REPOST from Data Science:
I've been toying with this idea for a while. I think there is probably some method in the text mining literature, but I haven't come across anything just right...
What is/are some methods for tackling a problem where the number of variables is its self a variable. This is not a missing data problem, but one where the nature of the problem fundamentally changes. Consider the following example:
Suppose I want to predict who will win a race, a simple multinomial classification problem. I have lots of past data on races, plenty to train on. Lets further suppose I have observed each contestant run multiple races. The problem however is that the number or racers is variable. Sometimes there are only 2 racers, sometimes there are as many as 100 racers.
One solution might be to train a separate model for each number or racers, resulting in 99 models in this case, using any method I choose. E.g. I could have 100 random forests.
Another solution might be to include an additional variable called 'number_of_contestants' and have input field for 100 racers and simply leave them blank when no racer is present. Intuitively, it seems that this method would have difficulties predicting the outcome of a 100 contestant race if the number of racers follows a Poisson distribution (which I didn't originally specify in the problem, but I am saying it here).