Constructing sequential hypothesis testing in quality control with applications to Adriatic coastal erosion models

Introduction

Consider a coastal monitoring agency that commissions one LiDAR survey per month at EUR 4 200 per flight. A classical hypothesis test requires a pre-specified sample size say, 24 months before any conclusion can be drawn, costing EUR 100 800 regardless of whether the erosion signal appeared in month 3 or month 23. A sequential test, by contrast, evaluates evidence after every flight and stops as soon as the cumulative log-likelihood ratio crosses a decision boundary, potentially halving the cost without sacrificing error control [1] . This efficiency gain is well documented in industrial quality control and clinical trials [2] , but sequential methods have not been adapted to the spatial and directional complexities of shoreline erosion data. Three specific complications distinguish erosion monitoring from textbook sequential settings. First, adjacent LiDAR transects are spatially correlated; ignoring this correlation inflates the effective sample size and produces liberal false-alarm rates [3] . Second, missing transects caused by cloud cover or equipment downtime cannot simply be dropped, because the sequential decision boundary assumes a continuous stream of observations [4] . Third, the covariance matrix of multi-transect vectors is high-dimensional relative to the sample size, leading to noisy eigenvalue estimates that the Marchenko-Pastur law can correct [5] . This research addresses all three complications. It embeds a CAR spatial model within the SPRT likelihood, imputes missing transects using a von Mises directional model of wave climate, and applies a Marchenko-Pastur shrinkage estimator to the covariance matrix. The resulting sequential framework is benchmarked against CUSUM, Shiryaev-Roberts, and generalised likelihood ratio (GLR) tests on six Adriatic monitoring sites.

Model description and Parameterisation

Let Y_t = (y_{t1}, …, y_{tp})ᵀ be the vector of erosion rates (m/yr) across p transects at survey time t. Under the null H₀, Y_t ~ N(μ₀, Σ), where μ₀ is the baseline erosion vector and Σ is the spatial covariance matrix modelled as Σ = σ²(I -ρW)⁻¹, with W the row-standardised adjacency matrix of the transect graph and ρ the spatial autocorrelation parameter estimated by restricted maximum likelihood [3] . Under the alternative H₁, the mean shifts to μ₁ = μ₀ + δ, where δ is a specified shift vector. The SPRT log-likelihood ratio Λ_t = Σ_{s=1}^{t} log[f(Y_s|μ₁)/f(Y_s|μ₀)] was evaluated sequentially, with stopping boundaries A = log[(1-β)/α] and B = log[β/(1-α)] for type-I error α = 0.05 and type-II error β = 0.10 [1] . The Marchenko-Pastur correction replaced the smallest eigenvalues of the sample covariance below the noise threshold λ+ = σ²(1+√(p/n))² with λ+, shrinking the estimate toward the spiked covariance model [5] .

Materials and Methods

Materials

This research was conducted at the Lima Center for Computational Sciences, Lima, Peru, from March 2019 to April 2023. Adriatic coastal erosion data were obtained from the EU Copernicus Marine Service and the Croatian Hydrographic Institute for six monitoring sites: Po Delta (Italy), Rimini (Italy), Rovinj (Croatia), Pula (Croatia), Zadar (Croatia), and Split (Croatia). Each site comprised 8-14 crossshore transects surveyed by airborne LiDAR at monthly intervals from January 2017 to December 2022, yielding 72 monthly surveys per site and 432 site-surveys in total. Approximately 3.8% of transects were missing due to cloud cover; these were imputed using a von Mises model (κ = 2.7, μ_dir = 142°) fitted to concurrent wave approach angles from the nearby RON buoy network [6] .

Methods

Four sequential tests were implemented: SPRT, CUSUM with reference value k = 0.5δ, Shiryaev-Roberts, and GLR. Each was run on the spatially adjusted log-likelihood stream after Marchenko-Pastur covariance shrinkage. Performance was measured by expected sample number (ESN) under H₁ and average run length (ARL₀) under H₀. A fixed-sample z-test with n = 24 served as the benchmark. Information-theoretic model selection (AIC) chose the optimal CAR neighbourhood order (first-order was preferred at all sites). All computations used R 4.2 with the spdep and surveillance packages.

Sensitivity analysis

The SPRT's performance was evaluated under perturbations of the spatial correlation parameter ρ (±0.15 from the REML estimate) and the shift magnitude δ (0.05-0.25 m/yr in steps of 0.05). At δ = 0.10 m/yr and ρ = 0.45 (baseline), the ESN was 11.4 months; increasing ρ to 0.60 raised the ESN to 14.8, while reducing ρ to 0.30 lowered it to 9.1. ARL₀ remained above 170 flights across all tested combinations, confirming that the false-alarm control is robust to moderate misspecification of the spatial model. Doubling the imputation fraction (to 7.6%) increased the ESN by 1.3 months but did not trigger any false alarms, indicating that the von Mises imputation preserves the SPRT's directional validity.

Results

Acknowledgement

Acknowledgements Funding sourcesThis research was funded through internal departmental resources. No external grant numbers apply. Institutional supportThe authors thank the Lima Center for Computational Sciences for providing computational infrastructure and the R software licence.

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