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Simulation of Uniform Distribution on SurfacesPowerPoint Presentation

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

Distribution A

Distribution of trees in a typical cultivated field.

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

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

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

- 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 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

- 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

- The region corresponds into the surface S to a region whose area can be approximated by
- We compute

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

- 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

- 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.

A uniform distribution on the surface S={(x,y,6exp{-x2-y2})}

Another example

- Let
f(x,y)=x2-y2

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

Again, 1000 points have been used.

Uniform distribution on the hyperboloidS = {(x,y, x2-y2)}

Uniform distribution on the surface S={(x,y,6arctan x)}

Under another perspective S={(x,y,6arctan x)}

Uniform distribution on the surface S={(x,y,(x2+y2)/2)}

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

- 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

- 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 onS={(x,y,6 exp{-x2-y2})} usingz1

A non uniform distribution onS={(x,y,6 exp{-x2-y2})} usingz2

Non uniform distribution onS = {(x,y, x2-y2)}

Non uniform distribution onS={(x,y,6arctan x)}

Another non uniform distribution onS={(x,y,6arctan x)}

Non uniform distribution onS={(x,y,(x2+y2)/2)}

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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