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People move through their days in a city in regular patterns. At certian times you see certian people in certian places. These patterns are periodic. So, to make this more abstract, suppose you are able to observe a small rectangular interval of the city for only 1 min each day. You can look at any location at any time but only for 1 min. This is long enough for you to establish which way a person is moving and how fast they are going.

The people in this city are perfectly regular, they do the same thing every day and there are no weekends.

Your task is that of a detective, you must find out where a person came from. You know this person is at location (2,7), for example, at time t.

So, you can observe that person and follow their trajectory back to what you guess is their previous location based on their velocity and direction they were going when last observed but doing this takes a whole day. Then from that location you repeat the process.

You want to trace the person back to their house. Is there any way to speed up this process.

(This is based on a real-world problem I'm dealing with. The answer may be "no")

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This problem isn't well-defined without any assumptions on the movements. The velocities could change arbitrarily abruptly, so there's no way to know what sort of guess to make about the past trajectory and how far to extrapolate it. If the velocities change often and abruptly, it might be faster to just scan the whole city in a raster; on the other hand, if the movements are very smooth, just a handful of extrapolations might lead to the house. – joriki Jun 10 '11 at 6:55
How would you define smoothness in this context? You're talking something more than the derivative, right? – futurebird Jun 10 '11 at 6:57
Yes; I didn't mean "smooth" in the sense of "infinitely differentiable", since that doesn't tell you anything quantitative about the changes. You could assume that the position function is analytic, but then the problem would be trivial, since one extrapolation would be enough. Or you could make some sort of quantitative assumptions about how much/how rapidly the function deviates from a polynomial or series approximation. – joriki Jun 10 '11 at 7:09
up vote 1 down vote accepted

This is more of a modelling question than a maths one, I suspect, and my answer is going to be more of a modeller's answer than a mathematician's.

The short answer, as joriki touched upon, is that it depends what prior information you've got on their movement behaviour - specifically, either their route choice algorithm, or the probability distribution of changes in their velocities.

If you have no such information (i.e. if the motion is completely random), then there will be few algorithmic shortcuts.

If you have some information (e.g. we do, for example, have statistical observations of human route-choice behaviour, as well as probabilistic models of them), then you could maybe start looking at a Bayesian approach to work backwards in leaps; maybe look at taking maximum-likelihood estimates of prior positions, and sampling there.

An information-theory specialist might be able to give you a better answer, if you can provide more details about the specific problem.

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