rank-preserving-calibration 0.11.0

Rank Preserving Calibration of Multiclass Probabilities

Survey statisticians and machine learning practitioners often need to adjust the predicted class probabilities from a classifier so that they match known population totals (column marginals). Simple post-hoc methods that apply separate logit shifts or raking to each class can scramble the ranking of individuals within a class when there are three or more classes. This package implements a rank-preserving calibration procedure that projects probabilities onto the intersection of two convex sets:

1. Row-simplex: each row sums to one and all entries are non-negative.
2. Isotonic column marginals: within each class, values are non-decreasing when instances are sorted by their original scores for that class, and the sum of each column equals a user-supplied target.

The algorithm uses Dykstra's alternating projection method in Euclidean geometry. When the specified column totals are feasible, the procedure returns a matrix that preserves cross-person discrimination within each class, matches the desired totals, and remains a valid probability distribution for each instance. That claim is checked against an independent convex solver in the test suite rather than asserted.

Feasibility is exact, not approximate. Every row sums to one, so the grand total is fixed at N and the targets must satisfy sum(M) == N. When they do, the intersection is never empty — the constant matrix Q[i, j] = M[j] / N satisfies all three constraint sets — so a failure is a conditioning or iteration-budget problem, never infeasibility. When they do not, no matrix satisfies both constraint sets and there is no "closest point satisfying both" to return: the solver warns, then raises CalibrationError. Rescale first:

import numpy as np

N = 3
M = np.array([1.2, 0.9, 1.1])  # any targets
M = M * N / M.sum()  # now sums to N exactly

New: Nearly Isotonic Calibration

This package now supports nearly isotonic constraints that allow small violations of strict monotonicity when appropriate:

• Epsilon-slack constraints: Allow z[i+1] ≥ z[i] - ε instead of strict z[i+1] ≥ z[i]
• Penalised rank violations: calibrate_soft (squared error) and calibrate_kl_soft (KL divergence) charge λ per unit of downward step instead of forbidding it

These relaxed constraints can provide better balance between rank preservation and probability calibration when strict isotonic constraints are too restrictive.

Which solver?

calibrate() picks for you, and the default is right in every case measured. The two are mathematically equivalent — they solve the same convex problem, and the test suite asserts they agree — so the choice is practical:

The default is not a close call. Measured at J=4 on feasible targets:

The reason is the algorithm class rather than the problem: the two constraint sets meet at a shallow angle, which is exactly where first-order methods crawl and where an interior-point method is unaffected. A sparse QP alone is not enough — OSQP, also first-order, is worse than Dykstra here, returning 592 rank violations at N=400 while reporting that it ran.

Above roughly N = 6400 the exact solve becomes expensive again, since interior-point factorisation cost grows steeply. Relax the constraint rather than wait:

import numpy as np
from rank_preserving_calibration import calibrate

P = np.array([[0.6, 0.3, 0.1], [0.2, 0.5, 0.3], [0.1, 0.2, 0.7]])
M = np.array([1.0, 1.0, 1.0])

result = calibrate(P, M, nearly={"mode": "epsilon", "eps": 0.05})

which permits adjacent within-class decreases of at most eps and is handled exactly, by lowering the isotonic bound rather than by penalising violations.

Installation

pip install rank-preserving-calibration

Runtime dependencies are numpy, scipy and clarabel (Apache-2.0, ~2.5 MB). The examples also need matplotlib and scikit-learn.

Usage

Basic Usage

import numpy as np
from rank_preserving_calibration import calibrate

P = np.array(
    [
        [0.6, 0.3, 0.1],
        [0.2, 0.5, 0.3],
        [0.1, 0.2, 0.7],
    ]
)

# Target column sums, e.g. population class frequencies. Must sum to the
# number of rows (3 in this example) for perfect feasibility.
M = np.array([1.0, 1.0, 1.0])

result = calibrate(P, M)

print("Adjusted probabilities:\n", result.Q)
print("Converged:", result.converged)
print("Iterations:", result.iterations)
print("Max row error:", result.max_row_error)
print("Max column error:", result.max_col_error)
print("Rank violations:", result.max_rank_violation)

Nearly Isotonic Usage

from rank_preserving_calibration import calibrate

# Epsilon-slack: allow small rank violations (recommended above N ~ 6400)
result = calibrate(P, M, nearly={"mode": "epsilon", "eps": 0.05})

# Penalise rank violations instead of bounding them. This changes the
# objective, not the constraint set, so it lives in its own function.
from rank_preserving_calibration import calibrate_soft

soft = calibrate_soft(P, M, lam_m=100.0, lam_r=1.0)

The returned CalibrationResult contains the calibrated matrix Q with the same shape as P. Each row of Q sums to one, the column sums match M, and within each column the entries are sorted in non-decreasing order according to the order implied by the original P.

Evaluation and Metrics

After calibration, it's important to validate that the constraints are satisfied and understand the impact on prediction quality. This package provides comprehensive metrics for evaluation:

Constraint Validation

from rank_preserving_calibration import feasibility_metrics, isotonic_metrics

# Check constraint satisfaction
feasibility = feasibility_metrics(result.Q, M)
print(f"Max row error: {feasibility['row']['max_abs_error']}")
print(f"Max column error: {feasibility['col']['max_abs_error']}")

# Check rank preservation
isotonic = isotonic_metrics(result.Q, P)
print(f"Max rank violation: {isotonic['max_rank_violation']}")
print(f"Violation mass: {isotonic['total_violation_mass']}")

Calibration Quality Assessment

import numpy as np
from sklearn.metrics import log_loss

from rank_preserving_calibration import distance_metrics

# True class labels for the three rows above, if you have them.
y_true = np.array([0, 1, 2])

# Measure calibration changes
distances = distance_metrics(result.Q, P)
print(f"Frobenius distance: {distances['frobenius']}")
print(f"Max change: {distances['max_abs']}")

# Evaluate with labeled data (if available)
if y_true is not None:
    original_nll = log_loss(y_true, P)
    calibrated_nll = log_loss(y_true, result.Q)
    print(f"NLL improvement: {original_nll - calibrated_nll}")

Available Metrics

For log loss and the Brier score, use sklearn.metrics.log_loss and sklearn.metrics.brier_score_loss; this package only adds the metrics that are specific to calibration.

Complete Evaluation Workflow

import numpy as np
from rank_preserving_calibration import (
    calibrate_dykstra,
    feasibility_metrics,
    isotonic_metrics,
    distance_metrics,
    top_label_ece,
)

# Calibrate
result = calibrate_dykstra(P, M)

# 1. Validate constraints
feasibility = feasibility_metrics(result.Q, M)
isotonic = isotonic_metrics(result.Q, P)
print(f"Converged: {result.converged}")
print(f"Row constraint satisfied: {feasibility['row']['max_abs_error'] < 1e-6}")
print(f"Rank preserved: {isotonic['max_rank_violation'] < 1e-6}")

# 2. Assess calibration impact
distances = distance_metrics(result.Q, P)
print(f"Average change per probability: {distances['mean_abs']:.4f}")

# 3. Evaluate predictive quality (if labels available)
if y_true is not None:
    ece_before = top_label_ece(y_true, P)
    ece_after = top_label_ece(y_true, result.Q)
    print(f"Calibration error before: {ece_before['ece']:.3f}")
    print(f"Calibration error after: {ece_after['ece']:.3f}")

Functions

calibrate_dykstra(P, M, **kwargs)

Calibrate using Dykstra's alternating projections (recommended). Supports both strict and nearly isotonic constraints.

calibrate_soft(P, M, lam_m, lam_r)

Minimise 0.5||Q - P||² + 0.5·lam_m·||Qᵀ1 - M||² + lam_r·Σ(downward steps) with rows on the simplex, as a quadratic program solved exactly by Clarabel.

calibrate_kl(P, M, R=None, A=None) and calibrate_kl_soft(P, M, lam, R=None, A=None)

Minimise KL divergence from a reference R with ranks taken from an anchor A, either strictly or with a λ penalty; both are solved exactly as exponential-cone programs.

create_test_case(case_type, N, J, **kwargs) (in tests.data_helpers)

Generate synthetic test data for various scenarios used in testing.

Arguments

Returns

CalibrationResult

calibrate, calibrate_qp and calibrate_dykstra return a CalibrationResult object with the following attributes:

• Q: NumPy array of shape [N, J] containing the calibrated probabilities. Each row sums to one, each column approximately sums to the corresponding entry of M, and within each column the values are non-decreasing according to the ordering induced by P.
• converged: boolean indicating whether the solver met the tolerance criteria.
• iterations: number of iterations performed.
• max_row_error: maximum absolute deviation of row sums from 1.
• max_col_error: maximum absolute deviation of column sums from M.
• max_rank_violation: maximum violation of monotonicity (should be 0 up to numerical tolerance).
• final_change: final relative change between iterations.

Algorithm Notes

• Dykstra's Method: Uses alternating projections with memory terms to ensure convergence to the intersection of constraint sets. Rows are projected onto the simplex via the algorithm of Duchi et al., and columns are projected via the pool-adjacent-violators algorithm followed by an additive shift to match column totals. This is the recommended method for most applications.

• Nearly Isotonic Extensions:

  • Epsilon-slack (Dykstra): Projects onto the convex set {z : z[i+1] ≥ z[i] - ε} using coordinate transformation. Maintains theoretical convergence guarantees.
  • Rank penalty (calibrate_soft): Minimises 0.5||Q - P||² + λ∑max(0, z[i] - z[i+1]) per column, alongside a marginal penalty, as one quadratic program solved by an interior-point method. Checked against an independent convex solver.

Examples

The examples in docs/examples are scripts that run on every docs build and are rendered, with their output and figures, in the documentation. Run one locally with uv run --group docs python docs/examples/plot_label_shift.py.

When to Use Nearly Isotonic Calibration

Use Nearly Isotonic When:

• Model predictions have good discrimination but need marginal calibration
• Some predictions are already well-calibrated
• Small rank violations are acceptable in your domain
• You want to preserve model confidence where possible

Use Strict Isotonic When:

• Rank order is critical (regulatory, safety applications)
• Model predictions have clear monotonic relationship
• Conservative approach is preferred

Testing

python -m pytest tests/ -v

License

This software is released under the terms of the MIT license.

🔗 Adjacent Repositories

• finite-sample/calibre — Advanced Calibration Models
• finite-sample/optimal-classification-cutoffs — Script for calculating the optimal cut-off for max. F1-score, etc.
• finite-sample/fairlex — Leximin Calibration
• finite-sample/pyppur — pyppur: Python Projection Pursuit Unsupervised (Dimension) Reduction To Min. Reconstruction Loss or DIstance DIstortion