如何在Python中执行分位数回归

在本文中，我们将了解如何在Python中执行分位数回归。

线性回归被定义为根据给定的变量集构建因变量和自变量之间的关系的统计方法。在执行线性回归时，我们对计算响应变量的平均值感到好奇。相反，我们可以使用一种称为分位数回归的机制来计算或估计响应值的分位数（百分位数）值。例如，第 30_个百分位、第 50_个百分位等。

分位数回归

分位数回归只是线性回归的扩展版本。分位数回归构建一组变量（也称为自变量）和分位数（也称为百分位数）因变量之间的关系。

在Python中执行分位数回归

计算分位数回归是一个循序渐进的过程。下面详细讨论所有步骤：

创建用于演示的数据集

现在让我们创建一个数据集。例如，我们正在创建一个数据集，其中包含 20 辆不同品牌汽车的总行驶距离和总排放量的信息。

Python3

# Python program to create a dataset
 
# Importing libraries
import numpy as np
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf
import matplotlib.pyplot as plt
 
np.random.seed(0)
 
# Specifying the number of rows
rows = 20
 
# Constructing Distance column
Distance = np.random.uniform(1, 10, rows)
 
# Constructing Emission column
Emission = 20 + np.random.normal(loc=0, scale=.25*Distance, size=20)
 
# Creating a dataframe
df = pd.DataFrame({'Distance': Distance, 'Emission': Emission})
 
df.head()

输出：

Distance    Emission
0    5.939322    22.218454
1    7.436704    19.618575
2    6.424870    20.502855
3    5.903949    18.739366
4    4.812893    16.928183

估计分位数回归

现在我们将借助以下方法构建分位数回归模型，

• 行驶距离：作为预测变量
• 达到的里程数：作为响应变量

现在，我们将利用这个模型来估计基于汽车总行驶距离产生的 70% 排放量。

Python3

# Python program to illustrate
# how to estimate quantile regression
 
# Importing libraries
import numpy as np
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf
import matplotlib.pyplot as plt
 
np.random.seed(0)
 
# Number of rows
rows = 20
 
# Constructing Distance column
Distance = np.random.uniform(1, 10, rows)
 
# Constructing Emission column
Emission = 40 + Distance + np.random.normal(loc=0,
                                            scale=.25*Distance,
                                            size=20)
 
# Creating the data set
df = pd.DataFrame({'Distance': Distance,
                   'Emission': Emission})
 
# fit the model
model = smf.quantreg('Emission ~ Distance',
                     df).fit(q=0.7)
 
# view model summary
print(model.summary())

根据该程序的输出，估计的回归方程可以推导出为，

 val = 39.5647 + 1.3042 * X (distance in km) 

 It implies that the 70th percentile of emission for all the cars that travel X km is expected to be val.

输出：

可视化分位数回归

为了可视化和理解分位数回归，我们可以使用散点图和拟合的分位数回归。

Python3

# Python program to visualize quantile regression
 
# Importing libraries
import numpy as np
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf
import matplotlib.pyplot as plt
 
np.random.seed(0)
 
# Number of rows
rows = 20
 
# Constructing Distance column
Distance = np.random.uniform(1, 10, rows)
 
# Constructing Emission column
Emission = 40 + Distance + np.random.normal(loc=0,
                                            scale=.25*Distance,
                                            size=20)
 
# Creating a dataset
df = pd.DataFrame({'Distance': Distance,
                   'Emission': Emission})
 
# #fit the model
model = smf.quantreg('Emission ~ Distance',
                     df).fit(q=0.7)
 
# define figure and axis
fig, ax = plt.subplots(figsize=(10, 8))
 
# get y values
y_line = lambda a, b: a + Distance
y = y_line(model.params['Intercept'],
           model.params['Distance'])
 
# Plotting data points with the help
# pf quantile regression equation
ax.plot(Distance, y, color='black')
ax.scatter(Distance, Emission, alpha=.3)
ax.set_xlabel('Distance Traveled', fontsize=20)
ax.set_ylabel('Emission Generated', fontsize=20)
 
# Save the plot
fig.savefig('quantile_regression.png')

输出：