梯度下降算法

手工实现方式

import numpy as np
import matplotlib.pyplot as plt

# 用来加载中文
import matplotlib

matplotlib.rcParams['font.sans-serif'] = ['SimHei']
matplotlib.rcParams['font.family'] = 'sans-serif'
matplotlib.rcParams['axes.unicode_minus'] = False  # 用来正常显示负号


def loadDataSet(filename):
    '''加载文件，将feature存在X中，y存在Y中'''
    X = []
    Y = []
    with open(filename, 'rb') as f:
        for idx, line in enumerate(f):
            line = line.decode('utf-8').strip()
            if not line:
                continue
            eles = line.split()
            if idx == 0:
                numFeature = len(eles)
            eles = list(map(float, eles))  # 将数据转换成float型
            X.append(eles[:-1])  # 除最后一列都是feature，append(list)
            Y.append([eles[-1]])  # 最后一列是实际值,同上
        return np.array(X), np.array(Y)  # 将X,Y列表转化成矩阵


def h(theta, X):
    '''定义模型函数'''
    return np.dot(X, theta)  # 此时的X为处理后的X


def J(theta, X, Y):
    '''定义代价函数'''
    m = len(X)
    return np.sum(np.dot((h(theta, X) - Y).T, (h(theta, X) - Y)) / (2 * m))


def bgd(alpha, maxloop, epsilon, X, Y):
    '''定义梯度下降公式，其中alpha为学习率控制步长，maxloop为最大迭代次数，epsilon为阈值控制迭代（判断收敛）'''
    m, n = X.shape  # m为样本数，n为特征数，在这里为2

    # 初始化参数为零
    theta = np.zeros((2, 1))

    count = 0  # 记录迭代次数
    converged = False  # 是否收敛标志
    cost = np.inf  # 初始化代价为无穷大
    costs = []  # 记录每一次迭代的代价值
    thetas = {0: [theta[0, 0]], 1: [theta[1, 0]]}  # 记录每一轮theta的更新

    while count <= maxloop:
        if converged:
            break
        # 更新theta
        count = count + 1

        # 单独计算
        # theta0 = theta[0,0] - alpha / m * (h(theta, X) - Y).sum()
        # theta1 = theta[1,0] - alpha / m * (np.dot(X[:,1][:,np.newaxis].T,(h(theta, X) - Y))).sum()   # 重点注意一下
        # 同步更新
        # theta[0,0] = theta0
        # theta[1,0] = theta1
        # thetas[0].append(theta0)
        # thetas[1].append(theta1)

        # 一起计算
        theta = theta - alpha / (1.0 * m) * np.dot(X.T, (h(theta, X) - Y))
        # X.T : n*m , h(theta, Y) : m*1 , np.dot(X.T, (h(theta, X)- Y)) : n*1
        # 同步更新
        thetas[0].append(theta[0])
        thetas[1].append(theta[1])

        # 更新当前cost
        cost = J(theta, X, Y)
        costs.append(cost)

        # 如果收敛，则不再迭代
        if cost < epsilon:
            converged = True
    return theta, costs, thetas


X, Y = loadDataSet('D:\python_project\HandlePythonExample\day01\data\ex1.txt')
print(X.shape)
print(Y.shape)

m, n = X.shape
X = np.concatenate((np.ones((m, 1)), X), axis=1)  # 将第一列为1的矩阵，与原X相连
print(X.shape)

alpha = 0.02  # 学习率
maxloop = 1500  # 最大迭代次数
epsilon = 0.01  # 收敛判断条件

result = bgd(alpha, maxloop, epsilon, X, Y)
theta, costs, thetas = result  # 最优参数保存在theta中，costs保存每次迭代的代价值，thetas保存每次迭代更新的theta值

print(theta)
# 到此，参数学习出来了，模型也就定下来了，若要预测新的实例，进行以下即可
# Y_predict = h(theta, X_predict)

# 以下为训练集的预测值
XCopy = X.copy()
XCopy.sort(0)  # axis=0 表示列内排序
# print(XCopy)
# print(Y)
yHat = h(theta, XCopy)
# print(XCopy[:,1].shape, yHat.shape, theta.shape)

# 绘制回归直线
plt.xlabel(u'城市人口(万)')
plt.ylabel(u'利润(万元)')
plt.plot(XCopy[:, 1], yHat, color='r')
plt.scatter(X[:, 1].flatten(), Y.T.flatten())  # 画散点图
plt.show()

调用库实现梯度下降法

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression

# 用来加载中文
import matplotlib

matplotlib.rcParams['font.sans-serif'] = ['SimHei']
matplotlib.rcParams['font.family'] = 'sans-serif'
matplotlib.rcParams['axes.unicode_minus'] = False  # 用来正常显示负号


def h(theta, X):
    '''定义模型函数'''
    return np.dot(X, theta)  # 此时的X为处理后的X


def loadDataSet(filename):
    '''加载文件，将feature存在X中，y存在Y中'''
    X = []
    Y = []
    with open(filename, 'rb') as f:
        for idx, line in enumerate(f):
            line = line.decode('utf-8').strip()
            if not line:
                continue
            eles = line.split()
            if idx == 0:
                numFeature = len(eles)
            eles = list(map(float, eles))  # 将数据转换成float型
            X.append(eles[:-1])  # 除最后一列都是feature，append(list)
            Y.append([eles[-1]])  # 最后一列是实际值,同上
        return np.array(X), np.array(Y)  # 将X,Y列表转化成矩阵


X, Y = loadDataSet('D:\python_project\HandlePythonExample\day01\data\ex1.txt')


print(X.shape)
print(Y.shape)
reg = LinearRegression().fit(X, Y)
print(reg.get_params())
print(reg.coef_.tolist()[0])
print(reg.intercept_)
print(reg.predict(np.array([[500]])))

m, n = X.shape
X = np.concatenate((np.ones((m, 1)), X), axis=1)  # 将第一列为1的矩阵，与原X相连
XCopy = X.copy()
XCopy.sort(0)  # axis=0 表示列内排序
a = (reg.coef_.tolist()[0])[0]
b = (reg.intercept_.tolist()[0])
theat = np.array([[b],[a]])
yHat = h(theat, XCopy)

plt.xlabel(u'城市人口(万)')
plt.ylabel(u'利润(万元)')


plt.plot(XCopy[:, 1], yHat, color='r')

plt.scatter(X[:, 1].flatten(), Y.T.flatten())  # 画散点图
plt.show()