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圖示：
# normalize the data 歸一化值 (x - mean) / (std)
pga.at = (pga.at - pga.at.mean()) / pga.at.std()
pga.v = (pga.v - pga.v.mean()) / pga.v.std()
pga.ap = (pga.ap - pga.ap.mean()) / pga.ap.std()
pga.rh = (pga.rh - pga.rh.mean()) / pga.rh.std()
pga.pe = (pga.pe - pga.pe.mean()) / pga.pe.std()
def cost(theta0, theta1, theta2, theta3, theta4, x1, x2, x3, x4, y):
j = 0
m = len(x1)
for i in range(m):
h=theta0+x1[i]*theta1+x2[i]*theta2+x3[i]*theta3+x4[i]*theta4
j += (h - y[i])**2
j /= (2*m)
return j
def partial_cost_theta4(theta0,theta1,theta2,theta3,theta4,x1,x2,x3,x4,y):
h = theta0 + x1 * theta1 + x2 * theta2 + x3 * theta3 + x4 * theta4
diff = (h - y) * x4
partial = diff.sum() / (x2.shape[0])
return partial
def partial_cost_theta3(theta0,theta1,theta2,theta3,theta4,x1,x2,x3,x4,y):
h = theta0 + x1 * theta1 + x2 * theta2 + x3 * theta3 + x4 * theta4
diff = (h - y) * x3
partial = diff.sum() / (x2.shape[0])
return partial
def partial_cost_theta2(theta0,theta1,theta2,theta3,theta4,x1,x2,x3,x4,y):
h = theta0 + x1 * theta1 + x2 * theta2 + x3 * theta3 + x4 * theta4
diff = (h - y) * x2
partial = diff.sum() / (x2.shape[0])
return partial
def partial_cost_theta1(theta0,theta1,theta2,theta3,theta4,x1,x2,x3,x4,y):
h = theta0 + x1 * theta1 + x2 * theta2 + x3 * theta3 + x4 * theta4
diff = (h - y) * x1
partial = diff.sum() / (x2.shape[0])
return partial
def partial_cost_theta0(theta0, theta1, theta2, theta3, theta4, x1, x2, x3, x4, y):
h = theta0 + x1 * theta1 + x2 * theta2 + x3 * theta3 + x4 * theta4
diff = (h - y)
partial = diff.sum() / (x2.shape[0])
return partial
def gradient_descent(x1,x2,x3,x4,y, alpha=0.1, theta0=0, theta1=0,theta2=0,theta3=0,theta4=0):
max_epochs = 1000 # maximum number of iterations 最大迭代次數
counter = 0 # intialize a counter 當前第幾次
c = cost(theta0, theta1, theta2, theta3, theta4, x1, x2, x3, x4, y) ## initial cost 當前代價函式
costs = [c] # lets store each update 每次損失值都記錄下來
convergence_thres = 0.000001
cprev = c + 10
theta0s = [theta0]
theta1s = [theta1]
theta2s = [theta2]
theta3s = [theta3]
theta4s = [theta4]
while (np.abs(cprev - c) > convergence_thres) and (counter < max_epochs):
cprev = c
update0 = alpha * partial_cost_theta0(theta0, theta1, theta2, theta3, theta4, x1, x2, x3, x4, y)
update1 = alpha * partial_cost_theta1(theta0, theta1, theta2, theta3, theta4, x1, x2, x3, x4, y)
update2 = alpha * partial_cost_theta2(theta0, theta1, theta2, theta3, theta4, x1, x2, x3, x4, y)
update3 = alpha * partial_cost_theta3(theta0, theta1, theta2, theta3, theta4, x1, x2, x3, x4, y)
update4 = alpha * partial_cost_theta4(theta0, theta1, theta2, theta3, theta4, x1, x2, x3, x4, y)
theta0 -= update0
theta1 -= update1
theta2 -= update2
theta3 -= update3
theta4 -= update4
c = cost(theta0, theta1, theta2, theta3, theta4, x1, x2, x3, x4, y)
#return
return
print(「costs =」, gradient_descent(pga.at, pga.v,pga.ap,pga.rh,pga.pe)[『costs』])
descend = gradient_descent(pga.at, pga.v,pga.ap,pga.rh,pga.pe, alpha=.01)
plt.scatter(range(len(descend[「costs」])), descend[「costs」])
plt.show()
損失函式隨迭代次數變換圖：
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