\[h_\theta(x) = {1 \over {1+e^{-x}}} \\
{\partial h(x)\over \partial x }= h(x)(1-h(x))\]
def sigmoid(x):
return 1/(1 + np.math.exp(-x))
def sigmoid_derive(x):
return sigmoid(x) * (1 - sigmoid(x))
\[cost(h_\theta(x),y)=\sum_{i=1}^m−y_ilog(h_θ(x_i))−(1−y_i)log(1−h_θ(x_i))\]
def cost(x, y, theta):
x = np.array(x)
y = np.array(y).astype('int')
sig = x.dot(theta)
return sum([-np.math.log(sigmoid(sig[i])) if y[i]==1 else -np.math.log(1 - sigmoid(sig[i])) for i in range(len(y))])
\[\begin{split}
{\partial cost(h(\theta,x), y)\over \partial\theta_i} &=
{\sum_{i=1}^m {y_i\over h(\theta,x)}{\partial h(\theta,x)\over\partial\theta} }
+{1-y_i\over{1-h(\theta,x)}}{\partial h(\theta,x)\over\partial\theta}\\
&=
{\sum_{i=1}^m {y_i\over h(\theta,x)}{(h(\theta x)(1-h(\theta x)))}x_i }
+{1-y_i\over{1-h(\theta,x)}}{(h(\theta x)(1-h(\theta x)))x_i}\\
&=
\sum_{i=1}^m y_i(1-h(\theta x))x_i + (1-y_i)h(\theta x)x_i\\
grad(cost(h_\theta(x),y))
&=({\partial cost(h(\theta,x), y)\over \partial\theta_1},{\partial cost(h(\theta,x), y)\over \partial\theta_2},...,
{\partial cost(h(\theta,x), y)\over \partial\theta_m})
\end{split}\]
def grad(x, y, theta):
x = np.array(x)
y = np.array(y).astype('int')
sig = np.array([sigmoid(i) for i in x.dot(theta)])
return np.array([(1-sig[i])*x[i] for i in range(len(y))]).sum(axix=0)
def logistic_grad_descent(x,y, num=1000):
x = np.array(x)
y = np.array(y).astype('int')
theta = np.zeros(x.shape[1])
for i in range(num):
g = grad(x, y, theta)
theta = theta - g
if i % (num//10) == 0:
print(cost(x, y, theta))
class LogisticClassfier():
def __init__(self):
pass
def fit(self, x, y):
x = np.array(x)
y = np.array(y).astype('int')
theta = np.zeros(x.shape[1])
for i in range(num):
g = grad(x, y, theta)
theta = theta - g
if i % (num//10) == 0:
print(cost(x, y, theta))
self.theta = theta
def predict(x):
return np.array([sigmoid(i) for i in np.dot(theta,x)] )
