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Contact rate between individuals of different probability density functions

  1. Sep 22, 2014 #1
    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!!
     
  2. jcsd
  3. Sep 22, 2014 #2

    mathman

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    Your question isn't too elementary. I can't understand it!
     
  4. Sep 23, 2014 #3

    Stephen Tashi

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    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 [itex] u(x,t) [/itex] 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 [itex] u(x,t) [/itex] 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 [itex]u(x,t) [/itex] at each discrete interval of time. Is your problem discrete?
     
  5. Sep 23, 2014 #4
    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).
     
    Last edited: Sep 23, 2014
  6. Sep 23, 2014 #5

    Stephen Tashi

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    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.
     
    Last edited: Sep 24, 2014
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