簡單直線回歸 (Simple Linear Regression)

套路27: 簡單直線回歸 (Simple Linear Regression)

1. 使用時機: 以單一變數(自變數)預測判斷依變數與自變數之間相關的方向(趨勢)和程度。

2. 分析類型: 母數分析(parametric analysis)。

3. 範例資料: 咪路測量手養鳥出生天數(day)與翅膀長度(cm)，資料如下：

X (day)	1	3	4	6	9	10	12	15	17	19	23	25	27
Y (翼長)	1.4	1.5	2.2	2.4	3.1	3.2	3.5	3.9	4.1	4.5	4.7	5.0	5.2

X，Y是否有線性關係?

4. 畫圖看資料分佈:

  v1 = [1,3,4,6,9,10,12,15,17,19,23,25,27]

  v2 = [1.4,1.5,2.2,2.4,3.1,3.2,3.5,3.9,4.1,4.5,4.7,5,5.2]

  dat = {'Day':v1,'Length':v2}  # Day, Length是資料標題

  import pandas as pd

  df = pd.DataFrame(dat)

  import seaborn as sb

  from matplotlib import pyplot as plt

  sb.lmplot(x = "Day", y = "Length", data = df)

  plt.show()

# 條帶區域為95%信賴區間。

5. 使用Python計算直線回歸方程式

方法:

  v1 = [1,3,4,6,9,10,12,15,17,19,23,25,27]

  v2 = [1.4,1.5,2.2,2.4,3.1,3.2,3.5,3.9,4.1,4.5,4.7,5,5.2]

  dat = {'Day':v1,'Length':v2}  # Day, Length是資料標題

  import pandas as pd

  df = pd.DataFrame(dat)

  import statsmodels.api as sm

  X = df['Day']

  y = df['Length']

  X2 = sm.add_constant(X)

  est = sm.OLS(y, X2)

  est2 = est.fit()

  print(est2.summary())

結果:

                            OLS Regression Results                           

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

Dep. Variable:                 Length   R-squared:                       0.965

Model:                            OLS   Adj. R-squared:                  0.962

Method:                 Least Squares   F-statistic:                     307.2

Date:                Tue, 23 Jul 2019   Prob (F-statistic):           2.19e-09

Time:                        16:25:41   Log-Likelihood:                0.75314

No. Observations:                  13   AIC:                             2.494

Df Residuals:                      11   BIC:                             3.624

Df Model:                           1                                        

Covariance Type:            nonrobust                                        

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

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

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

const          1.5260      0.129     11.829      0.000       1.242       1.810

Day            0.1454      0.008     17.528      0.000       0.127       0.164

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

Omnibus:                        1.649   Durbin-Watson:                   0.884

Prob(Omnibus):                  0.438   Jarque-Bera (JB):                1.191

Skew:                          -0.541   Prob(JB):                        0.551

Kurtosis:                       1.987   Cond. No.                         29.2

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