Simulation of Uniform Distribution on Surfaces Giuseppe Melfi

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

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

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

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

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

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 6

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. 7

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 8

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. 9

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

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 11

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. 12

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

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

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

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

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

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

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

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. 20

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 21

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) 22

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

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

… and with less points 25

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

With a normal vertical distribution 27

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

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

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

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. 31