98 C. A. Lindley et al. where each element in the structure has a finite VC dimensionhk, and can be ordered according to their respective complexities, h1 ≤h2 ≤ · · · ≤hk ≤. . . (9) Having defined the structure S, the optimal model selection boils down to two steps: 1. Selecting an element Sk from the structure. 2. Estimating the model from this element. For each element in the structure (step (1)), the bound (5) is evaluated after estimating the optimal parameters in the element that minimise the empirical risk (step (2)). The penalisation during the learning process is thus established by the denominator in (5), which is computed with respect to the VC dimensionhk corresponding to the element Sk. Therefore, as elements of higher complexity are selected, the empirical risk will likely diminish, but at expense of a higher penalisation as a result of bringing the denominator closer to zero. An optimal element in the structure will be that providing the minimal “guaranteed” risk. Case Study: Model Selection for Modelling a Sdof Impulse Response The response of a Single-Degree-of-Freedom(SDOF) mass-damper-spring system (Fig. 1) is considered in this simple case study. Here, the dynamical model can be defined by the following equation of motion, m¨x(t)+c˙x(t)+kx(t)=F(t) (10) where m, c and k are the dynamic coefficients corresponding to the mass, damping and stiffness of the system, x(t) is the response at a given time instance t, and F(t) is the input force. The dots over the variables in equation (10) denote the derivatives of the variable with respect to time. An impulse response, h(t), was simulated here by solving equation (10) for a given selection of coefficients. In particular, the mass, damping and stiffness coefficients were given values of m= 1, c =20, and k =1×106, respectively. Using the simulated target function, training samples were generated by the following, y(t)=h(t)+ϵ (11) where ϵ corresponds to additive noise term, which follows a normal distribution ϵ ∼ N(0,σ2). The noise is defined by a signal-to-noise ratio (SNR) equal to ten. Four training sample sets of varying sizes were created by selecting points at intervals of 16, 12, and4, whereby the input-training samples t are uniform in[0, 0.3]. Therefore, each training set ended up with sample sizes of n=63, n=126andn=251, respectively. m k c F(t) Figure 1: Mass-spring-damper SDOF system. Once the complexity is defined for given set of functions, estimates of the expected risk no longer depend distribution from which the sample is assumed to have been drawn. This implication is advantageous sin density estimation is ill-posed when no prior knowledge of the probability distribution is available [5]. Fig. 1 Mass-spring-damper SDOF system
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