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biJDeAڭ̥wqXӱ`ΪƾǹBAèϥίx}ӪܳoǹB⦡C

x}m]Transpose^UCG $$ (AB)^T=B^TA^T $$

x}ϯx}]Inverse^B⺡UCG $$ (AB)^{w1}=B^{-1}A^{-1} $$

@ӯ¶q $f(\mathbf{x})$ ס]Gradient^󦹨ƹC@ܼưLҧΦVqG $$ \nabla f(\mathbf{x}) = \left[ \begin{matrix} \partial f(\mathbf{x})/\partial x_1 \\ \vdots \\ \partial f(\mathbf{x})/\partial x_n \\ \end{matrix} \right], 䤤 \mathbf{x}=\left[\begin{matrix} x_1\\ \vdots \\ x_n \\ \end{matrix}\right] $$

Vqܼ $\mathbf{x}$ G]Quodratic Form^iHܦpUG $$ \mathbf{x}^TA\mathbf{x} = \sum_{i=1}^n a_{ii}x_i^2 + \sum_{i=1}^n \sum_{j=1, j \neq i}^n a_{ij} x_i x_j, $$ 䤤ڭ̥iH] $A$ O@ӹٯx}A]pG $A$ ١Aڭ̥iHϥ $(A+A^T)/2$ ӨNӤ|ܭӪGG $$ \mathbf{x}^TA\mathbf{x} = \mathbf{x}^T \left(\frac{A+A^T}{2} \right)\mathbf{x} $$

G $\mathbf{x}^TA\mathbf{x}$ ץiHܦpUG $$ \nabla (\mathbf{x}^TA\mathbf{x}) = \left(\frac{A+A^T}{2}\right) \mathbf{x} $$

XeƾǩwqAڭ̥iHoUCXӫ]]ҦקO $\mathbf{x}$ ӶiAð] $A$ ٯx}^G

FeoǫAڭ̴NiHNΦb̤pkɡC]ڭ̭nѪDO $$ A\mathbf{\theta}=\mathbf{y} $$ 䤤 $A$ O@ $m \times n$ wx}A$\mathbf{y}$ O@ $m \times 1$ wVqA $\mathbf{\theta}$ hO@ $n \times 1$ VqCڭ̰] $m>n$AbpUA{ӼƤj󥼪ƭӼơA]WLTѡAϤWߡA[W@~tVq $\mathbf{e}$G $$ A\mathbf{\theta}=\mathbf{y}+\mathbf{e} $$ ~thig

$$ E(\mathbf{\theta})=\|\mathbf{e}\|^2=\mathbf{e}^T\mathbf{e}= (A\mathbf{\theta}-\mathbf{y})^T(A\mathbf{\theta}-\mathbf{y}) $$

ѩTѨäsbA]ڭ̰hӨD䦸AאּMDϥ~t $E(\mathbf{\theta})$ ̤p $\mathbf{\theta}$ ȡCѩ $E(\mathbf{\theta})$ O $\mathbf{\theta}$ G{A]ڭ̥iH $E(\mathbf{\theta})$ i氾LAåO䵥sAYio@ $n$ @upߤ{ӸѥX̨Ϊ $\mathbf{\theta}$ ȡCyܻAڭ̥iHp $E(\mathbf{\theta})$ סG $$ \begin{array}{rcl} \nabla E(\mathbf{\theta}) & = & \nabla ( (A\mathbf{\theta}-\mathbf{y})^T(A\mathbf{\theta}-\mathbf{y}) )\\ & = & \nabla ( (\mathbf{\theta}^TA^T-\mathbf{y}^T)(A\mathbf{\theta}-\mathbf{y}) )\\ & = & \nabla ( \mathbf{\theta}^TA^TA\mathbf{\theta}-\mathbf{\theta}^TA^T\mathbf{y} - \mathbf{y}^TA\mathbf{\theta} + \mathbf{y}^T\mathbf{y} )\\ & = & \nabla ( \mathbf{\theta}^TA^TA\mathbf{\theta} - 2\mathbf{\theta}^TA^T\mathbf{y} + \mathbf{y}^T\mathbf{y} )\\ & = & \nabla (\mathbf{\theta}^TA^TA\mathbf{\theta}) - 2\nabla(\mathbf{\theta}^TA^T\mathbf{y}) + \nabla(\mathbf{y}^T\mathbf{y}) \\ & = & 2A^TA\mathbf{\theta}-2A^T\mathbf{y} \end{array} $$

OW׵sAYio $\mathbf{\theta}$ ̨έȡG $$ \hat{\mathbf{\theta}} = (A^TA)^{-1}A^T\mathbf{y} $$

Hint
̤pkUجʽAiѦҵ̥t@ۧ@GuNeuroVFuzzy and Soft ComputingvAPrentice HallA1997C


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