In this work, we introduce a new acquisition function for sequential sampling to efficiently quantify rare-event statistics of an input-to-response (ItR) system with given input probability and expensive function evaluations. Our acquisition is a generalization of the likelihood-weighted (LW) acquisition that was initially designed for the same purpose and then extended to many other applications. The improvement in our acquisition comes from the generalized form with two additional parameters, by varying which one can target and address two weaknesses of the original LW acquisition: (1) that the input space associated with rare-event responses is not sufficiently stressed in sampling; (2) that the surrogate model (generated from samples) may have significant deviation from the true ItR function, especially for cases with complex ItR function and limited number of samples. In addition, we develop a critical procedure in Monte-Carlo discrete optimization of the acquisition function, which achieves orders of magnitude acceleration compared to existing approaches for such type of problems. The superior performance of our new acquisition to the original LW acquisition is demonstrated in a number of test cases, including some cases that were designed to show the effectiveness of the original LW acquisition. We finally apply our method to an engineering example to quantify the rare-event roll-motion statistics of a ship in a random sea.

Clustering is a technique used in machine learning and data analysis to group similar data points together. The goal of clustering is to identify patterns and relationships in the data without any prior knowledge of the underlying structure. Clustering is commonly used in unsupervised learning, where the algorithm is not given any labeled data and must find its own structure in the data. There are numerous applications of clustering in various fields such as finance, marketing, biology, social networks, image and video processing, and many more. There are several different algorithms that can be used for clustering, including k-means, hierarchical clustering, and DBSCAN.