1. 1 Simulation of Uniform Distribution on Surfaces Giuseppe Melfi Université de Neuchâtel Espace de l’Europe, 4 2002 Neuchâtel

2. 2 Introduction Random distributions are quite usual in nature. In particular: Environmental sciences Geology Botanics Meteorology are concerned

3. 3 Distribution A Distribution of trees in a typical cultivated field.

4. 4 Distribution B Distribution of trees in a typical intensive production. For the same surface and the same minimal distance, there are 15% more trees.

5. 5 Distribution C Distribution of trees in a plane forest. Uniform random distribution on a plane.

6. 6 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

7. 7 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.

8. 8 Algorithm 1 Step 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

9. 9 Step 2: Random assignment We assign to each point in D a random number w in (0,1). We have that w 1, w 2, …,w N are drawn according to a uniform distribution. This will be employed to select points on the basis of a suitable probability of selection.

10. 10 Step 3: Uniformizer coefficient The region corresponds into the surface S to a region whose area can be approximated by We compute

11. 11 Step 4: Points selection The probability of (x i, y i, f(x i, y i )) to be selected must be proportional to the quantity The point (x i, y i, f(x i, y i )) is selected if

12. 12 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.

13. 13 Some examples Let f(x,y)=6exp{-(x 2 +y 2 )} 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.

14. 14 A uniform distribution on the surface S={(x,y,6exp{-x 2 -y 2 })}

15. 15 Another example Let f(x,y)=x 2 -y 2 Let D=(-1,1) x (-1,1) Again, 1000 points have been used.

16. 16 Uniform distribution on the hyperboloid S = {(x,y, x 2 -y 2 )}

17. 17 Uniform distribution on the surface S={(x,y,6arctan x)}

18. 18 Under another perspective S={(x,y,6arctan x)}

19. 19 Uniform distribution on the surface S={(x,y,(x 2 +y 2 )/2)}

20. 20 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.

21. 21 Algorithm 2 Step 1: Generation of random points in D Step 2: Random assignment Step 3: Compute Step 4: (x i,y i,f(x i,y i )) is selected if

22. 22 Non uniform distribution: an example Let f(x,y)=6 exp{-(x 2 +y 2 )} It is the surface considered in first example Let z 1 (x,y)=3-|3-f(x,y)| This corresponds to give more probability to points for which f(x,y)=3 Let z 2 (x,y)=exp{-f(x,y) 2 } In this case we give a probability of Gaussian type, depending on value of f(x,y)

23. 23 A non uniform distribution on S={(x,y,6 exp{-x 2 -y 2 })} using z 1

24. 24 A non uniform distribution on S={(x,y,6 exp{-x 2 -y 2 })} using z 2

25. 25 … and with less points

26. 26 Non uniform distribution on S = {(x,y, x 2 -y 2 )}

27. 27 With a normal vertical distribution

28. 28 Non uniform distribution on S={(x,y,6arctan x)}

29. 29 Another non uniform distribution on S={(x,y,6arctan x)}

30. 30 Non uniform distribution on S={(x,y,(x 2 +y 2 )/2)}

31. 31 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.