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The Slepian model based independent interval approximation of persistency and zero-level excursion distributions

Henrik Bengtsson, Krzysztof Podgorski

TL;DR

The paper tackles the long-standing problem of excursion-time distributions for stationary Gaussian processes by introducing a Slepian-process-based independent interval approximation (IIA). By matching the expected value function of a clipped Slepian process to a non-stationary switch process, the authors obtain a mathematically meaningful and practically implementable approximation, with an explicit Laplace-domain form $\Psi_{\widehat{T}}(s)=\dfrac{1-s\mathcal{L}(E_0)(s)}{1+s\mathcal{L}(E_0)(s)}$ and a stochastic representation via a 2-geometric divisor. This framework yields valid excursion distributions for a large class of Gaussian processes with monotone covariances and provides direct avenues to estimate the persistency coefficient, including Monte Carlo sampling from the divisor and connections to the covariance via $E_0(t)=-\frac{1}{\sqrt{-r''(0)}}\frac{r'(t)}{\sqrt{1-r(t)^2}}$. The work also clarifies when the ordinary IIA may fail, particularly for non-monotone covariances, and demonstrates the method on diffusion, random acceleration, shifted Matérn covariances, and Matérn cases, highlighting both its accuracy and its limitations in practice.

Abstract

In physics and engineering literature, the distribution of the excursion time of a stationary Gaussian process has been approximated through a method based on a stationary switch process with independently distributed switching times. The approach matches the covariance of the clipped Gaussian process with that of the stationary switch process. By expressing the switching time distribution as a function of the covariance, the so-called independent interval approximation (IIA) is obtained for the excursions of Gaussian processes. This approach has successfully approximated the persistency coefficient for many vital processes in physics but left an unanswered question about when such an approach leads to a mathematically meaningful and proper excursion distribution. Here, we propose an alternative approximation: the Slepian-based IIA. This approach matches the expected values of the clipped Slepian process and the corresponding switch process initiated at the origin. It is shown that these two approaches are equivalent, and thus, the original question of the mathematical validity of the IIA method can be rephrased using the Slepian model setup. We show that this approach leads to valid approximations of the excursion distribution for a large subclass of the Gaussian processes with monotonic covariance. Within this class, the approximated excursion time distribution has a stochastic representation that connects directly to the covariance of the underlying Gaussian process. This representation is then used to approximate the persistency coefficient for several important processes to illustrate the Slepian-based IIA approach. Lastly, we argue that the ordinary IIA is ill-suited in certain situations, such as for Gaussian processes with a non-monotonic covariance.

The Slepian model based independent interval approximation of persistency and zero-level excursion distributions

TL;DR

The paper tackles the long-standing problem of excursion-time distributions for stationary Gaussian processes by introducing a Slepian-process-based independent interval approximation (IIA). By matching the expected value function of a clipped Slepian process to a non-stationary switch process, the authors obtain a mathematically meaningful and practically implementable approximation, with an explicit Laplace-domain form and a stochastic representation via a 2-geometric divisor. This framework yields valid excursion distributions for a large class of Gaussian processes with monotone covariances and provides direct avenues to estimate the persistency coefficient, including Monte Carlo sampling from the divisor and connections to the covariance via . The work also clarifies when the ordinary IIA may fail, particularly for non-monotone covariances, and demonstrates the method on diffusion, random acceleration, shifted Matérn covariances, and Matérn cases, highlighting both its accuracy and its limitations in practice.

Abstract

In physics and engineering literature, the distribution of the excursion time of a stationary Gaussian process has been approximated through a method based on a stationary switch process with independently distributed switching times. The approach matches the covariance of the clipped Gaussian process with that of the stationary switch process. By expressing the switching time distribution as a function of the covariance, the so-called independent interval approximation (IIA) is obtained for the excursions of Gaussian processes. This approach has successfully approximated the persistency coefficient for many vital processes in physics but left an unanswered question about when such an approach leads to a mathematically meaningful and proper excursion distribution. Here, we propose an alternative approximation: the Slepian-based IIA. This approach matches the expected values of the clipped Slepian process and the corresponding switch process initiated at the origin. It is shown that these two approaches are equivalent, and thus, the original question of the mathematical validity of the IIA method can be rephrased using the Slepian model setup. We show that this approach leads to valid approximations of the excursion distribution for a large subclass of the Gaussian processes with monotonic covariance. Within this class, the approximated excursion time distribution has a stochastic representation that connects directly to the covariance of the underlying Gaussian process. This representation is then used to approximate the persistency coefficient for several important processes to illustrate the Slepian-based IIA approach. Lastly, we argue that the ordinary IIA is ill-suited in certain situations, such as for Gaussian processes with a non-monotonic covariance.
Paper Structure (21 sections, 14 theorems, 76 equations, 5 figures, 3 tables)

This paper contains 21 sections, 14 theorems, 76 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The independent interval approximation
  3. The switch process
  4. The ordinary IIA
  5. The Slepian-based IIA
  6. The approximated excursion distribution
  7. Properties of the Slepian-based IIA
  8. Persistency coefficient through the IIA
  9. Applications and examples
  10. The $d$-dimensional diffusion
  11. Random acceleration process
  12. Shifted and non-shifted Gaussian covariance
  13. Matérn covariance
  14. Conclusion
  15. Acknowledgement
  16. ...and 6 more sections

Key Result

Proposition 1

Let $D(t)$ be a non-stationary switch process with the expected value function $E(t)=\text{\sf E} D(t)$. Then we have the relations for $s>0$ where $\Psi(s)=\mathcal{L}(f)(s)$.

Figures (5)

  • Figure 1: The Torsethaugen spectrum (left) and the corresponding correlation (right) that have been used for generating data that illustrate the introduced concepts.
  • Figure 2: The excursion intervals of a Gaussian process $X(t)$ together with the corresponding clipped process.
  • Figure 3: The non-stationary switch process (left) and the stationary version, (right) with $\delta=1$. The inspection paradox states that the interval that contains zero has the distribution $A+B$, which is not the same as the other interval distributions.
  • Figure 4: (right) The logarithm of $E_0$ for $d=1,...,10$. The steeper curves are associated with higher dimensions. (left) Logarithm of the empirical cdf from the approximated excursion distribution for the diffusion process, which is used for estimating the persistency coefficient.
  • Figure 5: The Matérn covariance case. Left: The covariances $r$ for $\nu=l/2$, $l=4,5,\dots, 20$, the larger $\nu$ the stronger dependence. Middle: The survival functions $E_0$ of the geometric divisor, the larger $\nu$ the heavier tail.

Theorems & Definitions (23)

  • Proposition 1
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Proposition 4
  • proof
  • Proposition 5
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • ...and 13 more