Contact rate between individuals of different probability density functions

[nigels]

Hi all,

I'm interested in obtaining some measure of contact (or encounter) likelihood between two individuals, each is spatially distributed with some probability density function at time $t$ such that,

Space use of individual 1 = $u(\mathbf{x}, t)$
Space use of individual 2 = $v(\mathbf{x}, t)$

Would I solve for the area overlap between $u(\mathbf{x},t)$ and $v(\mathbf{x},t)$, perhaps by taking the integral of their joint distribution? Or should I take the convolution of the two? Frankly, I'm not sure when convolution method is actually applicable. Furthermore, once I have the contact rates at time $t$ across the entire domain, to find its value within a certain region (say, $a \leq x \leq b$ in 1D domain), do I simply take the integral of the resultant contact PDF from $a$ to $b$?

Sorry if this question is too elementary. I really appreciate all your help!

 

[mathman]

Your question isn't too elementary. I can't understand it!

 

[Stephen Tashi]

I'm interested in obtaining some measure of contact (or encounter) likelihood between two individuals, each is spatially distributed with some probability density function at time $t$ such that,

Space use of individual 1 = $u(\mathbf{x}, t)$
Space use of individual 2 = $v(\mathbf{x}, t)$

This sounds like a description of a real world problem, but it needs refining. If we give a density function for the probability that a particle is "at position x at time t" and ask something about what happens in a "time interval" then we need more than the the density function to answer such a question.

The particle might be moving from place to place in a continuous fashion. A person who took someI (position,time) measurements could fit a probability density function $u(x,t)$ to the data, but that function isn't a model for how the particle is moving because it doesn't capture the requirement for continuous motion.

If we assume that a trajectory of the particle is generated by taking an independent random sample from the density $u(x,t)$ at each instant of time t, then we have a very jumpy discontinuous motion , more jumpy than "Brownian" motion. The mathematics of something moving in "randomly" in time needs to be described by a "stochastic process", not merely by a probability density function.

If you are dealing with events given by discrete intervals of space and time (like "person A is in room 25 during the hour 3 of the day") and you don't intend to subdivide these intervals then it might be possible to model movement by taking a random sample from a discrete density $u(x,t)$ at each discrete interval of time. Is your problem discrete?

 

[nigels]

Hi Stephen,

My question does pertain to a real-world problem, and the two dependent variables are actually steady-state solutions to a set of Fokker-Planck equations modeling an advection-diffusion process given static point-attractors. So the system is spatio-temporally continuous (time interval converges to zero in the derivation of the Fokker-Planck).

 

[Stephen Tashi]

If you have trajectories that are solutions to a deterministic set of equations and you want to introduce probability into the picture then you must be specific about how this probability arises. For example, you might have a distribution on the set of initial conditions and pick an initial condition at random and then pick the trajectory that is a solution for that initial condition. This is a random selection of an entire trajectory, not a random selection of a single point (x,t). If you dealing with data from an experiment then probability might enter the picture as a random error in measurement.

If you have specific trajectory x = f(t) and pick t at random from some distribution then you can find the value of x. This gives a random selection of (x,t). However, a probability density u(x,t) fit to such data is not a good model for continuous motion of given particle. In continuous motion, the particle's position at (x,t+h) is not independent of the position at (x,t). So independent random samples from u(x,t) "at each instant of time" do not model the trajectory of a single particle correctly.