scholarly journals Approximating distribution functions by iterated function systems

2005 ◽  
Vol 2005 (1) ◽  
pp. 33-46 ◽  
Author(s):  
Stefano Maria Iacus ◽  
Davide La Torre
Keyword(s):  
Monte Carlo ◽  
Distribution Function ◽  
Iterated Function System ◽  
Asymptotic Properties ◽  
Iterated Function Systems ◽  
Monte Carlo Analysis ◽  
Distribution Functions ◽  
Density Estimator ◽  
New Class ◽  
Iterated Function

An iterated function system (IFS) on the space of distribution functions is built with the aim of proposing a new class of distribution function estimators. One IFS estimator and its asymptotic properties are studied in detail. We also propose a density estimator derived from the IFS distribution function estimator by using Fourier analysis. Relative efficiencies of both estimators, for small and moderate sample sizes, are presented via Monte Carlo analysis.

scholarly journals Hölder-differentiability of Gibbs distribution functions

2009 ◽  
Vol 147 (2) ◽  
pp. 489-503 ◽  
Author(s):  
MARC KESSEBÖHMER ◽  
BERND O. STRATMANN
Keyword(s):  
Distribution Function ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Iterated Function Systems ◽  
Gibbs Distribution ◽  
Distribution Functions ◽  
Limit Sets ◽  
Iterated Function ◽  
Natural Home ◽  
Set Of Points

AbstractIn this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil's staircases) supported on limit sets of finitely generated conformal iterated function systems in ℝ. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not α-Hölder-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris and Xiao. In particular, our results clearly show that the results of these authors have their natural home within the thermodynamic formalism.

scholarly journals How Many Monte Carlo Simulations Does One Need to Do? II. Lognormal, Binomial, Cauchy, and Exponential Distributions

2005 ◽  
Vol 23 (6) ◽  
pp. 429-461
Author(s):  
Ian Lerche ◽  
Brett S. Mudford
Keyword(s):  
Monte Carlo ◽  
Distribution Function ◽  
Probability Distribution ◽  
General Procedure ◽  
Mean Value ◽  
Distribution Functions ◽  
Considerable Uncertainty ◽  
Free Parameters ◽  
Two Parameters ◽  
Achievable Accuracy

This article derives an estimation procedure to evaluate how many Monte Carlo realisations need to be done in order to achieve prescribed accuracies in the estimated mean value and also in the cumulative probabilities of achieving values greater than, or less than, a particular value as the chosen particular value is allowed to vary. In addition, by inverting the argument and asking what the accuracies are that result for a prescribed number of Monte Carlo realisations, one can assess the computer time that would be involved should one choose to carry out the Monte Carlo realisations. The arguments and numerical illustrations are carried though in detail for the four distributions of lognormal, binomial, Cauchy, and exponential. The procedure is valid for any choice of distribution function. The general method given in Lerche and Mudford (2005) is not merely a coincidence owing to the nature of the Gaussian distribution but is of universal validity. This article provides (in the Appendices) the general procedure for obtaining equivalent results for any distribution and shows quantitatively how the procedure operates for the four specific distributions. The methodology is therefore available for any choice of probability distribution function. Some distributions have more than two parameters that are needed to define precisely the distribution. Estimates of mean value and standard error around the mean only allow determination of two parameters for each distribution. Thus any distribution with more than two parameters has degrees of freedom that either have to be constrained from other information or that are unknown and so can be freely specified. That fluidity in such distributions allows a similar fluidity in the estimates of the number of Monte Carlo realisations needed to achieve prescribed accuracies as well as providing fluidity in the estimates of achievable accuracy for a prescribed number of Monte Carlo realisations. Without some way to control the free parameters in such distributions one will, presumably, always have such dynamic uncertainties. Even when the free parameters are known precisely, there is still considerable uncertainty in determining the number of Monte Carlo realisations needed to achieve prescribed accuracies, and in the accuracies achievable with a prescribed number of Monte Carol realisations because of the different functional forms of probability distribution that can be invoked from which one chooses the Monte Carlo realisations. Without knowledge of the underlying distribution functions that are appropriate to use for a given problem, presumably the choices one makes for numerical implementation of the basic logic procedure will bias the estimates of achievable accuracy and estimated number of Monte Carlo realisations one should undertake. The cautionary note, which is the main point of this article, and which is exhibited sharply with numerical illustrations, is that one must clearly specify precisely what distributions one is using and precisely what free parameter values one has chosen (and why the choices were made) in assessing the accuracy achievable and the number of Monte Carlo realisations needed with such choices. Without such available information it is not a very useful exercise to undertake Monte Carlo realisations because other investigations, using other distributions and with other values of available free parameters, will arrive at very different conclusions.

scholarly journals Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure

2012 ◽  
Vol 154 (2) ◽  
pp. 325-349 ◽  
Author(s):  
G. C. BOORE ◽  
K. J. FALCONER
Keyword(s):  
Directed Graph ◽  
Hausdorff Measure ◽  
Iterated Function Systems ◽  
New Class ◽  
Iterated Function ◽  
Separation Conditions

AbstractFor directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known.

scholarly journals Hölder differentiability of self-conformal devil's staircases

2014 ◽  
Vol 156 (2) ◽  
pp. 295-311 ◽  
Author(s):  
SASCHA TROSCHEIT
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Hausdorff Dimension ◽  
Probability Distribution Function ◽  
Iterated Function System ◽  
Gibbs Measures ◽  
General Sense ◽  
Multifractal Formalism ◽  
Iterated Function ◽  
Set Of Points

AbstractIn this paper we consider the probability distribution function of a Gibbs measure supported on a self-conformal set given by an iterated function system (devil's staircase) applied to a compact subset of ${\mathbb R}$. We use thermodynamic multifractal formalism to calculate the Hausdorff dimension of the sets Sα0, Sα∞ and Sα, the set of points at which this function has, respectively, Hölder derivative 0, ∞ or no derivative in the general sense. This extends recent work by Darst, Dekking, Falconer, Kesseböhmer and Stratmann, and Yao, Zhang and Li by considering arbitrary such Gibbs measures given by a potential function independent of the geometric potential.

scholarly journals On fractal properties of Weierstrass-type functions

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Claire David
Keyword(s):  
Real Number ◽  
Iterated Function System ◽  
Function System ◽  
Box Dimension ◽  
Intrinsic Properties ◽  
Weierstrass Function ◽  
Real Numbers ◽  
New Class ◽  
Fractal Properties ◽  
Iterated Function

In the sequel, starting from the classical Weierstrass function defined, for any real number $x$, by $ {\mathcal W}(x)=\displaystyle \sum_{n=0}^{+\infty} \lambda^n\,\cos \left(2\, \pi\,N_b^n\,x \right)$, where $\lambda$ and $N_b$ are two real numbers such that~\mbox{$0 <\lambda<1$},~\mbox{$ N_b\,\in\,\N$} and $ \lambda\,N_b > 1 $, we highlight intrinsic properties of curious maps which happen to constitute a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-differentiabilty of Weierstrass type functions.

scholarly journals Generalized Iterated Function Systems Containing Functions of Integral Type

2018 ◽  
Vol 7 (3.31) ◽  
pp. 126
Author(s):  
Minirani S ◽  
. .
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Product Space ◽  
Iterated Function System ◽  
Complete Metric Space ◽  
Iterated Function Systems ◽  
Function System ◽  
Finite Collection ◽  
Integral Type ◽  
Iterated Function

A finite collection of mappings which are contractions on a complete metric space constitutes an iterated function system. In this paper we study the generalized iterated function system which contain generalized contractions of integral type from the product space . We prove the existence and uniqueness of the fixed point of such an iterated function system which is also known as its attractor. 

scholarly journals Sustainable Construction of Earth Dams: Use of Heterogeneous Material from the Dam Site

2020 ◽  
Vol 12 (23) ◽  
pp. 9940
Author(s):  
Rubén Galindo ◽  
José Sánchez-Martín ◽  
Claudio Olalla Marañón
Keyword(s):  
Monte Carlo ◽  
Heterogeneous Material ◽  
Monte Carlo Analysis ◽  
Distribution Functions ◽  
Earth Dam ◽  
Flow Rates ◽  
The Core ◽  
Geotechnical Characteristics ◽  
Dam Site ◽  
Dam Core

The volumes of soil required for the construction of an earth dam are usually of importance, so that, consequently, a key aspect to mitigate the negative impacts of dam construction, both from an economic and environmental point of view, is the use of materials in the vicinity of the dam location. However, this is often complicated because the existence of good quality materials with homogeneous properties, necessary for the dam core, is scarce in sites near the dam and their presence in sufficient volume for its construction is not usual. Unfortunately, using and transporting soil with good geotechnical characteristics to construct the core in a faraway location is economically and environmentally unsustainable. Therefore, the possibility of using less suitable material at the dam site as part of the core must be studied. Thus, in the present research the use of a soil of great heterogeneity in its geotechnical properties with a great dispersion of permeability is analyzed. Considering permeability as a random variable, combinations of representative values of heterogeneous soils are analyzed using their mean permeability and coefficients of variation that allow generating different lognormal distribution functions to carry out a Monte Carlo analysis. By maintaining the soil’s global heterogeneity, it was possible to study an unlimited disposition of lifts of different permeability. The statistical formulation allowed the research of the variation of the seepage flows and maximum gradients produced as a function of the variability of their mean permeability, being able to detect the factors with the greatest influence on the generation of high flows and gradients. Thus, it was possible to verify how high gradients were obtained for situations in which the seepage flow rates were moderate and low; the highest maximum gradients were observed in the lowest lifts of the dam core. In addition, based on the results of the Monte Carlo analysis, design charts have been developed for flow rates and maximum gradients, dependent on the mean permeability and the coefficient of variation, which allow judging whether heterogeneous material can be used, under conditions of safety, for the construction of the core of a dam.

EXISTENCE AND BOX DIMENSION OF GENERAL RECURRENT FRACTAL INTERPOLATION FUNCTIONS

2020 ◽  
pp. 1-13
Author(s):  
HUO-JUN RUAN ◽  
JIAN-CI XIAO ◽  
BING YANG
Keyword(s):  
Iterated Function System ◽  
General Setting ◽  
Iterated Function Systems ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Recurrent System ◽  
Definition Of ◽  
Interpolation Functions

Abstract The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx.5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$ . We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

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