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zYk=1PuK`{Iyasp;@EYJXz-xfl0IvaF1H6Ww;YHUG@EQ*{2_;R=8sIg+Yk=1PuK`{Iyasp;@EYJXz-xfl0IvaF1H1-!4e%P^HNb0t*8r~pUIV-a zcn$Cx;5Fz$Su-^78WF@vm diff --git a/linear_regression/polynomial.org b/linear_regression/polynomial.org index 7201122..921d755 100644 --- a/linear_regression/polynomial.org +++ b/linear_regression/polynomial.org @@ -15,19 +15,21 @@ h_w(x) = w_1 + w_2x + w_3x^2 Then, we should define a cost function. A common approach is to use the *Mean Square Error* cost function: \begin{equation}\label{eq:cost} - J(w) = \frac{1}{2n} \sum_{i=0}^n (h_w(x^{(i)}) - \hat{y}^{(i)})^2 + J(w) = \frac{1}{2n} \sum_{i=0}^n (h_w(x^{(i)}) - y^{(i)})^2 \end{equation} -Note that in Equation \ref{eq:cost} we average by $2n$ and not $n$. This is because it get simplify -while doing the partial derivatives as we will see below. This is a pure cosmetic approach which do -not impact the gradient decent (see [[https://math.stackexchange.com/questions/884887/why-divide-by-2m][here]] for more informations). The next step is to $min_w J(w)$ -for each weight $w_i$ (performing the gradient decent). Thus we compute each partial derivatives: +With $n$ the number of observations, $x^{(i)}$ the value of the independant variable associated with +the observation $y^{(i)}$. Note that in Equation \ref{eq:cost} we average by $2n$ and not $n$. This +is because it simplify the partial derivatives expression as we will see below. This is a pure +cosmetic approach which do not impact the gradient decent (see [[https://math.stackexchange.com/questions/884887/why-divide-by-2m][here]] for more informations). The next +step is to $min_w J(w)$ for each weight $w_i$ (performing the gradient decent, see [[https://towardsdatascience.com/gradient-descent-demystified-bc30b26e432a][here]]). Thus we +compute each partial derivatives: \begin{align} \frac{\partial J(w)}{\partial w_1}&=\frac{\partial J(w)}{\partial h_w(x)}\frac{\partial h_w(x)}{\partial w_1}\nonumber\\ - &= \frac{1}{n} \sum_{i=0}^n (h_w(x^{(i)}) - \hat{y}^{(i)})\\ + &= \frac{1}{n} \sum_{i=0}^n (h_w(x^{(i)}) - y^{(i)})\\ \text{similarly:}\nonumber\\ - \frac{\partial J(w)}{\partial w_2}&= \frac{1}{n} \sum_{i=0}^n x(h_w(x^{(i)}) - \hat{y}^{(i)})\\ - \frac{\partial J(w)}{\partial w_3}&= \frac{1}{n} \sum_{i=0}^n x^2(h_w(x^{(i)}) - \hat{y}^{(i)}) + \frac{\partial J(w)}{\partial w_2}&= \frac{1}{n} \sum_{i=0}^n x(h_w(x^{(i)}) - y^{(i)})\\ + \frac{\partial J(w)}{\partial w_3}&= \frac{1}{n} \sum_{i=0}^n x^2(h_w(x^{(i)}) - y^{(i)}) \end{align} diff --git a/linear_regression/polynomial.pdf b/linear_regression/polynomial.pdf index ecf2f285fbaaec717c8deeaf256efdddcf375910..5f72883edae49ff0df04ad6fd8a1be59e8abbc08 100644 GIT binary patch delta 47219 zcmV)FK)=7->;}8<1|=m>L`E$!E;R}z(A34)3OFDzAa7!73OqatFHB`_XLM*FGB7oh 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