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Simulation of Uniform Distribution on Surfaces

Giuseppe Melfi

Université de Neuchâtel

Espace de l’Europe, 4

2002 Neuchâtel

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Introduction

Random distributions are quite usual in nature. In particular:

• Environmental sciences

• Geology

• Botanics

• Meteorology

are concerned

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Distribution A

Distribution of trees in a typical cultivated field.

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Distribution B

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

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Distribution C

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

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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

ekX k

k

!)Pr(

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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.

2,: RDRDf

DyxRyxfyxS ),(:),(,, 3

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Algorithm 1Step 1:

Generation of N points in D

• 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

).,(),( dcbaD

),( 212 nn uu )1,0()1,0(

).,),...(,( 11 NN yxyx

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Step 2: Random assignment

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

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

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

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Step 3: Uniformizer coefficient

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

• We compute

yxy

yxf

x

yxf

2

00

2

00 ),(),(1

22

1max

y

f

x

fM

D

)()( 00 yyxx

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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

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1

y

f

x

f

M

y

yxf

x

yxf

w

iiii

i

22),(),(

1

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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.

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1

y

f

x

f

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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.

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

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Another example

• Let

f(x,y)=x2-y2

• Let

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

Again, 1000 points have been used.

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

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

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

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

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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.

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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

22

1),(maxy

f

x

fyxzM

D

22),(),(

1),(

y

yxf

x

yxf

M

yxzw iiiiii

i

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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)

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A non uniform distribution on S={(x,y,6 exp{-x2-y2})} using z1

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A non uniform distribution on S={(x,y,6 exp{-x2-y2})} using z2

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… and with less points

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

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With a normal vertical distribution

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

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Another non uniform distribution on

S={(x,y,6arctan x)}

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

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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.