Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the use of Statistical Learning Theory for model selection in Structural Health Monitoring

C. A. Lindley, N. Dervilis, K. Worden

Abstract

Whenever data-based systems are employed in engineering applications, defining an optimal statistical representation is subject to the problem of model selection. This paper focusses on how well models can generalise in Structural Health Monitoring (SHM). Although statistical model validation in this field is often performed heuristically, it is possible to estimate generalisation more rigorously using the bounds provided by Statistical Learning Theory (SLT). Therefore, this paper explores the selection process of a kernel smoother for modelling the impulse response of a linear oscillator from the perspective of SLT. It is demonstrated that incorporating domain knowledge into the regression problem yields a lower guaranteed risk, thereby enhancing generalisation.

On the use of Statistical Learning Theory for model selection in Structural Health Monitoring

Abstract

Whenever data-based systems are employed in engineering applications, defining an optimal statistical representation is subject to the problem of model selection. This paper focusses on how well models can generalise in Structural Health Monitoring (SHM). Although statistical model validation in this field is often performed heuristically, it is possible to estimate generalisation more rigorously using the bounds provided by Statistical Learning Theory (SLT). Therefore, this paper explores the selection process of a kernel smoother for modelling the impulse response of a linear oscillator from the perspective of SLT. It is demonstrated that incorporating domain knowledge into the regression problem yields a lower guaranteed risk, thereby enhancing generalisation.
Paper Structure (10 sections, 1 theorem, 20 equations, 4 figures)

This paper contains 10 sections, 1 theorem, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The model-selection problem
  3. Managing complexity and minimisation of structural risk
  4. Case study: Model selection for modelling a SDOF impulse response
  5. Predictors for the regression problem
  6. Model selection strategy
  7. Model selection strategy
  8. Results and discussion
  9. Conclusions
  10. Acknowledgments

Key Result

Theorem 1

(Vapnik-Chervonenkis Vapnik2006) For any $\delta\in(0,1)$, with probability at least $(1-\delta)$ and $\forall \theta \in \Theta$, where when the set of loss functions$Q(\mathrm{z},\theta), \theta \in \Theta$is nonnegative, unbounded and contains an infinite number of elements. The measure $h$ quantifies the capacity of the function class, and the constants $a_1$, $a_2$ and $c$ are adjusted to sui

Figures (4)

  • Figure 1: Mass-spring-damper SDOF system.
  • Figure 2: Expected risk and predition error (MSE) computed for the SE and SDOF kernel function, with training sets of sizes (a) $n=63$, (b) $n=126$, and (c) $n=251$.
  • Figure 3: SE and SDOF kernel-smoother predictions inferred with training sets of sizes (a) $n=63$, (b) $n=126$, and (c) $n=251$. The true underlying signal was included for comparison.
  • Figure 4: Estimated complexities corresponding to the SE and SDOF kernel with training sets of sizes (a) $n=63$, (b) $n=126$, and (c) $n=251$.

Theorems & Definitions (4)

  • Definition 2.1: Key Theorem of Learning Theory Vapnik1999
  • Theorem 1
  • Definition 3.1: VC dimension Vapnik1999
  • Definition 4.1: SDOF kernel function Cross2021