# Understanding Quantile Regression with Scikit-Learn

I have a case where I want to predict a time value in minutes.

This is the problem of regression.

I also want to predict the upper bound and lower bound.

I can do it two ways:

1. Train 3 models: one for the main prediction, one for say a higher prediction and one for a lower prediction.

2. Use Quantile regression whcih gives a lower and upper bound.

However, I am not understanding how Quantile regression works.

Here is the code:

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(1)

#----------------------------------------------------------------------
#  First the noiseless case
X = np.atleast_2d(np.random.uniform(0, 10.0, size=100)).T
X = X.astype(np.float32)

# Observations
y = f(X).ravel()

dy = 1.5 + 1.0 * np.random.random(y.shape)
noise = np.random.normal(0, dy)
y += noise
y = y.astype(np.float32)

# Mesh the input space for evaluations of the real function, the prediction and
# its MSE
xx = np.atleast_2d(np.linspace(0, 10, 1000)).T
xx = xx.astype(np.float32)

alpha = 0.95

n_estimators=250, max_depth=3,
learning_rate=.1, min_samples_leaf=9,
min_samples_split=9)

clf.fit(X, y)

# Make the prediction on the meshed x-axis
y_upper = clf.predict(xx)

clf.set_params(alpha=1.0 - alpha)
clf.fit(X, y)

# Make the prediction on the meshed x-axis
y_lower = clf.predict(xx)

clf.set_params(loss='ls')
clf.fit(X, y)

# Make the prediction on the meshed x-axis
y_pred = clf.predict(xx)

# Plot the function, the prediction and the 90% confidence interval based on
# the MSE
fig = plt.figure()
plt.plot(X, y, 'b.', markersize=10, label=u'Observations')
plt.plot(xx, y_pred, 'r-', label=u'Prediction') # pred
plt.plot(xx, y_upper, 'k-') #
plt.plot(xx, y_lower, 'k-') #
plt.fill(np.concatenate([xx, xx[::-1]]),
np.concatenate([y_upper, y_lower[::-1]]),
alpha=.5, fc='b', ec='None', label='90% prediction interval')
plt.xlabel('$x$')
plt.ylabel('$f(x)$')
plt.ylim(-10, 20)
plt.legend(loc='upper left')
plt.show()


My questions are:

1. How does quantile regression work here i.e. how is the model trained?
2. How to use a quantile regression mode at prediction time, does it give 3 predictions, what is y_lower and y_upper?
• Please elaborate on what you intend the "upper bound and lower bound" to represent: that will help us determine whether you even need quantile regression.
– whuber
Jun 24 '18 at 15:35
• @whuber I am predicting Estimated Time of Arrival for consumers. I want to give them a range i.e. instead of saying your order will arrive in 74 hours, I will say your order will arrive between 68-78 hours. Quantile regression gives an upper bound and lower bound..from there I guessed it fits my problem..is any other algorithm possible too? Jun 24 '18 at 15:39
• You appear to be asking for a prediction interval. See stats.stackexchange.com/….
– whuber
Jun 24 '18 at 15:41
• @whuber yes, that 's where quantile regression is used, right? scikit-learn.org/stable/auto_examples/ensemble/… I see only one method for getting prediction interval..if quantile regression can be used for it, why not use it? It seems like a great method..howvere please take a look at the code..I am not getting how the intervals are getting predicted? Jun 24 '18 at 15:43
• That's a possible use of quantile regression. It's not necessarily an appropriate use, though: it depends on your statistical assumptions. But assuming that quantile regression is what you want to do, then it's unclear what you're trying to ask. Is your question "how do I do quantile regression" or would it be something more focused than that?
– whuber
Jun 24 '18 at 15:45

When creating the classifier, you've passed loss='quantile' along with alpha=0.95. You are optimizing quantile loss for 95th percentile in this situation. You can read up more on how quantile loss works here and here.
In your code, you have created one classifier. You're first fitting and predicting for alpha=0.95, then using clf.set_params() you're using the same classifier to fit and predict for alpha=0.05.