实现线性回归中的梯度下降法
这篇博客主要是具体编程实现线性回归中使用梯度下降法。
之后,我们将以上函数封装进我们之前写的 LinearRegression 中,封装的函数名为 fit_gd:
# LinearRegression.py
import numpy as np
from metrics import r2_score
class LinearRegression:
def __init__(self):
"""初始化 Linear Regression"""
self.coef_ = None # 系数
self.interception_ = None # 截距
self._theta = None # θ
def fit_normal(self, X_train, y_train):
"""根据训练数据集X_train,y_train训练Linear Regression模型"""
assert X_train.shape[0] == y_train.shape[0], \
"the size of X_train must be equal to the size of y_train"
X_b = np.hstack([np.ones((len(X_train), 1)), X_train]) # 在 X_train 前加一列 1
self._theta = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y_train)
self.interception_ = self._theta[0] #截距
self.coef_ = self._theta[1:] #系数
return self
def fit_gd(self, X_train, y_train, eta=0.01, n_iters=1e4):
"""根据训练数据集X_train,y_train,使用梯度下降法训练Linear Regression模型"""
assert X_train.shape[0] == y_train.shape[0], \
"the size of X_train, y_train must be equal to the size of y_train"
def J(theta, X_b, y):
try:
return np.sum((y - X_b.dot(theta)) ** 2) / len(X_b)
except:
return float('inf')
def dJ(theta, X_b, y):
res = np.empty(len(theta))
res[0] = np.sum(X_b.dot(theta) - y)
for i in range(1, len(theta)):
res[i] = np.sum((X_b.dot(theta) - y).dot(X_b[:, i]))
return res * 2 / len(X_b)
def gradient_descent(X_b, y, initial_theta, eta, n_iters=1e4, epsilon=1e-8):
theta = initial_theta
i_iter = 0
while i_iter < n_iters:
gradient = dJ(theta, X_b, y)
last_theta = theta
theta = theta - eta * gradient
if(abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
break
i_iter += 1
return theta
X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
initial_theta = np.zeros(X_b.shape[1])
self._theta = gradient_descent(X_b, y_train, initial_theta, eta)
self.interception_ = self._theta[0]
self.coef_ = self._theta[1:]
return self
def predict(self, X_predict):
"""给定待预测数据集X_predict,返回表示X_predict的结果向量"""
assert self.interception_ is not None and self.coef_ is not None, \
"must fit before predict!"
assert X_predict.shape[1] == len(self.coef_), \
"the feature number of X_predict must be equal to X_train"
X_b = np.hstack([np.ones((len(X_predict), 1)), X_predict])
return X_b.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"""
y_predict = self.predict(X_test)
return r2_score(y_test, y_predict)
def __repr__(self):
return "LinearRegression()"
封装完成,下面就开始进行测试:
在这篇博客中使用了一个相对比较笨的方法封装好了我们的线性回归算法,使用梯度下降法进行训练。之前我们说过,当我们的算法有很多计算的时候,如果能进行向量化是最好的。我将在下一篇博客中讨论在梯度下降法的过程中是否可以用向量化的方式来进行提速。与此同时,我们也要看看,当我们使用真实的数据来运用我们的梯度下降法训练线性回归模型的时候,我们还有哪些需要注意的地方。
具体代码见 23 实现线性回归中的梯度下降法.ipynb