An Interpretable Correlation-Driven Virtual Sensing Framework for Multi-Point Deformation Reconstruction of Hydraulic Structures

Reliable deformation reconstruction supports structural health monitoring when a target sensor is unavailable, unreliable, or requires independent verification. The proposed framework contributes an interpretable correlation-driven representation for virtual sensing rather than new X-means, Gaussian process, LSSVR, or MK-LSSVR algorithms. BIC-guided X-means adaptively identifies monitoring-point groups with statistically similar standardized deformation histories. A fixed-covariance Gaussian process equipped witha squared exponential kernel, calibrated exclusively from training-period measurements and used solely as a probabilistic representation of inter-point dependence without explicit residual extraction, transforms the associated-point measurements into the time-varying synergistic expectation μsyn(t). This quantity is combined with conventional water pressure, temperature, and time-effect factors as the dynamic M3 input. The same kernel formulation defines fixed synergistic-variance and mutual-information diagnostics for configuration-level interpretation; these quantities are neither dynamic regressors nor evidence used to establish the reported performance conclusions. Controlled M1-M2-M3 comparisons under identical LSSVR and MK-LSSVR backbones isolate the effects of conventional factors, raw associated-point measurements, and the correlation-derived representation. The framework was evaluated at M-α/C2, M-μ/C1, and M-ρ/C3 using horizontal-displacement records from 20 monitoring points in a hydraulic concrete structure in Sichuan Province, China. For M-α, M3 + MK-LSSVR achieved an RMSE of 0.1016 mm, an MAE of 0.0388 mm, and an R2 of 0.9890; relative to M2 + MK-LSSVR, RMSE and MAE decreased by 68.4% and 69.3%, respectively. The same configuration gave the lowest RMSE and MAE and the highest R2 for M-μ and M-ρ, reproducing the within-target ranking in all three X-means groups. The framework is intended for virtual sensing, sensor-replacement support, data verification, and monitoring–consistency checking, not as a direct damage or safety indicator.