In geosciences, the use of classical Euclidean methods is unsuitable for treating and analyzing some types of data, as this may not belong to a vector space. This is the case for correlation matrices, belonging to a subfamily of symmetric positive definite matrices, which in turn form a cone shape Riemannian manifold. We propose two novel applications for dealing with the problem of accounting with the non-linear behavior usually presented on multivariate geological data by exploiting the manifold features of correlations matrices. First, we employ an extension for the linear model of coregionalization (LMC) that alters the linear mixture, which is assumed fixed on the domain, and making it locally varying according to the local strength in the dependency of the coregionalized variables. The main challenge, once this relaxation on the LMC is assumed, is to solve appropriately the interpolation of the different known correlation matrices throughout the domain, in a reliable and coherent fashion. The present work adopts the non-euclidean framework to achieve our objective by locally averaging and interpolating the correlations between the variables, retaining the intrinsic geometry of correlation matrices. A second application deals with the problem of clustering of multivariate data.
翻译:在地球科学中,使用古典的欧几里德方法不适合处理和分析某些类型的数据,因为这可能不属于矢量空间,对属于对称正确定基质子组的对应矩阵就属于这种情况,该基质反过来形成一个锥体形状的里曼尼方形。我们提出两种新的应用方法,通过利用相关基质的多重特征来处理通常在多轨地质数据上出现的非线性行为会计问题。首先,我们采用一个扩展,用于改变线性混合物的线性区域化线性模型(LMC),该模型假定在域上固定的线性混合物,并根据共同区域化变量的当地依赖性强弱,使该基质在本地有差异。一旦假定LMC的放松,主要挑战是以可靠和连贯的方式适当解决整个域内已知的不同相关基质的相互推算问题。目前的工作采用了非欧几里德框架,通过本地平均和对各变量之间的相关性进行相互调,从而实现我们的目标,保持相关基质矩阵的内在几里基质测量,并保持相关基质矩阵的内在几里位性。