34 On Digital Twins, Mirrors and Virtualisations 289 The simulation might provide the whole density function for MCt , or just low-order moments. In the first case, suppose that the model returns the predictive mean of the process mC(t) =E[MCt ] (where Eis an expectation), then, mC(t) can be used to determine whether MhC(Dtr) is an -mirror in themean. Alternatively, suppose that the model returns enough information to determine confidence intervals on the prediction. In this case, then if rC(t) ∈ ([mC(t) −ασC r (t)], [mC(t) +ασC r (t)]) with probability determined by α, and for all schedules in Dt , then one can define MhC(Dtr) as an α-mirror. Note that a given stochastic model can be both an -mirror and an α-mirror. It would be possible to define various metrics for comparison in the uncertain case; the one based on low-order moments described above is related to the reliability metric discussed in [5], which is in turn related to a formulation of validation as an outlier analysis problem, as discussed in [14]. If the comparison were made on the whole predictive or parameter density functions, one might define a distance measure defined in terms of Kullback-Liebler divergence, for example, and this would lead to the definition of a KL-mirror etc. 34.2.3 The Environment and Virtualisation Raising the question of uncertainty means that one must reconsider the status of the environment. Recall that the environment is comprised of all those variables which can have a causal influence on S, the structure of interest. In general, this set will be composed of variables that can be controlled (e.g. forces applied to the structure) and variables that can not (or can not be controlled with any precision). In an operational modal analysis context for example, even the forces may not be controllable. It is therefore necessary to separate the variables (in context) accordingly into eC u andeC c (uncontrolled and controlled, respectively). This distinction is very important if one wishes to use the model to make true predictions i.e. to determine what the structure might do at some point in the future, under a given (controlled) forcing, but when the eC u are unknown. In this situation, what is needed is a generative model MEC u , that will make some best estimate of e C u(t), ˆe C u =MEC u (t) (34.5) This model itself will need to be validated appropriately, as far as possible. Given training data for the eC u, it might be possible to establish a nonparametric black-box model that is an - or α-mirror, or one could substitute mean values for the variables and treat variations as uncertainty that needs to be propagated. In any case, one can now make predictions (in the given context), pC(t) =M[eC c (t), ˆe C u =MEC u (t)] (34.6) It is now possible to make another important definition: a virtualisation for a given context Cis a pair, VC =(MhC 1 ,MEC 2 ) (34.7) where the two models concerned are -mirrors with the fidelities specified. The importance of the virtualisation is that it can be used to examine what-if scenarios for the structure of interest in previously unseen circumstances. Of course, one can make a similar definition with α-mirrors. Finally, it is important to note that a virtualisation, is itself a model, and as such can also be an - or α-mirror; this will prove to be of interest later, when the use of virtualisations for design is discussed. The problem of the ‘environment’ is discussed in [6]; however, there it appears to have been condensed into the estimation/calibration of a further parameter set. 34.2.4 The Turing Mirror One can also think of a semi-philosophical means of defining a mirror; this parallels the Turing test in the field of artificial intelligence, which is a test of the ability of a machine to perform in a manner indistinguishable from a human [15]. The test will involve two protagonists: an interrogator and an oracle. The two people can only interact in a very limited way, the interrogator is allowed to present questions to the oracle about the structure of interest via a remote interface. The
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