## 10-5 ?炵??ц看姝革?浣跨敤璁婂舰娉?

礛癸计眔

$$\ln y = \ln a + bx$$

$\ln a$ の $b$ 跑Θ絬┦把计иノ¨程キよ猭〃тㄤ叫ǎㄏノ跑猭絛ㄒ

Example 1: 10-Ρ絬览籔癹耴だ猂/transformFit01.mload data2.txt x = data2(:, 1); % 戈翴 x 畒夹 y = data2(:, 2); % 戈翴 y 畒夹 A = [ones(size(x)) x]; theta = A\log(y); subplot(2,1,1) plot(x, log(y), 'o', x, A*theta); xlabel('x'); ylabel('ln(y)'); title('ln(y) vs. x'); legend('Actual value', 'Predicted value'); a = exp(theta(1)) % 侩醚眔ぇ把计 b = theta(2) % 侩醚眔ぇ把计 y2 = a*exp(b*x); subplot(2,1,2); plot(x, y, 'o', x, y2); xlabel('x'); ylabel('y'); legend('Actual value', 'Predicted value'); title('y vs. x'); fprintf('粇畉キよ㎝ = %d\n', sum((y-y2).^2)); a = 4.3282 b = -1.8235 粇畉キよ㎝ = 8.744185e-01

瓃瓜い材瓜琌 $\ln y$ 癸 $x$ 瓜τ材玥琌 $y$ 癸 $x$ 瓜叫疭猔種竒パ跑猭ぇ程キよ猭┮眔程羆キよ粇畉琌 $$E'=\sum_{i=1}^m (\ln y_i - \ln a - bx_i)^2$$

τぃ琌家羆キよ粇畉 $$E=\sum_{i=1}^m (y_i - ae^{bx_i})^2$$

Example 2: 10-Ρ絬览籔癹耴だ猂/transformFit02.mload data2.txt x = data2(:, 1); % 戈翴 x 畒夹 y = data2(:, 2); % 戈翴 y 畒夹 A = [ones(size(x)) x]; theta = A\log(y); a = exp(theta(1)) % 侩醚眔ぇ把计 b = theta(2) % 侩醚眔ぇ把计 theta0 = [a, b]; % fminsearch 币﹍把计 theta = fminsearch(@(x)errorMeasure3(x, data2), theta0); x = data2(:, 1); y = data2(:, 2); y2 = theta(1)*exp(theta(2)*x); plot(x, y, 'o', x, y2); xlabel('x'); ylabel('y'); legend('Actual value', 'Predicted value'); title('y vs. x'); fprintf('粇畉キよ㎝ = %d\n', sum((y-y2).^2)); a = 4.3282 b = -1.8235 粇畉キよ㎝ = 1.680455e-01

パ瓃絛ㄒиㄏノ跑猭т菠把计礛ノ fminsearch ㄓ癸粇畉キよ㎝秈︽程て眔粇畉キよ㎝穦ゑ玡絛ㄒノ跑猭临璶讽礛硂絛ㄒいи﹚竡ゲ斗砆程て粇畉羆㎝ㄧΑ errorMeasure3.m竊絞碩ㄤず甧ぃ弄叫︽把σ盒い郎

ㄆ龟癬ㄓ琌獶絬┦癹耴家竒筁ㄇ锣传碞穦跑Θ絬┦癹耴家碞ㄏノи硂竊┮弧跑猭ㄓ秈︽絬┦癹耴把计―琌ㄇ跑猭ノ獶絬┦家の闽锣传よ猭

 絪腹 獶絬┦家 锣传絬┦家 把计锣传そΑ 1 $y=\frac{ax}{1+bx}$ $\underbrace{\frac{1}{y}}_Y=\underbrace{\frac{1}{a}}_\alpha \frac{1}{x}+\underbrace{\frac{b}{a}}_\beta$ $a=\frac{1}{\alpha}$, $b=\frac{\beta}{\alpha}$ 2 $y=\frac{a}{x+b}$ $\underbrace{\frac{1}{y}}_Y=\underbrace{\frac{1}{a}}_\alpha x+\underbrace{\frac{b}{a}}_\beta$ $a=\frac{1}{\alpha}$, $b=\frac{\beta}{\alpha}$ 3 $y=\frac{ax}{x^2+b^2}$ $\underbrace{\frac{1}{y}}_Y=\underbrace{\frac{1}{a}}_\alpha x+\underbrace{\frac{b^2}{a}}_\beta \frac{1}{x}$ $a=\frac{1}{\alpha}$, $b=\pm\sqrt{\frac{\beta}{\alpha}}$ 4 $y=ax^b$ $\underbrace{\ln y}_Y = \underbrace{b}_\alpha \ln x + \underbrace{\ln a}_\beta$ $a=e^\beta, b=\alpha$ 5 $y=\frac{1}{1+ax^b}$ $\underbrace{\ln \frac{1-y}{y}}_Y = \underbrace{b}_\alpha \ln x + \underbrace{\ln a}_\beta$ $a=e^\beta, b=\alpha$ 6 $y=\frac{1}{1+exp\left(\frac{ax}{b+x}\right)}$ $\underbrace{\left[\ln \frac{1-y}{y}\right]^{-1}}_Y=\underbrace{\frac{b}{a}}_\alpha \frac{1}{x}+\underbrace{\frac{1}{a}}_\beta$ $a=\frac{1}{\beta}, b=\frac{\alpha}{\beta}$ 7 $y=\ln a + x - \ln (e^x+b)$ $\underbrace{e^{-y}}_Y=\underbrace{\frac{b}{a}}_\alpha e^x+\underbrace{\frac{1}{a}}_\beta$ $a=\frac{1}{\beta}, b=\frac{\alpha}{\beta}$ 8 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $\underbrace{y^2}_Y=\underbrace{-\frac{b^2}{a^2}}_\alpha x^2+\underbrace{b^2}_\beta$ $a^2=-\frac{\beta}{\alpha}, b^2=\beta$ 9 $(x-a)^2+(y-b)^2=c^2$ $\underbrace{x^2+y^2}_Y=\underbrace{2a}_\alpha x+\underbrace{2b}_\beta y+\underbrace{c^2-a^2-b^2}_\gamma$ $a=\alpha/2, b=\beta/2\\c=\gamma+\frac{\alpha^2}{4}+\frac{\beta^2}{4}$ 10 $y=a e^{\left[-\left(\frac{x-c}{b}\right)^2\right]}$ $\underbrace{\ln y}_Y = \underbrace{-\frac{1}{b^2}}_\alpha x^2+ \underbrace{\frac{2c}{b^2}}_\beta x+\underbrace{\ln a-\frac{c^2}{b^2}}_\gamma$ $a=exp \left( \gamma - \frac{\beta^2}{4}\right)\\b=\pm \sqrt{-\frac{1}{\alpha}}\\c=-\frac{\beta}{2\alpha}$ 11 $y=\frac{a}{\sqrt{(1+bx^2)^2+c}}$ $\underbrace{\frac{1}{y^2}}_Y=\underbrace{\frac{b^2}{a^2}}_\alpha x^4+\underbrace{\frac{2b}{a^2}}_\beta x^2+\underbrace{\frac{c+1}{a^2}}_\gamma$ $a=\pm\frac{\sqrt{4\alpha}}{\beta}\\b=\frac{2\alpha}{\beta}\\c=\frac{4\alpha\gamma}{\beta^2}-1$

MATLAB祘Α砞璸秈顶絞