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Slepian model based independent interval approximation for the level excursion distributions

Henrik Bengtsson, Krzysztof Podgorski

TL;DR

This work addresses the problem of characterizing level-excursion distributions for Gaussian processes beyond zero-level crossings by extending the Slepian-model–based IIA. The authors formulate a probabilistic foundation by clipping the Slepian process and matching it to a non-stationary switch process, yielding explicit non-zero-level excursion representations via Laplace-transformed expected-value functions. They derive conditions ensuring the resulting distributions are valid and demonstrate the approach on non-zero crossings for Gaussian diffusions, notably in two dimensions, where persistency coefficients and excursion tails align well with direct simulations. The method offers a practical, tractable framework for estimating excursion-time distributions and related tail properties, with potential impact on physics-inspired applications where non-zero threshold exceedances are of interest.

Abstract

The independent interval approximation of the excursion time distributions for Gaussian processes has been used in physics and engineering. A new but related approach matches the expected value of the clipped Slepian to the expected value of a non-stationary binary stochastic process. This approach is extended to non-zero crossings and provides a probabilistic foundation for the validity of the approximations for a large class of processes. Both the above and below distributions are approximated. While the Slepian-based method was shown to be equivalent to the classical IIA for the zero-level, this is not the case for non-zero excursions.

Slepian model based independent interval approximation for the level excursion distributions

TL;DR

This work addresses the problem of characterizing level-excursion distributions for Gaussian processes beyond zero-level crossings by extending the Slepian-model–based IIA. The authors formulate a probabilistic foundation by clipping the Slepian process and matching it to a non-stationary switch process, yielding explicit non-zero-level excursion representations via Laplace-transformed expected-value functions. They derive conditions ensuring the resulting distributions are valid and demonstrate the approach on non-zero crossings for Gaussian diffusions, notably in two dimensions, where persistency coefficients and excursion tails align well with direct simulations. The method offers a practical, tractable framework for estimating excursion-time distributions and related tail properties, with potential impact on physics-inspired applications where non-zero threshold exceedances are of interest.

Abstract

The independent interval approximation of the excursion time distributions for Gaussian processes has been used in physics and engineering. A new but related approach matches the expected value of the clipped Slepian to the expected value of a non-stationary binary stochastic process. This approach is extended to non-zero crossings and provides a probabilistic foundation for the validity of the approximations for a large class of processes. Both the above and below distributions are approximated. While the Slepian-based method was shown to be equivalent to the classical IIA for the zero-level, this is not the case for non-zero excursions.
Paper Structure (10 sections, 10 theorems, 56 equations, 5 figures, 2 tables)

This paper contains 10 sections, 10 theorems, 56 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Slepian process
  3. Stationary and non-stationary switch processes
  4. Clipping Slepian and Gaussian processes
  5. IIA for a non-zero crossing level
  6. The stationary IIA
  7. The Slepian-based IIA
  8. Non-zero level persistency coefficients for Gaussian diffusion
  9. Conclusion
  10. Acknowledgement

Key Result

Proposition 1

Let $D(t)$ be a non-stationary switch process then the Laplace transform of $P_\delta(t)=\text{\sf P}{\left(D(t)=1 | \delta\right)}$, $t>0$, is given by Additionally, the Laplace transform of the expected value function $E_\delta(t)=\text{\sf E}{( D(t) |\delta)}$, $t>0$ is given by

Figures (5)

  • Figure 1: The Slepian model for three Gaussian diffusions: 1-dimensional (top), 2-dimensional (middle), 3-dimensional (bottom). Right: Three components of the decomposition; Left: Components put together to create ten samples from the Slepian model at five different levels.
  • Figure 2: The definition of the switch process attached to the origin, with the initial state $\delta=-1$.
  • Figure 3: The excursion intervals of a process $X(t)$ together with the corresponding clipped process (left) and a sample of the switch process under the IIA (right) with exponential distributions having the means matched those in the top graph.
  • Figure 4: The expected values of the Slepian model based on the diffusion covariance, for an $u$ up-crossing (left) and an $u$ down-crossing (right), for $u=0,1/2,1,5/4$.
  • Figure 5: Normalized histograms of the approximated excursion distribution for the Slepian model based on the diffusion covariance, for an $u$ up-crossing (left) and an $u$ down-crossing (right), for $u=1/2,5/4$.

Theorems & Definitions (16)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Remark 1
  • Theorem 2
  • Lemma 1
  • proof
  • Corollary 1
  • proof : Proof of Theorem \ref{['th:expctSlep']}
  • ...and 6 more