1.5 欠拟合和过拟合
2024-04-13 07:25:52
1.5 欠拟合和过拟合
欠拟合（Underfitting）：选择的模型过于简单，以致于模型对训练集和未知数据的预测都很差的现象。
过拟合（Overfitting）：选择的模型过于复杂（所包含的参数过多），以致于模型对训练集的预测很好，但对未知数据预测很差的现象（泛化能力差）。
过拟合可以通过观测训练集和验证集的loss变化判断：随着epoch或step的增加，训练集 loss不断下降，而验证集loss先下降后抬升。
解决过拟合和欠拟合的方法
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解决欠拟合的方法
增加特征数
高斯特征
组合特征
增加模型复杂度
或者采用拟合能力更强的模型
或者采用拟合能力更强的模型
神经网络
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解决过拟合的方法
增加训练数据量
减小模型复杂度
在目标函数中添加正则项
提前终止 EARLY STOPPING
DROP OUT
过拟合常见解决方法
1. 增加训练样本数目
从理论上说，只要有足够多的样本数量，任何模型都不会出现过拟合的情况。
生成200个训练样本
# 设置随机种子
np.random.seed(34)sample_num = 200# 从-10到10中随机抽取100个浮点数
x_train = np.random.uniform(-5, 5, size=sample_num)# 将x从shape为(sample_num,)变为(sample_num,1)
X_train = x_train.reshape(-1,1)# 生成y值的实际函数
y_train_real = 0.5 * x_train ** 3 + x_train ** 2 + 2 * x_train + 1# 生成误差值
err_train = np.random.normal(0, 5, size=sample_num)# 真实y值加上误差值，得到样本的y值
y_train = y_train_real + err_train# 画出样本的散点图
plt.scatter(x_train, y_train, marker='o', color='g')# 画出实际函数曲线
plt.plot(np.sort(x_train), y_train_real[np.argsort(x_train)], color='b', label='real curve')
plt.legend()
plt.xlabel('x')
plt.ylabel('y')
plt.show()
30阶多项式模型训练
# 生成多项式数据
poly = PolynomialFeatures(degree=30, include_bias=False)
X_train_poly = poly.fit_transform(X_train)# 数据标准化（减均值除标准差）
scaler = StandardScaler()
X_train_poly_scaled = scaler.fit_transform(X_train_poly)# 线性回归模型训练
reg30 = LinearRegression()
reg30.fit(X_train_poly_scaled, y_train)# 模型预测
y_train_pred30 = reg30.predict(X_train_poly_scaled)# 画出样本的散点图
plt.scatter(x_train, y_train, marker='o', color='g')# 画出实际函数曲线
plt.plot(np.sort(x_train), y_train_real[np.argsort(x_train)], color='b', label='real curve')# 画出预测函数曲线
plt.plot(np.sort(x_train), y_train_pred30[np.argsort(x_train)], color='r', label='prediction curve')
plt.legend()
plt.xlabel('x')
plt.ylabel('y')
plt.show()# 计算MSE
mse = mean_squared_error(y_train_pred30, y_train)
print('MSE: {}'.format(mse))
MSE: 24.92469385931622
明显过拟合的情况相较于100个数据点时候，要好很多，MSE也小了不少
30阶多项式模型预测
# 生成多项式数据
poly = PolynomialFeatures(degree=30, include_bias=False)
X_test_poly = poly.fit_transform(X_test)# 数据标准化（减均值除标准差）
scaler = StandardScaler()
X_test_poly_scaled = scaler.fit_transform(X_test_poly)# 模型预测
y_test_pred30 = reg30.predict(X_test_poly_scaled)# 画出样本的散点图
plt.scatter(x_test, y_test, marker='o', color='c', label='test dataset')# 画出预测函数曲线
plt.plot(np.sort(x_test), y_test_pred30[np.argsort(x_test)], color='r', label='30 order')
plt.legend()
plt.xlabel('x')
plt.ylabel('y')
plt.show()
计算MSE
mse30 = mean_squared_error(y_test_pred30, y_test)# 打印结果
print('MSE:')
print('30 order polynomial: {:.2f}'.format(mse30))
输出结果：
MSE:
30 order polynomial: 32.32
2. 在目标函数中增加正则项
多项式拟合：
y=w0+w1x+w2x2+w3x3+⋯+wnxny=w_0+w_1x+w_2x^2+w_3x^3+\cdots+w_nx^ny=w0+w1x+w2x2+w3x3+⋯+wnxn
查看回归系数www
将结果转换为pd.DataFrame表格形式
coef1 = pd.DataFrame(reg1.coef_, index=['w1'], columns=['coef'])
coef3 = pd.DataFrame(reg3.coef_, index=['w1', 'w2', 'w3'], columns=['coef'])
coef10 = pd.DataFrame(reg10.coef_, index=['w'+str(i) for i in range(1,11)], columns=['coef'])
coef30 = pd.DataFrame(reg30.coef_, index=['w'+str(i) for i in range(1,31)], columns=['coef'])
1阶多项式模型参数
coef1
| coef1 | |
|---|---|
| w1 | 9.900252 |
3阶多项式模型参数
coef3
| coef3 | |
|---|---|
| w1 | 7.789175 |
| w2 | 7.000036 |
| w3 | 25.295452 |
10阶多项式模型参数
coef10
| coef10 | |
|---|---|
| w1 | 7.998547 |
| w2 | 4.203915 |
| w3 | 20.728305 |
| w4 | 15.694967 |
| w5 | 10.679321 |
| w6 | -53.302415 |
| w7 | -5.051154 |
| w8 | 72.956004 |
| w9 | -1.464603 |
| w10 | -32.583643 |
30阶多项式模型参数
coef30
| coef | |
|---|---|
| w1 | 1.274825e+01 |
| w2 | -1.515071e+02 |
| w3 | -2.784062e+02 |
| w4 | 9.881947e+03 |
| w5 | 8.313355e+03 |
| w6 | -2.294226e+05 |
| w7 | -1.443295e+05 |
| w8 | 2.805398e+06 |
| w9 | 1.540533e+06 |
| w10 | -2.094909e+07 |
| w11 | -1.036432e+07 |
| w12 | 1.035820e+08 |
| w13 | 4.594186e+07 |
| w14 | -3.558672e+08 |
| w15 | -1.390464e+08 |
| w16 | 8.741809e+08 |
| w17 | 2.938075e+08 |
| w18 | -1.558118e+09 |
| w19 | -4.373421e+08 |
| w20 | 2.020758e+09 |
| w21 | 4.563577e+08 |
| w22 | -1.888828e+09 |
| w23 | -3.264643e+08 |
| w24 | 1.240156e+09 |
| w25 | 1.523610e+08 |
| w26 | -5.429771e+08 |
| w27 | -4.173929e+07 |
| w28 | 1.424089e+08 |
| w29 | 5.084136e+06 |
| w30 | -1.693219e+07 |
可以发现30阶的模型系数www绝对值已经到了10810^8108量级
模型太复杂（阶数过多），发生过拟合时，系数www绝对值往往会很大，输入xxx的很小变化都可能带来输出yyy较大的变化，导致函数变化剧烈。
添加正则项的实质就是限制系数www的大小，使得输入xxx的变化尽量小的影响输出yyy的变化
训练Ridge回归（加L2正则）
使用到的api：
加L2正则的线性回归sklearn.linear_model.Ridge
用到的参数：
- alpha：惩罚项，默认1.0。
# 生成多项式数据
poly = PolynomialFeatures(degree=30, include_bias=False)
X_train_poly = poly.fit_transform(X_train)# 数据标准化（减均值除标准差）
scaler = StandardScaler()
X_train_poly_scaled = scaler.fit_transform(X_train_poly)# 线性回归模型训练
ridge30 = Ridge(alpha=1e-5)
ridge30.fit(X_train_poly_scaled, y_train)# 模型预测
y_train_pred30 = ridge30.predict(X_train_poly_scaled)# 画出样本的散点图
plt.scatter(x_train, y_train, marker='o', color='g')# 画出实际函数曲线
plt.plot(np.sort(x_train), y_train_real[np.argsort(x_train)], color='b', label='real curve')# 画出预测函数曲线
plt.plot(np.sort(x_train), y_train_pred30[np.argsort(x_train)], color='r', label='prediction curve')
plt.legend()
plt.xlabel('x')
plt.ylabel('y')
plt.show()# 计算MSE
mse = mean_squared_error(y_train_pred30, y_train)
print('MSE: {}'.format(mse))
查看加正则后的回归系数
将结果转换为pd.DataFrame表格形式
coef30 = pd.DataFrame(ridge30.coef_, index=['w'+str(i) for i in range(1,31)], columns=['coef'])
coef30
| coef | |
|---|---|
| w1 | 10.011475 |
| w2 | 3.557561 |
| w3 | -4.055929 |
| w4 | 7.173610 |
| w5 | 100.444037 |
| w6 | 26.619616 |
| w7 | -75.014554 |
| w8 | -90.566477 |
| w9 | -182.926719 |
| w10 | -16.985566 |
| w11 | 172.986261 |
| w12 | 101.786157 |
| w13 | 257.743920 |
| w14 | 114.148736 |
| w15 | 14.865359 |
| w16 | 11.446064 |
| w17 | -256.410161 |
| w18 | -117.329198 |
| w19 | -313.210520 |
| w20 | -169.444870 |
| w21 | -113.923504 |
| w22 | -95.081605 |
| w23 | 205.527434 |
| w24 | 73.135599 |
| w25 | 413.318846 |
| w26 | 222.862258 |
| w27 | 261.405745 |
| w28 | 183.223254 |
| w29 | -458.772049 |
| w30 | -248.055649 |
可以发现，系数www的确小了很多
模型检验
# 生成多项式数据
poly = PolynomialFeatures(degree=30, include_bias=False)
X_test_poly = poly.fit_transform(X_test)# 数据标准化（减均值除标准差）
scaler = StandardScaler()
X_test_poly_scaled = scaler.fit_transform(X_test_poly)# 模型预测
y_test_pred30 = ridge30.predict(X_test_poly_scaled)# 画出样本的散点图
plt.scatter(x_test, y_test, marker='o', color='c', label='test dataset')# 画出预测函数曲线
plt.plot(np.sort(x_test), y_test_pred30[np.argsort(x_test)], color='r', label='30 order')
plt.legend()
plt.xlabel('x')
plt.ylabel('y')
plt.show()
计算MSE
mse30 = mean_squared_error(y_test_pred30, y_test)# 打印结果
print('MSE:')
print('30 order polynomial: {:.2f}'.format(mse30))
MSE:
30 order polynomial: 43.50
添加正则项是有效的
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