Simulation of Uniform Distribution on Surfaces

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Simulation of Uniform Distribution on Surfaces

• Giuseppe Melfi
• Université de Neuchâtel
• Espace de lEurope, 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

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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.

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Algorithm 1Step 1Generation 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

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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

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

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Some examples

• Let
• f(x,y)6exp-(x2y2)
• 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,(x2y2)/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

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Non uniform distribution an example

• Let f(x,y)6 exp-(x2y2)
• 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,(x2y2)/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.