whsmooth

Whittaker-Henderson smoothing for P&C ratemaking and actuarial graduation --- 1D, 2D, automatic lambda selection.

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What is whsmooth?

Whittaker-Henderson smoothing is a workhorse technique across actuarial practice: smoothing claim severity by exposure cell, graduating loss development factors, stabilizing rate relativities, and graduating mortality tables. The method has been used since the 1920s; it minimizes a weighted sum of fidelity to the data and roughness of the smoothed curve, controlled by a single parameter lambda.

Despite its centrality, no dedicated Python implementation exists. R has MortalitySmooth and ungroup, both life-actuarial. Python practitioners --- especially in P&C --- are left writing ad-hoc scripts.

whsmooth fills this gap with:

• P&C-first design --- built for severity, frequency, LDF, and rate relativity smoothing, with mortality graduation as a natural special case.
• Sparse-matrix solver --- banded penalty system solved with scipy.sparse for O(n) performance.
• Principled lambda selection --- REML, GCV, and AIC, so you stop guessing your smoothing parameter.
• 1D and 2D --- smooth a vector by vehicle age, or an entire age × territory grid in one call.
• Sklearn-style API --- fit(y).fitted_, no surprises.

Installation

pip install whsmooth

# With plotting helpers
pip install "whsmooth[plot]"

# With pandas integration
pip install "whsmooth[pandas]"

Quickstart --- Smoothing claim severity by vehicle age

import numpy as np
from whsmooth import WH1D

# Observed mean severity by vehicle age (0-25), with claim counts as weights
ages = np.arange(0, 26)
severity_obs = np.array([
    8050, 8120, 8210, 8290, 8390, 8520, 8680, 8850, 9050, 9270,
    9510, 9770, 10050, 10350, 10670, 11010, 11380, 11760, 12300, 12800,
    13100, 14200, 13400, 15100, 14000, 16500,
])
counts = np.array([
    5200, 4500, 3900, 3400, 2950, 2550, 2200, 1900, 1640, 1420,
    1230, 1060, 920, 790, 685, 590, 510, 440, 380, 330,
    285, 245, 210, 180, 155, 135,
])

# Fit with GCV-selected lambda; weights = exposure
wh = WH1D(lam='gcv', order=2).fit(severity_obs, weights=counts)

print(f"Selected lambda: {wh.lambda_:.2f}")
print(f"Effective degrees of freedom: {wh.edf_:.2f}")
smoothed = wh.fitted_

The high-credibility cells (low ages, large counts) are tracked closely; the sparse tail (older vehicles) is stabilized toward the underlying trend instead of chasing noise.

Quickstart --- 2D smoothing on age × territory

from whsmooth import WH2D

# Pure premium grid: 10 age bands × 8 territories
pure_premium = ...  # shape (10, 8)
exposure = ...      # shape (10, 8)

wh2 = WH2D(lam=('gcv', 'gcv'), order=(2, 2)).fit(pure_premium, weights=exposure)
pp_smooth = wh2.fitted_

Other use cases

• LDF smoothing: stabilize loss development factors before extrapolation.
• Frequency relativities: smooth claim frequency by exposure class.
• Mortality graduation (life): graduate qx by age --- same algorithm, drop in your qx_crude and exposure.

# Mortality graduation example
wh = WH1D(lam='reml', order=2).fit(qx_crude, weights=exposure)
qx_grad = wh.fitted_

Features

Module	Description
WhittakerHenderson1D / WH1D	1D smoothing, any difference order, weighted
WhittakerHenderson2D / WH2D	2D smoothing on grids (Kronecker penalties)
lam='gcv' / 'reml' / 'aic'	Automatic lambda selection
whsmooth.diagnostics	gcv_score, reml_score, aic_score, edf
whsmooth._penalties	Difference penalty matrices, composite penalties

How it works

The Whittaker-Henderson smoother solves:

min_a   sum_i w_i (y_i - a_i)^2  +  lambda * ||D_d a||^2

where D_d is the d-th order difference matrix and lambda controls the smoothness-fidelity tradeoff. The system has a banded coefficient matrix and is solved via sparse Cholesky factorization --- O(n) instead of O(n^3).

For 2D problems, the penalty extends to Kronecker products of row and column difference matrices, following Currie, Durban & Eilers (2004).

References

P&C / general

• Werner, G. & Modlin, C. (2016). Basic Ratemaking (5th ed.). Casualty Actuarial Society.
• Verrall, R. J. (1996). "Claims Reserving and Generalised Additive Models." Insurance: Mathematics and Economics, 19(1), 31-43.
• Eilers, P. H. C. & Marx, B. D. (1996). "Flexible Smoothing with B-splines and Penalties." Statistical Science, 11(2), 89-121.

Original

• Whittaker, E. T. (1922). "On a New Method of Graduation." Proc. Edinburgh Math. Soc., 41, 63-75.
• Henderson, R. (1924). "A New Method of Graduation." Trans. Actuarial Society of America, 25, 29-40.

Life / mortality

• Currie, I. D., Durban, M. & Eilers, P. H. C. (2004). "Smoothing and Forecasting Mortality Rates." Statistical Modelling, 4(4), 279-298.
• Camarda, C. G. (2012). "MortalitySmooth: An R Package for Smoothing Poisson Counts with P-Splines." J. Statistical Software, 50(1), 1-24.

Part of the actuarial Python ecosystem

whsmooth is part of a 6-library P&C actuarial stack:

• actudist --- severity & frequency distributions
• burncost --- burning cost analysis
• actuarcredibility --- credibility methods
• whsmooth --- smoothing (this library)
• pyratemaking --- rating tables (coming)
• pyinsurancerating --- rating engine (coming)

Contributing

See CONTRIBUTING.md. PRs welcome --- open an issue first to discuss substantive changes.

Citation

If you use whsmooth in academic work, see CITATION.cff or:

López, I. (2026). whsmooth: Whittaker-Henderson smoothing for P&C ratemaking in Python (v0.1.0). https://github.com/CosmikArt/whsmooth

Author

Isaac López

License

MIT