scholarly journals Deterministic Sampling from Uniform Distributions with Sierpinski Space-Filling Curves

2020 ◽  
Author(s):  
Hime Aguiar e O. Jr
Keyword(s):  
Probability Distributions ◽  
Real Interval ◽  
Space Filling ◽  
Space Filling Curve ◽  
Space Filling Curves ◽  
Uniform Distributions ◽  
Deterministic Sampling ◽  
Discrete Manner ◽  
Filling Curve ◽  
Computational Properties

In this paper the problem of sampling from uniform probability distributions is approached by means of space-filling curves (SFCs), a topological concept that has found a number of important applications in recent years. Departing from the theoretical fact that they are surjective but not necessarilly injective, the investigation focused upon the structure of the distributions obtained when their domain is swept in a uniform and discrete manner, and the corresponding values used to build histograms, that are approximations of their true PDFs. This work concentrates on the real interval [0,1], and the Sierpinski space-filling curve was chosen because of its favorable computational properties. In order to validate the results, the Kullback-Leibler distance is used when comparing the obtained distributions in several levels of granularity with other already established sampling methods. In truth, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. In this fashion, sequences resulting from this sort of process are not truly random.

scholarly journals Deterministic Sampling from Univariate Normal Distributions with Sierpinski Space-Filling Curves

2021 ◽  
Author(s):  
Hime Oliveira
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Number ◽  
Probability Distributions ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Space Filling ◽  
Space Filling Curves ◽  
Deterministic Sampling ◽  
Standard Normal

This work addresses the problem of sampling from Gaussian probability distributions by means of uniform samples obtained deterministically and directly from space-filling curves (SFCs), a purely topological concept. To that end, the well-known inverse cumulative distribution function method is used, with the help of the probit function,which is the inverse of the cumulative distribution function of the standard normal distribution. Mainly due to the central limit theorem, the Gaussian distribution plays a fundamental role in probability theory and related areas, and that is why it has been chosen to be studied in the present paper. Numerical distributions (histograms) obtained with the proposed method, and in several levels of granularity, are compared to the theoretical normal PDF, along with other already established sampling methods, all using the cited probit function. Final results are validated with the Kullback-Leibler and two other divergence measures, and it will be possible to draw conclusions about the adequacy of the presented paradigm. As is amply known, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. That said, sequences resulting from this kind of procedure are not truly random. Even so, and to be coherent with the literature, the expression ”random number” will be used along the text to mean ”pseudo-random number”.

Follow your curiosity, along a space-filling curve

2020 ◽  
pp. 265-272
Author(s):  
Susan D'Agostino
Keyword(s):  
Mathematics Students ◽  
German Mathematician ◽  
Space Filling ◽  
Space Filling Curve ◽  
Isaac Newton ◽  
Space Filling Curves ◽  
Filling Curve

“Follow your curiosity, along a space-filling curve” tells the story of Italian mathematician Giuseppe Peano’s quest for and discovery of a space-filling curve—a curve that completely fills a space such as a square—that most mathematicians and scientists at the time did not believe existed. For example, Isaac Newton, in his Philosophiae Naturalis Principia Mathematica, tried to ban space-filling curves. The discussion of space-filling curves is enhanced with numerous hand-drawn sketches showing how to construct German mathematician David Hilbert’s space-filling curve. Mathematics students and enthusiasts are encouraged to foster a Peano-like curiosity in mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

scholarly journals Multiplicativity of the uniform norm and independent functions

1990 ◽  
Vol 42 (1) ◽  
pp. 153-155
Author(s):  
A. Guyan Robertson
Keyword(s):  
Recent Work ◽  
Lebesgue Measure ◽  
Uniform Norm ◽  
Continuous Functions ◽  
Space Filling ◽  
Stochastic Independence ◽  
Space Filling Curve ◽  
Space Filling Curves ◽  
Independent Functions ◽  
Filling Curve

It has long been known that there is a close connection between stochastic independence of continuous functions on an interval and space-filling curves [9]. In fact any two nonconstant continuous functions on [0, 1] which are independent relative to Lebesgue measure are the coordinate functions of a space filling curve. (The results of Steinhaus [9] have apparently been overlooked in more recent work in this area [3, 5, 6].)

scholarly journals A Matrix Iterative Approach to Systematically Generate Hilbert-type Space-filling Curves

2015 ◽  
Vol 14 (12) ◽  
pp. 6281-6294
Author(s):  
Ruisong Ye ◽  
Li Liu
Keyword(s):  
Point Of View ◽  
Iterative Approach ◽  
Matrix Approach ◽  
Space Filling ◽  
Type Space ◽  
Space Filling Curve ◽  
Space Filling Curves ◽  
The Matrix ◽  
Filling Curve ◽  
Hilbert Type

Hilbert-type space-filling curve has attracted much interest thanks to its mathematical importance and extensive applications in signal processing. In this paper, we construct the complete six Hilbert-type space-filling curves form amatrix point of view. The address matrix for each considered Hilbert-type space-filling curve can be easily generated by a recursive manner. Besides the six Hilbert-type space-filling curves, we also construct their corresponding variation versions. The merit of the matrix approach is that the iterative algorithm is easy to implement and can be generalized to produce any other Hilbert-type space-filling curves and their variation versions.

Region of Interest Based MRI Brain Image Compression Using Peano Space Filling Curve

2016 ◽  
Vol 11 (2) ◽  
pp. 114-120 ◽  
Author(s):  
C. Peter Devadoss ◽  
Balasubramanian Sankaragomathi ◽  
Thirugnanasambantham Monica
Keyword(s):  
Image Compression ◽  
Region Of Interest ◽  
Space Filling ◽  
Brain Image ◽  
Space Filling Curve ◽  
Mri Brain ◽  
Filling Curve ◽  
Mri Brain Image

A Space Filling Curve

1983 ◽  
Vol 90 (4) ◽  
pp. 283
Author(s):  
Liu Wen
Keyword(s):  
Space Filling ◽  
Space Filling Curve ◽  
Filling Curve
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