线性回归¶
- 任务： 预测房价
- 输入：五维特征
- 标签：房价
1. 数据准备¶
In [1]:
import numpy as np
from sklearn import preprocessing
# 加载数据
lines=np.loadtxt('../input/USA_Housing.csv', delimiter=',', dtype='str')
print("输入")
for i in range(lines.shape[1]-1):
print(lines[0, i])
print("标签")
print(lines[0,-1])
x_total = lines[1:, :5].astype('float')
y_total = lines[1:, 5:].astype('float').flatten()
# 数据预处理和切分
x_total = preprocessing.scale(x_total)
y_total = preprocessing.scale(y_total)
x_train = x_total[:4000]
x_test = x_total[4000:]
y_train = y_total[:4000]
y_test = y_total[4000:]
print('训练集大小: ', x_train.shape[0])
print('测试集大小: ', x_test.shape[0])
2. Normal Equation¶
$\mu = (X^T X)^{-1}X^Ty$
In [2]:
X_train = np.hstack([x_train, np.ones((x_train.shape[0], 1))])
NE_solution = np.dot(np.dot(np.linalg.inv(np.dot(np.transpose(X_train), X_train)), np.transpose(X_train)), y_train.reshape([-1, 1]))
print(NE_solution)
X_test = np.hstack([x_test, np.ones((x_test.shape[0], 1))])
y_pred_test = np.dot(X_test, NE_solution).flatten()
rmse_loss = np.sqrt(np.square(y_test - y_pred_test).mean())
print('rmse_loss:', rmse_loss)
3. Sklearn库¶
In [3]:
from sklearn import linear_model
linreg = linear_model.LinearRegression()
linreg.fit(x_train, y_train)
print(linreg.coef_)
print(linreg.intercept_)
y_pred_test = linreg.predict(x_test)
rmse_loss = np.sqrt(np.square(y_test - y_pred_test).mean())
print('rmse_loss:', rmse_loss)
4. Gradient Descent¶
$ \begin{aligned} \frac{\partial}{\partial \theta_j} J(\theta) &= \frac{\partial}{\partial \theta_j} \frac{1}{2} (h_\theta(x) - y)^2\\ &= 2 \cdot \frac{1}{2} (h_\theta(x) - y) \cdot \frac{\partial}{\partial \theta_j} (h_\theta(x) - y) \\ &= (h_\theta(x) - y) \cdot \frac{\partial}{\partial \theta_j} (\sum_{i=0}^n \theta_i x_i - y) \\ &= (h_\theta(x) - y) x_j \end{aligned} $
In [4]:
def shuffle_aligned_list(data):
num = data[0].shape[0]
shuffle_index = np.random.permutation(num)
return [d[shuffle_index] for d in data]
def batch_generator(data, batch_size, shuffle=True):
batch_count = 0
while True:
if batch_count * batch_size + batch_size >= data[0].shape[0]:
batch_count = 0
if shuffle:
data = shuffle_aligned_list(data)
start = batch_count * batch_size
end = start + batch_size
batch_count += 1
yield [d[start:end] for d in data]
In [5]:
import matplotlib.pyplot as plt
num_steps = 600
learning_rate = 0.01
batch_size = 40
weight = np.zeros(6)
np.random.seed(0)
batch_g = batch_generator([x_train, y_train], batch_size, shuffle=True)
x_test_concat = np.hstack([x_test, np.ones([x_test.shape[0], 1])])
loss_list = []
for i in range(num_steps):
rmse_loss = np.sqrt(np.square(np.dot(x_test_concat, weight) - y_test).mean())
loss_list.append(rmse_loss)
x_batch, y_batch = batch_g.__next__()
x_batch = np.hstack([x_batch, np.ones([batch_size, 1])])
y_pred = np.dot(x_batch, weight)
w_gradient = (x_batch * np.tile((y_pred - y_batch).reshape([-1, 1]), 6)).mean(axis=0)
weight = weight - learning_rate * w_gradient
print('weight:', weight)
print('rmse_loss:', rmse_loss)
loss_array = np.array(loss_list)
plt.plot(np.arange(num_steps), loss_array)
plt.show()
