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Tensor Field Visualization,
Topological Data Analysis
S. Gerber, O. Reubel, P.-T. Bremer, V. Pascucci, R.T. Whitaker. In Journal of Computational and Graphical Statistics, Vol. 22, No. 1, pp. 193--214. 2013.
This paper introduces a novel partition-based regression approach that incorporates topological information. Partition-based regression typically introduce a quality-of-fit-driven decomposition of the domain. The emphasis in this work is on a topologically meaningful segmentation. Thus, the proposed regression approach is based on a segmentation induced by a discrete approximation of the Morse-Smale complex. This yields a segmentation with partitions corresponding to regions of the function with a single minimum and maximum that are often well approximated by a linear model. This approach yields regression models that are amenable to interpretation and have good predictive capacity. Typically, regression estimates are quantified by their geometrical accuracy. For the proposed regression, an important aspect is the quality of the segmentation itself. Thus, this paper introduces a new criterion that measures the topological accuracy of the estimate. The topological accuracy provides a complementary measure to the classical geometrical error measures and is very sensitive to over-fitting. The Morse-Smale regression is compared to state-of-the-art approaches in terms of geometry and topology and yields comparable or improved fits in many cases. Finally, a detailed study on climate-simulation data demonstrates the application of the Morse-Smale regression. Supplementary materials are available online and contain an implementation of the proposed approach in the R package msr, an analysis and simulations on the stability of the Morse-Smale complex approximation and additional tables for the climate-simulation study.
Comments on the “Meshless Helmholtz-Hodge decomposition”|
H. Bhatia, G. Norgard, V. Pascucci, P.-T. Bremer. In IEEE Transactions on Visualization and Computer Graphics, Vol. 19, No. 3, pp. 527--528. 2013.
The Helmholtz-Hodge decomposition (HHD) is one of the fundamental theorems of fluids describing the decomposition of a flow field into its divergence-free, curl-free and harmonic components. Solving for an HDD is intimately connected to the choice of boundary conditions which determine the uniqueness and orthogonality of the decomposition. This article points out that one of the boundary conditions used in a recent paper \"Meshless Helmholtz-Hodge decomposition\"  is, in general, invalid and provides an analytical example demonstrating the problem. We hope that this clarification on the theory will foster further research in this area and prevent undue problems in applying and extending the original approach.
enRoute: Dynamic Path Extraction from Biological Pathway Maps for In-Depth Experimental Data Analysis|
C Partl, A Lex, M Streit, D Kalkofen, K Kashofer, D Schmalstieg. In Proceedings of the IEEE Symposium on Biological Data Visualization (BioVis '12), IEEE, pp. 107--114. 2012.
Pathway maps are an important source of information when analyzing functional implications of experimental data on biological processes. However, associating large quantities of data with nodes on a pathway map and allowing in depth-analysis at the same time is a challenging task. While a wide variety of approaches for doing so exist, they either do not scale beyond a few experiments or fail to represent the pathway appropriately. To remedy this, we introduce enRoute, a new approach for interactively exploring experimental data along paths that are dynamically extracted from pathways. By showing an extracted path side-by-side with experimental data, enRoute can present large amounts of data for every pathway node. It can visualize hundreds of samples, dozens of experimental conditions, and even multiple datasets capturing different aspects of a node at the same time. Another important property of this approach is its conceptual compatibility with arbitrary forms of pathways. Most notably, enRoute works well with pathways that are manually created, as they are available in large, public pathway databases. We demonstrate enRoute with pathways from the well-established KEGG database and expression as well as copy number datasets from humans and mice with more than 1,000 experiments. We validate enRoute using case studies with domain experts, who used enRoute to explore data for glioblastoma multiforme in humans and a model of steatohepatitis in mice.
StratomeX: Visual Analysis of Large-Scale Heterogeneous Genomics Data for Cancer Subtype Characterization |
A. Lex, M. Streit, H. Schulz, C. Partl, D. Schmalstieg, P.. Park, N. Gehlenborg. In Computer Graphics Forum (EuroVis '12), Vol. 31, No. 3, pp. 1175--1184. 2012.
dentification and characterization of cancer subtypes are important areas of research that are based on the integrated analysis of multiple heterogeneous genomics datasets. Since there are no tools supporting this process, much of this work is done using ad-hoc scripts and static plots, which is inefficient and limits visual exploration of the data. To address this, we have developed StratomeX, an integrative visualization tool that allows investigators to explore the relationships of candidate subtypes across multiple genomic data types such as gene expression, DNA methylation, or copy number data. StratomeX represents datasets as columns and subtypes as bricks in these columns. Ribbons between the columns connect bricks to show subtype relationships across datasets. Drill-down features enable detailed exploration. StratomeX provides insights into the functional and clinical implications of candidate subtypes by employing small multiples, which allow investigators to assess the effect of subtypes on molecular pathways or outcomes such as patient survival. As the configuration of viewing parameters in such a multi-dataset, multi-view scenario is complex, we propose a meta visualization and configuration interface for dataset dependencies and data-view relationships. StratomeX is developed in close collaboration with domain experts. We describe case studies that illustrate how investigators used the tool to explore subtypes in large datasets and demonstrate how they efficiently replicated findings from the literature and gained new insights into the data.
Visualizing Uncertainty in Biological Expression Data|
C. Holzhüter, A. Lex, D. Schmalstieg, H. Schulz, H. Schumann, M. Streit. In Proceedings of the SPIE Conference on Visualization and Data Analysis (VDA '12), Vol. 8294, pp. 82940O-82940O-11. 2012.
Expression analysis of ~omics data using microarrays has become a standard procedure in the life sciences. However, microarrays are subject to technical limitations and errors, which render the data gathered likely to be uncertain. While a number of approaches exist to target this uncertainty statistically, it is hardly ever even shown when the data is visualized using for example clustered heatmaps. Yet, this is highly useful when trying not to omit data that is "good enough" for an analysis, which otherwise would be discarded as too unreliable by established conservative thresholds. Our approach addresses this shortcoming by first identifying the margin above the error threshold of uncertain, yet possibly still useful data. It then displays this uncertain data in the context of the valid data by enhancing a clustered heatmap. We employ different visual representations for the different kinds of uncertainty involved. Finally, it lets the user interactively adjust the thresholds, giving visual feedback in the heatmap representation, so that an informed choice on which thresholds to use can be made instead of applying the usual rule-of-thumb cut-offs. We exemplify the usefulness of our concept by giving details for a concrete use case from our partners at the Medical University of Graz, thereby demonstrating our implementation of the general approach.
Model-Driven Design for the Visual Analysis of Heterogeneous Data|
M. Streit, H. Schulz, A. Lex, D. Schmalstieg, H. Schumann. In IEEE Transactions on Visualization and Computer Graphics, Vol. 18, No. 6, IEEE, pp. 998--1010. 2012.
As heterogeneous data from different sources are being increasingly linked, it becomes difficult for users to understand how the data are connected, to identify what means are suitable to analyze a given data set, or to find out how to proceed for a given analysis task. We target this challenge with a new model-driven design process that effectively codesigns aspects of data, view, analytics, and tasks. We achieve this by using the workflow of the analysis task as a trajectory through data, interactive views, and analytical processes. The benefits for the analysis session go well beyond the pure selection of appropriate data sets and range from providing orientation or even guidance along a preferred analysis path to a potential overall speedup, allowing data to be fetched ahead of time. We illustrate the design process for a biomedical use case that aims at determining a treatment plan for cancer patients from the visual analysis of a large, heterogeneous clinical data pool. As an example for how to apply the comprehensive design approach, we present Stack'n'flip, a sample implementation which tightly integrates visualizations of the actual data with a map of available data sets, views, and tasks, thus capturing and communicating the analytical workflow through the required data sets.
A Practical Workflow for Making Anatomical Atlases in Biological Research|
Y. Wan, A.K. Lewis, M. Colasanto, M. van Langeveld, G. Kardon, C.D. Hansen. In IEEE Computer Graphics and Applications, Vol. 32, No. 5, pp. 70--80. 2012.
An anatomical atlas provides a detailed map for medical and biological studies of anatomy. These atlases are important for understanding normal anatomy and the development and function of structures, and for determining the etiology of congenital abnormalities. Unfortunately, for biologists, generating such atlases is difficult, especially ones with the informative content and aesthetic quality that characterize human anatomy atlases. Building such atlases requires knowledge of the species being studied and experience with an art form that can faithfully record and present this knowledge, both of which require extensive training in considerably different fields. (For some background on anatomical atlases, see the related sidebar.)
With the latest innovations in data acquisition and computing techniques, atlas building has changed dramatically. We can now create atlases from 3D images of biological specimens, allowing for high-quality, faithful representations. Labeling of structures using fluorescently tagged antibodies, confocal 3D scanning of these labeled structures, volume rendering, segmentation, and surface reconstruction techniques all promise solutions to the problem of building atlases.
However, biology researchers still ask, \"Is there a set of tools we can use or a practical workflow we can follow so that we can easily build models from our biological data?\" To help answer this question, computer scientists have developed many algorithms, tools, and program codes. Unfortunately, most of these researchers have tackled only one aspect of the problem or provided solutions to special cases. So, the general question of how to build anatomical atlases remains unanswered.
For a satisfactory answer, biologists need a practical workflow they can easily adapt for different applications. In addition, reliable tools that can fit into the workflow must be readily available. Finally, examples using the workflow and tools to build anatomical atlases would demonstrate these resources' utility for biological research.
To build a mouse limb atlas for studying the development of the limb musculoskeletal system, University of Utah biologists, artists, and computer scientists have designed a generalized workflow for generating anatomical atlases. We adapted it from a CG artist's workflow of building 3D models for animated films and video games. The tools we used to build the atlas were mostly commercial, industry-standard software packages. Having been developed, tested, and employed for industrial use for decades, CG artists' workflow and tools, with certain adaptations, are the most suitable for making high-quality anatomical atlases, especially under strict budgetary and time limits. Biological researchers have been largely unaware of these resources. By describing our experiences in this project, we hope to show biologists how to use these resources to make anatomically accurate, high-quality, and useful anatomical atlases.
The ViSUS Visualization Framework|
V. Pascucci, G. Scorzelli, B. Summa, P.-T. Bremer, A. Gyulassy, C. Christensen, S. Philip, S. Kumar. In High Performance Visualization: Enabling Extreme-Scale Scientific Insight, Chapman and Hall/CRC Computational Science, Ch. 19, Edited by E. Wes Bethel and Hank Childs (LBNL) and Charles Hansen (UofU), Chapman and Hall/CRC, 2012.
The ViSUS software framework was designed with the primary philosophy that the visualization of massive data need not be tied to specialized hardware or infrastructure. In other words, a visualization environment for large data can be designed to be lightweight, highly scalable and run on a variety of plat- forms or hardware. Moreover, if designed generally such an infrastructure can have a wide variety of applications, all from the same code base. Figure 19.1 details example applications and the major components of the ViSUS infrastructure. The components can be grouped into three major categories. First, a lightweight and fast out-of-core data management framework using multi- resolution space filling curves. This allows the organization of information in an order that exploits the cache hierarchies of any modern data storage architectures. Second, a data flow framework that allows data to be processed during movement. Processing massive datasets in their entirety would be a long and expensive operation which hinders interactive exploration. By designing new algorithms to fit within this framework, data can be processed as it moves. Third, a portable visualization layer which was designed to scale from mobile devices to powerwall displays with same code base. In this chapter we will describe the ViSUS infrastructure, as well as give practical examples of its use in real world applications.
Design Study Methodology: Reflections from the Trenches and the Stacks|
M. Sedlmair, M.D. Meyer, T. Munzner. In IEEE Transactions on Visualization and Computer Graphics, Vol. 18, No. 12, Note: Honorable Mention for Best Paper Award., pp. 2431--2440. 2012.
Design studies are an increasingly popular form of problem-driven visualization research, yet there is little guidance available about how to do them effectively. In this paper we reflect on our combined experience of conducting twenty-one design studies, as well as reading and reviewing many more, and on an extensive literature review of other field work methods and methodologies. Based on this foundation we provide definitions, propose a methodological framework, and provide practical guidance for conducting design studies. We define a design study as a project in which visualization researchers analyze a specific real-world problem faced by domain experts, design a visualization system that supports solving this problem, validate the design, and reflect about lessons learned in order to refine visualization design guidelines. We characterize two axes—a task clarity axis from fuzzy to crisp and an information location axis from the domain expert’s head to the computer—and use these axes to reason about design study contributions, their suitability, and uniqueness from other approaches. The proposed methodological framework consists of 9 stages: learn, winnow, cast, discover, design, implement, deploy, reflect, and write. For each stage we provide practical guidance and outline potential pitfalls. We also conducted an extensive literature survey of related methodological approaches that involve a significant amount of qualitative field work, and compare design study methodology to that of ethnography, grounded theory, and action research.
The Four-Level Nested Model Revisited: Blocks and Guidelines|
M.D. Meyer, M. Sedlmair, T. Munzner. In Workshop on BEyond time and errors: novel evaLuation methods for Information Visualization (BELIV), IEEE VisWeek 2012, 2012.
We propose an extension to the four-level nested model for design and validation of visualization systems that defines the term \"guidelines\" in terms of blocks at each level. Blocks are the outcomes of the design process at a specific level, and guidelines discuss relationships between these blocks. Within-level guidelines provide comparisons for blocks within the same level, while between-level guidelines provide mappings between adjacent levels of design. These guidelines help a designer choose which abstractions, techniques, and algorithms are reasonable to combine when building a visualization system. This definition of guideline allows analysis of how the validation efforts in different kinds of papers typically lead to different kinds of guidelines. Analysis through the lens of blocks and guidelines also led us to identify four major needs: a definition of the meaning of block at the problem level; mid-level task taxonomies to fill in the blocks at the abstraction level; refinement of the model itself at the abstraction level; and a more complete set of guidelines that map up from the algorithm level to the technique level. These gaps in visualization knowledge present rich opportunities for future work.
Aggregate Gaze Visualization with Real-Time Heatmaps|
A. Duchowski, M. Price, M.D. Meyer, P. Orero. In Proceedings of the ACM Symposium on Eye Tracking Research and Applications (ETRA), pp. 13--20. 2012.
A GPU implementation is given for real-time visualization of aggregate eye movements (gaze) via heatmaps. Parallelization of the algorithm leads to substantial speedup over its CPU-based implementation and, for the first time, allows real-time rendering of heatmaps atop video. GLSL shader colorization allows the choice of color ramps. Several luminance-based color maps are advocated as alternatives to the popular rainbow color map, considered inappropriate (harmful) for depiction of (relative) gaze distributions.
Gaussian Mixture Model Based Volume Visualization|
S. Liu, J.A. Levine, P.-T. Bremer, V. Pascucci. In Proceedings of the IEEE Large-Scale Data Analysis and Visualization Symposium 2012, Note: Received Best Paper Award, pp. 73--77. 2012.
Representing uncertainty when creating visualizations is becoming more indispensable to understand and analyze scientific data. Uncertainty may come from different sources, such as, ensembles of experiments or unavoidable information loss when performing data reduction. One natural model to represent uncertainty is to assume that each position in space instead of a single value may take on a distribution of values. In this paper we present a new volume rendering method using per voxel Gaussian mixture models (GMMs) as the input data representation. GMMs are an elegant and compact way to drastically reduce the amount of data stored while still enabling realtime data access and rendering on the GPU. Our renderer offers efficient sampling of the data distribution, generating renderings of the data that flicker at each frame to indicate high variance. We can accumulate samples as well to generate still frames of the data, which preserve additional details in the data as compared to either traditional scalar indicators (such as a mean or a single nearest neighbor down sample) or to fitting the data with only a single Gaussian per voxel. We demonstrate the effectiveness of our method using ensembles of climate simulations and MRI scans as well as the down sampling of large scalar fields as examples.
Keywords: Uncertainty Visualization, Volume Rendering, Gaussian Mixture Model, Ensemble Visualization
A Multiscale Approach to Network Event Identification using Geolocated Twitter Data|
C. Yang, I. Jensen, P. Rosen. In First IMC Workshop on Internet Visualization (WIV 2012), pp. (accepted). 2012.
The large volume of data associated with social networks hinders the unaided user from interpreting network content in real time. This problem is compounded by the fact that there are limited tools available for enabling robust visual social network exploration. We present a network activity visualization using a novel aggregation glyph called the clyph. The clyph intuitively combines spatial, temporal, and quantity data about multiple network events. We also present several case studies where major network events were easily identified using clyphs, establishing them as a powerful aid for network users and owners.
Data management and analysis with WRF and SFIRE|
J. Beezley, M. Martin, P. Rosen, J. Mandel, A. Kochanski. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, Note: UCD CCM Report 312, 2012.
We introduce several useful utilities in development for the creation and analysis of real wildland fire simulations using WRF and SFIRE. These utilities exist as standalone programs and scripts as well as extensions to other well known software. Python web scrapers automate the process of downloading and preprocessing atmospheric and surface data from common sources. Other scripts simplify the domain setup by creating parameter files automatically. Integration with Google Earth allows users to explore the simulation in a 3D environment along with real surface imagery. Postprocessing scripts provide the user with a number of output data formats compatible with many commonly used visualization suites allowing for the creation of high quality 3D renderings. As a whole, these improvements build toward a unified web application that brings a sophisticated wildland fire modeling environment to scientists and users alike.
Extending the SCIRun Problem Solving Environment to Large-Scale Applications|
J. Knezevic, R.-P. Mundani, E. Rank, A. Khan, C.R. Johnson. In Proceedings of Applied Computing 2012, IADIS, pp. 171--178. October, 2012.
To make the most of current advanced computing technologies, experts in particular areas of science and engineering should be supported by sophisticated tools for carrying out computational experiments. The complexity of individual components of such tools should be hidden from them so they may concentrate on solving the specific problem within their field of expertise. One class of such tools are Problem Solving Environments (PSEs). The contribution of this paper refers to the idea of integration of an interactive computing framework applicable to different engineering applications into the SCIRun PSE in order to enable interactive real-time response of the computational model to user interaction even for large-scale problems. While the SCIRun PSE allows for real-time computational steering, we propose extending this functionality to a wider range of applications and larger scale problems. With only minor code modifications the proposed system allows each module scheduled for execution in a dataflow-based simulation to be automatically interrupted and re-scheduled. This rescheduling allows one to keep the relation between the user interaction and its immediate effect transparent independent of the problem size, thus, allowing for the intuitive and interactive exploration of simulation results.
Visualizing Network Traffic to Understand the Performance of Massively Parallel Simulations|
A.G. Landge, J.A. Levine, A. Bhatele, K.E. Isaacs, T. Gamblin, S. Langer, M. Schulz, P.-T. Bremer, V. Pascucci. In IEEE Transactions on Visualization and Computer Graphics, Vol. 18, No. 12, IEEE, pp. 2467--2476. Dec, 2012.
The performance of massively parallel applications is often heavily impacted by the cost of communication among compute nodes. However, determining how to best use the network is a formidable task, made challenging by the ever increasing size and complexity of modern supercomputers. This paper applies visualization techniques to aid parallel application developers in understanding the network activity by enabling a detailed exploration of the flow of packets through the hardware interconnect. In order to visualize this large and complex data, we employ two linked views of the hardware network. The first is a 2D view, that represents the network structure as one of several simplified planar projections. This view is designed to allow a user to easily identify trends and patterns in the network traffic. The second is a 3D view that augments the 2D view by preserving the physical network topology and providing a context that is familiar to the application developers. Using the massively parallel multi-physics code pF3D as a case study, we demonstrate that our tool provides valuable insight that we use to explain and optimize pF3D’s performance on an IBM Blue Gene/P system.
Uncertainty in the Development and Use of Equation of State Models|
V.G. Weirs, N. Fabian, K. Potter, L. McNamara, T. Otahal. In International Journal for Uncertainty Quantification, pp. 255--270. 2012.
In this paper we present the results from a series of focus groups on the visualization of uncertainty in Equation-Of-State (EOS) models. The initial goal was to identify the most effective ways to present EOS uncertainty to analysts, code developers, and material modelers. Four prototype visualizations were developed to presented EOS surfaces in a three-dimensional, thermodynamic space. Focus group participants, primarily from Sandia National Laboratories, evaluated particular features of the various techniques for different use cases and discussed their individual workflow processes, experiences with other visualization tools, and the impact of uncertainty to their work. Related to our prototypes, we found the 3D presentations to be helpful for seeing a large amount of information at once and for a big-picture view; however, participants also desired relatively simple, two-dimensional graphics for better quantitative understanding, and because these plots are part of the existing visual language for material models. In addition to feedback on the prototypes, several themes and issues emerged that are as compelling as the original goal and will eventually serve as a starting point for further development of visualization and analysis tools. In particular, a distributed workflow centered around material models was identified. Material model stakeholders contribute and extract information at different points in this workflow depending on their role, but encounter various institutional and technical barriers which restrict the flow of information. An effective software tool for this community must be cognizant of this workflow and alleviate the bottlenecks and barriers within it. Uncertainty in EOS models is defined and interpreted differently at the various stages of the workflow. In this context, uncertainty propagation is difficult to reduce to the mathematical problem of estimating the uncertainty of an output from uncertain inputs.
The EpiCanvas infectious disease weather map: an interactive visual exploration of temporal and spatial correlations|
P.H. Gesteland, Y. Livnat, N. Galli, M.H. Samore, A.V. Gundlapalli. In J. Amer. Med. Inform. Assoc., Vol. 19, Note: Awarded 1st place for Outstanding Research Article at ISDS 2012 and the Homer R. Warner Award at the AMIA Annual Symposium 2012, pp. 954--959. 2012.
Advances in surveillance science have supported public health agencies in tracking and responding to disease outbreaks. Increasingly, epidemiologists have been tasked with interpreting multiple streams of heterogeneous data arising from varied surveillance systems. As a result public health personnel have experienced an overload of plots and charts as information visualization techniques have not kept pace with the rapid expansion in data availability. This study sought to advance the science of public health surveillance data visualization by conceptualizing a visual paradigm that provides an 'epidemiological canvas' for detection, monitoring, exploration and discovery of regional infectious disease activity and developing a software prototype of an 'infectious disease weather map'. Design objectives were elucidated and the conceptual model was developed using cognitive task analysis with public health epidemiologists. The software prototype was pilot tested using retrospective data from a large, regional pediatric hospital, and gastrointestinal and respiratory disease outbreaks were re-created as a proof of concept.
Biomedical Visual Computing: Case Studies and Challenges|
C.R. Johnson. In IEEE Computing in Science and Engineering, Vol. 14, No. 1, pp. 12--21. 2012.
PubMed ID: 22545005
PubMed Central ID: PMC3336198
Computer simulation and visualization are having a substantial impact on biomedicine and other areas of science and engineering. Advanced simulation and data acquisition techniques allow biomedical researchers to investigate increasingly sophisticated biological function and structure. A continuing trend in all computational science and engineering applications is the increasing size of resulting datasets. This trend is also evident in data acquisition, especially in image acquisition in biology and medical image databases.
For example, in a collaboration between neuroscientist Robert Marc and our research team at the University of Utah's Scientific Computing and Imaging (SCI) Institute (www.sci.utah.edu), we're creating datasets of brain electron microscopy (EM) mosaics that are 16 terabytes in size. However, while there's no foreseeable end to the increase in our ability to produce simulation data or record observational data, our ability to use this data in meaningful ways is inhibited by current data analysis capabilities, which already lag far behind. Indeed, as the NIH-NSF Visualization Research Challenges report notes, to effectively understand and make use of the vast amounts of data researchers are producing is one of the greatest scientific challenges of the 21st century.
Visual data analysis involves creating images that convey salient information about underlying data and processes, enabling the detection and validation of expected results while leading to unexpected discoveries in science. This allows for the validation of new theoretical models, provides comparison between models and datasets, enables quantitative and qualitative querying, improves interpretation of data, and facilitates decision making. Scientists can use visual data analysis systems to explore \"what if\" scenarios, define hypotheses, and examine data under multiple perspectives and assumptions. In addition, they can identify connections between numerous attributes and quantitatively assess the reliability of hypotheses. In essence, visual data analysis is an integral part of scientific problem solving and discovery.
As applied to biomedical systems, visualization plays a crucial role in our ability to comprehend large and complex data-data that, in two, three, or more dimensions, convey insight into many diverse biomedical applications, including understanding neural connectivity within the brain, interpreting bioelectric currents within the heart, characterizing white-matter tracts by diffusion tensor imaging, and understanding morphology differences among different genetic mice phenotypes.
Interactive visualization of probability and cumulative density functions|
K. Potter, R.M. Kirby, D. Xiu, C.R. Johnson. In International Journal of Uncertainty Quantification, Vol. 2, No. 4, pp. 397--412. 2012.
PubMed ID: 23543120
PubMed Central ID: PMC3609671
The probability density function (PDF), and its corresponding cumulative density function (CDF), provide direct statistical insight into the characterization of a random process or field. Typically displayed as a histogram, one can infer probabilities of the occurrence of particular events. When examining a field over some two-dimensional domain in which at each point a PDF of the function values is available, it is challenging to assess the global (stochastic) features present within the field. In this paper, we present a visualization system that allows the user to examine two-dimensional data sets in which PDF (or CDF) information is available at any position within the domain. The tool provides a contour display showing the normed difference between the PDFs and an ansatz PDF selected by the user, and furthermore allows the user to interactively examine the PDF at any particular position. Canonical examples of the tool are provided to help guide the reader into the mapping of stochastic information to visual cues along with a description of the use of the tool for examining data generated from a uncertainty quantification exercise accomplished within the field of electrophysiology.
Keywords: visualization, probability density function, cumulative density function, generalized polynomial chaos, stochastic Galerkin methods, stochastic collocation methods