时间序列、趋势和与趋势的偏差

Question

我有一个时间序列是这样的：

 2000 0.000
 2001 -0.174
 2002 -0.131
 2003 0.127
 2004 0.566
 2005 0.723
 2006 0.675
 2007 1.171
 2008 2.338
 2009 2.625
 2010 3.746
 2011 3.612
 2012 4.729
 2013 8.156
 2014 16.330
 2015 27.584

估计此序列的线性趋势，然后计算趋势线和序列之间的差距的最有效方法是什么？

提前谢谢你！

Answer 1

使用scikit-learn中的简单线性回归模型

import pandas as pd
import numpy as np
from sklearn.linear_model import LinearRegression
a = {'year':[2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015],'y_true':[0,-0.174,-0.131,0.127,0.566,0.723,0.675,1.171,2.338,2.625,3.746,3.612,4.729,8.156,16.330,27.584]}
df = pd.DataFrame(a)
x = np.array(df['year']).reshape(-1,1)
y_true = df['y_true']
linear_reg = LinearRegression().fit(x,y_true)
y_pred = linear_reg.predict(x)
df['y_pred'] = y_pred
df['difference'] = y_true - y_pred
print(df)

输出：

    year  y_true     y_pred  difference
0   2000   0.000  -4.366596    4.366596
1   2001  -0.174  -3.183741    3.009741
2   2002  -0.131  -2.000887    1.869887
3   2003   0.127  -0.818032    0.945032
4   2004   0.566   0.364822    0.201178
5   2005   0.723   1.547676   -0.824676
6   2006   0.675   2.730531   -2.055531
7   2007   1.171   3.913385   -2.742385
8   2008   2.338   5.096240   -2.758240
9   2009   2.625   6.279094   -3.654094
10  2010   3.746   7.461949   -3.715949
11  2011   3.612   8.644803   -5.032803
12  2012   4.729   9.827657   -5.098657
13  2013   8.156  11.010512   -2.854512
14  2014  16.330  12.193366    4.136634
15  2015  27.584  13.376221   14.207779