# Combining multiple measures into a single measure

I have a situation where I have a set of j distinct categories, let's call them C. I have measured C in i different ways for a number of observations. I want to find the category Cj that best represents each observation based on the i measures of C. So, I basically want to go from a set of (# of items in C) * i variables on each observation, to only the category Cj which is the best representation of that particular observation.

Because for each measurement i in my list, there is a benefit of having a larger value of i in the category, I have though about the option of ranking the categories for each observation based on i, and do this for all i, and then combine the rankings. So in this case I get i lists with values ranging from 1 to (# of items in C) for each observation. If I rank them, so that rank 1 is the largest value in terms of the measure i, I can combine the elements of the i lists with for instance an average, and the smallest value of the combination, with position j, will then be used as the best category Cj, representing the observation.

I want a good way to find the category Cj that best represents each observation. Is there a general method to approach this problem?

Thanks.