rank-preserving-calibration 0.9.0

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]
• Lambda-penalty approach: Penalize isotonicity violations with a tunable parameter

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

An ADMM optimization implementation is also provided as an alternative solver that minimizes ||Q - P||² subject to the same constraints.

Which solver?

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

method	how	when
"qp" (default)	sparse quadratic program, interior point	always
"dykstra"	alternating projections, numpy only	reference implementation; useful if you cannot take the solver dependency
"admm"	augmented Lagrangian, then projection	when you want primal/dual residual traces

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

N	dykstra	qp
25	0.03 s	0.002 s
50	9.6 s	0.004 s
100	10.5 s	0.01 s
400	does not converge in 60,000 iterations	0.12 s
1600	—	1.4 s
6400	—	50 s

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

# For JIT acceleration of the numpy-only Dykstra path
pip install rank-preserving-calibration[performance]

Runtime dependencies are numpy, scipy and clarabel (Apache-2.0, ~2.5 MB). Optional extras:

• [performance]: Adds numba (JIT compilation)
• [docs]: Documentation building dependencies
• Examples require matplotlib

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})

# Lambda-penalty: soft isotonic constraint, ADMM only (it changes the
# objective rather than the constraint set, so it is not a projection)
result = calibrate(P, M, method="admm", nearly={"mode": "lambda", "lam": 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.

Performance Features

from rank_preserving_calibration import calibrate

# JIT acceleration applies to the numpy-only Dykstra path, not the QP solver.
result = calibrate(P, M, method="dykstra", use_jit=False)

result = calibrate(
    P, M,
    method="dykstra",
    max_iters=5000,
    use_jit=True,       # 2-10x speedup on the projection path
)

With the [performance] extras installed:

• JIT Compilation: Automatically accelerates hot loops using Numba
• Progress Bars: Shows calibration progress with iteration count, convergence metrics, and ETA
• Typical speedup: 2-3x for moderate problems (N=500, J=10), up to 10x for larger problems

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 rank_preserving_calibration import distance_metrics, nll, brier

# 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 = nll(y_true, P)
    calibrated_nll = nll(y_true, result.Q)
    print(f"NLL improvement: {original_nll - calibrated_nll}")

Available Metrics

Function	Purpose
feasibility_metrics(Q, M)	Validate row (simplex) and column (marginal) constraints
isotonic_metrics(Q, P)	Check rank preservation and measure violations
distance_metrics(Q, P)	Quantify changes between original and calibrated probabilities
tie_group_variance(Q, P)	Assess handling of tied predictions (useful for ties='group')
nll(y, probs)	Negative log-likelihood (requires true labels)
brier(y, probs)	Brier score (requires true labels)
top_label_ece(y, probs)	Expected calibration error for top predictions
classwise_ece(y, probs)	Per-class calibration error analysis
sharpness_metrics(probs)	Prediction confidence and entropy analysis
auc_deltas(y, P, Q)	One-vs-rest AUC changes after calibration

Complete Evaluation Workflow

import numpy as np
from rank_preserving_calibration import (
    calibrate_dykstra, feasibility_metrics, isotonic_metrics,
    distance_metrics, nll, 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_admm(P, M, **kwargs)

Calibrate using ADMM optimization with penalty parameter rho. Supports lambda-penalty nearly isotonic constraints.

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

Generate synthetic test data for various scenarios used in testing.

Arguments

Parameter	Type	Description
P	ndarray of shape [N, J]	Base multiclass probabilities or non-negative scores. Rows will be projected to the simplex.
M	ndarray of shape [J]	Target column totals (e.g. population class frequencies). The sum of M should equal the number of rows N for exact feasibility.
max_iters	int	Maximum number of projection iterations (default 3000 for Dykstra, 1000 for ADMM).
tol	float	Relative convergence tolerance (default 1e-7 for Dykstra, 1e-6 for ADMM).
verbose	bool	If True, prints convergence diagnostics.
nearly	dict	Nearly isotonic parameters: {"mode": "epsilon", "eps": 0.05} or {"mode": "lambda", "lam": 1.0}.
rho	float	ADMM penalty parameter (default 1.0, ADMM only).

Returns

CalibrationResult

Both functions 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.

ADMMResult

The ADMM function returns an ADMMResult object with additional convergence history:

• All CalibrationResult attributes plus:
• objective_values: objective function values over iterations.
• primal_residuals: primal residual norms over iterations.
• dual_residuals: dual residual norms over 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.
  • Lambda-penalty (ADMM): Uses proximal operator to minimize ||Q - P||² + λ∑max(0, z[i] - z[i+1]). More experimental but provides soft constraints.

• ADMM: Solves the constrained optimization problem using the Alternating Direction Method of Multipliers. May converge faster for some problems but requires tuning the penalty parameter rho. The algorithm minimizes the sum of squared differences 0.5 * ||Q - P||²_F subject to the calibration constraints.

Examples

See our comprehensive documentation examples at https://finite-sample.github.io/rank_preserving_calibration/examples.html:

• Medical Diagnosis: Breast cancer risk calibration across populations
• Financial Risk: Credit scoring with regulatory compliance
• Text Classification: Sentiment analysis with domain adaptation
• Computer Vision: OCR deployment across applications
• Survey Research: Demographic reweighting for representative samples

Each example uses real datasets and provides complete analysis workflows with business context and performance evaluation.

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/adaptive-eb — Adaptive Entropy Balancing via Multiplicative Weights
• finite-sample/pyppur — pyppur: Python Projection Pursuit Unsupervised (Dimension) Reduction To Min. Reconstruction Loss or DIstance DIstortion