模块

sigmoid: 映射到概率的函数
model: 返回预测结果值(预测函数)
cost: 根据参数计算损失
gradient: 计算每个参数的梯度方向
descent: 梯度下降，参数更新
accuracy: 计算精度

sigmoid函数

g ( z ) = 1 1 + e − z g(z) = \frac{1}{1 + e^{-z}} g(z)=1+e−z1

def sigmoid(z):"""sigmoid函数"""return 1 / (1 + np.exp(-z))

预测函数

h θ ( x ) = g ( θ T x ) = 1 1 + e − θ T x h_\theta(x) = g(\theta^Tx) = \frac{1}{1 + e^{-\theta^Tx}} hθ(x)=g(θTx)=1+e−θTx1

def model(X, theta):"""预测函数"""return sigmoid(np.dot(X, theta.T))

损失函数

损失函数，将对数似然函数去负号

D ( h θ ( x ) , y ) = − y log ⁡ ( h θ ( x ) ) − ( 1 − y ) log ⁡ ( 1 − h θ ( x ) ) D(h_\theta(x), y) = -y\log(h_\theta(x)) - (1-y)\log(1-h_\theta(x)) D(hθ(x),y)=−ylog(hθ(x))−(1−y)log(1−hθ(x))

求平均损失

J ( θ ) = 1 n ∑ i = 1 n D ( h θ ( x i ) , y i ) J(\theta) = \frac{1}{n}\sum\limits_{i=1}^{n}D(h_\theta(x^i), y^i) J(θ)=n1i=1∑nD(hθ(xi),yi)

def cost(X, y, theta):"""损失函数"""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))

梯度计算

即求偏导

∂ J ∂ θ j = − 1 m ∑ i = 1 n ( y i − h θ ( x i ) ) x j i \frac{\partial J}{\partial \theta_j} = - \frac{1}{m}\sum\limits_{i=1}^n (y^i-h_\theta(x^i))x^i_j ∂θj∂J=−m1i=1∑n(yi−hθ(xi))xji

def gradient(X, y, theta):"""梯度函数"""# 梯度和θ参数一一对应grad = np.zeros(theta.shape)error = (model(X, theta) - y).ravel()for j in range(len(theta.ravel())):term = np.multiply(error, X[:, j])  # 都是一维grad[0, j] = np.sum(term) / len(X)return grad

梯度下降

import numpy.random
def shuffleData(data):"""洗牌，打乱数据"""np.random.shuffle(data)cols = data.shape[1]X = data[:, 0:cols-1]y = data[:, cols-1:]return X, y

import timedef descent(data, batchSize, stopType, thresh, alpha):"""梯度下降求解"""# 初始化init_time = time.time()  # 起始时间i = 0  # 迭代次数k = 0  # batchX, y = shuffleData(data)  # 洗牌theta = np.zeros([1, X.shape[1]])  # θ参数costs = [cost(X, y, theta)]  # 损失值while True:grad = gradient(X[k: k+batchSize], y[k: k+batchSize], theta)k += batchSizeif k >= n:k = 0X, 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# 三种停止策略if ((stopType == 'iter' and i > thresh) or(stopType == 'cost' and abs(costs[-1] - costs[-2]) < thresh) or(stopType == 'grad' and np.linalg.norm(grad) < threshold)):breakreturn theta, i-1, costs, grad, time.time()-init_time

def runExpe(data, batchSize, stopType, thresh, alpha):theta, iter, costs, grad, dur = descent(data, batchSize, stopType, thresh, alpha)# 以下可视化处理name = "Original" if (data[:, 1] > 2).sum() > 0 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 == 'iter':strStop = "{} iterations".format(thresh)elif stopType == 'cost':strStop = "costs change < {}".format(thresh)else:strStop = "gradient norm < {}".format(thresh)name += strStopprint("***{}\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. iterations')fig.show()return theta

精度

def predict(X, theta):"""预测值"""# 设定阈值xreturn [1 if x >= 0.5 else 0 for x in model(X, theta)]

# 数据准备
path = 'LogiReg_data.txt'
pdData = pd.read_csv(path, header=None, names=['Exam 1', 'Exam 2', 'Admitted'])
# 插入全为1的列
pdData.insert(0, 'Ones', 1)
# dataframe转换成array类型
orig_data = pdData.values
# 数据数量
n = orig_data.shape[0]# 数据标准化处理
from sklearn import preprocessing as pp
scaled_data = orig_data.copy()
scaled_data[:, 1:3] = pp.scale(orig_data[:, 1:3])# 执行逻辑回归
theta = runExpe(scaled_data, n, 'iter', 5000, 0.01)scaled_X = scaled_data[:, :3]
y = scaled_data[:, 3]
# 预测值
predictions = predict(scaled_X, theta)
# 计算准确率
correct = 0
for num, i in enumerate(predictions):if i == y[num]:correct += 1
print('accuracy = {}%'.format(correct / len(predictions)))

完整代码

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import numpy.random
import timedef show_data():"""显示原始数据"""# pdData.head()  # 显示头5条positive = pdData[pdData['Admitted'] == 1]negative = pdData[pdData['Admitted'] == 0]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')fig.show()def sigmoid(z):"""sigmoid函数"""return 1 / (1 + np.exp(-z))def model(X, theta):"""预测函数"""return sigmoid(np.dot(X, theta.T))def cost(X, y, theta):"""损失函数"""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))def gradient(X, y, theta):"""计算梯度函数"""# 梯度和θ参数一一对应grad = np.zeros(theta.shape)error = (model(X, theta) - y).ravel()for j in range(len(theta.ravel())):term = np.multiply(error, X[:, j])  # 都是一维grad[0, j] = np.sum(term) / len(X)return graddef shuffleData(data):"""洗牌，打乱数据"""np.random.shuffle(data)cols = data.shape[1]X = data[:, 0:cols-1]y = data[:, cols-1:]return X, ydef descent(data, batchSize, stopType, thresh, alpha):"""梯度下降求解"""# 初始化init_time = time.time()  # 起始时间i = 0  # 迭代次数k = 0  # batch   X, y = shuffleData(data)  # 洗牌Xtheta = np.zeros([1, X.shape[1]])  # θ参数costs = [cost(X, y, theta)]  # 损失值while True:grad = gradient(X[k: k+batchSize], y[k: k+batchSize], theta)k += batchSizeif k >= n:k = 0X, 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# 三种停止策略if ((stopType == 'iter' and i > thresh) or(stopType == 'cost' and abs(costs[-1] - costs[-2]) < thresh) or(stopType == 'grad' and np.linalg.norm(grad) < threshold)):breakreturn theta, i-1, costs, grad, time.time()-init_timedef runExpe(data, batchSize, stopType, thresh, alpha):theta, iter, costs, grad, dur = descent(data, batchSize, stopType, thresh, alpha)# 可视化处理name = "Original" if (data[:, 1] > 2).sum() > 0 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 == 'iter':strStop = "{} iterations".format(thresh)elif stopType == 'cost':strStop = "costs change < {}".format(thresh)else:strStop = "gradient norm < {}".format(thresh)name += strStopprint("***{}\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. iterations')fig.show()return thetadef predict(X, theta):"""预测值"""# 设定阈值xreturn [1 if x >= 0.5 else 0 for x in model(X, theta)]path = 'LogiReg_data.txt'
pdData = pd.read_csv(path, header=None, names=['Exam 1', 'Exam 2', 'Admitted'])
# 插入全为1的列
pdData.insert(0, 'Ones', 1)
# dataframe转换成array类型
orig_data = pdData.values
# 数据数量
n = orig_data.shape[0]# 数据标准化处理
from sklearn import preprocessing as pp
scaled_data = orig_data.copy()
scaled_data[:, 1:3] = pp.scale(orig_data[:, 1:3])theta = runExpe(scaled_data, n, 'iter', 5000, 0.01)scaled_X = scaled_data[:, :3]
y = scaled_data[:, 3]
# 预测值
predictions = predict(scaled_X, theta)
# 计算准确率
correct = 0
for num, i in enumerate(predictions):if i == y[num]:correct += 1
print('accuracy = {}%'.format(correct / len(predictions)))

附录

梯度下降三种比较方法

目标函数 J ( θ ) = 1 2 m ∑ i = 1 m ( y i − h θ ( x i ) ) 2 J(\theta) = \frac{1}{2m}\sum\limits_{i=1}^{m} (y^i - h_\theta(x^i))^2 J(θ)=2m1i=1∑m(yi−hθ(xi))2

批量梯度下降： ∂ ∂ θ j J ( θ ) = − 1 m ∑ i = 1 m ( y i − h θ ( x i ) ) x j i \frac{\partial}{\partial\theta_j} J(\theta) = -\frac{1}{m} \sum\limits_{i=1}^{m} (y^i - h_\theta(x^i))x^{i}_{j} ∂θj∂J(θ)=−m1i=1∑m(yi−hθ(xi))xji

(容易得到最优解，每次考虑所有样本，速度慢)

随机梯度下降： θ j ′ = θ j + ( y i − h θ ( x i ) ) x j i \theta_j^{'} = \theta_j + (y^i - h_\theta(x^i)) x^i_j θj′=θj+(yi−hθ(xi))xji

(每次一个样本，迭代速度快，但不一定每次朝收敛方向)

小批量梯度下降： θ j : = θ j − α 1 10 ∑ k = i i + 9 ( h θ ( x ( k ) ) − y ( k ) ) x j ( k ) \theta_j:= \theta_j - \alpha \frac{1}{10} \sum\limits_{k=i}^{i+9} (h_\theta(x^{(k)}) - y^{(k)}) x^{(k)}_j θj:=θj−α101k=i∑i+9(hθ(x(k))−y(k))xj(k)

(每次更新选择一小部分数据来算，实用！)

似然函数与对数似然函数

似然函数： L ( θ ) = ∏ i = 1 m P ( y i ∣ x i ; θ ) = ∏ i = 1 m ( h θ ( x i ) ) y i ( 1 − h θ ( x i ) ) 1 − y i L(\theta) = \prod_{i=1}^{m}P(y^i | x^i; \theta) = \prod_{i=1}^{m}(h\theta(x^i))^{y^i}(1-h\theta(x^i))^{1-y^i} L(θ)=∏i=1mP(yi∣xi;θ)=∏i=1m(hθ(xi))yi(1−hθ(xi))1−yi

对数似然函数： l ( θ ) = log ⁡ L ( θ ) = ∑ i = 1 m ( y i log ⁡ h θ ( x i ) + ( 1 − y i ) log ⁡ ( 1 − h θ ( x i ) ) ) l(\theta) = \log L(\theta) = \sum\limits_{i=1}^m(y^i \log h_\theta(x^i) + (1-y^i) \log(1-h_\theta(x^i))) l(θ)=logL(θ)=i=1∑m(yiloghθ(xi)+(1−yi)log(1−hθ(xi)))

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模块