The traditional ellipsoid convex set is a kind of basic non-probabilistic model to measure uncertainties. However, it is difficult or inaccurate to quantify the uncertainties of variables with multimodal distributed samples. In this paper, a more generalized non-probabilistic ellipsoid model named multimodal ellipsoid model is proposed to effectively deal with the multimodal distributed samples. The samples with one or more similar properties are clustered together, and the principal directions of the samples and characteristic matrix are appropriately found through the Gaussian mixture model. Then, the multimodal ellipsoid model can be constructed by using the elliptical contour features of the Gaussian model to measure the uncertainties of variables. The proposed multimodal ellipsoid model can not only establish traditional ellipsoid model, but also establish multi-ellipsoid model for uncertain variables with multimodal samples. Furthermore, combining with the multimodal ellipsoid model and performance measure approach, the uncertain propagation results of system are obtained accurately. Three numerical examples and one engineering application are provided to demonstrate the effectiveness and accuracy of the proposed multimodal ellipsoid model.

传统的椭球凸集是一种衡量不确定性的基本非概率模型。然而，用多峰分布样本量化变量的不确定性是困难的或不准确的。本文提出了一种更广义的非概率椭球模型——多模态椭球模型，以有效处理多模态分布样本。将具有一个或多个相似性质的样本聚类在一起，通过高斯混合模型适当地找到样本的主方向和特征矩阵。然后，利用高斯模型的椭圆轮廓特征来衡量变量的不确定性，可以构建多模态椭球模型。所提出的多模态椭球模型不仅可以建立传统的椭球模型，还可以建立具有多模态样本的不确定变量的多椭球模型。此外，结合多模态椭球模型和性能测量方法，准确地获得了系统的不确定传播结果。提供了三个数值例子和一个工程应用来证明所提出的多模态椭球模型的有效性和准确性。