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。假设你是一个大学系的管理员,你想根据两次考试的结果来决定每个申请人的录取机会。你有以前的申请人的历史数据,你可以用它作为逻辑回归的训练集。对于每一个培训例子,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。
1.数据集:下载链接: https://pan.baidu.com/s/1IHqaJqDkSJ--igVV7kDsrg 密码: 2jbg
源码链接: https://pan.baidu.com/s/1uk2_zfaJH29SnU8buvyOwg 密码: k9vd
2.代码:
#三大件 import numpy as np import pandas as pd import matplotlib.pyplot as plt %matplotlib inline
import os #输出数据的前五行 path = 'data' + os.sep + 'LogiReg_data.txt' pdData = pd.read_csv(path, header=None, names=['Exam 1', 'Exam 2', 'Admitted']) pdData.head()
#用点图显示数据
positive = pdData[pdData['Admitted'] == 1] # returns the subset of rows such Admitted = 1, i.e. the set of *positive* examples
negative = pdData[pdData['Admitted'] == 0] # returns the subset of rows such Admitted = 0, i.e. the set of *negative* examples fig, ax = plt.subplots(figsize=(10,5)) ax.scatter(positive['Exam 1'], positive['Exam 2'], s=30, c='b', marker='o', label='Admitted') ax.scatter(negative['Exam 1'], negative['Exam 2'], s=30, c='r', marker='x', label='Not Admitted') ax.legend() ax.set_xlabel('Exam 1 Score') ax.set_ylabel('Exam 2 Score')
plt.show()
The logistic regression
def sigmoid(z): #定义sigmoid函数 return 1 / (1 + np.exp(-z))
nums = np.arange(-10, 10, step=1) #显示sigmoid函数 fig, ax = plt.subplots(figsize=(12,4)) ax.plot(nums, sigmoid(nums), 'r') plt.show()
def model(X, theta): return sigmoid(np.dot(X, theta.T))
pdData.insert(0, 'Ones', 1) # in a try / except structure so as not to return an error if the block si executed several times # set X (training data) and y (target variable) orig_data = pdData.as_matrix() # convert the Pandas representation of the data to an array useful for further computations cols = orig_data.shape[1] X = orig_data[:,0:cols-1] y = orig_data[:,cols-1:cols] # convert to numpy arrays and initalize the parameter array theta #X = np.matrix(X.values) #y = np.matrix(data.iloc[:,3:4].values) #np.array(y.values) theta = np.zeros([1, 3])
损失函数
将对数似然函数去负号
D(hθ(x),y)=−ylog(hθ(x))−(1−y)log(1−hθ(x))
D(hθ(x),y)=−ylog(hθ(x))−(1−y)log(1−hθ(x))
求平均损失:
D(hθ(x),y)=−ylog(hθ(x))−(1−y)log(1−hθ(x))
J(θ)=1n∑i=1nD(hθ(xi),yi)
J(θ)=1n∑i=1nD(hθ(xi),yi)
J(θ)=1n∑i=1nD(hθ(xi),yi)
def cost(X, y, theta): #计算平均损失 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)
计算梯度:
def gradient(X, y, theta): grad = np.zeros(theta.shape) error = (model(X, theta)- y).ravel() for j in range(len(theta.ravel())): #for each parmeter term = np.multiply(error, X[:,j]) grad[0, j] = np.sum(term) / len(X) return grad
比较3中不同梯度下降方法
STOP_ITER = 0 STOP_COST = 1 STOP_GRAD = 2 def stopCriterion(type, value, threshold): #设定三种不同的停止策略 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
import numpy.random #洗牌 def shuffleData(data): np.random.shuffle(data) cols = data.shape[1] X = data[:, 0:cols-1] y = data[:, cols-1:] return X, y
def descent(data, theta, batchSize, stopType, thresh, alpha): #梯度下降求解 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) #重新洗牌 theta = theta - alpha*grad # 参数更新 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): #import pdb; pdb.set_trace(); 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(figsize=(12,4)) ax.plot(np.arange(len(costs)), costs, 'r') ax.set_xlabel('Iterations') ax.set_ylabel('Cost') ax.set_title(name.upper() + ' - Error vs. Iteration') return theta
设定迭代次数
#选择的梯度下降方法是基于所有样本的 n=100 runExpe(orig_data, theta, n, STOP_ITER, thresh=5000, alpha=0.000001)
设定阈值 1E-6, 差不多需要110 000次迭代
runExpe(orig_data, theta, n, STOP_COST, thresh=0.000001, alpha=0.001) plt.show()
精度计算:
#设定阈值 def predict(X, theta): return [1 if x >= 0.5 else 0 for x in model(X, theta)]
scaled_X = scaled_data[:, :3] y = scaled_data[:, 3] predictions = predict(scaled_X, theta) correct = [1 if ((a == 1 and b == 1) or (a == 0 and b == 0)) else 0 for (a, b) in zip(predictions, y)] accuracy = (sum(map(int, correct)) % len(correct)) print ('accuracy = {0}%'.format(accuracy))