Bobo老师机器学习笔记第六课-梯度下降法在线性回归中的应用
在上一篇博客中大概介绍了一下梯度下降法,那么梯度下降法在线性回归中如何更好的应用了,本篇博客做一介绍。
在BoBo老师的课程中,梯度下降法主要讲了2中,批量梯度下降法(Batch Gradient Descent)和随机梯度下降法(Stochastic Gradient Descent)。
一、理论介绍
1、批量梯度下降法(Batch Gradient Descent)
损失函数以及未使用向量化的方程:
进行向量化后的梯度方程:
从上图方程中可以看出,每一次求theta的值都要把所有的样本遍历一遍,所以这是为什么成为批量梯度下降法。
2、随机梯度下降法(Stonastic Gradient Descent)
损失函数不变,但计算计算梯度的方法如下:
从上图的公式可以看出,计算梯度是随机取出其中一个样本进行计算的。此外还要注意学习率的区别:
a一般取5,b为50,i_iters表示当前迭代的次数。 而这个值在批量梯度学习算法中是一个常量,一般是0.01
二、在线性回归中的应用
# -*- coding: utf-8 -*-
import numpy as np
from metrics import r2_score
class LinearRegression(object):
def __init__(self):
self.coef_ = None # 表示系数
self.intercept_ = None # 表示截距
self._theta = None # 过程计算值,不需要暴露给外面
def fit_normal(self, X_train, y_train):
"""根据训练数据集X_train, y_train, 利用正规方程进行训练Linear Regression模型,利用正规方程训练的时候就不需要对数据进行向量化"""
assert X_train is not None and y_train is not None, "训练集X和Y不能为空"
assert X_train.shape[0] == y_train.shape[0], "训练集X和Y的样本数要相等"
# np.linalg.inv(X) 表示求X的逆矩阵
# 不能忘了X要增加一列,第一列数据为0
ones = np.ones(shape=(len(X_train), 1))
X_train = np.hstack((ones, X_train))
self._theta = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train)
self.intercept_ = self._theta[0]
self.coef_ = self._theta[1:]
def fit_gd(self, X_train, y_train, eta=0.01, n_iters=1e4):
"""
用批量梯度下降法训练模型
:param X_train: 经过向量化的特征数据
:param y_train:
:param eta: 步长
:param n_iters: 迭代次数
:return:
"""
def J(theta, X_b, y):
"""
损失函数
此处要注意:X_b相对于原特征矩阵多了一列 n * 1的列向量, 所以X_b是(m, n+1)的矩阵
:return:
"""
return np.sum((y - X_b.dot(theta)) ** 2) / len(y)
def DJ(theta, X_b, y):
"""
获取梯度
:return:
"""
# 注释掉的算法是不用向量计算的实现
# res = np.empty(len(theta))
# res[0] = np.sum(X_b.dot(theta) - y)
# for i in range(1, len(theta)):
# res[i] = (X_b.dot(theta) - y).dot(X_b[:, i])
# return res * 2 / len(X_b)
return X_b.T.dot(X_b.dot(theta) - y) * 2 / len(X_b)
def gredient_descent(X_b, y, theta, n_inters=1e4, eta=0.01, epsilon=1e-8):
"""
利用批量梯度下降法训练线性回归
:param X_b: 是(m, n+1)的矩阵
:param y:
:param init_ethta: etha初始化值
:param n_inters: 迭代次数
:param eta: 变化率步长, 默认是0.01
:param epsilon: 精度,用来比较当前etha和上一次etha差值
:return:
"""
cur_inter = 0
while cur_inter < n_inters:
last_theta = theta
theta = theta - eta * DJ(theta, X_b, y)
if abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon:
break
cur_inter += 1
return theta
X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
init_ethta = np.zeros(X_b.shape[1])
self._theta = gredient_descent(X_b, y_train, init_ethta, eta=eta, n_inters=n_iters)
self.coef_ = self._theta[1:]
self.intercept_ = self._theta[0]
return self
def fit_sgd(self, X_train, y_train, n_iters=5, t0=5.0, t1=50.0):
"""
利用随机梯度下降法训练线性回归
:param X_train: 向量化的特征值
:param y_train:
:param n_iters: 迭代次数
:param t0: 用来计算学习率
:param t1: 用来计算学习率
:return:
"""
def DJ(theta, X_b_i, y_i):
"""
获取梯度
X_b_i: 是X_b向量中的一个样本
:return:
"""
return X_b_i.T.dot(X_b_i.dot(theta) - y_i) * 2
def sgd(X_b, y, theta, n_inters, t0, t1):
def learning_rate(t):
return t0 / (t + t1)
m = len(X_b)
for cur_index in range(n_inters):
indexs = np.random.permutation(m)
X_b_new = X_b[indexs]
y_new = y[indexs]
for i in range(m):
gradient = DJ(theta, X_b_new[i], y_new[i])
theta = theta - learning_rate(cur_index * m + i) * gradient
return theta
X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
initial_theta = np.random.randn(X_b.shape[1])
self._theta = sgd(X_b, y_train, initial_theta, n_iters, t0, t1)
self.coef_ = self._theta[1:]
self.intercept_ = self._theta[0]
def predict(self, X_test):
"""给定待预测数据集X_test,返回表示X_test的结果向量"""
assert X_test.shape[1] == self.coef_.shape[0], '测试集X的特征值个数不对'
ones = np.ones(shape=(len(X_test), 1))
X_test = np.hstack((ones, X_test))
return X_test.dot(self._theta)
def score(self, X_test, y_test):
"""根据测试数据集 X_test 和 y_test 确定当前模型的准确度"""
assert X_test.shape[0] == y_test.shape[0], '测试集X和Y的个数不相等'
return r2_score(y_test, self.predict(X_test))
def __repr__(self):
return '%s(coef_:%s, intercept_:%s)' % (self.__class__.__name__, self.coef_, self.intercept_)
def test_regession_using_gd():
from linearregression import LinearRegression
# step 1: 创建训练数据
m = 10000 # 假设10000个样本
x = np.random.normal(size=m)
X = x.reshape(-1, 1)
y = 4. * x + 3. + np.random.normal(0, 3, size=m)
# step 2: 数据标准化
X_train, X_test, y_train, y_test = train_test_split(X, y)
standardscaler = StandardScaler()
standardscaler.fit(X_train)
x_train_standard = standardscaler.transform(X_train)
# step 3: 训练模型
lrg = LinearRegression()
# 批量梯度下降法
# lrg.fit_gd(x_train_standard, y_train, eta=0.001, n_iters=1e6)
# 随机梯度下降法
lrg.fit_sgd(x_train_standard, y_train)
# step 4: 获取评分
x_test_standard = standardscaler.transform(X_test)
print 'score:', lrg.score(x_test_standard, y_test)
运行结果:
批量梯度下降法
LinearRegression(coef_:[4.01485425], intercept_:3.0079330321024687)
score: 0.6569687845397328随机梯度下降法
LinearRegression(coef_:[3.96601229], intercept_:3.030732598906986) score: 0.644882902625483
三、在Sklearn中的实现
def test_sklearn():
# step 1: 创建训练数据
m = 10000 # 假设10000个样本
x = np.random.normal(size=m)
X = x.reshape(-1, 1)
y = 4. * x + 3. + np.random.normal(0, 3, size=m)
# step 2: 数据标准化
X_train, X_test, y_train, y_test = train_test_split(X, y)
standardscaler = StandardScaler()
standardscaler.fit(X_train)
x_train_standard = standardscaler.transform(X_train)
# step 3:调用sklearn模型
from sklearn.linear_model import SGDRegressor
# 随机梯度下降法
sgr = SGDRegressor()
sgr.fit(x_train_standard, y_train)
print sgr
# step 4: 获取评分
x_test_standard = standardscaler.transform(X_test)
print 'score:', sgr.score(x_test_standard, y_test)
运行结果:
SGDRegressor(alpha=0.0001, average=False, epsilon=0.1, eta0=0.01, fit_intercept=True, l1_ratio=0.15, learning_rate='invscaling', loss='squared_loss', max_iter=None, n_iter=None, penalty='l2', power_t=0.25, random_state=None, shuffle=True, tol=None, verbose=0, warm_start=False) score: 0.6465744359869433
要是你在西安,感兴趣一起学习AIOPS,欢迎加入QQ群 860794445