Convergence-Aware Pareto Selection of Covariate Scaling Transformations for Markov Deterioration Hazard Models: Evidence from Bridge Inspection Data

Takato Yasuno; Keita Kobayashi; Ryuta Sakaguchi

stat.MESep 24, 2026

Convergence-Aware Pareto Selection of Covariate Scaling Transformations for Markov Deterioration Hazard Models: Evidence from Bridge Inspection Data

Authors: Takato Yasuno, Keita Kobayashi, Ryuta Sakaguchi

Organizations: Digital Business Promotion Department, Yachiyo Engineering Co., Ltd.

Abstract

Maximum likelihood estimation of Markov deterioration hazard models for infrastructure asset management is sensitive to the numerical conditioning of explanatory covariates, yet production pipelines often adopt a single scaling convention without systematic justification. We study four covariate scaling transformations---baseline max scaling, min-max scaling, z-score scaling, and Box-Cox transformation with a training-derived positivity shift---applied to an Exponential Hazard Markov (EHM) model estimated via L-BFGS-B on bridge inspection transition data.

Figures & tables

Member case	Total rows	Train	Validation	Golden test
Main girder	2,409	624	217	1,568
Pier	3,087	1,158	285	1,644
Bearing	4,269	1,634	640	1,995
Deck slab	3,478	1,491	325	1,662
Cross girder	604	216	113	275
Longitudinal girder	272	86	41	145

Table 1: Member-case sample sizes under the replicated baseline conditions (seed

20260924

, 4 covariates, unweighted likelihood).

Method	Converged	Selected	Mean accuracy	Median time (s)
Max scaling (baseline)	9 / 10	0	0.5174	0.928
Min-max scaling	9 / 10	1	0.5174	0.988
Z-score scaling	10 / 10	7	0.5255	0.358
Box-Cox transformation	4 / 10	3	0.4546	0.110

Table 2: Method-level summary across 10 member cases (40 trials total; 32 converged, 11 case-level Pareto selections).

Figure 1: Accuracy–time trade-off (left) and convergence/selection coverage (right) across all 40 trials. Black-outlined markers indicate Pareto-selected trials. Z-score scaling (green) converges for every case and occupies the low-time, high-accuracy region for most cases.

Member case	Selected method(s)
Main girder	z-score
Pier	min-max
Bearing	Box-Cox
Deck slab	z-score
Cross girder	z-score, Box-Cox
Longitudinal girder	z-score

Table 3: Pareto-selected scaling method(s) per member case.

Figure 2: Per-case, per-method golden-test accuracy (left) and $\log_{10}$ optimizer time in seconds (right). Gray cells indicate non-convergence; black-outlined cells indicate the Pareto-selected trial(s) for that case.

Member case	Box-Cox accuracy	Box-Cox log loss	Alternatives’ log loss
Bearing	0.4832	14.2609	1.9758–2.0454
Cross girder	0.3964	16.6358	2.9707–3.4432
Slope protection	0.4737	13.8499	$\approx$ 1.2810

Table 4: Box-Cox golden-test accuracy and log loss versus alternative methods, for the three cases where Box-Cox is Pareto-selected.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Member case	Max	Min-max	Z-score	Box-Cox
Main girder	0.6665 / 0.408	0.6665 / 0.498	0.6665 / 0.260	—
Pier	0.4848 / 2.638	0.4848 / 1.705	0.4757 / 2.413	—
Bearing	0.4591 / 5.842	0.4596 / 4.902	0.4596 / 2.079	0.4832 / 0.393
Deck slab	0.4958 / 0.928	0.4958 / 0.988	0.4958 / 0.358	0.4651 / 2.525
Cross girder	0.3927 / 0.257	0.3927 / 0.199	0.3927 / 0.105	0.3964 / 0.110
Longitudinal girder	—	—	0.6069 / 0.126	—

Appendix

Table 5: Golden-test accuracy / optimizer time (seconds) for every trial. Bold indicates the Pareto-selected method(s) for that case.

Markov Models

Markov models describe systems that evolve probabilistically, with future behavior determined by the current state and transition rules.

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FedAvg-Based CTMC Hazard Model for Federated Bridge Deterioration Assessment

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Markov Chains Structural Health Monitoring Federated Learning

Open paper AskPDF arXiv

Bridge periodic inspection records contain sensitive information about public infrastructure, making cross-organizational data sharing impractical under existing data governance constraints. We propose a federated framework for estimating a Continuous-Time Markov Chain (CTMC) hazard model of bridge deterioration, enabling municipalities to collaboratively train a shared benchmark model without transferring raw inspection records. Each User holds local inspection data and trains a log-linear hazard model over three deterioration-direction transitions -- Good

\to

Minor, Good

\to

Severe, and Minor

\to

Severe -- with covariates for bridge age, coastline distance, and deck area. Local optimization is performed via mini-batch stochastic gradient descent on the CTMC log-likelihood, and only a 12-dimensional pseudo-gradient vector is uploaded to a central server per communication round. The server aggregates User updates using sample-weighted Federated Averaging (FedAvg) with momentum and gradient clipping. All experiments in this paper are conducted on fully synthetic data generated from a known ground-truth parameter set with region-specific heterogeneity, enabling controlled evaluation of federated convergence behaviour. Simulation results across heterogeneous Users show consistent convergence of the average negative log-likelihood, with the aggregated gradient norm decreasing as User scale increases. Furthermore, the federated update mechanism provides a natural participation incentive: Users who register their local inspection datasets on a shared technical-standard platform receive in return the periodically updated global benchmark parameters -- information that cannot be obtained from local data alone -- thereby enabling evidence-based life-cycle planning without surrendering data sovereignty.

Abstract

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