I would most appreciate names of methods/techniques. Again, I don't have the terminology to describe the problem very well -- I'll edit as needed.
There is a variable (x,y,z) where x is a timeframe label, y is time remaining till end of the timeframe (y is "t minus") and z is the quantity of interest. For every x, the y-z plot traces a very similar function -- so similar that one outer factor might be separating all of them (they look like the same function scaled).
For example, x is a year, y is days till new years eve of that year, and z is the number of new year's resolutions for that year (fulfilled or intended). At 370-360 people go nuts making them, and at 30 people realize that with only December left, they're not gonna go scuba diving for the first time this year.
The question is about predicting z for (x_(n+1), 0, z) once you have a short sequence of y's. I don't know if it's reasonable to expect that as y->0, the interval for this prediction shrinks. Here are my ideas:
- Fix y and run linear regression. This decimates the sample, but fits great. It's effective, but I don't think it's right. Running MLR on, for example, the last three measurements also seems like abuse.
- Come up with the function family where one parameter separates one x from another. Use the sequence of y's to get increasing precision on the parameter estimate. This appeals to me, but I don't know how to produce a prediction interval for z at y=0.
- Transform the whole thing into a straight line and run linear regression -- force y to be a good predictor. I guess this would also involve finding a function family to fit the data.
- Is this a classic application for non-linear regression? (It would be first for me, hence the other ideas)
If you have a tag suggestion, it could probably be an answer.
I'm now reasonably sure that this is a classic application for nonlinear regression (as much as I can be given that it's my first application of it).
I wish I had more guidance on how to report the error introduced when the predictive function is first fitted to past data. The method used and the placement of parameters seem to be the sensitive part. I also wonder if there is a more specific method for estimating the value of the fitted function for one given input.