20 Towards Population-Based Structural Health Monitoring, Part IV: Heterogeneous Populations, Transfer and Mapping 193 [15–17]. Graph-based methods that define the relationships between source domains by stating them graphically using a transferability metric have been developed [15]. This provides similar motivation for the Irreducible Element and Attributed Graph approach [3, 4]; however, in an SHM context these graphical representations can be formed from a physics-based viewpoint, aiding the strength of knowledge about similarity. Other approaches in avoiding negative transfer have sought to weight each source domain based on its relevance to the target domain; this is known as instance weighting [18]. Local cluster-based weighting has also been proposed, meaning that for each class on a source domain, an individual weighting is provided, stating that informativeness may not be globally shared in a particular source domain [16]. As a result, it is important to consider and account for the possibility of negative transfer when identifying what structures, and their corresponding datasets, to use in transfer learning, as well as developing algorithms to reduce or remove the possibility of negative transfer within PBSHM. 20.3 Domain Adaptation Domain adaptation is one form of transfer learning that seeks to transfer feature spaces between source and target domains, assuming that their marginal distributions over the finite sample set are not equal p(Xs) = p(Xt ). These techniques are primarily designed for homogeneous transfer learning, where the feature space and label space are consistent [7, 11, 19, 20]; however, heterogeneous transfer learning forms of domain adaptation do exist [21–23]. This section outlines domain adaptation and its assumptions, before demonstrating its applicability to homogenous populations, and a heterogeneous population; that contain both geometric and material differences. These case studies motivate what aspects of PBSHM are currently achievable, highlighting the required areas of further research in making PBSHM applicable across the complete range of problems outlined in Sect. 20.2.3. Domain adaptation is formally defined (for a single source and target domain) as: Definition 6 Domain adaptation applies when a given inference is required for a target domain Dt and task Tt , and is the process of improving the target predictive function ft (·) in Tt given a source domain Ds and task Ts, whilst assuming Xs =Xt and Ys =Yt but that p(Xs) =p(Xt ), and one can further assume p(Ys | Xs) =p(Yt | Xt ). Various algorithms have been developed for this scenario [11, 19, 20], and several have been applied in a PBSHM context [24]. One approach is implemented in this section, Joint Domain Adaption (JDA) [11], in order to visually motivate the PBSHM problems in which a domain adaptation approach is applicable, although it is noted that other techniques may offer different levels of classification performance. Briefly, JDA is a technique that assumes the joint distributions between the source and target are different p(Ys,Xs) = p(Yt ,Xt ), and finds the optimal latent mapping in whichp(φ(Xs)) ≈p(φ(Xt )) andp(φ(Xs)| Ys =c) ≈p(φ(Xt )| Yt =c) for each class c ∈ {1, . . . ,C} in Y (the class conditionals are matched as the conditionals are often challenging and computationally expensive to compute). This match is performed by leveraging the empirical form of the Maximum Mean Discrepancy (MMD) distance [25]; the difference between two mean embeddings in a Reproducing Kernel Hilbert Space (RKHS), induced by a kernel K[26] (where a variety of kernels can be implemented). The summation of the marginal and class conditional distributions can be formed as, Dist(p(φ(Xs)), p(φ(Xt ))) +Dist(p(φ(Xs)| Ys), p(φ(Xt )| Yt )) ≈tr W T KMcKW (20.1) whereWare weights associated with the empirical MMD distance andMc is the MMD matrix, defining the mean embedding; this summation can be minimised to find the optimal latent mapping. It is noted that JDA assumes an unlabelled target domain and utilises a simplistic form of semi-supervised learning, using a classifier trained on the source domain projection to predict pseudo-labels ˆYt for the projected target domain. The MMD matrix is defined as, Mc(i,j) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 N (c) s N (c) s , xi,xj ∈D(c) s 1 N (c) t N (c) t , xi,xj ∈D (c) t −1 N (c) s N (c) t , 4xi ∈D(c) s xj ∈D (c) t xi ∈D(c) s xj ∈D (c) t 0, otherwise (20.2)
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