Simulation of Uniform Distribution on Surfaces

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

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

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

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

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. 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. 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

9. 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.

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

11. 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

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. 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.

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

15. Another example • Let f(x,y)=x2-y2 • Let D=(-1,1)x(-1,1) Again, 1000 points have been used.

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

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

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

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

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. 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. 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)

23. A non uniform distribution onS={(x,y,6 exp{-x2-y2})} usingz1

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

25. … and with less points

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

27. With a normal vertical distribution

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

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

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

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.