## Sammanfattning

This work is to popularize the method of computing the distribution of the excursion times

for a Gaussian process that involves extended and multivariate Rice's formula.

The approach was used in numerical implementations of the high-dimensional integration routine and in earlier work it was shown that the computations are more effective and thus more precise than those based on Rice expansions.

The joint distribution of successive excursion times is clearly related to the distribution

of the number of level crossings, a problem that can be attacked via the Rice series expansion,

based on the moments of the number of crossings.

Another point of attack is the

``Independent Interval Approximation'' (IIA)

intensively studied for the persistency of physical systems.

It treats the lengths of successive crossing intervals as

statistically independent. Under IIA, a renewal type argument

leads to an expression that provides the approximate interval

distribution via its Laplace transform.

However, the independence is not valid in most

typical situations.

Even if it leads to acceptable results for the persistency exponent of the long excursion time distribution or some classes of processes, rigorous assessment of the approximation error is not readily available.

Moreover, we show that the IIA approach cannot deliver properly defined probability distributions and thus the method is limited only to persistence studies.

The ocean science community favours a third approach, in which a class of

parametric marginal distributions, either fitted to excursion data or

derived from a narrow band approximation, is extended by a copula technique

to bivariate and higher order distributions.

This paper presents an alternative approach that is both more general, more accurate

and relatively unknown. It is based on exact expressions for the probability density for

one and for two successive excursion lengths. The numerical routine {\sf RIND} computes

the densities using recent advances in scientific computing and is easily accessible for

a general covariance function, via simple {\sc Matlab} interface.

The result solves the problem of two step excursion dependence for a general stationary

differentiable Gaussian process, both in theoretical sense and in practical numerical sense.

The work offers also some analytical results that explain the effectiveness

of the implemented method.

for a Gaussian process that involves extended and multivariate Rice's formula.

The approach was used in numerical implementations of the high-dimensional integration routine and in earlier work it was shown that the computations are more effective and thus more precise than those based on Rice expansions.

The joint distribution of successive excursion times is clearly related to the distribution

of the number of level crossings, a problem that can be attacked via the Rice series expansion,

based on the moments of the number of crossings.

Another point of attack is the

``Independent Interval Approximation'' (IIA)

intensively studied for the persistency of physical systems.

It treats the lengths of successive crossing intervals as

statistically independent. Under IIA, a renewal type argument

leads to an expression that provides the approximate interval

distribution via its Laplace transform.

However, the independence is not valid in most

typical situations.

Even if it leads to acceptable results for the persistency exponent of the long excursion time distribution or some classes of processes, rigorous assessment of the approximation error is not readily available.

Moreover, we show that the IIA approach cannot deliver properly defined probability distributions and thus the method is limited only to persistence studies.

The ocean science community favours a third approach, in which a class of

parametric marginal distributions, either fitted to excursion data or

derived from a narrow band approximation, is extended by a copula technique

to bivariate and higher order distributions.

This paper presents an alternative approach that is both more general, more accurate

and relatively unknown. It is based on exact expressions for the probability density for

one and for two successive excursion lengths. The numerical routine {\sf RIND} computes

the densities using recent advances in scientific computing and is easily accessible for

a general covariance function, via simple {\sc Matlab} interface.

The result solves the problem of two step excursion dependence for a general stationary

differentiable Gaussian process, both in theoretical sense and in practical numerical sense.

The work offers also some analytical results that explain the effectiveness

of the implemented method.

Originalspråk | engelska |
---|---|

Utgivare | arXiv.org |

Antal sidor | 46 |

Status | Published - 2020 juli 29 |

## Ämnesklassifikation (UKÄ)

- Oceanografi, hydrologi, vattenresurser