Uncertainty Visualization Using Copula-Based Analysis in Mixed Distribution Models

Distributions are often used to model uncertainty in many scientific datasets. To preserve the correlation among the spatially sampled grid locations in the dataset, various standard multivariate distribution models have been proposed in visualization literature. These models treat each grid location as a univariate random variable which models the uncertainty at that location. Standard multivariate distributions (both parametric and nonparametric) assume that all the univariate marginals are of the same type/family of distribution. But in reality, different grid locations show different statistical behavior which may not be modeled best by the same type of distribution. In this paper, we propose a new multivariate uncertainty modeling strategy to address the needs of uncertainty modeling in scientific datasets. Our proposed method is based on a statistically sound multivariate technique called Copula, which makes it possible to separate the process of estimating the univariate marginals and the process of modeling dependency, unlike the standard multivariate distributions. The modeling flexibility offered by our proposed method makes it possible to design distribution fields which can have different types of distribution (Gaussian, Histogram, KDE etc.) at the grid locations, while maintaining the correlation structure at the same time. Depending on the results of various standard statistical tests, we can choose an optimal distribution representation at each location, resulting in a more cost efficient modeling without significantly sacrificing on the analysis quality. To demonstrate the efficacy of our proposed modeling strategy, we extract and visualize uncertain features like isocontours and vortices in various real world datasets. We also study various modeling criterion to help users in the task of univariate model selection.

1 Introduction

Most of the numerical simulations which are used to model complex real world physical phenomenon generate uncertain data. The lack of a proper ground truth and/or simulation parameter knowledge are some of the common causes of uncertainty. In order to avoid making erroneous decisions using such data, it is important to incorporate the uncertainty into the analysis process itself. For example, tasks like feature extraction and visualization should reflect the effect of uncertainty in the data. With recent advances in computing power and resources, scientists are able to model the uncertainty by running ensemble of simulations with varying experiment parameters, thus, generating multiple realizations of the same physical phenomenon. These multiple realizations/values at each of the spatially sampled points (grid locations) represent the uncertainty in that location and are often modeled as stochastic random variables. Various approaches have been proposed [2], [32], [37] to extract probabilistic/uncertain features from such a field of random variables using standard statistical tools.