This brings to mind the Viterbi algorithm that calculates the most likely sequence of events in a hidden Markov model (Markov model itself is just a state machine on a graph with weighed edges, where each weight on a state transition is represented as a probability of that transition).
It's essentially bases to the same algorithm: you can eliminate sequences if they lead to the same event that is already part of a solution, but with lesser probability. The amount of paths would be exponential, but the ability to eliminate them keeps it polynomial. That really brings forth the beauty of dynamic programming.
One really nice application of the Viterbi algorithm is map-matching, i.e. taking a GPS trail and "snapping" it to the road network to determine the actual route taken. It's difficult because GPS trails are often sparse and noisy, so you can't do something simple like "project onto the closest road segment". If you take the "states" of the model to be road segments and derive the transition probabilities from the shortest route between segments, you can apply Viterbi and get a very accurate result most of the time. Of course, calculating shortest routes involves Dijkstra, another famous dynamic algorithm.
Unfortunately it's not a simple project to get started on because you already need to have a way to import road network data (e.g. from openstreetmap), build an internal graph model, do some geographic indexing and perform fast many-to-many routing calculations.
yep - look up 'Hidden Markov Map Matching through Noise and Sparseness' by Krumm and Newson
it calculates the final probability by combining 'emission' probabilities (the probability that a GPS observation was on a particular road) by the 'transition' probability that if an observation was on a particular road at one point, what is the probability that it is now on this other road segment. By combining these two the final probability incorporates both the nearness of the GPS signals to the roads and the connectivity of the road network itself.
I've found the formulas applied in this paper are good in practice only if the GPS updates are relatively frequent
I've also found that it can be tricky to map "convoluted" routes that don't necessary go simply from point A to pointB.
If you don't mind me asking, roughly what frequency threshold have you found the algorithm to perform badly above, and are you aware of any algorithms or formulae which perform better in these situations?
it should be in seconds - the problem is that the paper assumes that the 'great circle' (straight line) distance between two points should be almost the same as the 'route' distance between those points, with an exponential probability distribution.
This means that if the path between two points is not simple (around a corner) the probability drops off very quickly. If the time between measurements is in minutes, this heuristic is pretty useless (and you should really use log-scale for your numbers!)
edit: this is actually shown in figure 8 of the paper where they explore different 'sampling periods'
edit 2: I have not explored other methods yet, but it would probably make sense to start by deriving the formula the way they do, by exploring ground-truth data.
edit 3: I just noticed that my comments are largely repeating what you're saying - sorry!
Ah, that rings a bell now. You can vary a parameter they call "beta" to allow for more convoluted routes, and I think a larger value gives a little leeway for less frequent fixes.
Agreed, the log scale is really important to avoid arithmetic underflow =] I believe OSRM and Graphhopper both do it that way. In my implementation I've flipped from thinking of measurement/transition "probabilities" to "disparities", and I choose the final route that has the least disparity from the trail. It seems to handle trails with around a 30-60s frequency over a 5-10hr period with decent accuracy.
actually, beta is less useful than that! I think it represents the median difference between the two distances, its not a tolerance (at least as far as I can recall after experimenting with tuning this value).
As with you, I have found that it still often gives ok results with slower frequencies, as long as the transition probabilities are still relatively in the same scale as each other for a particular observation pair. However it means that there's no point trying to 'tune' using the gamma and beta parameters
A lot of problems on sequences are amendable to dynamic programming. Two other famous ones are Smith-Waterman and Needleman-Wunsch for finding optimal global and local alignments of sequences–a typical problem in genetics.
It's essentially bases to the same algorithm: you can eliminate sequences if they lead to the same event that is already part of a solution, but with lesser probability. The amount of paths would be exponential, but the ability to eliminate them keeps it polynomial. That really brings forth the beauty of dynamic programming.