ⵜⵓⵖⴰⵍⵉⵏ ⵜⴰⵣⴻⴳⵣⴰⵡⵜ

ⴰⵙⵎⴰⵢⵏⵓ ⵅⴼ October 18, 2024 [ⵜ_ⵉ_ⵎ_ⵎ_ⵉ] ⵏⴷⵇⵉⵇⴰ

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 :

$$ \hat{y} = a*{0} + a*{1}.x*{1} + \dots + a*{p}.x_{p} $$

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. $p=1$.

• 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:

$$ \hat{y} = a*{0} + a*{1}.x $$

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.

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

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

ⴰⴷ ⵏⴻⵙⵙⴻⵔⵙ:

$$ L = \sum\limits*{i=1}^{N} (y*{i} - (a*{0} + a*{1}.x_{i}))^2 $$

ⵉⵡⴰⴽⴽⴻⵏ ⴰⴷ ⵏⴰⴼ ⴰⵣⴰⵍ ⴰⵎⴻⵥⵢⴰⵏ, ⵢⴻⵙⵙⴻⴼⴽ ⴰⴷ ⵏⴻⴼⵔⵓ ⵜⵉⴳⵏⴰⵜⵉⵏⴰ :

$$ \begin{cases} \frac{\partial L}{\partial a_{0}} = 0\ \frac{\partial L}{\partial a_{1}} = 0 \end{cases} $$

$$ \begin{cases} \sum\limits_{i=1}^{N} -2(y_{i} - (a_{0} + a_{1}.x_{i})) = 0\ \sum\limits_{i=1}^{N} -2x_{i}(y_{i} - (a_{0} + a_{1}.x_{i})) = 0 \end{cases} $$

ⵏⴻⴱⴷⴰ ⵙ ⵓⵙⵏⴻⵔⵏⵉ ⵏ ⵜⴰⴳⴷⴰⵍⵜ ⵜⴰⵎⴻⵣⵡⴰⵔⵓⵜ:

$$ \sum\limits_{i=1}^{N} y_{i} - \sum\limits_{i=1}^{N}a_{0} + \sum\limits_{i=1}^{N} a_{1}.x_{i} = 0\ $$

$$ \sum\limits_{i=1}^{N} y_{i} - Na_{0} + \sum\limits_{i=1}^{N} a_{1}.x_{i} = 0\ $$

$$ a_{0} = \frac{\sum\limits_{i=1}^{N} y_{i}}{N} - \frac{\sum\limits_{i=1}^{N} x_{i}}{N}a_{1} $$

$$ a_{0} = Y - Xa_{1} $$

ⵏⴻⵙⴱⴻⴷⴷⴻⵍ ⴷⴻⴳ ⵜⴰⴳⴷⴰⵍⵜ ⵜⵉⵙ ⵙⵏⴰⵜ:

$$ \sum\limits_{i=1}^{N} x_{i}(y_{i} - Y + Xa_{1} - a_{1}x_{i}) = 0 $$

$$ \sum\limits_{i=1}^{N} (y_{i} - Y) + a_{1}(X - x_{i}) = 0 $$

$$ \sum\limits_{i=1}^{N} (y_{i} - Y) - \sum\limits_{i=1}^{N}a_{1}(x_{i} - X) = 0 $$

$$ a_{1} = \frac{\sum\limits_{i=1}^{N} (y_{i} - Y)}{\sum\limits_{i=1}^{N}(x_{i} - X)} = \frac{\sum\limits_{i=1}^{N} (y_{i} - Y)(x_{i} - X)}{\sum\limits_{i=1}^{N}(x_{i} - X)^2} = \frac{COV(X, Y)}{VAR(X)} $$

ⵏⴻⴱⴷⴻⵍⴷ ⵜⵓⵖⴰⵍⵉⵏ ⴷⴻⴳ $a_{0}ɣ:

$$ \begin{cases} a_{0} = Y - X\frac{COV(X, Y)}{VAR(X)}\ a_{1} = \frac{COV(X, Y)}{VAR(X)} \end{cases} $$

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 $p$:

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

ⵉⵀⵉ, ⴰⵎⵣⵓⵏ ⵢⴻⵜⵜⵡⴰⵔⵓ ⴰⴽⴽⴻⵏ ⵉ ⴷⵉⵜⴻⴷⴷⵓⵏ:

$$ \hat{y} = a*{0} + a*{1}x*{1} + a*{2}x*{2} + \dots + a*{p}x_{p} $$

ⵏⴻⵖ, ⵢⴻⵣⵎⴻⵔ ⴰⴷ ⵢⴻⵜⵜⵡⴰⵔⵓ ⵙ ⵜⴰⵍⵖⴰ ⵏ ⵎⴰⵜⵔⵉⵅ:

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

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 :

$$ \hat{W} = \underset{W}{\operatorname{argmin}} \sum\limits*{i=1}^{N} (y*{i} - \hat{y_{i}})^2 $$

ⵉ ⵜⵉⴽⴽⴻⵍⵜ ⵏⵏⵉⴹⴻⵏ ⴰⴷ ⵏⴻⵙⵙⴻⵔⵙ:

$$ L = \sum\limits*{i=1}^{N} (y*{i} - \hat{y_{i}})^2 $$

$$ = (Y-XW)^{T}(Y-XW) $$

$$ = Y^TY-Y^TXW-W^TX^TY+W^TX^TXW $$

$$ = Y^TY-2W^TX^TY+W^TX^TXW $$

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:

$$ \frac{\partial (-2W^TX^TY+W^TX^TXW)}{\partial W} = 0 $$

$$ -2X^TY+2X^TX\hat{W} = 0 $$

$$ \hat{W} = (X^TX)^{-1}X^TY $$

ⴰⵙⴻⵇⴷⴻⵛ ⵏ ⵓⵖⴻⵍⵍⵓⵢ ⵏ ⵜⵖⴻⵔⵖⴻⵔⵜ

ⴰⵜⴰⵏ ⵓⵙⵏⵓⵍⴼⵓ ⵏ algorithme ⵏ ⵓⵖⴻⵍⵍⵓⵢ ⵏ ⵜⵖⴻⵔⵖⴻⵔⵜ:

$$ w*{n+1} = w*{n} - lr \times \frac{\partial f}{\partial w_{n}} $$

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):

$$ \begin{cases} a_{0}^{(n+1)} = a_{0}^{(n)} - lr \times \frac{\partial L}{\partial a_{0}}\ a_{1}^{(n+1)} = a_{1}^{(n)} - lr \times \frac{\partial L}{\partial a_{1}} \end{cases} $$

ⵓ ⵏⴻⵥⵔⴰ ⴱⴻⵍⵍⵉ:

$$ \begin{cases} \frac{\partial L}{\partial a_{0}} = \sum\limits_{i=1}^{N} -2(y_{i} - (a_{0} + a_{1}.x_{i}))\ \frac{\partial L}{\partial a_{1}} = \sum\limits_{i=1}^{N} -2x_{i}(y_{i} - (a_{0} + a_{1}.x_{i})) \end{cases} $$

ⵙ ⵓⴱⴻⴷⴷⴻⵍ:

$$ \begin{cases} a_{0}^{(n+1)} = a_{0}^{(n)} + 2 \times lr \times \sum\limits_{i=1}^{N} (y_{i} - (a_{0}^{(n)} + a_{1}^{(n)}.x_{i}))\ a_{1}^{(n+1)} = a_{1}^{(n)} + 2 \times lr \times \sum\limits_{i=1}^{N} x_{i}(y_{i} - (a_{0}^{(n)} + a_{1}^{(n)}.x_{i})) \end{cases} $$

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.