62 K. Worden and E. J. Cross one selects a model and a particle from the previous weighted set of particles and perturbs it via a predefined kernel; again the selection of the kernel is a matter of choice. A widely used option is to define the perturbation kernel as a multivariate Gaussian centred on the mean of the particle population, with a covariance matrix set to the covariance of the population obtained in the previous iteration. For a deep discussion of various schemes for specifying the perturbation kernels, the reader is referred to [36]. In this study, the particle perturbation distribution is uniform and symmetric around zero, with the interval width (for each parameter) taken as the range of the parameter in the previous population. One then calculates the distance . (D∗, D), compares to the new tolerance threshold and accepts the new particle if . (D∗, D) ≤ε2; otherwise the particle is rejected. This process is repeated until a new set of N particles is assembled. One then updates the particle’s weight according to the kernel. The entire procedure is repeated until convergence is met. One way to accept convergence is to impose a target threshold close to zero; another is to control the acceptance ratio, which is measured at each iteration. This ratio is the quotient of the number of proposals to the full number of proposed particles at every step. When this ratio falls below some given limit, the algorithm is halted. There are actually numerous meaningful ways to establish convergence [15]. When the algorithm halts, the approximate marginal posterior distribution for model candidate .Mi is estimated by: .P(Mi|D∗) ≈ Accepted particles for Mi Total number of particles N (8.21) As one can see, the algorithm does require the selection of a number of hyperparameters—the sequence of acceptance thresholds. A careful choice of those hyperparameters is important, since they will affect performance of the algorithm. A bad choice may lead to computational expense and/or biased parameter estimates. 8.6 Case Study in Equation Discovery: Hysteretic Systems Hysteretic systems—or systems with memory—have always presented problems in terms of system modelling. The models often contain terms which are highly (and/or discontinuously) nonlinear in the parameters, and they may also evolve via unmeasured states. Even parameter estimation can be challenging. One compact and versatile model class which is popular for hysteretic SI is the Bouc-We n (BW) class [37, 38]. The general single-degree-of-freedom (SDOF) hysteretic system described in the terms of Wen [38] is represented below, where.g(y, ˙y) is the polynomial part of the internal restoring force, .z(y, ˙y) is the hysteretic part and.x(t) is the excitation, as usual: .m¨y +g(y, ˙y) +z(y, ˙y) =x(t) (8.22) where mis the system mass. In the following discussion, the polynomial part of the internal force will be considered linear: i.e. .g(y, ˙y) =c˙y +ky. The hysteretic component is defined by Wen [38], via the additional equation of motion: . ˙z = −α|˙y|zn −β˙y|zn|+A˙y, for n odd −α|˙y|zn−1| z|−β˙y|zn|+A˙y, for n even (8.23) The parameters. α, . βand n govern the shape and the smoothness of the hysteretic loop. The equations offer a simplification from the point of view of parameter estimation, in that the linear stiffness term from g in Eq.(8.22) can be combined with the .A˙y term in the state equation for z. The reader can refer to [14] for full details. In this illustration, the response output will be assumed to be displacement. Data were simulated in the same manner as for the Duffing example earlier. The sampling interval was taken as 0.001s, corresponding to a sampling frequency of 1000 Hz. Here, as in the first example, the excitation is Gaussian with zero mean, but with a standard deviation of 10. The exact parameter values used to generate the training data and the parameter ranges are summarised in Table 8.3. Figure 8.5 shows the BW model response with .n =2. The training data used here were composed of 1001 points, corresponding to a record duration of 1s.
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