32 Towards Population-Based Structural Health Monitoring, Part I: Homogeneous Populations and Forms 293 It should be observed from Fig. 32.3 that, while the normal-condition FRFs are similar, population variation can still be observed within the associated FRFs. 32.3.2 Gaussian Process Regression of the FRF as the Population Form Considering Sect. 32.2.2, the GP prior is set as zero-mean; therefore, the real and imaginary parts of the FRF (H) are regressed independently, with two distinct GPs, such that, f(x) + =Re H(ω) , x ω (32.10) or, f(x) + =Im H(ω) , x ω (32.11) This approach is adopted – rather than regressing the phases and magnitudes – as it is better suited to the proposed, general formulation of the GP. Modelling the FRF with two regressors fails to capture covariance between the outputs; in fact, for a linear system, one function completely determines the other [17]; thus, it is only really necessary to model one output. Importantly, only systems {S1, . . . ,S10} within the population contribute training data to learn the form. The remaining 10 systems {S11, . . . ,S20} (nine simulated systems and the experimental rig) are held-out of the training process. This choice is made to test the generalisation of the form, when applied tonewmembers within the strongly-homogeneous case. To reduce the computational load, the GP regression is trained using a random sub-sample of 5000 inputs and outputs from the simulated normal-condition data. A more rigorous approach for dealing with large datasets, such as sparse GPs [18], is being considered for future work. The resulting GP representation of the form, and the data used to train it, are illustrated in Fig. 32.4. Novelty Detection via the Form The form can now be used to monitor future data and inform damage detection. In this example, test FRF data fromall the members in the population are compared to the form; however, to reiterate, of the twenty members, only {S1, . . . ,S10} Fig. 32.4 Gaussian process regression of the FRF as the population-form. Blue markers indicate the training-set. The red line indicates the mean prediction, E[y∗], and the shaded area indicates 3-σ uncertainty, corresponding to the variance, V[y∗]
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