ⵜⵓⵖⴰⵍⵉⵏ ⵜⴰⵣⴻⴳⵣⴰⵡⵜ

ⵜⵓⵙⵏⴰⴽⵜ
ⵜⵓⵖⴰⵍⵉⵏ ⵜⴰⵣⴻⴳⵣⴰⵡⵜ
ⵜⵓⵖⴰⵍⵉⵏ ⵜⴰⵣⴻⴳⵣⴰⵡⵜ cover image

ⵜⴰⵣⵙⴰⵔⴻⵜ

ⵢⴻⵜⵜⵓⵏⴻⴼⴽⴷ ⵢⵉⵡⴻⵏ ⵏ ⵓⵙⵎⴻⵍ ⵏ ⵢⵉⵙⴻⴼⴽⴰ ɣD = {(X_{1}, Y_{2}), \ⵜⵉⵏⵇⵉⴹⵉⵏ,(X_{N}, Y_{N})}$ ⴰⵎ ɣX_{ⵉ}ⵖ ⴷ ɣY_{ⵉ }ⵖ ⵜⵜⴽⴻⵎⵎⵉⵍⴻⵏ, Iswi ⵏ "Linear Regression" ⴷ ⵜⵓⴼⴼⵖⴰ ⵏ ⵜⴼⴻⵍⵡⵉⵜ ⵉⴳⴻⵔⵔⵣⴻⵏ ⴰⴽⴽ ⵉ ⴷⵢⴻⵜⵜⴰⵡⵉⵏ ⵉⵙⴻⴼⴽⴰⴰⴳⵉ.

ⵙ ⵡⴰⵡⴰⵍⴻⵏ ⵏⵏⵉⴹⴻⵏ, ⵏⴻⴱⵖⴰ ⴰⴷ ⴷⵏⴻⵙⵏⵓⵍⴼⵓ ⴰⵎⵣⵓⵏ :

y^=a∗0+a∗1.x∗1+⋯+a∗p.x_p\hat{y} = a*{0} + a*{1}.x*{1} + \dots + a*{p}.x\_{p}

ⴰⵏⴷⴰ ɣpɣ ⴷ ⴰⵎⴹⴰⵏ ⵏ ⵜⵎⵉⴹⵔⴰⵏⵜ ⵏ ⵓⵎⴳⵉⵔⴻⴷ ⵖⵅⵖ.

ⴷⴻⴳ ⵓⵎⴰⴳⵔⴰⴷⴰ ⴰⴷ ⵏⵡⴰⵍⵉ ⴰⵎⴻⴽ ⴰⵔⴰ ⵏⴻⴼⵔⵓ ⵓⴳⵓⵔⴰ ⴷⴻⴳ ⴽⵔⴰⴹ ⵏ ⵜⵎⵓⵖⵍⵉⵡⵉⵏ :

  • ⵎⵉ ⴰⵔⴰ ⵢⵉⵍⵉ X ⴷ ⵢⵉⵡⴻⵏ ⵏ ⵓⵙⵡⵉⵔ ⵉ.ⴻ. p=1p=1.

  • ⵎⵉ ⴰⵔⴰ ⵢⵉⵍⵉ X ⴷ ⴰⵟⴰⵙ ⵏ ⵜⵖⴰⵡⵙⵉⵡⵉⵏ ⵉ.ⴻ. ɣp>1ɣ.

  • ⴰⵙⴻⵇⴷⴻⵛ ⵏ ⵜⵎⴻⵥⵥⵓⵖⵜ ⵏ ⵜⵎⴻⵥⵥⵓⵖⵜ.

ⵖⵅⵖ ⴷ ⵢⵉⵡⴻⵏ ⵏ ⵓⵙⵡⵉⵔ (ⴰⵎⴽⵓⵥ ⴰⵎⴻⵥⵢⴰⵏ ⵓⵙⵍⵉⴳ)

ⴰⵎⵣⵓⵏ ⵉ ⵏⴻⴱⵖⴰ ⴰⴷ ⴷⵏⴻⵙⵏⵓⵍⴼⵓ ⴷ ⵡⵉⵏ ⵏ ⵜⴰⵍⵖⴰ:

y^=a∗0+a∗1.x\hat{y} = a*{0} + a*{1}.x

ⵛⴼⵓ ⴱⴻⵍⵍⵉ ⵉⵙⵡⵉ ⵏ ⵜⴳⴻⵔⵎⴰⵏⵜ ⵏ ⵜⴼⴻⵍⵡⵉⵜ ⴷ ⵜⵓⴼⴼⵖⴰ ⵏ ⵜⴼⴻⵍⵡⵉⵜ ⵉ ⴷⵢⴻⵜⵜⴰⵡⵉⵏ ⵙ ⵡⴰⵣⴰⵍⵉⵙ ⵉ ⵢⵉⵙⴻⴼⴽⴰ. ⵙ ⵡⴰⵡⴰⵍⵏⵏⵉⴹⴻⵏ, ⵢⴻⵙⵙⴻⴼⴽ ⴰⴷ ⵏⴻⵙⵙⴻⵎⵥⵉ ⴰⵣⴰⵍ ⵢⴻⵍⵍⴰⵏ ⴳⴰⵔ ⵜⵏⴻⵇⴹⵉⵏ ⵏ ⵢⵉⵙⴻⴼⴽⴰ ⴷ ⵜⵖⴻⵔⵖⴻⵔⵜ.

(a∗0^,a∗1^)=argmin⁡(a∗0,a∗1)∑∗i=1N(y∗i−y∗i^)2(\hat{a*{0}}, \hat{a*{1}}) = \underset{(a*{0}, a*{1})}{\operatorname{argmin}} \sum\limits*{i=1}^{N} (y*{i} - \hat{y*{i}})^2

=argmin⁡(a∗0,a∗1)∑∗i=1N(y∗i−(a∗0+a∗1.x∗i))2= \underset{(a*{0}, a*{1})}{\operatorname{argmin}} \sum\limits*{i=1}^{N} (y*{i} - (a*{0} + a*{1}.x*{i}))^2

ⴰⴷ ⵏⴻⵙⵙⴻⵔⵙ:

L=∑∗i=1N(y∗i−(a∗0+a∗1.x_i))2L = \sum\limits*{i=1}^{N} (y*{i} - (a*{0} + a*{1}.x\_{i}))^2

ⵉⵡⴰⴽⴽⴻⵏ ⴰⴷ ⵏⴰⴼ ⴰⵣⴰⵍ ⴰⵎⴻⵥⵢⴰⵏ, ⵢⴻⵙⵙⴻⴼⴽ ⴰⴷ ⵏⴻⴼⵔⵓ ⵜⵉⴳⵏⴰⵜⵉⵏⴰ :

{∂L∂a0=0∂L∂a1=0\begin{cases} \frac{\partial L}{\partial a_{0}} = 0\\ \frac{\partial L}{\partial a_{1}} = 0 \end{cases}
{∑i=1N−2(yi−(a0+a1.xi))=0∑i=1N−2xi(yi−(a0+a1.xi))=0\begin{cases} \sum\limits_{i=1}^{N} -2(y_{i} - (a_{0} + a_{1}.x_{i})) = 0\\ \sum\limits_{i=1}^{N} -2x_{i}(y_{i} - (a_{0} + a_{1}.x_{i})) = 0 \end{cases}

ⵏⴻⴱⴷⴰ ⵙ ⵓⵙⵏⴻⵔⵏⵉ ⵏ ⵜⴰⴳⴷⴰⵍⵜ ⵜⴰⵎⴻⵣⵡⴰⵔⵓⵜ:

∑i=1Nyi−∑i=1Na0+∑i=1Na1.xi=0\sum\limits_{i=1}^{N} y_{i} - \sum\limits_{i=1}^{N}a_{0} + \sum\limits_{i=1}^{N} a_{1}.x_{i} = 0\\
∑i=1Nyi−Na0+∑i=1Na1.xi=0\sum\limits_{i=1}^{N} y_{i} - Na_{0} + \sum\limits_{i=1}^{N} a_{1}.x_{i} = 0\\
a0=∑i=1NyiN−∑i=1NxiNa1a_{0} = \frac{\sum\limits_{i=1}^{N} y_{i}}{N} - \frac{\sum\limits_{i=1}^{N} x_{i}}{N}a_{1}
a0=Y−Xa1a_{0} = Y - Xa_{1}

ⵏⴻⵙⴱⴻⴷⴷⴻⵍ ⴷⴻⴳ ⵜⴰⴳⴷⴰⵍⵜ ⵜⵉⵙ ⵙⵏⴰⵜ:

∑i=1Nxi(yi−Y+Xa1−a1xi)=0\sum\limits_{i=1}^{N} x_{i}(y_{i} - Y + Xa_{1} - a_{1}x_{i}) = 0
∑i=1N(yi−Y)+a1(X−xi)=0\sum\limits_{i=1}^{N} (y_{i} - Y) + a_{1}(X - x_{i}) = 0
∑i=1N(yi−Y)−∑i=1Na1(xi−X)=0\sum\limits_{i=1}^{N} (y_{i} - Y) - \sum\limits_{i=1}^{N}a_{1}(x_{i} - X) = 0
a1=∑i=1N(yi−Y)∑i=1N(xi−X)=∑i=1N(yi−Y)(xi−X)∑i=1N(xi−X)2=COV(X,Y)VAR(X)a_{1} = \frac{\sum\limits_{i=1}^{N} (y_{i} - Y)}{\sum\limits_{i=1}^{N}(x_{i} - X)} = \frac{\sum\limits_{i=1}^{N} (y_{i} - Y)(x_{i} - X)}{\sum\limits_{i=1}^{N}(x_{i} - X)^2} = \frac{COV(X, Y)}{VAR(X)}

ⵏⴻⴱⴷⴻⵍⴷ ⵜⵓⵖⴰⵍⵉⵏ ⴷⴻⴳ $a_{0}ɣ:

{a0=Y−XCOV(X,Y)VAR(X)a1=COV(X,Y)VAR(X)\begin{cases} a_{0} = Y - X\frac{COV(X, Y)}{VAR(X)}\\ a_{1} = \frac{COV(X, Y)}{VAR(X)} \end{cases}

ⵖⵅⵖ ⴷ ⴰⵟⴰⵙ ⵏ ⵢⵉⵃⵔⵉⵛⴻⵏ (ⴰⵎⴽⵓⵥ ⴰⵎⴻⵥⵢⴰⵏ ⵓⵙⵍⵉⴳ)

ⴷⴻⴳ ⵜⴰⵍⵓⴼⵜⴰ, ɣX_{ⵉ}ⵖ ⵓⵔ ⵢⴻⵍⵍⵉ ⴰⵔⴰ ⴷ ⴰⵎⴹⴰⵏ ⵏ ⵜⵉⴷⴻⵜ, ⵎⴰⵛⴰ ⴷⴻⴳ ⵓⵎⴽⴰⵏⵉⵙ ⴷ ⴰⴼⴻⵔⴷⵉⵙ ⵏ ⵜⵖⴻⵔⵖⴻⵔⵜ pp:

X∗i=(X∗i1,X∗i2,…,X∗ip)X*{i} = (X*{i1},X*{i2},\dots,X*{ip})

ⵉⵀⵉ, ⴰⵎⵣⵓⵏ ⵢⴻⵜⵜⵡⴰⵔⵓ ⴰⴽⴽⴻⵏ ⵉ ⴷⵉⵜⴻⴷⴷⵓⵏ:

y^=a∗0+a∗1x∗1+a∗2x∗2+⋯+a∗px_p\hat{y} = a*{0} + a*{1}x*{1} + a*{2}x*{2} + \dots + a*{p}x\_{p}

ⵏⴻⵖ, ⵢⴻⵣⵎⴻⵔ ⴰⴷ ⵢⴻⵜⵜⵡⴰⵔⵓ ⵙ ⵜⴰⵍⵖⴰ ⵏ ⵎⴰⵜⵔⵉⵅ:

Y^=X.W\hat{Y} = X.W

ⴰⵏⴷⴰ:

  • ⵖⵢⵖ ⵉⴳⴰ ⵜⴰⵍⵖⴰ ⵖ(ⵏ, 1)ⵖ.

  • ⵖⵅⵖ ⵉⴳⴰ ⵜⴰⵍⵖⴰ ⵖ(ⵏ, p)ⵖ.

  • ⵖⵡⵖ ⴷ ⵜⴰⵍⵖⴰ ⵖ(p, 1)ⵖ: ⴷ ⴰⴼⴻⵔⴷⵉⵙ ⵏ ⵢⵉⴼⴻⵔⴷⵉⵙⴻⵏ ⵖ(ⵡ_{1}, ⵖ_{2}, \ⵜⵉⵏⵇⵉⴹⵉⵏ, ⵡ_{p})ⵖ.

ⴰⵎ ⵡⴰⴽⴽⴻⵏ ⵉ ⴷⵢⴻⵍⵍⴰ ⴷⴻⴳ ⵜⴰⵍⵓⴼⵜ ⵜⴰⵎⴻⵣⵡⴰⵔⵓⵜ, ⵏⴻⵙⵄⴰ ⵉⵙⵡⵉ ⵏ ⵓⵙⵏⴻⴼⵍⵉ ⵏ ⵜⵖⴰⵡⵙⴰⴰⴳⵉ ⵉ ⴷⵉⵜⴻⴷⴷⵓⵏ :

W^=argmin⁡W∑∗i=1N(y∗i−y_i^)2\hat{W} = \underset{W}{\operatorname{argmin}} \sum\limits*{i=1}^{N} (y*{i} - \hat{y\_{i}})^2

ⵉ ⵜⵉⴽⴽⴻⵍⵜ ⵏⵏⵉⴹⴻⵏ ⴰⴷ ⵏⴻⵙⵙⴻⵔⵙ:

L=∑∗i=1N(y∗i−y_i^)2L = \sum\limits*{i=1}^{N} (y*{i} - \hat{y\_{i}})^2

=(Y−XW)T(Y−XW)= (Y-XW)^{T}(Y-XW)
=YTY−YTXW−WTXTY+WTXTXW= Y^TY-Y^TXW-W^TX^TY+W^TX^TXW
=YTY−2WTXTY+WTXTXW= Y^TY-2W^TX^TY+W^TX^TXW

ⵉⵎⵉ ⵏⴻⴱⵖⴰ ⴰⴷ ⵏⴻⵙⵙⴻⵎⵥⵉ ⵖⵍⵖ ⴷⴻⴳ ⵡⴰⵢⴻⵏ ⵢⴻⵔⵣⴰⵏ ⵖⵡⵖ, ⵉⵀⵉ ⵏⴻⵣⵎⴻⵔ ⴰⴷ ⵏⵃⴻⵇⴻⵔ ⴰⵡⴰⵍ ⴰⵎⴻⵣⵡⴰⵔⵓ "ⵖⵢ^ⵜⵢⵖ" ⴰⵛⴽⵓ ⴷ ⵉⵍⴻⵍⵍⵉ ⵙⴻⴳ ⵖⵡⵖ ⵢⴻⵔⵏⴰ ⴰⴷ ⵏⴻⴼⵔⵓ ⵜⴰⴳⴳⴰⵢⵜⴰ:

∂(−2WTXTY+WTXTXW)∂W=0\frac{\partial (-2W^TX^TY+W^TX^TXW)}{\partial W} = 0
−2XTY+2XTXW^=0-2X^TY+2X^TX\hat{W} = 0
W^=(XTX)−1XTY\hat{W} = (X^TX)^{-1}X^TY

ⴰⵙⴻⵇⴷⴻⵛ ⵏ ⵓⵖⴻⵍⵍⵓⵢ ⵏ ⵜⵖⴻⵔⵖⴻⵔⵜ

ⴰⵜⴰⵏ ⵓⵙⵏⵓⵍⴼⵓ ⵏ algorithme ⵏ ⵓⵖⴻⵍⵍⵓⵢ ⵏ ⵜⵖⴻⵔⵖⴻⵔⵜ:

w∗n+1=w∗n−lr×∂f∂w_nw*{n+1} = w*{n} - lr \times \frac{\partial f}{\partial w\_{n}}

ⵜⵓⵔⴰ ⴰⵢⴻⵏ ⴰⴽⴽ ⵉ ⵉⵍⴰⵇ ⴰⴷ ⵜⵏⴻⵅⴷⴻⵎ ⴷ ⴰⵙⴻⵇⴷⴻⵛⵉⵙ ⵖⴻⴼ ⵙⵉⵏ ⵏ ⵢⵉⴼⴻⵔⴷⵉⵙⴻⵏ ⵖⴰ_{0}ⵖ ⴷ ⵖⴰ_{1}ⵖ (ⴷⴻⴳ ⵜⴻⴳⵏⵉⵜ ⵏ ⵢⵉⵡⴻⵏ ⵏ ⵓⵎⴳⵉⵔⴻⴷ ⵖⵅⵖ):

{a0(n+1)=a0(n)−lr×∂L∂a0a1(n+1)=a1(n)−lr×∂L∂a1\begin{cases} a_{0}^{(n+1)} = a_{0}^{(n)} - lr \times \frac{\partial L}{\partial a_{0}}\\ a_{1}^{(n+1)} = a_{1}^{(n)} - lr \times \frac{\partial L}{\partial a_{1}} \end{cases}

ⵓ ⵏⴻⵥⵔⴰ ⴱⴻⵍⵍⵉ:

{∂L∂a0=∑i=1N−2(yi−(a0+a1.xi))∂L∂a1=∑i=1N−2xi(yi−(a0+a1.xi))\begin{cases} \frac{\partial L}{\partial a_{0}} = \sum\limits_{i=1}^{N} -2(y_{i} - (a_{0} + a_{1}.x_{i}))\\ \frac{\partial L}{\partial a_{1}} = \sum\limits_{i=1}^{N} -2x_{i}(y_{i} - (a_{0} + a_{1}.x_{i})) \end{cases}

ⵙ ⵓⴱⴻⴷⴷⴻⵍ:

{a0(n+1)=a0(n)+2×lr×∑i=1N(yi−(a0(n)+a1(n).xi))a1(n+1)=a1(n)+2×lr×∑i=1Nxi(yi−(a0(n)+a1(n).xi))\begin{cases} a_{0}^{(n+1)} = a_{0}^{(n)} + 2 \times lr \times \sum\limits_{i=1}^{N} (y_{i} - (a_{0}^{(n)} + a_{1}^{(n)}.x_{i}))\\ a_{1}^{(n+1)} = a_{1}^{(n)} + 2 \times lr \times \sum\limits_{i=1}^{N} x_{i}(y_{i} - (a_{0}^{(n)} + a_{1}^{(n)}.x_{i})) \end{cases}

ⵉⵙⴻⵇⵙⵉⵢⴻⵏ

  • ⵎⴰⵜⵜⴰ ⵜⴰⵍⵖⴰ ⵏ ⵓⴼⴻⵔⴷⵉⵙ ⵏ ⵢⵉⴼⴻⵔⴷⵉⵙⴻⵏ ⵉⴳⴻⵔⵔⵣⴻⵏ ⴷⴻⴳ ⵜⴰⵍⵓⴼⵜ ⵏ ⵜⴳⴻⵔⵎⴰⵏⵜ ⵜⴰⵣⴻⴳⵣⴰⵡⵜ ⵏ ⵡⴰⵟⴰⵙ ⵏ ⵢⵉⵃⵔⵉⵛⴻⵏ:

  • ⵖ \ ⴼⵔⴰⵛ {ⴽⵓⴱ (ⵅ, ⵢ)} {VAR (ⵢ)}ɣ

  • ⵖ \ ⴼⵔⴰⵛ {ⴽⵓⴱ (ⵅ, ⵢ)} {VAR (ⵅ)} $

  • ⵖ(ⵅ^ⵜⵅ)^{-1}ⵅ^ⵜⵢⵖ "ⴰⵙⵡⵉⵔ"

  • ⴰⵢⵖⴻⵔ ⴰⵔⴰ ⵏⴻⵙⵙⴻⵔⵙ ⵜⴰⵙⴻⴽⴽⵉⵔⵜ ⵖⴻⵔ 0 ?

  • ⴰⴽⴽⴻⵏ ⴰⴷ ⵏⴰⴼ ⵜⴰⴳⴳⴰⵔⴰ. "ⴷ ⴰⵢⴻⵏ ⵢⴻⵍⵀⴰⵏ"

  • ⴰⴷ ⵜⴻⵙⵏⴻⵇⵙⴻⴹ ⴰⵙⴻⵏⵜⴻⵍ.

  • ⴰⴷ ⵢⴻⵇⵇⵉⵎ ⴽⴰⵏ ⴰⵃⵔⵉⵛ ⵏ ⵜⵉⴷⴻⵜ ⵏ ⵜⵎⴻⵥⵔⵉ.

  • ⴷ ⴰⵛⵓ ⵉ ⴷ ⵉⵙⵡⵉ ⵏ ⵜⴳⴻⵔⵎⴰⵏⵜ ⵏ ⵜⴼⴻⵍⵡⵉⵜ ?

  • ⴰⴷ ⵏⴰⴼ ⵜⴰⵣⵔⵉⵔⵜ ⵉ ⵉⵣⵔⵉⵏ ⴰⴽⴽⵯ ⵜⵏⵇⵇⵉⴹⵉⵏ.

  • ⴰⴽⴽⴻⵏ ⴰⴷ ⵏⴰⴼ ⵜⴰⵖⴻⵛⵜ ⵉ ⴷⵢⴻⵙⵎⴻⴽⵜⴰⵢⴻⵏ ⵉⵙⴻⴼⴽⴰ ⴰⴽⴽⴻⵏ ⵉⵡⴰⵜⴰ."correct"

  • ⴰⴷ ⵏⴰⴼ ⵜⴰⵖⴻⵛⵜ ⵉ ⴷⵢⴻⴱⴹⴰⵏ ⵉⵙⴻⴼⴽⴰ ⴰⴽⴽⴻⵏ ⵉⵍⴰⵇ.

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ⴰⴷ ⵏⴻⵇⵇⵉⵎ ⴷⴻⴳ ⵓⵙⵉⵡⴰⴹ

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