极大稳定过程的精确模拟
Exact simulation of max-stable processes
摘要：
最大稳定过程作为空间极端事件的模型具有重要作用。它们作为无限多个随机函数上的逐点最大值的复杂结构使得模拟非常重要。在实践中使用的基于有限近似的算法通常是不精确的，并且计算效率低。我们将提出两个算法，在有限数量的位置精确模拟的最大稳定过程。第一种算法是对Brown-Resnick过程的\citet{DM-2014}方法的推广，它是基于谱测度的模拟。第二种算法依赖于只模拟极值函数的思想，即在构造最大稳定过程中有效地对逐点最大值做出贡献的那些函数。我们研究了这两种算法的复杂性，并证明了第二个过程总是更有效。此外，我们提供了封闭的表达式，他们的实施，涵盖最流行的最大稳定过程和极值copula模型。对于密集网格上的模拟，提出了第二种算法的自适应设计。
英文摘要：
Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes simulation highly nontrivial. Algorithms based on finite approximations that are used in practice are often not exact and computationally inefficient. We will present two algorithms for exact simulation of a max-stable process at a finite number of locations. The first algorithm generalizes the approach by \citet{DM-2014} for Brown--Resnick processes and it is based on simulation from the spectral measure. The second algorithm relies on the idea to simulate only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We study the complexity of both algorithms and prove that the second procedure is always more efficient. Moreover, we provide closed expressions for their implementation that cover the most popular models for max-stable processes and extreme value copulas. For simulation on dense grids, an adaptive design of the second algorithm is proposed.
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