我们将建立一个逻辑回归模型来预测一个学生是否被大学录取。假设你是一个大学的管理员，你想根据两次考试的结果来决定每个申请人的录取机会。你有以前申请人的历史数据，你可以用它作为逻辑回归的训练集，对于每一个训练例子，你有两个考试的申请人的分数和录取决定。为了做到这一点，我们将建立一个分类模型，根据考试成绩估计入学概率。
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
plt.rcParams['font.sans-serif']=['SimHei'] # 用来正常显示中文标签
plt.rcParams['axes.unicode_minus']=False # 用来正常显示负号
data = pd.read_csv("grade.csv")
Pass = data[data["Admitted"] == 1] # 获取及格的数据
noPass = data[data["Admitted"] == 0] # 获取不及格的数据
fig, ax = plt.subplots()
ax.scatter(Pass["EXAM 1"], Pass["EXAM 2"], s = 30, c = 'b', marker = 'o', label = 'PASS')
ax.scatter(noPass["EXAM 1"], noPass["EXAM 2"], s = 30, c = 'r', marker = 'x', label = 'noPASS')
ax.legend(loc = 2)
ax.set_xlabel('EXAM 1 score')
ax.set_ylabel('EXAM 2 score')
ax.set_title('逻辑回归案例')
plt.show()
接下来就是算法的实现
目标：建立分类器（求解出三个参数e1,e2,e3）
设定阈值，根据阈值判断录取结果
要完成的模块
1.sigmoid：映射到概率的函数
2.model：返回预测结果值
3.cost：根据参数计算损失
4.gradient：计算每个参数的梯度方向
5.descent：进行参数更新
6.accuracy：计算精度
# sigmoid：映射到概率的函数
def sigmoid(z):
return 1 / (1 + np.exp(-z))
# model：返回预测结果值
def model(X, theta):
return sigmoid(np.dot(X, theta.T))
data.insert(0, 'Ones', 1)
orig_data = data.as_matrix()
cols = orig_data.shape[1]
X = orig_data[:, 0:cols-1]
y = orig_data[:, cols-1:cols]
theta = np.zeros([1, 3])
# cost：根据参数计算损失
def cost(X, y, theta):
left = np.multiply(-y, np.log(model(X, theta)))
right = np.multiply(1-y, np.log(1 - model(X, theta)))
return np.sum(left - right) / (len(X))
# gradient：计算每个参数的梯度方向
def gradient(X, y, theta):
grad = np.zeros(theta.shape)
error = (model(X, theta) - y).ravel()
for j in range(len(theta.ravel())):
term = np.multiply(error, X[:, j])
grad[0, j] = np.sum(term) / len(X)
return grad
STOP_ITER = 0
STOP_COST = 1
STOP_GRAD = 2
# 设定三种不同的停止策略
def stopCriterion(type, value, threshold):
if type == STOP_ITER:
return value > threshold
elif type == STOP_COST:
return abs(value[-1] - value[-2]) < threshold
elif type == STOP_GRAD:
return np.linalg.norm(value) < threshold
# 将数据打乱
def shuffleData(data1):
shuffle(data1)
cols = data1.shape[1]
X = data1[:, 0:cols-1]
y = data1[:, cols-1:]
return X, y
# 梯度下降求解
def descent(data, theta, batchSize, stopType, thresh, alpha):
init_time = time.time()
i = 0 # 迭代次数
k = 0 # batch
X, y = shuffleData(data)
grad = np.zeros(theta.shape) # 计算的梯度
costs = [cost(X, y, theta)] # 损失值
while True:
grad = gradient(X[k:k+batchSize], y[k:k+batchSize], theta)
k += batchSize # 取batch数量个数据
if k >= n:
k = 0
X, y =shuffleData(data) # 重新打乱
theta = theta - alpha * grad # 更新参数
costs.append(cost(X, y, theta)) # 计算新的损失
i += 1
if stopType == STOP_ITER:
value = i
elif stopType == STOP_COST:
value = costs
elif stopType == STOP_GRAD:
value = grad
if stopCriterion(stopType, value, thresh):
break
return theta, i-1, costs, grad, time.time() - init_time
def runExpe(data, theta, batchSize, stopType, thresh, alpha):
theta, iter, costs, grad, dur = descent(data, theta, batchSize, stopType, thresh, alpha)
name = "Original" if (data[:,1] > 2).sum() > 1 else "Scaled"
name += "data - learning rate: {} - ".format(alpha)
if batchSize == n:
strDescType = "Gradient"
elif batchSize == 1:
strDescType = "Stochastic"
else:
strDescType = "MiNi-batch({})".format(batchSize)
name += strDescType + "descent - Stop:"
if stopType == STOP_ITER:
strStop = "{} iterations".format(thresh)
elif stopType == STOP_COST:
strStop = "costs change < {}".format(thresh)
else:
strStop = "gradient norm < {}".format(thresh)
name += strStop
print("***{}\nTheta:{} - Iter: {} - Last cost: {:03.2f} - Duration: {:03.2f}s".format(name,theta,iter,costs[-1],dur))
fig, ax = plt.subplots()
ax.plot(np.arange(len(costs)), costs, 'r')
ax.set_xlabel('Iterations')
ax.set_ylabel('Cost')
ax.set_title(name.upper() + '- Error vs. Iteration')
plt.show()
return theta
if __name__ == '__main__':
n = 100
# runExpe(orig_data,theta,n,STOP_ITER,thresh=5000,alpha=0.0001)
# 根据损失值停止
# runExpe(orig_data, theta, n, STOP_COST, thresh=0.000001, alpha=0.001)
# 根据梯度变化停止
runExpe(orig_data, theta, n, STOP_GRAD, thresh=0.05, alpha=0.001)
