Open-source implementation of the HUSK framework for probabilistic unfolding of hedonic data (liking, purchase intent, acceptability).
HUSK applies the Ennis-Johnson (1993) Thurstone-Shepard kernel with three innovations:
- L2 shrinkage on log-variance parameters to prevent degeneracy
- Dimensionality stability via sigma_sq=1.0 initialization for 3D analyses
- Adaptive response transforms (per-consumer bias or per-consumer affine)
from husk import fit_husk
import numpy as np
# Your ratings matrix: consumers x products, NaN for missing
ratings = np.array([
[0.8, 0.3, np.nan, 0.6],
[0.2, 0.9, 0.7, np.nan],
[0.5, 0.4, 0.6, 0.8],
])
result = fit_husk(ratings, n_dims=3, sigma_sq=1.0, response_transform='per_consumer_bias')
print(result['product_positions']) # J x 3 array
print(result['consumer_positions']) # I x 3 array
print(result['predicted_ratings']) # I x J array
print(result['rating_corr']) # Pearson correlation| Consumers | Config | Time |
|---|---|---|
| < 150 | fit_husk(ratings, optimizer='bfgs', response_transform='per_consumer_bias') |
0.02s |
| >= 150 | fit_husk(ratings, optimizer='adam', response_transform='per_consumer_affine') |
0.6s |
Both use n_dims=3, sigma_sq=1.0, shrinkage=0.005 by default.
from husk.evaluate import evaluate_cv
result = evaluate_cv(ratings, optimizer='bfgs', response_transform='per_consumer_bias',
n_dims=3, k=5, seed=42)
print(f"Held-out correlation: {result['out_corr_mean']:.4f}")The results/ directory contains 1,790 cross-validated experiments reproducible with fit_husk (BFGS and Adam optimizers with none, per_consumer_bias, and per_consumer_affine transforms). The paper describes a broader evaluation of 2,237 experiments that also includes LSA (Ennis) and other optimizers for comparison. See results/results_v2.tsv (33-column structured TSV).
The HUSK kernel is the expected value of Shepard's (1987) generalization gradient under Thurstonian noise:
E[L_ij] = kappa^(-d/2) * exp(-D^2_ij / kappa)
kappa = 1 + 2(tau^2_i + sigma^2_j)
where D is the distance between consumer i and product j in d-dimensional latent space, tau^2 is per-consumer variance, and sigma^2 is per-product variance. This formula is the moment generating function of a noncentral chi-squared distribution evaluated at the appropriate point (Ennis & Johnson, 1993).
HUSK builds directly on Landscape Segmentation Analysis (Ennis, 1993), which solved the degeneracy problem that had made multidimensional unfolding impractical for applied work. LSA introduced three ideas that proved critical: sigmoid-bounded predictions, per-consumer biases, and multi-start optimization. These innovations made preference mapping viable for the food and consumer products industries, where LSA has been the standard tool for three decades (Ennis & Ennis, 2013).
HUSK takes the theoretical kernel that underlies LSA (Ennis & Johnson, 1993), derived from Coombs's (1964) ideal point theory, Thurstone's (1927) discriminal process, and Shepard's (1987) universal law of generalization, and adds three modern techniques: L2 shrinkage on log-variance parameters to prevent degeneracy, a dimensionality stability correction for 3D analyses, and adaptive response transforms that can approximate LSA's built-in bias structure when the data calls for it.
The relationship between HUSK and LSA is one of generalization. LSA's structural commitments (sigmoid bounding, built-in biases) are well-chosen defaults validated by decades of applied success. HUSK makes those commitments configurable, confirming their effectiveness while extending the model to conditions where different choices prove optimal. For details, see Ennis and Ennis (in preparation), "HUSK: An open-source model for hedonic unfolding with shrinkage kernel."
@article{ennis2026husk,
title={HUSK: An Open-Source Model for Hedonic Unfolding with Shrinkage Kernel},
author={Ennis, John M. and Ennis, Daniel M.},
note={In preparation},
year={2026}
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