Bottom Line Up Front (BLUF):
I am new to Hidden Markov Models and am trying to learn by using them. I want to learn by working.
- How do I use binned counts of states (described below) and a matrix of state transition probabilities to estimate most likely complete trajectories in time?
- How do I use a matrix of state transition probabilities and expected trajectories in time to set binned counts of states collection such that using only the matrix and bins I can reconstruct the states with error below a set threshold?
A set of 1d coupled spring-masses are set in motion from an unknown initial point. This physical system was measured, and for each second the measure of interest, the location of one of the masses, fell within a bin, the counter for that bin was incremented. At an arbitrary time the "histogram" so far was stored, the counters reset, and the recording resumed. The bin count and definitions don't change.
The masses, the springs, and the rest-state geometry are known. They are such that the characteristic time of the system in on the order of a few seconds.
It is also known that the amount of time used to store the data in the histogram changes each time, as do the initial conditions. The range of the variation of initial conditions is relatively bounded. The histograms sampling time, while larger than the characteristic times of the system is typically less than 100x the characteristic time.
The "most likely sequence of states over time" from the histograms and system physics using a Hidden Markov Model. That is to say, given the physics, and the histogram, tell the initial positions and velocities and from them the entire path during the time-window.
Describe the requirements on the histogram window to reduce the maximum error in the estimation below a particular threshold.
Some of the time-bins for the histograms are so large that exact extraction of the sequence of states is unlikely, but estimation of likely sequence should still be possible. There are some of the histograms that are short duration and have just a few states, so exact extraction might be possible. With knowledge of the initial state, and a sequence of transition likelihoods, it should be possible to estimate the system behavior over time.