Visual Exploration of High Dimensional Scalar Functions

Abstract

An important goal of scientific data analysis is to understand thebehavior of a system or process based on a sample of the system.In manyinstances it is possible to observe both input parameters and system outputs, and characterize the system as a high-dimensional function. Such data setsarise, for instance, in large numerical simulations, as energy landscapes inoptimization problems, or in the analysis of image data relating to biological ormedical parameters.This paper proposes an approach to analyze and visualizing such datasets.The proposed method combines topological and geometric techniques toprovide interactive visualizations of discretely sampled high-dimensionalscalar fields.The method relies on a segmentation of the parameter space usingan approximate Morse-Smale complex on the cloud of point samples. For eachcrystal of the Morse-Smale complex, a regression of the system parameters withrespect to the output yields a curve in the parameter space. The result is asimplified geometric representation of the Morse-Smale complex in the highdimensional input domain.Finally, the geometric representation is embeddedin 2D, using dimension reduction, to provide a visualization platform. Thegeometric properties of the regression curves enable the visualization ofadditional information about each crystal such as local and global shape, width,length, and sampling densities. The method is illustrated on several synthetic examples of two dimensionalfunctions. Two use cases, using data sets from the UCI machine learning repository, demonstrate the utility of the proposed approach on real data.Finally, in collaboration with domain experts the proposed method is applied totwo scientific challenges. The analysis of parameters of climate simulationsand their relationship to predicted global energy flux and the concentrationsof chemical species in a combustion simulation and their integration withtemperature.