PSI - Issue 78

Michele Mattiacci et al. / Procedia Structural Integrity 78 (2026) 1159–1166

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2. Background

2.1. Nonlinear Cointegration empowered by Neural Networks

Nonlinear cointegration theory enables the combination of nonstationary time series sharing the same order of integration to derive stationary quantities, free from potential drifts a ff ecting the original time series Perman (1991). Unlike linear cointegration approaches Johansen (1988), nonlinear cointegration allows for time series with an order of integration di ff erent from one, and can accommodate nonlinear dependencies between variables. In the context of SHM applications, let y 1 ( t ) , y 2 ( t ) , . . . , y n ( t ) represent a set of nonstationary time series of monitoring data, measured by n sensors installed on a structure, all having the same integration order I ( d ) , with d ≥ 1. These series are said to be nonlinearly cointegrated if a stationary nonlinear combination of them exists. For each sensor i , the stationary residual r i ( t ) is defined as r i ( t ) = y i ( t ) − z ( y − i ( t )), where y i ( t ) is the dependent monitoring data value and y − i ( t ) is the vector of all signals excluding the i -th one, i.e., y − i ( t ) = [ y 1 ( t ) , . . . , y i − 1 ( t ) , y i + 1 ( t ) , . . . , y n ( t )]. The function z ( · ) is a nonlinear regression function trained to estimate y i ( t ) from the remaining sensor data. In this framework, z ( · ) may be represented by any suitable machine learning algorithm capable of modeling the nonlinear interactions among the considered nonstationary time series Cross and Worden (2011). The resulting residual r i ( t ) serves as a new monitor ing feature associated with the i -th sensor, designed to be stationary under undamaged conditions and sensitive to damage-induced changes in the structural response. As such, this residual is expected to remain stable over time in the absence of structural anomalies, while its statistical properties will vary when damage occurs. The core of the proposed framework is the integration of temporal dependencies within a feedforward multilayer perceptron (MLP) architecture, which is employed to model the nonlinear nature and EOV influence on the time series data. Consider the regressor’s conventional input matrix X ∈ R T × n , where each element y i ( t j ) denotes the data value measured by the i -th sensor at time t j , for T time instants and n sensors. Instead of using X directly, a temporally enriched input matrix X ′ is constructed using a sliding-window mechanism that captures V successive time steps for each sensor: X ′ =     y ( t V ) y ( t V − 1 ) . . . y ( t 1 ) y ( t V + 1 ) y ( t V ) . . . y ( t 2 ) . . . . . . . . . . . . y ( t T ) y ( t T − 1 ) . . . y ( t T − V + 1 )     , (1) where y ( t j ) = [ y 1 ( t j ) , . . . , y n ( t j )] is the vector of strain values from all sensors at time t j . The resulting matrix X ′ ∈ R ( T − V + 1) × ( V · n ) represents a time-lagged embedding of the original data. To ensure proper alignment between input and output sequences, V − 1 entries of the target sequence are discarded. Model training is carried out using a grid search-based hyperparameter tuning, supported by k -fold cross-validation to enhance generalization performance and avoid overfitting. 2.2. Damage Detection via Change-Point Detection Change-point detection (CPD) refers to the process of identifying abrupt changes in the statistical properties of sequential data, which may reflect significant events such as the onset of structural damage Nigro et al. (2014). CPD algorithms aim to detect the most probable points in a time series where the underlying generative process shifts. The core assumption is that the monitored data can be modeled by a probability distribution that remains constant until a change occurs. When a change-point is reached, the distribution governing the data shifts, and CPD methods attempt to distinguish between the pre-change and post-change distributions. These methods operate on sequences of observations, which may be processed one point at a time or in batches. CPD algorithms are typically categorized as either online or o ffl ine. Online methods process the data sequentially and make real-time decisions based on sliding windows or incremental evaluation strategies Thompson et al. (2024). O ffl ine methods, in contrast, analyze the entire dataset retrospectively and are typically more computationally de manding Li et al. (2021). In the context of continuous SHM, where data is collected and analyzed in real-time, online CPD techniques o ff er a valuable tool for immediate damage detection within the proposed framework.