Derived Fields - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 137

Fig. 14.5 Multifield-graph computed for the hurricane Isabel data set. The size and color of the
disks represent the degree of correlation/similarity between the fields. Image courtesy of Sauber
et al. [ 22 ]. © IEEE Reprinted, with permission, from IEEE Transactions on Visualization and
Computer Graphics 12(5)

14.3.2 Local Statistical Complexity

Multifield data have also been studied using statistical and information theoretic
methods. Jänicke et al. [ 10 ] adapt the notion of local statistical complexity to the
context of time-varying fields and apply it to study data available from PDE simula-
tions [ 11 ]. The local statistical complexity is a measure of the amount of information
required from the past to predict the field in the current time step a specific point. It
is computed using the notion of entropy and mutual information as a time-varying
scalar field. Consider a time-varying field. All points that could possibly influence the
value of the field at a point p are arranged into a light cone. The size of the region of
influence increases by one for each time step away from p . A light cone ( l + ) into the
future time steps is also considered, similar to the light cone in the past ( l − ), for the
computation. A conditional distribution P

l + | l − )

can be defined on the light cones.
The local statistical complexity is computed as the mutual information between the
distribution represented by a particular light cone and the equivalence class of past
light cones that have similar conditional distribution. Jänicke et al. [ 10 , 11 ] describe
efficient algorithms to compute the local statistical complexity. Features are iden-
tified as complex if the probability that they occur again is low. They demonstrate
applications of this derived field to a wide range of data from diffusion, flow, and
weather simulations.

(

14.3.3 Multifield Comparison Measure

Nagaraj et al. introduced a gradient-based comparison measure for multiple scalar
fields [ 14 ]. The measure is defined as the norm of a matrix comprising the gradient
vectors of the different functions. Let A be a m

× n matrix of real numbers. The norm

of the matrix A , denoted as A , is defined as

=

n

,

A max Ax

x = 1, x ∈R