最小二乘曲線擬合matlab實現
阿新 • • 發佈：2019-02-12
clear;
close all;
%% sample
x_all = 0:0.1:10;
[y_truth_all,y_all]=unknown_model1(x_all);
N = length(x_all);
%use the first half for training
x = x_all(1:N/2);
y = y_all(1:N/2);
y_truth = y_truth_all(1:N/2);
plot(x,y_truth,'g-x','LineWidth',1.5);
hold on;
plot(x,y,'m-x','LineWidth',1.5);
legend('model truth','observation');
title('training');
%% curve fitting, try linear model
X = [x;ones(1,length(x))]';
Y = y';
lambda1 = (X'*X)\X'*Y; ％最小二乘擬合 (least-square fitting)
% evaluate the esimated model
ye1 = X*lambda1;
hold on;
plot(x,ye1,'c-o','LineWidth',1.5);
%% curve fitting, try order-3 order polynomial
X = [x.^3;x.^2;x;ones(1,length(x))]';
Y = y';
lambda2 = (X'*X)\X'*Y;％最小二乘擬合
% evaluate the esimated model
ye2 = X*lambda2;
hold on;
plot(x,ye2,'r-o','LineWidth',1.5);
%% curve fitting, try order-5 order polynomial
X = [x.^5;x.^4;x.^3;x.^2;x;ones(1,length(x))]';
Y = y';
lambda3 = (X'*X)\X'*Y;％最小二乘擬合
% evaluate the esimated model
ye3 = X*lambda3;
hold on;
plot(x,ye3,'b-o','LineWidth',1.5);
%%
figure;
plot(x,y,'g-x',x,ye1,'c-o',x,ye2,'r-o',x,ye3,'b-o','LineWidth',1.5);
legend('model truth', 'order-1','order-3', 'order-5');
以上是訓練的程式碼。由圖可見，３階和5階與真實模型最為接近，一階的誤差較大。
下面根據估計的模型對未來未知的資料進行預測，比較各模型與真實模型的誤差大小
%% show future trends
close all;
%% sample
x_all = 0:0.1:10;
[y_truth_all,y_all]=unknown_model1(x_all);
N = length(x_all);
%use the first half for training
x = x_all(1:N/2);
y = y_all(1:N/2);
y_truth = y_truth_all(1:N/2);
plot(x,y_truth,'g-x','LineWidth',1.5);
hold on;
plot(x,y,'m-x','LineWidth',1.5);
legend('model truth','observation');
title('training');
%% curve fitting, try linear model
X = [x;ones(1,length(x))]';
Y = y';
lambda1 = (X'*X)\X'*Y; ％最小二乘擬合 (least-square fitting)
% evaluate the esimated model
ye1 = X*lambda1;
hold on;
plot(x,ye1,'c-o','LineWidth',1.5);
%% curve fitting, try order-3 order polynomial
X = [x.^3;x.^2;x;ones(1,length(x))]';
Y = y';
lambda2 = (X'*X)\X'*Y;％最小二乘擬合
% evaluate the esimated model
ye2 = X*lambda2;
hold on;
plot(x,ye2,'r-o','LineWidth',1.5);
%% curve fitting, try order-5 order polynomial
X = [x.^5;x.^4;x.^3;x.^2;x;ones(1,length(x))]';
Y = y';
lambda3 = (X'*X)\X'*Y;％最小二乘擬合
% evaluate the esimated model
ye3 = X*lambda3;
hold on;
plot(x,ye3,'b-o','LineWidth',1.5);
%%
figure;
plot(x,y,'g-x',x,ye1,'c-o',x,ye2,'r-o',x,ye3,'b-o','LineWidth',1.5);
legend('model truth', 'order-1','order-3', 'order-5');
以上是訓練的程式碼。由圖可見，３階和5階與真實模型最為接近，一階的誤差較大。
下面根據估計的模型對未來未知的資料進行預測，比較各模型與真實模型的誤差大小
%% show future trends