10-1 線性迴歸?????

@ӨAuX]Curve Fitting^ҫإߪƾǼҫOuJBXv]Single-input Single-outputA²SISO^AҥHSʥiΤ@uӪܡCYإߨ㦳ӿJƾǼҫAhSʥiΤ@ӦӪܡADi٬X]Surface Fitting^CۦPAiH@몺uhJBXv]MISO^ ΡuhJBhXv]MIMO^ƾǼҫAҦpg]Artificial Neural Networks^C

L׬OuXάOX]άO䥦hJҫXD^AbƤRW٬jkR]Regression Analysis^Aκ٬X]Data Fitting^AoA쪺ƾDzz׻PRޥ۷sxCbAڭ̥uдXس̰򥻪kAHΦpϥ MATLAB ӹ{oǰ򥻪kC

jkRPҨϥΪƾǼҫܤjYApGҨϥΪҫOuʼҫAhD٬uʰjk]Linear Regression^FϤAYϥΫDuʼҫAh٬Duʰjk]Nonlinear Regression^C`NнuʰjkAU`hЫDuʰjuC

uʰjkObjkṞ`ΪkAзǸѪkάƾǩʽi@έpθƤRЬѡA]wzפμƾDZɸԭzAŪ̥iۦѦҬyCHUNH²dҨӻpϥ MATLAB ӶiuʰjkC

]ڭ̪[ƬO 1790 1990 ~]H 10 ~@^`HfAƥiѸJɮ census.mat oApUG

Example 1: 10-uXPjkR/censusPlot01.mload census.mat % JHf plot(cdate, pop, 'o'); % cdate N~סApop NHf` xlabel('~'); ylabel('Hf`');

䤤 cdate ~סApop Hf`ơA̳O׬ 21 VqC]ڭ̭nwb褸 2000 ~ 2010 ~Hf`ơAڭ̴NھڤWzƨӫإߤ@ƾǼҫAè̦ҫӶiwC

Hint
oӡuHf~תơvdҡAO@ӫܦѪdҡ] MATLAB ٦ѡ^A̦X{bѤTjv ForsytheBMalcolmBMoler ҼgѡuComputer Methods for Mathematical ComputationsvA 1977 ~ Prentice-Hall XҥXC

ѤWϸIաAڭ̥iH[oAqLoIuiO@GߪuAҥHڭ̥iH]һݪƾǼҫ $$ y = f(x; a_0, a_1, a_2) = a_0+a_1x+a_2x^2 $$

䤤 $y$]Hf`ơ^ҫXA$x$]~ס^ҫJA$[a_0,a_1, a_2]$ hҫѼơ]Parameters^CѩoǰѼƬ۹X y OeuYAҥHҫ٬u㦳uʰѼơ]Linear-in-the-parameters^vҫAڭ̪ȫhOX̦nѼƭȡAϱoҫXPڸƶVVnAL{Y٬uʰjk]Linear Regression^C

]ڭ̪[ƥig $(x_i, y_i), i = 1 - 21$CJ $x_i$ ɡAڿX $y_i$AҫwȬ $f(x_i; a_0, a_1, a_2) = a_0+a_1x_i+a_2x_i^2$A]~t $[y_i-f(x_i)]^2$A`~t hiܦpUG $$ E(a_0, a_1, a_2) = \sum_{i=1}^{21}[y_i-f(x_i)]^2 = \sum_{i=1}^{21}[y_i-(a_0+a_1x_i+a_2x_i^2)]^2 $$

Wz~t $E$ OѼ $a_0$B$a_1$B$a_2$ ơA]Ѱ򥻷LnoAYnDo $E$ ̤pȡAڭ̥iHDX $E$ $a_0$B$a_1$B$a_2$ ɦAO䬰sAYiѥX $a_0$B$a_1$B$a_2$ ̨έȡCi@B[Aڭ̪DҫuʰѼơAҥH~t $E$ $a_0$B$a_1$B$a_2$ GAӾɦ $\frac{\partial E}{\partial a_0}$B$\frac{\partial E}{\partial a_1}$ $\frac{\partial E}{\partial a_2}$ $a_0$B$a_1$B$a_2$@A]bOɦsAڭ̥iHo@դT@upߤ{ANiHѥXѼ $a_0$B$a_1$B$a_2$ ̨έȡC

qt@譱ӬݡA] 21 [IqLߪuAhڭ̥iHNo 21 IaJߪu{AoUC 21 ӵG

$$ \left\{ \begin{matrix} a_0 + a_1 x_1 + a_2 x_1^2 & = & y_1\\ a_0 + a_1 x_2 + a_2 x_2^2 & = & y_1\\ \vdots & = & \vdots \\ a_0 + a_1 x_{21} + a_2 x_{21}^2 & = & y_{21}\\ \end{matrix} \right. $$

ig

$$ \underbrace{ \left[ \begin{matrix} 1 & x_1 & x_1^2\\ 1 & x_2 & x_2^2\\ \vdots & \vdots & \vdots\\ 1 & x_{21} & x_{21}^2\\ \end{matrix} \right] }_A \underbrace{ \left[ \begin{matrix} a_1\\ a_2\\ a_3\\ \end{matrix} \right] }_\theta = \underbrace{ \left[ \begin{matrix} y_1\\ y_2\\ \vdots\\ y_{21}\\ \end{matrix} \right] }_y $$

䤤 $A$B$y$ wA$\theta$ VqCܩ㪺AWz{էt 21 Ӥ{Aou 3 ӥ $\theta=\left[\theta_1, \theta_2, \theta_3 \right]^T$AҥHq`sb@ոѨӺo 21 Ӥ{Cb@뱡pUAڭ̥u@ $\theta$Aϱo䪺t̤pAtig

$$ \left\| y-A\theta \right\|^2 = (y-A\theta)^T(y-A\theta) $$

Yez`~t $E$C

ѩDbuʰjkg`IAҥH MATLAB Ѥ@²Kuv]\^OAӸѥX̨Ϊ $\theta$C]ڭ̦b̫@p`|pHx}ާ@ӺX $\theta$ ̨έȡC^bHUdҤAڭ̧QΡuvӺX̨Ϊ $\theta$ ȡAæPɵeX㦳̤p~tGuApUG

Example 2: 10-uXPjkR/census01.mload census.mat % JHf plot(cdate, pop, 'o'); % cdate N~סApop NHf` A = [ones(size(cdate)), cdate, cdate.^2]; y = pop; theta = A\y; % QΡuvAX̨Ϊ theta plot(cdate, pop, 'o', cdate, A*theta, '-'); legend('ڤHf', 'wHf'); xlabel('~'); ylabel('Hf`');

Hint
  • AiOЦpUG{O A*theta = yAڭ̥iN AܵkAӱo theta = A\yCpߪOG A bĤ@AҥH쵥kAA MOĤ@C
  • Yڭ̭nѪ{O theta*A = yAhP˪io̤p theta = A/yC
  • uvԲӻAiѡuuʥNơvĥ|`up߽uʤ{vC

ѤWzdҡAڭ̥iHX̨Ϊ $\theta=[a_0, a_1, a_2]=[21130, 23.51, 0.00654]$A]㦳̤p~tߪuiHgG $$ y = f(x) = a_0 + a_1 x + a_2 x^2 = 21130 + 23.51 x + 0.00654 x^2 $$

ھڹϧΡA{wg۷Gҵ 21 [ICھڦߪuƾǼҫAڭ̥iHwb 2000 M 2010 ~Hf`ƬG

Example 3: 10-uXPjkR/census02.mload census.mat % JHf A = [ones(size(cdate)), cdate, cdate.^2]; theta = A\pop; % QΡuvAX̨Ϊ theta t=2000; pop2000 = [1, t, t^2]*theta; % b 2000 ~Hfuƹw t=2010; pop2010 = [1, t, t^2]*theta; % b 2010 ~Hfuƹw fprintf('Hfb2000~w = %g ]ʸUH^\n', pop2000); fprintf('Hfb2010~w = %g ]ʸUH^\n', pop2010);Hfb2000~w = 274.622 ]ʸUH^ Hfb2010~w = 301.824 ]ʸUH^

Hint
йڰݬݬ Google jAݬݬb 2000 M 2010 ~HfUOh֡Hڭ̳oӹwҫǶܡH

bWzҤlAڭ̪ƾǼҫO@ӤGߪuAڭ̥iNsAo@ n hG

$$y = f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$$

QΦhƾǼҫӶi榱uXAq٬uhX]Polynomial Fitting^vA]ObƤR`Ψ쪺ޥCwhXAMATLAB ѤF polyfit OӧX̨ΰѼơAĪGMuv]\^O@PC@X̨ΰѼƫAڭ̥iHϥ polyval OӭphȡA{X[²AШUCdҡG

Example 4: 10-uXPjkR/census03.mload census.mat % JHf theta = polyfit(cdate, pop, 2); % iGhXAX theta fprintf('b2000~w = %g ]ʸUH^\n', polyval(theta, 2000)); fprintf('b2010~w = %g ]ʸUH^\n', polyval(theta, 2010));b2000~w = 274.622 ]ʸUH^ b2010~w = 301.824 ]ʸUH^

bWzdҤAupolyfit(cdate, pop, 2)v 2 Nڭ̥Ψ쪺ҫO 2 hCϥ polyfit M polyvalAOO{X²uܦhOH polyfitBpolyval Ψ䥦hOAiѡuhBzΤRv`C

uʰjk\P_APҿҫܤjYCҫҧtѼƶVhA~t|VpAYѼƭӼƵIӼơAhb@뱡pUA~t|sAoäܹw|̷ǡA]ItTAkXƪҫNҫTvT̤jAwǽTפ]|tC]Auҫסv]YiܰѼƪӼơ^MuwǽTסvOۤܿŪӦ]ALk̭ݱoA]ڭ̥gѹDAѡAӧ@ӥIC

Yb MATLAB UJucensusvANiHݨϥΤPƪhAӹ census ƶi榱uXGApUG

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