ⵜⵓⵖⴰⵍⵉⵏ ⵜⴰⵣⴻⴳⵣⴰⵡⵜ

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.