We consider the problem of describing excursion regions as $Gamma = { x inD : f(x) geq au}$, where $auinmathbb{R}$, $f$ is a real valued functiondefined on $D$, a compact subset of $mathbb{R}^d$, $dge2$. We introduceprofile extrema functions defined as a solution of a constrained optimizationproblem over a subset of D indexed by a matrix of size $dimes p$. Therelationships between $Gamma$ and profile extrema functions convey a simple,although intrinsically partial, visualization of the set. The function $f$considered here is expensive to evaluate and only a very limited number offunction evaluations is available. In a Bayesian approach we approximate $f$with a posterior functional of a Gaussian process (GP) $(Z_x)_{x in D}$. Wepresent a plug-in approach where we consider the profile extrema functions ofthe posterior mean given n evaluations of $f$. We quantify the uncertainty onthe estimates by studying the distribution of the profile extrema of Z withposterior quasi-realizations. We provide a probabilistic bound for thequantiles of such objects based on the sample quantiles of thequasi-realizations. The technique is applied to a real 5--dimensional coastalflooding test case where the response is the total flooded area on a sitelocated on the Atlantic French coast as a function of the offshore conditionsand $Gamma$ is the set of conditions that lead to a high hazard level.
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机译：我们考虑作为$ gamma = {x ind：f（x） geq tau } $，其中$ tau in mathbb {r} $，$ f $是一个真实的问题在$ d $的volding函数函数，一个紧凑的子集 mathbb {r} ^ d $，$ d ge $。我们介绍了Extrema函数，定义为由Dize $ d times p $矩阵索引的D索引的D索引的受限优化问题的解决方案。 $ Gamma $和Profile Extrema函数之间的主题传达简单，虽然本质上部分地，可视化集合。这里考虑的函数$ F $昂贵，以评估昂贵，并且只有非常有限的数字漏分评估。在贝叶斯方法中，我们将$ F $大致用高斯过程（GP）$（Z_X）_ {x IN d} $的后函数。 Wepresent一种插件方法，我们考虑后部意味着在$ f $的N个评估中的extrema函数。我们通过研究Z的Z雌激素的曲线结构的分布来量化估计的不确定性。基于Quasi-实现的样本量级，我们为这些对象的样本量提供了概率。该技术适用于真正的5维沿海地上飞机测试案例，响应是在大西洋法国海岸上的Siteelocated作为近海条件和$ Gamma $的函数的总洪水区域，并且是导致a的条件高危险水平。
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