How to predict when next event occurs based on previous events?
Basically, I have a reasonably large list (a year's worth of data) of times that a single discrete event occurred (for my current project, a list of times that someone printed something). Based on this list, I would like to construct a statistical model of some sort that will predict the most likely time for the next event (the next print job) given all of the previous event times.
I've already read this, but the responses don't exactly help out with what I have in mind for my project. I did some additional research and found that a Hidden Markov Model would likely allow me to do so accurately, but I can't find a link on how to generate a Hidden Markov Model using just a list of times. I also found that using a Kalman filter on the list may be useful but basically, I'd like to get some more information about it from someone who's actually used them and knows their limitations and requirements before just trying something and hoping it works.
Thanks a bunch!
EDIT: So by Amit's suggestion in the comments, I also posted this to the Statistics StackExchange, CrossValidated. If you do know what I should do, please post either here or there
Answer
I'll admit it, I'm not a statistics kind of guy. But I've run into these kind of problems before. Really what we're talking about here is that you have some observed, discrete events and you want to figure out how likely it is you'll see them occur at any given point in time. The issue you've got is that you want to take discrete data and make continuous data out of it.
The term that comes to mind is density estimation. Specifically kernel density estimation. You can get some of the effects of kernel density estimation by simple binning (e.g. count the number events in a time interval such as every quarter hour or hour.) Kernel density estimation just has some nicer statistical properties than simple binning. (The produced data is often 'smoother'.)
That only takes care of one of your problems, though. The next problem is still the far more interesting one -- how do you take a time line of data (in this case, only printer data) and produced a prediction from it? First thing's first -- the way you've set up the problem may not be what you're looking for. While the miracle idea of having a limited source of data and predicting the next step of that source sounds attractive, it's far more practical to integrate more data sources to create an actual prediction. (e.g. maybe the printers get hit hard just after there's a lot of phone activity -- something that can be very hard to predict in some companies) The Netflix Challenge is a rather potent example of this point.
Of course, the problem with more data sources is that there's extra legwork to set up the systems that collect the data then.
Honestly, I'd consider this a domain-specific problem and take two approaches: Find time-independent patterns, and find time-dependent patterns.
An example time-dependent pattern would be that every week day at 4:30 Suzy prints out her end of the day report. This happens at specific times every day of the week. This kind of thing is easy to detect with fixed intervals. (Every day, every week day, every weekend day, every Tuesday, every 1st of the month, etc...) This is extremely simple to detect with predetermined intervals -- just create a curve of the estimated probability density function that's one week long and go back in time and average the curves (possibly a weighted average via a windowing function for better predictions).
If you want to get more sophisticated, find a way to automate the detection of such intervals. (Likely the data wouldn't be so overwhelming that you could just brute force this.)
An example time-independent pattern is that every time Mike in accounting prints out an invoice list sheet, he goes over to Johnathan who prints out a rather large batch of complete invoice reports a few hours later. This kind of thing is harder to detect because it's more free form. I recommend looking at various intervals of time (e.g. 30 seconds, 40 seconds, 50 seconds, 1 minute, 1.2 minutes, 1.5 minutes, 1.7 minutes, 2 minutes, 3 minutes, .... 1 hour, 2 hours, 3 hours, ....) and subsampling them via in a nice way (e.g. Lanczos resampling) to create a vector. Then use a vector-quantization style algorithm to categorize the "interesting" patterns. You'll need to think carefully about how you'll deal with certainty of the categories, though -- if your a resulting category has very little data in it, it probably isn't reliable. (Some vector quantization algorithms are better at this than others.)
Then, to create a prediction as to the likelihood of printing something in the future, look up the most recent activity intervals (30 seconds, 40 seconds, 50 seconds, 1 minute, and all the other intervals) via vector quantization and weight the outcomes based on their certainty to create a weighted average of predictions.
You'll want to find a good way to measure certainty of the time-dependent and time-independent outputs to create a final estimate.
This sort of thing is typical of predictive data compression schemes. I recommend you take a look at PAQ since it's got a lot of the concepts I've gone over here and can provide some very interesting insight. The source code is even available along with excellent documentation on the algorithms used.
You may want to take an entirely different approach from vector quantization and discretize the data and use something more like a PPM scheme. It can be very much simpler to implement and still effective.
I don't know what the time frame or scope of this project is, but this sort of thing can always be taken to the N-th degree. If it's got a deadline, I'd like to emphasize that you worry about getting something working first, and then make it work well. Something not optimal is better than nothing.
This kind of project is cool. This kind of project can get you a job if you wrap it up right. I'd recommend you do take your time, do it right, and post it up as function, open source, useful software. I highly recommend open source since you'll want to make a community that can contribute data source providers in more environments that you have access to, will to support, or time to support.
Best of luck!