8,657 research outputs found
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Teaching and Learning Data Visualization: Ideas and Assignments
This article discusses how to make statistical graphics a more prominent
element of the undergraduate statistics curricula. The focus is on several
different types of assignments that exemplify how to incorporate graphics into
a course in a pedagogically meaningful way. These assignments include having
students deconstruct and reconstruct plots, copy masterful graphs, create
one-minute visual revelations, convert tables into `pictures', and develop
interactive visualizations with, e.g., the virtual earth as a plotting canvas.
In addition to describing the goals and details of each assignment, we also
discuss the broader topic of graphics and key concepts that we think warrant
inclusion in the statistics curricula. We advocate that more attention needs to
be paid to this fundamental field of statistics at all levels, from
introductory undergraduate through graduate level courses. With the rapid rise
of tools to visualize data, e.g., Google trends, GapMinder, ManyEyes, and
Tableau, and the increased use of graphics in the media, understanding the
principles of good statistical graphics, and having the ability to create
informative visualizations is an ever more important aspect of statistics
education
Analytical learning and term-rewriting systems
Analytical learning is a set of machine learning techniques for revising the representation of a theory based on a small set of examples of that theory. When the representation of the theory is correct and complete but perhaps inefficient, an important objective of such analysis is to improve the computational efficiency of the representation. Several algorithms with this purpose have been suggested, most of which are closely tied to a first order logical language and are variants of goal regression, such as the familiar explanation based generalization (EBG) procedure. But because predicate calculus is a poor representation for some domains, these learning algorithms are extended to apply to other computational models. It is shown that the goal regression technique applies to a large family of programming languages, all based on a kind of term rewriting system. Included in this family are three language families of importance to artificial intelligence: logic programming, such as Prolog; lambda calculus, such as LISP; and combinatorial based languages, such as FP. A new analytical learning algorithm, AL-2, is exhibited that learns from success but is otherwise quite different from EBG. These results suggest that term rewriting systems are a good framework for analytical learning research in general, and that further research should be directed toward developing new techniques
Principal manifolds and graphs in practice: from molecular biology to dynamical systems
We present several applications of non-linear data modeling, using principal
manifolds and principal graphs constructed using the metaphor of elasticity
(elastic principal graph approach). These approaches are generalizations of the
Kohonen's self-organizing maps, a class of artificial neural networks. On
several examples we show advantages of using non-linear objects for data
approximation in comparison to the linear ones. We propose four numerical
criteria for comparing linear and non-linear mappings of datasets into the
spaces of lower dimension. The examples are taken from comparative political
science, from analysis of high-throughput data in molecular biology, from
analysis of dynamical systems.Comment: 12 pages, 9 figure
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