机器学习：多项式回归（Python）
多元线性回归闭式解：
closed_form_sol.py
import numpy as np
import matplotlib.pyplot as plt
class LRClosedFormSol:
def __init__(self, fit_intercept=True, normalize=True):
"""
:param fit_intercept: 是否训练bias
:param normalize: 是否标准化数据
"""
self.theta = None # 训练权重系数
self.fit_intercept = fit_intercept # 线性模型的常数项。也即偏置bias，模型中的theta0
self.normalize = normalize # 是否标准化数据
if normalize:
self.feature_mean, self.feature_std = None, None # 特征的均值，标准方差
self.mse = np.infty # 训练样本的均方误差
self.r2, self.r2_adj = 0.0, 0.0 # 判定系数和修正判定系数
self.n_samples, self.n_features = 0, 0 # 样本量和特征数
def fit(self, x_train, y_train):
"""
模型训练，根据是否标准化与是否拟合偏置项分类讨论
:param x_train: 训练样本集
:param y_train: 训练目标集
:return:
"""
if self.normalize:
self.feature_mean = np.mean(x_train, axis=0) # 按样本属性计算样本均值
self.feature_std = np.std(x_train, axis=0) + 1e-8 # 样本方差，为避免零除，添加噪声
x_train = (x_train - self.feature_mean) / self.feature_std # 标准化
if self.fit_intercept:
x_train = np.c_[x_train, np.ones_like(y_train)] # 添加一列1，即偏置项样本
# 训练模型
self._fit_closed_form_solution(x_train, y_train) # 求闭式解
def _fit_closed_form_solution(self, x_train, y_train):
"""
线性回归的闭式解，单独函数，以便后期扩充维护
:param x_train: 训练样本集
:param y_train: 训练目标集
:return:
"""
# pinv伪逆，即(A^T * A)^(-1) * A^T
self.theta = np.linalg.pinv(x_train).dot(y_train) # 非正则化
# xtx = np.dot(x_train.T, x_train) + 0.01 * np.eye(x_train.shape[1]) # 按公式书写
# self.theta = np.dot(np.linalg.inv(xtx), x_train.T).dot(y_train)
def get_params(self):
"""
返回线性模型训练的系数
:return:
"""
if self.fit_intercept: # 存在偏置项
weight, bias = self.theta[:-1], self.theta[-1]
else:
weight, bias = self.theta, np.array([0])
if self.normalize: # 标准化后的系数
weight = weight / self.feature_std.reshape(-1, 1) # 还原模型系数
bias = bias - weight.T.dot(self.feature_mean)
return weight, bias
def predict(self, x_test):
"""
测试数据预测，x_test：待预测样本集，不包括偏置项1
:param x_test:
:return:
"""
try:
self.n_samples, self.n_features = x_test.shape[0], x_test.shape[1]
except IndexError:
self.n_samples, self.n_features = x_test.shape[0], 1 # 测试样本数和特征数
if self.normalize:
x_test = (x_test - self.feature_mean) / self.feature_std # 测试数据标准化
if self.fit_intercept:
x_test = np.c_[x_test, np.ones(shape=x_test.shape[0])] # 存在偏置项，添加一列1
return x_test.dot(self.theta)
def cal_mse_r2(self, y_pred, y_test):
"""
计算均方误差，计算拟合优度的判定系数R方和修正判定系数
:param y_pred: 模型预测目标真值
:param y_test: 测试目标真值
:return:
"""
self.mse = ((y_test - y_pred) ** 2).mean() # 均方误差
# 计算测试样本的判定系数和修正判定系数
self.r2 = 1 - ((y_test - y_pred) ** 2).sum() / ((y_test - y_test.mean()) ** 2).sum()
self.r2_adj = 1 - (1 - self.r2) * (self.n_samples - 1) / (self.n_samples - self.n_features - 1)
return self.mse, self.r2, self.r2_adj
def plt_predict(self, y_pred, y_test, is_show=True, is_sort=True):
"""
绘制预测值与真实值对比图
:return:
"""
if self.mse is np.infty:
self.cal_mse_r2(y_pred, y_test)
if is_show:
plt.figure(figsize=(7, 5))
if is_sort:
idx = np.argsort(y_test)
plt.plot(y_pred[idx], "r:", lw=1.5, label="Predictive Val")
plt.plot(y_test[idx], "k--", lw=1.5, label="Test True Val")
else:
plt.plot(y_pred, "r:", lw=1.5, label="Predictive Val")
plt.plot(y_test, "k--", lw=1.5, label="Test True Val")
plt.xlabel("Test sample observation serial number", fontdict={"fontsize": 12})
plt.ylabel("Predicted sample value", fontdict={"fontsize": 12})
plt.title("The predictive values of test samples \n MSE = %.5e, R2 = %.5f, R2_adj = %.5f"
% (self.mse, self.r2, self.r2_adj), fontdict={"fontsize": 14})
plt.legend(frameon=False)
plt.grid(ls=":")
if is_show:
plt.show()
多项式构造
polynomial_feature.py
import numpy as np
class PolynomialFeatureData:
"""
生成特征多项式数据
"""
def __init__(self, x, degree, with_bias=False):
"""
参数初始化
:param x: 采用数据，向量形式
:param degree: 多项式最高阶次
:param with_bias: 是否需要偏置项
"""
self.x = np.asarray(x)
self.degree = degree
self.with_bias = with_bias
if with_bias:
self.data = np.zeros((len(x), degree + 1))
else:
self.data = np.zeros((len(x), degree))
def fit_transform(self):
"""
构造多项式特征数据
:return:
"""
if self.with_bias:
self.data[:, 0] = np.ones(len(self.x))
self.data[:, 1] = self.x.reshape(-1)
for i in range(2, self.degree + 1):
self.data[:, i] = (self.x ** i).reshape(-1)
else:
self.data[:, 0] = self.x.reshape(-1)
for i in range(1, self.degree):
self.data[:, i] = (self.x ** (i + 1)).reshape(-1)
return self.data
if __name__ == '__main__':
x = np.random.randn(5)
feat_obj = PolynomialFeatureData(x, 5, with_bias=True)
data = feat_obj.fit_transform()
print(data)
多项式回归
test_polynomial_fit.py
import numpy as np
import matplotlib.pyplot as plt
from polynomial_feature import PolynomialFeatureData
from closed_form_sol import LRClosedFormSol
def objective_fun(x):
"""
目标函数，假设一个随机二次多项式
:param x: 采样数据，向量
:return:
"""
return 0.5 * x ** 2 + x + 2
np.random.seed(42) # 随机种子，以便结果可再现
n = 30 # 样本量
raw_x = np.sort(6 * np.random.rand(n, 1) - 3) # 采样数据[-3, 3]，均匀分布
raw_y = objective_fun(raw_x) + 0.5 * np.random.randn(n, 1) # 目标值，添加噪声
degree = [1, 2, 5, 10, 15, 20] # 拟合多项式的最高阶次
plt.figure(figsize=(15, 8))
for i, d in enumerate(degree):
feature_obj = PolynomialFeatureData(raw_x, d, with_bias=False) # 特征数据对象
X_samples = feature_obj.fit_transform() # 生成特征多项式数据
lr_cfs = LRClosedFormSol() # 采用线性回归求解多项式回归
lr_cfs.fit(X_samples, raw_y) # 求解多项式回归系数
theta = lr_cfs.get_params() # 获得系数
print("degree: %d, theta is " %d, theta[0].reshape(-1)[::-1], theta[1])
y_train_pred = lr_cfs.predict(X_samples) # 在训练集上的预测
# 测试样本采样
X_test_raw = np.linspace(-3, 3, 150) # 测试数据
y_test = objective_fun(X_test_raw) # 测试数据的真值
feature_obj = PolynomialFeatureData(X_test_raw, d, with_bias=False) # 特征数据对象
X_test = feature_obj.fit_transform() # 生成特征多项式数据
y_test_pred = lr_cfs.predict(X_test) # 模型在测试样本上的预测值
# 可视化不同阶次下的多项式拟合曲线
plt.subplot(231 + i)
plt.scatter(raw_x, raw_y, edgecolors="k", s=15, label="Raw Data")
plt.plot(X_test_raw, y_test, "k-", lw=1, label="Objective Fun")
plt.plot(X_test_raw, y_test_pred, "r--", lw=1.5, label="Polynomial Fit")
plt.legend(frameon=False)
plt.grid(ls=":")
plt.xlabel("$x$", fontdict={"fontsize": 12})
plt.ylabel("$y(x)$", fontdict={"fontsize": 12})
test_ess = (y_test_pred.reshape(-1) - y_test) ** 2 # 误差平方
test_mse, test_std = np.mean(test_ess), np.std(test_ess)
train_ess = (y_train_pred - raw_y) ** 2 # 误差平方
train_mse, train_std = np.mean(train_ess), np.std(train_ess)
plt.title("Degree {} Test Mse = {:.2e}(+/-{:.2e}) \n Train Mse = {:.2e}(+/-{:.2e})"
.format(d, test_mse, test_std, train_mse, train_std))
plt.axis([-3, 3, 0, 9])
plt.tight_layout()
plt.show()
输出结果：
学习曲线
learning_curve1.py
import numpy as np
import matplotlib.pyplot as plt
from polynomial_feature import PolynomialFeatureData
from closed_form_sol import LRClosedFormSol
from sklearn.model_selection import train_test_split
def objective_fun(x):
"""
目标函数，假设一个随机二次多项式
:param x: 采样数据，向量
:return:
"""
return 0.5 * x ** 2 + x + 2
np.random.seed(42) # 随机种子，以便结果可再现
n = 100 # 样本量
raw_x = np.sort(6 * np.random.rand(n, 1) - 3) # 采样数据[-3, 3]，均匀分布
raw_y = objective_fun(raw_x) + 0.5 * np.random.randn(n, 1) # 目标值，添加噪声
degree = [1, 2, 5, 10] # 拟合阶次
plt.figure(figsize=(10, 7)) # 图像尺寸
for i, d in enumerate(degree):
# 生成特征多项式对象，包含偏置项
feta_obj = PolynomialFeatureData(raw_x, d, with_bias=False)
X_sample = feta_obj.fit_transform() # 生成特征多项式数据
X_train, X_test, y_train, y_test = \
train_test_split(X_sample, raw_y, test_size=0.2, random_state=0)
train_mse, test_mse = [], [] # 随样本量的增加，训练误差和测试误差
for j in range(1, 80):
lr_cfs = LRClosedFormSol() # 线性回归闭式解
theta = lr_cfs.fit(X_train[:j, :], y_train[:j]) # 拟合多项式
y_test_pred = lr_cfs.predict(X_test) # 测试样本预测
y_train_pred = lr_cfs.predict(X_train[:j, :]) # 训练样本预测
train_mse.append(np.mean((y_train_pred.reshape(-1) - y_train[:j].reshape(-1)) ** 2))
test_mse.append(np.mean((y_test_pred.reshape(-1) - y_test.reshape(-1)) ** 2))
# 可视化多项式拟合曲线
plt.subplot(221 + i)
plt.plot(train_mse, "k-", lw=1, label="Train")
plt.plot(test_mse, "r--", lw=1.5, label="Test")
plt.legend(frameon=False)
plt.grid(ls=":")
plt.xlabel("Train Size", fontdict={"fontsize": 12})
plt.ylabel("MSE", fontdict={"fontsize": 12})
plt.title("Learning Curve by Degree {}".format(d))
if i == 0:
plt.axis([0, 80, 0, 4])
if i == 1:
plt.axis([0, 80, 0, 1])
if i in [2, 3]:
plt.axis([0, 80, 0, 2])
plt.tight_layout()
plt.show()
学习曲线（采用10折交叉验证）
learning_curve2.py
import numpy as np
import matplotlib.pyplot as plt
from polynomial_feature import PolynomialFeatureData
from closed_form_sol import LRClosedFormSol
from sklearn.model_selection import KFold
def objective_fun(x):
"""
目标函数，假设一个随机二次多项式
:param x: 采样数据，向量
:return:
"""
return 0.5 * x ** 2 + x + 2
np.random.seed(42) # 随机种子，以便结果可再现
n = 300 # 样本量
raw_x = np.sort(6 * np.random.rand(n, 1) - 3) # 采样数据[-3, 3]，均匀分布
raw_y = objective_fun(raw_x) + 0.5 * np.random.randn(n, 1) # 目标值，添加噪声
k_fold = KFold(n_splits=10) # 划分为10折
degree = [1, 2, 4, 6, 8, 10] # 拟合阶次
plt.figure(figsize=(15, 8)) # 图像尺寸
for i, d in enumerate(degree):
# 生成特征多项式对象，包含偏置项
feta_obj = PolynomialFeatureData(raw_x, d, with_bias=False)
X_sample = feta_obj.fit_transform() # 生成特征多项式数据
train_mse, test_mse = [], [] # 随样本量的增加，训练误差和测试误差
for j in range(1, 270):
train_mse_, test_mse_ = 0.0, 0.0 # 交叉验证
for idx_train, idx_test in k_fold.split(raw_x, raw_y):
X_train, y_train = X_sample[idx_train], raw_y[idx_train]
X_test, y_test = X_sample[idx_test], raw_y[idx_test]
lr_cfs = LRClosedFormSol() # 线性回归闭式解
theta = lr_cfs.fit(X_train[:j, :], y_train[:j]) # 拟合多项式
y_test_pred = lr_cfs.predict(X_test) # 测试样本预测
y_train_pred = lr_cfs.predict(X_train[:j, :]) # 训练样本预测
train_mse_ += np.mean((y_train_pred.reshape(-1) - y_train[:j].reshape(-1)) ** 2)
test_mse_ += np.mean((y_test_pred.reshape(-1) - y_test.reshape(-1)) ** 2)
train_mse.append(train_mse_ / 10)
test_mse.append(test_mse_ / 10)
# 可视化多项式拟合曲线
plt.subplot(231 + i)
plt.plot(train_mse, "k-", lw=1, label="Train")
plt.plot(test_mse, "r--", lw=1.5, label="Test")
plt.legend(frameon=False)
plt.grid(ls=":")
plt.xlabel("Train Size", fontdict={"fontsize": 12})
plt.ylabel("MSE", fontdict={"fontsize": 12})
plt.title("Learning Curve by Degree {}".format(d))
if i == 0:
plt.axis([0, 250, 0, 4])
else:
plt.axis([0, 250, 0, 1])
plt.tight_layout()
plt.show()
