Probability Distributome: A Web Computational Infrastructure for Exploring the Properties, Interrelations, and Applications of Probability Distributions

Abstract

Probability distributions are useful for modeling, simulation, analysis, and inference on varieties of natural processes and physical phenomena. There are uncountably many probability distributions. However, a few dozen families of distributions are commonly defined and are frequently used in practice for problem solving, experimental applications, and theoretical studies. In this paper, we present a new computational and graphical infrastructure, the Distributome, which facilitates the discovery, exploration and application of diverse spectra of probability distributions. The extensible Distributome infrastructure provides interfaces for (human and machine) traversal, search, and navigation of all common probability distributions. It also enables distribution modeling, applications, investigation of inter-distribution relations, as well as their analytical representations and computational utilization. The entire Distributome framework is designed and implemented as an open-source, community-built, and Internet-accessible infrastructure. It is portable, extensible and compatible with HTML5 and Web2.0 standards (http://Distributome.org). We demonstrate two types of applications of the probability Distributome resources: computational research and science education. The Distributome tools may be employed to address five complementary computational modeling applications (simulation, data-analysis and inference, model-fitting, examination of the analytical, mathematical and computational properties of specific probability distributions, and exploration of the inter-distributional relations). Many high school and college science, technology, engineering and mathematics (STEM) courses may be enriched by the use of modern pedagogical approaches and technology-enhanced methods. The Distributome resources provide enhancements for blended STEM education by improving student motivation, augmenting the classical curriculum with interactive webapps, and overhauling the learning assessment protocols.

Keywords: Distributome; Probability distributions; applications; graphical user interface; inference; models; transformations.

Figure 1

Components of the decision making process – observable natural processes, modeling approaches, and analytic tools, and the corresponding model assumptions, algorithmic approximations, and scientific inference.

Figure 2

The Distributome Navigator provides an interactive web-based interface for traversal, search and exploration of the properties of distributions, as nodes, and their interrelations, as edges in the graph. The Navigator graphical interface is mobile device compatible, software platform agnostic and runs directly in the browser. User can keyword search for distributions, properties or relations, or navigate the graph with the mouse. The top-right corner accordion panels may be expanded to show or edit the appropriate meta data (distribution properties, invoke distribution actions, inter-distributional relations, and scientific publications).

Figure 3

Core Distributome applications (clockwise starting at the top-left) – sampling and simulation, inter-distribution relations navigator, distribution properties explorer, model fitting tools, and distribution calculators.

Figure 4

The Distributome Game is a game-interface where players aim to quickly identify the correspondences between pairs of processes (represented as problems) and probability distributions (as models). The Cartesian plane represents the game-board where rows and columns show problems/processes and distribution models, respectively. Correct and incorrect matches are green and red colored. Various optional hints and help mechanisms are provided for the players. Green and red cells indicate correct and incorrect pairing of the problems and model distributions, respectively. The last column indicated the number of guesses for each problem.

Figure 5

Distributome Colorblindness activity focuses on the estimation of the probability of female colorblindness using data for males.

Figure 6

The Distributome Extreme-Value distribution calculator is used to estimate the likelihood of the event of observing fewer than 27 homicides in Columbus, Ohio in a given year.