通過 1至10 階來擬合對比 均方誤差及R評分，可以確定最優的“最大階數”。

            
              import numpy as np
import matplotlib.pyplot as plt
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression,Perceptron
from sklearn.metrics import mean_squared_error,r2_score
from sklearn.model_selection import train_test_split

X = np.array([-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10]).reshape(-1, 1)
y = np.array(2*(X**4) + X**2 + 9*X + 2)
#y = np.array([300,500,0,-10,0,20,200,300,1000,800,4000,5000,10000,9000,22000]).reshape(-1, 1)

x_train, x_test, y_train, y_test = train_test_split(X, y, test_size=0.3)
rmses = []
degrees = np.arange(1, 10)
min_rmse, min_deg,score = 1e10, 0 ,0

for deg in degrees:
    # 生成多項式特征集(如根據degree=3 ,生成 [[x,x**2,x**3]] )
    poly = PolynomialFeatures(degree=deg, include_bias=False)
    x_train_poly = poly.fit_transform(x_train)

    # 多項式擬合
    poly_reg = LinearRegression()
    poly_reg.fit(x_train_poly, y_train)
    #print(poly_reg.coef_,poly_reg.intercept_) #系數及常數
    
    # 測試集比較
    x_test_poly = poly.fit_transform(x_test)
    y_test_pred = poly_reg.predict(x_test_poly)
    
    #mean_squared_error(y_true, y_pred) #均方誤差回歸損失,越小越好。
    poly_rmse = np.sqrt(mean_squared_error(y_test, y_test_pred))
    rmses.append(poly_rmse)
    # r2 范圍[0，1]，R2越接近1擬合越好。
    r2score = r2_score(y_test, y_test_pred)
    
    # degree交叉驗證
    if min_rmse > poly_rmse:
        min_rmse = poly_rmse
        min_deg = deg
        score = r2score
    print('degree = %s, RMSE = %.2f ,r2_score = %.2f' % (deg, poly_rmse,r2score))
        
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(degrees, rmses)
ax.set_yscale('log')
ax.set_xlabel('Degree')
ax.set_ylabel('RMSE')
ax.set_title('Best degree = %s, RMSE = %.2f, r2_score = %.2f' %(min_deg, min_rmse,score))  
plt.show()
            
          

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因為因變量?Y = 2*(X**4) + X**2 + 9*X + 2 ，自變量和因變量是完整的公式，看圖很明顯，degree >=4 的都符合，擬合函數都正確。 （RMSE 最小，R平方非負且接近于1，則模型最好）

如果將 Y 值改為如下：

            y = np.array([300,500,0,-10,0,20,200,300,1000,800,4000,5000,10000,9000,22000]).reshape(-1, 1)
          

degree=3 是最好的，且 r 平方也最接近于1（注意：如果 R 平方為負數，則不準確，需再次測試。因樣本數據較少，可能也會判斷錯誤）。

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