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

1.1 Background and Motivation

• Production pipelines often default to max scaling, although covariates span very different physical units and magnitudes.The study focuses on bridge deterioration models estimated by maximum likelihood, where scaling affects optimization conditioning.Source: §1.1 · p. 2

1.2 Problem Statement and Challenges

• The comparison addresses whether scaling changes golden-test accuracy and convergence time, and whether one default suits heterogeneous member cases.The authors identify three concerns: method-dependent convergence, probability miscalibration that accuracy can miss, and trade-offs between accuracy and time.Source: §1.2 · p. 2

1.3 Proposed Approach

• The study formalizes scaling as training-fitted operators, estimates an EHM model by maximum likelihood, and reports non-dominated methods using accuracy and optimizer time.The comparison applies each fitted transformation unchanged to the golden test and measures optimizer-only wall-clock time.Source: §1.3 · pp. 2–3

1.4 Key Contributions

• The contributions include a tested, production-embedded comparison pipeline and evidence that fast convergence alone does not establish probabilistic model quality.Box-Cox scaling is associated with 7–13× worse golden-test log loss in three of four converged cases, while the recommended decision rule adds a log-loss guardrail.Source: §1.4 · p. 3
• Z-score scaling converges in all 10 member cases, has the highest mean converged accuracy, and takes less than half the production baseline’s median optimizer time.The authors recommend it as the default candidate, with a log-loss guardrail for faster alternatives and independent-test confirmation.Source: §1.4 · p. 3

1.5 Paper Organization

• The paper proceeds from related work and formal methodology to dataset and implementation details, empirical results, interpretation, and conclusions.Source: §1.5 · p. 3

2.1 Multi-State and Markov Hazard Models

• The EHM study builds on survival and multi-state hazard models, focusing on numerical estimation behavior under alternative covariate preprocessing choices.Infrastructure applications often use monotone, absorbing state spaces to represent irreversible deterioration.Source: §2.1 · p. 3

2.2 Markov Deterioration Modeling in Civil Infrastructure Health Monitoring

• Prior bridge deterioration work develops Markov transition and hazard models, but does not address how covariate scaling affects their maximum-likelihood estimation or calibration.The authors identify a lack of systematic quantitative comparisons of scaling methods for this model class.Source: §2.2 · p. 4

2.3 Covariate Scaling and Optimizer Conditioning

• Scaling can improve optimizer conditioning, but prior guidance does not establish which scaling family to prefer for a given model class.The reviewed literature spans optimization, regression, multivariate statistics, and hazard models; this paper evaluates alternatives empirically for EHM.Source: §2.3 · pp. 4–5

2.4 Probability Calibration and Proper Scoring Rules

• Accuracy alone can obscure poor probability calibration, motivating proper scoring rules such as log loss alongside accuracy.The cited literature documents classifiers with strong accuracy but poorly calibrated probability estimates.Source: §2.4 · p. 5

2.5 Multi-Objective Model Selection

• Pareto selection exposes accuracy–cost trade-offs without fixing an arbitrary scalarization weight, retaining the full non-dominated set for practitioner review.Here the objectives are golden-test accuracy and optimizer convergence time.Source: §2.5 · p. 5

3 Methodology

• The methodology formalizes the statistical model, scaling operators, and multiobjective selection rule with notation, definitions, an algorithm, and stated assumptions.Supporting remarks and computational notes are collected in Appendix A.1.Source: §3 · p. 6

3.1 Problem Setup and Notation

• Each record represents a forward deterioration transition with an inspection interval and covariates; the study uses four covariates and infers the state count from training data.The data are restricted to monotone, non-repairable degradation by upstream cleaning.Source: §3.1 · p. 6

3.2 Exponential Hazard Markov (EHM) Model

• The EHM model assigns state-specific hazards in a continuous-time Markov chain with adjacent forward transitions and an absorbing final state.Parameters are estimated by unweighted maximum likelihood via L-BFGS-B, with at most 500 iterations.Source: §3.2 · p. 6

3.3 Covariate Scaling Operator Family

• Scaling operators are fitted on training covariates and reused unchanged on golden-test rows; Box-Cox uses a training-derived positivity shift and profile-likelihood estimate.Constant training columns map to zero under all four methods.Source: §3.3 · p. 7

3.4 Convergence Event and Time Measurement

• A trial counts as converged only when the solver reports success and parameters and log-likelihood are finite; optimizer time excludes preprocessing and post-fit work.Failed trials are excluded from Pareto selection but retained in reported comparisons.Source: §3.4 · p. 7

3.5 Golden-Test Risk Functionals

• Golden-test accuracy uses the arg-max prediction, while log loss and macro recall characterize calibration and class-level behavior after fitting.Only accuracy and convergence time enter formal selection.Source: §3.5 · p. 7

3.6 Multi-Objective Selection via Pareto Dominance

• The selection rule retains converged methods not dominated on accuracy and negative convergence time, so it can return multiple methods rather than one winner.The algorithm fits and evaluates each method per case, then computes the selected set using tolerance-controlled comparisons.Source: §3.6 · pp. 7–8

3.7 Assumptions and Threats

• The model assumes monotone degradation and may be misspecified when repairs or measurement-driven downward transitions occur.Golden-test selection also makes post-selection accuracy optimistic; the training-fitted Box-Cox shift does not guarantee positivity for test covariates outside the training range, which can trigger trial failure.Source: §3.7 · p. 8

4.1 Dataset Description

• The experiment uses third-to-fourth inspection transitions for training and bridge-level fourth-to-fifth transitions for validation and golden testing, capped at 20,000 rows per case.The validation partition is not used; 80% of the held-out transitions form the golden test.Source: §4.1 · pp. 8–9

4.2 Explanatory Covariates

• All ten cases use age, bridge length, bridge area, and traffic mean, whose native units differ by several orders of magnitude.Traffic mean averages the third- and fourth-inspection traffic volumes.Source: §4.2 · p. 9

4.3 Experimental Conditions

• Across 40 trials, only the scaling method varies; likelihood weighting, partitions, seed, row cap, and covariates match the production baseline.Each trial resets the estimation seed immediately before fitting, so optimizer trajectories differ with scaled covariates rather than differing initializations.Source: §4.3 · p. 9

4.4 Implementation and Software

• The implementation uses Python and SciPy's L-BFGS-B optimizer, records fit and timing metadata, and tests scaler behavior and Pareto selection edge cases.Tests include constant columns, Box-Cox positivity shifts, ties, and complete failure.Source: §4.4 · pp. 9–10

4.5 Reproducibility

• Results come from one deterministic run; convergence time is hardware-dependent, and the code and anonymized dataset are retained internally rather than distributed.The authors identify relative method rankings, not absolute timing, as the object of interest.Source: §4.5 · p. 10

5.4 Summary of Aggregate Outcomes

• 32 of 40 trials converged; z-score supplied 7 of 11 case-level Pareto selections and was the only method with zero convergence failures.The authors describe it as the most broadly reliable candidate under the replicated study conditions.Source: Table 2 · p. 10, §5.4 · p. 12

5.1 Convergence, Accuracy, and Timing by Method

• Z-score was the only method to converge in all ten cases, with median time below 40% of max or min-max scaling despite mean accuracy differences of at most 0.008.Box-Cox had the lowest median time among its converged trials, but that estimate reflects only four cases.Source: §5.1 · p. 10

5.2 Per-Case Pareto Selection

• Z-score was selected in 7 of 10 cases, Box-Cox in 3, min-max in 1, and max scaling never alone; Cross girder co-selected z-score and Box-Cox.In Cross girder, Box-Cox was slightly more accurate and no slower.Source: §5.2 · p. 10, Table 3 · p. 11

5.3 Box-Cox Convergence and Calibration

• In all three cases where Box-Cox was Pareto-selected, its accuracy was within 0.03 of alternatives but its log loss was 7–13× worse.Log loss was not part of the formal accuracy-and-time selection criterion, so Pareto selection alone did not detect this miscalibration.Source: §5.3 · pp. 11–12, Table 4 · p. 12

6.1 Why Z-Score Scaling Outperforms Range-Based Scaling

• Z-score scaling centers covariates and equalizes their variance, addressing the conditioning problem; it converged in all 10 cases with less than half the median time of max or min-max scaling.Max and min-max rescale ranges without centering, leaving covariates with different native units on substantially different effective scales.Source: §6.1 · pp. 12–13

6.2 Why Box-Cox Scaling Is Fast but Fragile

• Box-Cox was Pareto-selected in three cases and often fast when it converged, but its nonlinear fit from small training samples may be fragile on golden-test covariates outside the training range.It converged in only 4/10 cases, and the text identifies this extrapolation as a plausible mechanism for the observed log-loss deterioration.Source: §6.2 · p. 13

6.3 Accuracy and Convergence Time Are an Incomplete Selection Basis

• Box-Cox was Pareto-selected in three cases despite 7–13× worse log loss, showing that accuracy-and-time selection can retain severely miscalibrated models.Accuracy measures only the arg-max prediction, whereas log loss is sensitive to the predicted probability vector.Source: §6.3 · p. 13

6.4 Small-Sample Fragility

• The two smallest cases, with 27 and 86 training rows, had elevated convergence failure rates, making single-run comparisons exploratory and motivating repeated resampling or crossvalidation.Source: §6.4 · p. 13

6.5 Practical Recommendation

• The recommended decision rule starts with z-score as the default candidate, retains alternatives only when accuracy, log loss, and convergence stability pass guardrails, and uses Pareto comparison as a final tie-break.It also calls for confirmation on an untouched independent test partition.Source: §6.5 · pp. 13–14

6.6 Limitations

• Golden-test-based selection makes reported post-selection accuracy optimistic, so an independent untouched test is needed for a final performance claim.Source: §6.6 · p. 14
• Convergence time was measured with one random seed and one hardware configuration, making absolute timings hardware-dependent.The authors expect the qualitative ranking to be more robust, based on the conditioning argument.Source: §6.6 · p. 14
• The formal Pareto rule uses accuracy and convergence time but excludes log loss and macro recall, so it can select a miscalibrated method.Source: §6.6 · p. 14

• Z-score converged in all ten cases, had the highest mean converged accuracy, required less than half the baseline median optimizer time, and was Pareto-selected in 7 of 10 cases.Source: §7 · p. 14
• Box-Cox converged in only 4 of 10 cases yet was often fast and Pareto-selected, while its golden-test log loss was 7–13× worse despite comparable accuracy.Accuracy-and-time selection does not account for this calibration difference.Source: §7 · p. 14
• The paper recommends z-score as the default candidate, a log-loss guardrail for faster alternatives, and confirmation on an independent test partition.Source: §7 · p. 14
• Future work includes repeated-resampling validation, direct integration of a log-loss guardrail into selection, and testing whether the conditioning argument extends beyond the exponential-hazard model.Source: §7 · p. 15

• The EHM is described as a discretized multi-state extension of proportional hazards for ordered, absorbing, non-repairable deterioration states, not a model permitting recovery or competing risks.Source: §A.1 · p. 17
• All trials use uniform likelihood weights to match the unweighted production baseline; the available class-weighted likelihood variant is out of scope.Source: §A.1 · p. 17
• Accuracy scores only the arg-max transition, while log loss scores the full predicted probability vector and can reveal miscalibration that accuracy misses.Both are reported, although only accuracy and convergence time enter formal selection.Source: §A.1 · p. 17
• The model assumes monotone degradation enforced by upstream cleaning, so genuine repairs or measurement-driven downward transitions would misspecify its generator.Source: §A.1 · p. 17
• Scaling parameters are fit on training data only, but the golden test is used for method selection, making post-selection accuracy optimistic relative to a fully held-out evaluation.Source: §A.1 · p. 17
• The Box-Cox positivity shift is fit on training data; if test covariates fall outside the training range, positivity is not guaranteed and the implementation treats this as trial failure.Trials use SciPy's L-BFGS-B optimizer and Box-Cox implementation; the production codebase is retained for confidentiality.Source: §A.1 · p. 17

• Among converged methods, golden-test accuracy differences were small, so accuracy alone was an insufficient basis for method selection.Source: §A.2 · p. 17
• Cases with fewer than 100 training rows showed a higher proportion of non-converged trials than cases with more than 500 rows.Source: §A.2 · p. 18
• The Pareto rule retains every eligible non-dominated method, where eligibility requires convergence and finite likelihood and parameters; it does not force one winner.Source: §A.2 · p. 18
• The golden test was used directly for selection, making post-selection accuracy optimistic rather than an independent generalization estimate.Source: §A.2 · p. 18
• Cases with under 100 training rows produced noisy accuracy and log-loss estimates, and their conclusions are exploratory; each fit also used one random initialization seed.Source: §A.2 · p. 18
• All five hypotheses were empirically supported, including universal z-score convergence, its selection in 7/10 cases, Box-Cox calibration deterioration, and higher failure rates in cases with fewer than 100 rows.Source: §A.2 · p. 18

A.3 Full Per-Trial Comparison Table

• Table 5 reports golden-test accuracy and optimizer time for each converged trial across ten member cases and four scaling methods; non-convergence is marked “—”.Bold identifies the Pareto-selected method or methods for each case.Source: §A.3 · p. 18, Table 5 · p. 19

A.4 Software Versions

• The reported software versions are Python 3.13, SciPy 1.16.3, pandas 2.2.3, NumPy 2.3.5, and numdifftools 0.9.42.numdifftools was used for Hessian-based standard errors in the underlying model class, not scaling-method selection.Source: §A.4 · p. 18

A.5 Relationship to Prior Production Baseline

• The experiments replicate the prior production baseline’s member cases, covariates, unweighted likelihood, random seed, and row cap, while adding three scaling methods and Pareto selection.The study’s Max trials use the same conditions without changing the baseline’s recorded outputs.Source: §A.5 · pp. 18–19