多項式回歸分析 (Polynomial Regression)

套路29: 多項式回歸分析 (Polynomial Regression)

1. 使用時機: 以自變項的多項式預測一個因變項。

2. 分析類型: 回歸分析(regression analysis)。

3. 範例資料: 咪路以河口為起點測量不同距離採樣點河水中的鉛含量，資料如下。

Sample	Dist. (km)	Conc. (mg/L)
1	1.22	40.9
2	1.34	41.8
3	1.51	42.4
4	1.66	43.0
5	1.72	43.4
6	1.93	43.9
7	2.14	44.3
8	2.39	44.7
9	2.51	45.0
10	2.78	45.1
11	2.97	45.4
12	3.17	46.2
13	3.32	47.0
14	3.50	48.6
15	3.53	49.0
16	3.85	49.7
17	3.95	50.0
18	4.11	50.8
19	4.18	51.1

求多項式回歸方程式。

4. 建立資料

import numpy as np

import pandas as pd

x = np.array([1.22, 1.34, 1.51, 1.66, 1.72, 1.93, 2.14, 2.39, 2.51, 2.78, 2.97, 3.17, 3.32, 3.5, 3.53, 3.85, 3.95, 4.11, 4.18])

y = np.array([40.9, 41.8, 42.4, 43, 43.4, 43.9, 44.3, 44.7, 45, 45.1, 45.4, 46.2, 47, 48.6, 49, 49.7, 50, 50.8, 51.1])

df = pd.DataFrame(columns=['y', 'x'])

df['x'] = x

df['y'] = y

5. 執行回歸

import statsmodels.formula.api as smf

d1 = 2

w1 = np.polyfit(x, y, d1)

model1 = np.poly1d(w1)

res = smf.ols(formula='y ~ model(x)', data=df).fit()

res.summary()

結果:

OLS Regression Results                           

==============================================================================

Dep. Variable:                      y   R-squared:                       0.982

Model:                            OLS   Adj. R-squared:                  0.981

Method:                 Least Squares   F-statistic:                     917.5

Date:                Mon, 29 Jul 2019   Prob (F-statistic):           3.11e-16

Time:                        15:26:23   Log-Likelihood:                -10.198

No. Observations:                  19   AIC:                             24.40

Df Residuals:                      17   BIC:                             26.29

Df Model:                           1                                        

Covariance Type:            nonrobust                                        

==============================================================================

                 coef    std err          t      P>|t|      [0.025      0.975]

------------------------------------------------------------------------------

Intercept  -1.634e-13      1.519  -1.08e-13      1.000      -3.205       3.205

model(x)       1.0000      0.033     30.291      0.000       0.930       1.070

==============================================================================

Omnibus:                        2.026   Durbin-Watson:                   0.745

Prob(Omnibus):                  0.363   Jarque-Bera (JB):                0.901

Skew:                           0.521   Prob(JB):                        0.637

Kurtosis:                       3.230   Cond. No.                         697.

==============================================================================