# What machine learning algorithm is appropriate for predicting one time-series from another?

I have eye-tracking data on two subjects -- a teacher, and a student. It's in the form (x, y, time), so there is a series of these for each subject. What the teacher looks at influences what the student looks at. What method would I use to predict what the student is looking at, using only teacher data? Lets say I can train some learning algorithm using a gold standard set of student and teacher data.

I was thinking hidden markov model would be appropriate, given the definition in Wikipedia, but I am not sure how to put this into practice over my dataset.

More detail: I have data about how a teacher and student each look at a map and some readings. I have 40 of these datasets, which look like [(366,234,0), (386,234,5), ...] which means the teacher looked at point (366,234) at time 0 and then 5 seconds later moved up to look at coordinate (386, 234). I can to learn a model to understand the relationship between how a teacher looks at content, to predict how a student will look at the same content. So maybe the student looks at the content in the same order as the teacher but slower. Or perhaps the student doesn't look around as much but the teacher scans more of the content. I have both sets of data and want to see how accurate of a model I can get -- would I be able to predict the student's looking behavior within 50px of the teacher's looking behavior?

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No matter what you do, the first step is to figure out and formalize the plausible kinds of behavior you want to detect. Markov model would require the student to have only limited memory about what happened in the past (and not too long one, otherwise the number of possible states will be too large to handle) and time invariance. If the student rule is like "look at $x$ at the moment $n$ if the teacher looked at it at the moment $f(n)$" where $f$ is some simple but unknown function with $f(n)\le n$ like $f(n)=[n/2]$, then no Markov model will be of any use to detect such pattern. –  fedja Feb 16 '13 at 12:41
I imagine that both the teacher and the student are more likely to look at interesting things then they are to look at boring things. Then areas the teacher looks at more are more likely to be looked at by the student, but not necessarily at the same time. This is impossible to capture in a Markov model because it is time-independent. On the other hand, it is space-dependent, so density estimation on the complete teacher time sequence could capture this dependence. –  Carl Feynman Feb 16 '13 at 16:49
This is not a math question; it's a question about how a teacher's behavior affects a student's behavior. I think it is off-topic here. –  Steven Landsburg Feb 16 '13 at 17:16
@Steven It can be converted to a math question, but that conversion is 80% of the fun. If you have ever talked to engineers and other applied people, you know that the starting point is to figure out what they are really after. Whether this process is "proper math." or not is a matter of opinion. I believe it is but I've heard many arguments to the contrary too... –  fedja Feb 16 '13 at 20:32
Thanks everyone, I realized from posting here that the problem is a bit confusing. I reformulated the problem as a submarine/ship story, and posted to stats.stackexchange: stats.stackexchange.com/questions/50172/… I think fedja's first comment is most helpful. I think maybe I have to narrow down the possible behavioral patterns that may be occurring in the data. It'd be nice to have a systematic way of doing this other than "watching a bunch of data and noting down some patterns I observe". –  James Waythorn Feb 17 '13 at 8:18