Simulation of uniform distribution on surfaces
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Simulation of Uniform Distribution on Surfaces. Giuseppe Melfi Université de Neuchâtel Espace de l’Europe, 4 2002 Neuchâtel. Introduction. Random distributions are quite usual in nature. In particular: Environmental sciences Geology Botanics Meteorology are concerned .

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Simulation of uniform distribution on surfaces l.jpg

Simulation of Uniform Distribution on Surfaces

Giuseppe Melfi

Université de Neuchâtel

Espace de l’Europe, 4

2002 Neuchâtel


Introduction l.jpg
Introduction

Random distributions are quite usual in nature. In particular:

  • Environmental sciences

  • Geology

  • Botanics

  • Meteorology

    are concerned


Distribution a l.jpg
Distribution A

Distribution of trees in a typical cultivated field.


Distribution b l.jpg
Distribution B

Distribution of trees in a typical intensive production. For the same surface and the same minimal distance, there are 15% more trees.


Distribution c l.jpg
Distribution C

Distribution of trees in a plane forest. Uniform random distribution on a plane.


Problem how to simulate a distribution of points l.jpg
Problem: How to simulate a distribution of points

  • In a nonplanar surface

  • Such that points are distributed according to a random uniform distribution, namely the quantity of points for distinct unities of surface area (independently of gradient) follows a Poisson distribution X


Input and tools l.jpg
Input and tools

  • The input of such a problem is a function

    D compact, f supposed to be differentiable. This function describes the surface

  • The basic tool is a (pseudo-) random number generator.


Algorithm 1 step 1 generation of n points in d l.jpg
Algorithm 1Step 1:Generation of N points inD

  • D is bounded, so

  • Random points in the box

    can be partly inbedded in D.

  • This procedure allows us to simulate an arbitrary number of uniformily distributed points in D, say N, denoted


Step 2 random assignment l.jpg
Step 2: Random assignment

  • We assign to each point in D a random number w in (0,1).

  • We have that w1, w2, …,wNare drawn according to a uniform distribution.

  • This will be employed to select points on the basis of a suitable probability of selection.


Step 3 uniformizer coefficient l.jpg
Step 3: Uniformizer coefficient

  • The region corresponds into the surface S to a region whose area can be approximated by

  • We compute


Step 4 points selection l.jpg
Step 4: Points selection

  • The probability of (xi, yi, f(xi, yi)) to be selected must be proportional to the quantity

  • The point (xi, yi, f(xi, yi)) is selected if


Remarks l.jpg
Remarks

  • If S does not come from a bivariate function, but is simply a compact surface (e.g., a sphere), this approach is possible by Dini’s theorem.

  • If D is bounded but not necessarily compact, it suffices that

    is bounded.


Some examples l.jpg
Some examples

  • Let

    f(x,y)=6exp{-(x2+y2)}

  • Let

    D=(-3,3)x(-3,3)

  • We apply the preceding algorithm. We have 1000 points in D. A selection of these points will appear in simulation.



Another example l.jpg
Another example

  • Let

    f(x,y)=x2-y2

  • Let

    D=(-1,1)x(-1,1)

    Again, 1000 points have been used.




Under another perspective s x y 6arctan x l.jpg
Under another perspective S={(x,y,6arctan x)}



How to simulate non uniform distributions on surfaces l.jpg
How to simulate non uniform distributions on surfaces

Density can depend on

  • slope

  • orientation

  • punctual function

    These factors correspond to a positive function z(x,y) describing their punctual influence.


Algorithm 2 l.jpg
Algorithm 2

  • Step 1: Generation of random points in D

  • Step 2: Random assignment

  • Step 3: Compute

  • Step 4: (xi,yi,f(xi,yi)) is selected if


Non uniform distribution an example l.jpg
Non uniform distribution: an example

  • Let f(x,y)=6 exp{-(x2+y2)}

    It is the surface considered in first example

  • Let z1(x,y)=3-|3-f(x,y)|

    This corresponds to give more probability to points for which f(x,y)=3

  • Let z2(x,y)=exp{-f(x,y)2}

    In this case we give a probability of Gaussian type, depending on value of f(x,y)


A non uniform distribution on s x y 6 exp x 2 y 2 using z 1 l.jpg
A non uniform distribution onS={(x,y,6 exp{-x2-y2})} usingz1


A non uniform distribution on s x y 6 exp x 2 y 2 using z 2 l.jpg
A non uniform distribution onS={(x,y,6 exp{-x2-y2})} usingz2



Non uniform distribution on s x y x 2 y 2 l.jpg
Non uniform distribution onS = {(x,y, x2-y2)}



Non uniform distribution on s x y 6arctan x l.jpg
Non uniform distribution onS={(x,y,6arctan x)}



Non uniform distribution on s x y x 2 y 2 2 l.jpg
Non uniform distribution onS={(x,y,(x2+y2)/2)}


Further ideas l.jpg
Further ideas

  • A quantity of interest Q can depend on reciprocal distance of points

  • on disposition of points in a neighbourood of each point

  • A suitable model for an estimation of Q by Monte Carlo methods could be imagined.


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