Cleaning

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}
 
 

linear_regression/polynomial.py

@@ -27,13 +27,17 @@ def dh3():
     return(1/len(x)*np.sum((h(x)-y)*(x**2)))
 
 # Perform the gradient decent
-fig, ax = plt.subplots()
-frame=0 # Current frame (plot animation)
+fig, ax = plt.subplots(dpi=300)
+ax.set_xlim([0, 7])
+ax.set_ylim([0, 5])
+ax.plot(x,y,"ro")
+h_data,=ax.plot(x,h(x))
 alpha=0.005 # Proportion of the gradient to take into account
 accuracy=0.000001 # Accuracy of the decent
 done=False
 def decent(i):
-    global w1,w2,w3,x,y,frame
+    global w1,w2,w3,x,y
+    skip_frame=0 # Current frame (plot animation)
     while True: 
         w1_old=w1
         w1_new=w1-alpha*dh1()
@@ -47,14 +51,9 @@ def decent(i):
 
         if abs(w1_new-w1_old) <= accuracy and abs(w2_new-w2_old) <= accuracy and abs(w2_new-w2_old) <= accuracy:
             done=True
-        frame+=1
-        if frame >=1000:
-            frame=0
-            ax.clear()
-            ax.set_xlim([0, 7])
-            ax.set_ylim([0, 5])
-            ax.plot(x,y,"ro")
-            ax.plot(x,h(x))
+        skip_frame+=1
+        if skip_frame >=1000:
+            h_data.set_ydata(h(x))
             break
 
 def IsDone():
@@ -65,5 +64,5 @@ def IsDone():
         yield i
         
 anim=FuncAnimation(fig,decent,frames=IsDone,repeat=False)
-anim.save('polynomial.gif',dpi=80,writer="imagemagick")
+anim.save('polynomial.gif',writer="imagemagick",dpi=300)