Handling Quantile Crossing¶

The problem¶

When quantile models are fitted independently for each tau, predictions can cross: the 90th percentile prediction may be lower than the 10th. This violates the definition of quantiles and produces nonsensical intervals.

Joint constraints vs post-hoc rearrangement¶

QuantileRegression prevents crossing in two ways:

LP constraints during fitting enforce Q(tau_j) <= Q(tau_{j+1}) on training data
Row-wise rearrangement at prediction time fixes any remaining violations on new data

If you're using external models (LightGBM, XGBoost, etc.) that don't enforce non-crossing, the postprocess module provides tools to detect and fix crossings.

Detecting crossings¶

from quantile_guard.postprocess import check_crossing, crossing_summary
import numpy as np

# predictions: (n_samples, n_quantiles), one column per tau
taus = [0.1, 0.5, 0.9]
predictions = np.array([
    [1.0, 2.0, 3.0],  # ok
    [3.0, 2.0, 1.0],  # crossed
    [1.5, 1.5, 2.0],  # ok (equal is fine)
])

mask = check_crossing(predictions, taus)
# array([False, True, False])

summary = crossing_summary(predictions, taus)
# {
#     'crossing_rate': 0.333,
#     'crossing_magnitude': 0.667,
#     'max_magnitude': 2.0,
#     'n_crossing_samples': 1,
#     'n_total_samples': 3,
#     'worst_row_index': 1,
# }

Fixing crossings with rearrangement¶

The simplest fix: sort each row so predictions are non-decreasing.

from quantile_guard.postprocess import rearrange_quantiles

fixed = rearrange_quantiles(predictions, taus)
# array([[1., 2., 3.],
#        [1., 2., 3.],
#        [1.5, 1.5, 2.]])

This implements the rearrangement approach of Chernozhukov, Fernández-Val, and Galichon (2010). It guarantees monotonicity but may change the marginal distribution of each quantile's predictions.

When to use what¶

Situation	Recommendation
Fitting with this package	Non-crossing is automatic
Evaluating external predictions	Use `crossing_summary` to diagnose
Fixing external predictions	Use `rearrange_quantiles`
Need guaranteed non-crossing	Fit with `QuantileRegression(tau=[...])`