[Swarm-Modelling] Re: Non-integer timesteps

[Rick Lightburn]

I find this topic very interesting.  I will note this thread carefully.

I fear that the issue of "non-integer timesteps" might proceed from a 
misappreciation of the theory and will produce malformed models of the 
underlying phenomena.

In theory, in a stochastic process the probabilities of the system occupying 
any one state now depend on where the system was in prior periods.  It's 
easiest to understand this in terms of discrete times, but extensions to 
continuous time are possible -- at least on the chalkboard.  (I bet that these 
approaches can't be implemented computationally, but I don't know about 
computation to really say.)  

So sticking to the discrete-time case, the time-steps are ordinal, not 
cardinal:  first, second, third, ... etc.  There is no such thing as "first and 
two-fifths" -- the thing that follows "first" is "second."  In practice, we 
sometimes identify these time-steps with "days" or "hours" or "years" because 
the model -- i.e., mathematical formula -- we use describes what will happen 
"next" given where things are "now."  Because of this identification between, 
e.g., "first" and "today," we can wonder what will happen when "next" means, 
not "tomorrow" but "later the same day."

If the model we use describes what happens "tomorrow" on the basis of "today," 
then asking what will be the state 5 or 12 hours from now requires entirely 
different models -- different (at least formally) state-spaces, and wholly 
redefined stochastic processes.  A biological model of inheritance, e.g., is 
silent about embryology.

While there might be some software fix to trick you into thinking that you're 
modeling on non-integer timesteps, I'm afraid that you will have extended the 
underlying model in some way that you don't (or even can't) justify.  Before 
you implement any such fix, make sure that your model makes sense for those 
time-steps.  Sometimes it will, sometimes it won't.