一元線性迴歸-python
阿新 • • 發佈：2018-12-10
思路：
1、從0~10，生成等間距20個數作為x，
2、利用迴歸公式 y=5 + 2x +
3、計算y值
4、對資料進行估計
#生成從0到10之間選20個等間距的數
import numpy as np
import matplotlib.pyplot as plt
import statsmodels.api as sm
nsample = 20
#從0到10之間選20個等間距的數
x=np.linspace(0,10,nsample)
xarray([ 0. , 0.52631579, 1.05263158, 1.57894737, 2.10526316,
2.63157895, 3.15789474, 3.68421053, 4.21052632, 4.73684211,
5.26315789, 5.78947368, 6.31578947, 6.84210526, 7.36842105,
7.89473684, 8.42105263, 8.94736842, 9.47368421, 10. ])#使用最小二乘法，需要在陣列的前面新增一列 1，目的是與常數項組合
X=sm.add_constant(x)
Xarray([[ 1. , 0. ],
[ 1. , 0.52631579],
[ 1. , 1.05263158],
[ 1. , 1.57894737],
[ 1. , 2.10526316],
[ 1. , 2.63157895],
[ 1. , 3.15789474],
[ 1. , 3.68421053],
[ 1. , 4.21052632],
[ 1. , 4.73684211],
[ 1. , 5.26315789],
[ 1. , 5.78947368],
[ 1. , 6.31578947],
[ 1. , 6.84210526],
[ 1. , 7.36842105],
[ 1. , 7.89473684],
[ 1. , 8.42105263],
[ 1. , 8.94736842],
[ 1. , 9.47368421],
[ 1. , 10. ]])#構造y值，β0=2，β1=5
bate = np.array([2,5])
batearray([2, 5])
#設計誤差資料，構造高斯分佈
e=np.random.normal(size=nsample)
earray([-0.08130226, -0.99898515, -0.46717904, -0.52487297, -0.85998302,
1.00102852, 0.61557834, 0.4359724 , 1.36966089, -0.17069984,
0.33877027, -1.602145 , -0.1940928 , 1.58914167, -2.09103106,
-0.87802483, -0.46069062, -2.32511203, -1.42386623, -0.22494043])#實際值，y=β0 + x*β1 + e，構造出來的用於測試的真實值
y=np.dot(X,bate)+e
yarray([ 1.91869774, 3.6325938 , 6.79597886, 9.36986387, 11.66633277,
16.15892325, 18.40505202, 20.85702504, 24.42229247, 25.51351069,
28.65455974, 29.34522342, 33.38485457, 37.79966799, 36.75107421,
40.59565938, 43.64457254, 44.41173008, 47.94455482, 51.77505957])
資料構造完畢，計算迴歸方程
#最小二乘法
model=sm.OLS(y,X)
#擬合數據
res=model.fit()
#迴歸係數，即β0、β2
res.params
array([2.15061173, 4.90034992])
#檢視全部評估結果資料
res.summary()| Dep. Variable: | y | R-squared: | 0.996 |
|---|---|---|---|
| Model: | OLS | Adj. R-squared: | 0.995 |
| Method: | Least Squares | F-statistic: | 4072. |
| Date: | Thu, 13 Sep 2018 | Prob (F-statistic): | 1.15e-22 |
| Time: | 10:44:47 | Log-Likelihood: | -28.152 |
| No. Observations: | 20 | AIC: | 60.30 |
| Df Residuals: | 18 | BIC: | 62.30 |
| Df Model: | 1 | ||
| Covariance Type: | nonrobust |
| coef | std err | t | P>|t| | [0.025 | 0.975] | |
|---|---|---|---|---|---|---|
| const | 2.1506 | 0.449 | 4.788 | 0.000 | 1.207 | 3.094 |
| x1 | 4.9003 | 0.077 | 63.815 | 0.000 | 4.739 | 5.062 |
| Omnibus: | 0.468 | Durbin-Watson: | 1.957 |
|---|---|---|---|
| Prob(Omnibus): | 0.791 | Jarque-Bera (JB): | 0.572 |
| Skew: | 0.274 | Prob(JB): | 0.751 |
| Kurtosis: | 2.378 | Cond. No. | 11.5 |
Warnings: [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
#擬合估計值
y_=res.fittedvalues
y_array([ 2.15061173, 4.72974327, 7.30887481, 9.88800634, 12.46713788,
15.04626942, 17.62540096, 20.2045325 , 22.78366403, 25.36279557,
27.94192711, 30.52105865, 33.10019019, 35.67932172, 38.25845326,
40.8375848 , 43.41671634, 45.99584788, 48.57497942, 51.15411095])
#繪圖
fig,ax=plt.subplots(figsize=(8,6))
ax.plot(x,y,'o',label='data')#原始資料
ax.plot(x,y_,'r--',label='test')#擬合數據
ax.legend(loc='best')
plt.show()