10-3 ������������������������ fminsearch

۹ӨADuʰjk]Nonlinear Regression^O@ӤxDA]pUG

HƾǨӴyzA]ҥΪƾǼҫO $y=f(\mathbf{x}, \mathbf{\theta})$A䤤 $f(\mathbf{x}$ OJVqA$\mathbf{\theta})$ OiܫDuʨơA$y$ OXܼơAh`~t $$ E(\mathbf{\theta}) = \sum_{i=1}^n (y_i - f(\mathbf{x}_i, \mathbf{\theta}))^2 $$

䤤 $(\mathbf{x}_i, y_i)$ O $i$ ӤwICѩ $\mathbf{\theta}$ O $f$ DuʰѼơAҥH $E(\mathbf{\theta})$ äO $\mathbf{\theta}$ GA]ڭ̨õLk $\mathbf{\theta}$ $\mathbf{\theta}$ ɦsӸѥX̨Ϊ $\mathbf{\theta}$ ȡChӨD䦸Aڭ̥Τ@̨Τơ]Optimization^kAӧX $E(\mathbf{\theta})$ ̤pȡAҦpפUk]Gradient Descent^AάO Simplex UYjM]Simplex Downhill search^C

|ҨӻA]ҥΪƾǼҫ

$$ y= a_1 e^{\lambda_1 x} + a_2 e^{\lambda_2 x} $$

䤤A$a_1$B$a_2$ uʰѼơA $\lambda_1$B$\lambda_2$ DuʰѼơAhҫDuʡA`~tiܦpUG $$ E(a_1, a_2, \lambda_1, \lambda_2) = \sum_{i=1}^{m} (y_i - a_1 e^{\lambda_1 x_i} - a_2 e^{\lambda_2 x_i})^2 $$

X $E(a_1, a_2, \lambda_1, \lambda_2)$ ̤p $a_1$B$a_2$B$\lambda_1$ $\lambda_2$Aڭ̻ݱN $E$ g@ MATLAB 禡AåѨ䥦̨ΤƪkӨDX禡̤pȡC]禡 errorMeasure1.mA䤺eiܦpUG

Example 1: 10-uXPjkR/errorMeasure1.mfunction squaredError = errorMeasure1(theta, data) if nargin<1; return; end x = data(:,1); y = data(:,2); y2 = theta(1)*exp(theta(3)*x)+theta(2)*exp(theta(4)*x); squaredError = sum((y-y2).^2);

䤤 theta OѼƦVqA]tF $a_1$B$a_2$B$\lambda_1$ $\lambda_2$Adata hO[쪺IAǦ^ȫhO`~tCDX禡̤pȡAڭ̥iϥ fminsearch OAШUCdҡG

Example 2: 10-uXPjkR/nonlinearFit01.mload data.txt theta0 = [0 0 0 0]; tic theta = fminsearch(@(x)errorMeasure1(x, data), theta0); fprintf('pɶ = %g\n', toc); x = data(:, 1); y = data(:, 2); y2 = theta(1)*exp(theta(3)*x)+theta(2)*exp(theta(4)*x); plot(x, y, 'ro', x, y2, 'b-'); legend('Sample data', 'Regression curve'); fprintf('~tM = %d\n', sum((y-y2).^2));pɶ = 0.0762827 ~tM = 5.337871e-01

WϪuY fminsearch OҲͪjkuCbWz{Adata x}]tܼơ]Ydata(:,1)^M]ܼơ]Y data(:,2)^AHKNǤJ禡 errorMeasure1.mCTheta0 hOiܰѼ theta _lȡCfminsearch OhO@Өϥ Simplex UYjMk]Downhill Simplex Search^̨ΤƤkAΨӧX errorMeasure1 pȡAöǦ^ theta ̨έȡCԪkӸ`AiԾ\̪t@ۧ@uNeural-Fuzzy and Soft Computing V A Computational Approach to Learning and Machine IntelligencevAPrentice Hall A1997C


MATLAB{]pGig