线性模型-回归

学习资料

• 李宏毅《机器学习》回归

线性回归

• 数据：工资（特征）
• 目标：预测银行会贷款给我多少钱（标签）

Linear model：y = b + \sum w_ix_i,其中x_i叫作fature，w_i叫作weight，b叫作bias
用Loss function L，定义一个function的好坏

回归演示

import numpy as np
import matplotlib.pyplot as plt

x_data = [338., 333., 328., 207., 226., 25., 179., 60., 208., 606.]
y_data = [640., 633., 619., 393., 428., 27., 193., 66., 226., 1591.]
x_d = np.asarray(x_data)
y_d = np.asarray(y_data)

x = np.arange(-200, -100, 1)
y = np.arange(-5, 5, 0.1)
Z = np.zeros((len(x), len(y)))
X, Y = np.meshgrid(x, y)
# loss
for i in range(len(x)):
    for j in range(len(y)):
        b = x[i]
        w = y[j]
        Z[j][i] = 0  # meshgrid吐出结果：y为行，x为列
        for n in range(len(x_data)):
            Z[j][i] += (y_data[n] - b - w * x_data[n]) ** 2
        Z[j][i] /= len(x_data)

b=-120
w=-4
lr = 0.0000001
iteration = 100000

b=-120
w=-4
lr = 3
iteration = 10000


b_history = [b]
w_history = [w]

lr_b = 0
lr_w=0

loss_history = []
import time
start = time.time()
for i in range(iteration):
    m = float(len(x_d))
    y_hat = w * x_d  +b
    loss = np.dot(y_d - y_hat, y_d - y_hat) / m
    grad_b = -2.0 * np.sum(y_d - y_hat) / m
    grad_w = -2.0 * np.dot(y_d - y_hat, x_d) / m
    
    
    lr_b = lr_b + grad_b **2
    lr_w = lr_w  + grad_w **2
    
    # update param
    b -= lr/np.sqrt(lr_b) * grad_b
    w -= lr/np.sqrt(lr_w) * grad_w

    b_history.append(b)
    w_history.append(w)
    loss_history.append(loss)
    if i % 1000 == 0:
        print("Step %i, w: %0.4f, b: %.4f, Loss: %.4f" % (i, w, b, loss))
end = time.time()
print("大约需要时间：",end-start)

# plot the figure
plt.contourf(x, y, Z, 50, alpha=0.5, cmap=plt.get_cmap('jet'))  # 填充等高线
plt.plot([-188.4], [2.67], 'x', ms=12, mew=3, color="orange")
plt.plot(b_history, w_history, 'o-', ms=3, lw=1.5, color='black')
plt.xlim(-200, -100)
plt.ylim(-5, 5)
plt.xlabel(r'$b$')
plt.ylabel(r'$w$')
plt.title("linear regression")
plt.show()

主成分回归

from sklearn.decomposition import PCA
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import train_test_split
from sklearn.metrics import r2_score
import numpy as np

# 生成示例数据
X = np.random.rand(100, 10)  # 特征矩阵，假设有10个特征
y = np.random.rand(100)  # 目标变量

# 将数据集拆分为训练集和测试集
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

# 使用PCA进行降维
pca = PCA(n_components=5)  # 选择前5个主成分
X_train_pca = pca.fit_transform(X_train)
X_test_pca = pca.transform(X_test)

# 使用线性回归模型进行训练和预测
regression = LinearRegression()
regression.fit(X_train_pca, y_train)
y_pred = regression.predict(X_test_pca)

# 计算R2分数（拟合优度）
r2 = r2_score(y_test, y_pred)
print("R2 score:", r2)