[-.447,1.978,3.11,5.25,5.02,4.66,4.01,4.58,3.45,5.35,9.22];
%%求拟合多项式
n=3;%假设数据对(x0,y0)一共有m对，则一般状况下，应当保证n<m.观察数据图形的曲线，有助于适当阶数的肯定。
%%求得x0,y0数组所给数据的n阶拟合多项式系数向量p,为了保证较好的拟合效果，多项式的阶数要取得适当，太低，残差较大；太高,拟合模型将包含噪声影响。
P=polyfit(x0,y0,n)
%图示拟合状况
xx=0:0.01:1;
yy=polyval(P,xx);
% y = polyval(p,x) returns the value of a polynomial of degree n evaluated
% at x. The input argument p is a vector of length n+1 whose elements are
% the coefficients in descending powers of the polynomial to be evaluated.
% 
% y = p1xn + p2xn–1 + … + pnx + pn+1
plot(xx,yy,'-b',x0,y0,'.r','MarkerSize',20)
legend('拟合曲线','原始数据','Location','SouthEast')
xlabel('x')

最小二乘

Least squares problems fall into two categories: linear or ordinary least squares and non-linear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regression analysis; it has a closed-form solution. The non-linear problem is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, and thus the core calculation is similar in both cases.

实验二

%%给定自变量x数据组
x0=(0:0.1:1);
%%给定自变量y数据组
y0=[-.447,1.978,3.11,5.25,5.02,4.66,4.01,4.68,4.45,5.35,9.22]';
m=length(x0);
n=3;
%多项式阶数
%%初始化数据阵X
X=zeros(m,n+1);
%%将X设置成各列等比的形式
for k=1:n
    X(:,n-k+1)=(x0.^k);
end
X(:,n+1)=ones(m,1);
%%利用最小二乘原理求解多项式系数
aT=(X\y0)'

aT =    51.9713  -80.2234   37.7844   -0.8078