How to Perform Quantile Regression in Python

In this article, we are going to see how to perform quantile regression in Python.

Linear regression is defined as the statistical method that constructs a relationship between a dependent variable and an independent variable as per the given set of variables. While performing linear regression we are curious about computing the mean value of the response variable. Instead, we can use a mechanism known as quantile regression in order to compute or estimate the quantile (percentile) value of the response value. For example, 30_th percentile, 50_th percentile, etc. 

Quantile regression

Quantile regression is simply an extended version of linear regression. Quantile regression constructs a relationship between a group of variables (also known as independent variables) and quantiles (also known as percentiles) dependent variables. 

Perform quantile regression in Python

Calculation quantile regression is a step-by-step process. All the steps are discussed in detail below:

Creating a dataset for demonstration

Let us create a dataset now. As an example, we are creating a dataset that contains the information of the total distance traveled and total emission generated by 20 cars of different brands.  

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()

Output:

    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

Estimating Quantile Regression

Now we will construct a quantile regression model with the help of,

• Distance traveled: As a predictor variable
• Mileage achieved: As a response variable

Now, We will make use of this model to estimate the 70th percentile of emission generated based on the total distance traveled by cars.

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())

From the output of this program, the estimated regression equation can be deduced as,

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.

Output:

Visualization quantile regression

In order to visualize and understand the quantile regression, we can use a scatterplot along with the fitted quantile regression. 

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')

Output: