Imagine yourself as an unsuspecting spectator watching people enjoying in a water body. All of a sudden there is an impetus given to you, that catches you unawares, and the next moment you are surrounded by the very elixir of life - except that this time it is threatening to take the very life it can sustain. Scientifically - and now, probably religiously too - everybody does a short stint at staying in water for almost 9 months till he walks the land and is surrounded by air. But then you would immediately realise that this experience does not mean much without the loving care and protection of the mother. Also her womb was not as deep as her love!
The above scenario, quite approximately (!) sums up the predicament I found myself in - in a water body called Research .
My doctoral research was an attempt at proposing a General Framework towards Lossless reduction of the number of rules in a MISO Fuzzy Rule Based Systems. The usage of the word attempt might remind you of the various media reports on some assassination, or more precisely, an assassination that went awry. Of course, my attempt is way different from it - I actually managed to succeed.
The following gives in brief the problem I attempted to solve and also my approach:
| The Issue | A Fuzzy System consists of a set of Fuzzy If-Then rules of the form Here, X and Y are the input parameters / variables while Z is the output variable. A,B,C are the (linguistic) values they can assume. Assume for a moment that X can assume one of 5 values A1 - A5 while Y can assume one of B1 - B3. Now, X can assume a value Ai independent of Y assuming a value Bj. Thus, taking all the Combinations, we will have a total of 15 different scenarios and an equivalent number of rules. Generalising from the above, if X can take m different values and Y another n different values we will have m x n number of rules covering all the possible scenarios. In the case we have more than 2 input variables, for example, let there be r input variables each capable of assuming any of m_i different values we are looking at a total of m_1 x m_2 x ... x m_r number of rules! For a better appreciation of the above issue, consider the case when we have 3 input variables X_1,X_2 and X_3 and a single output variable Y each of which can assume 5 different (linguistic) values. Then we are looking at a whooping 5 x 5 x 5 = 125 rules! |
| My Approach | But in all these one thing has remained a constant - that of the number of values that the lone output variable can assume. In the event that any of the above scenarios finally lead to one of these countably few values, it is only natural to ask whether we can combine the different input scenarios that lead to the same conclusion. In the above example, even though they are 125 rules the number of distinct conclusions they can lead to are only 5. Thus should it not be possible to combine the rules (actually the antecednts - the parts that precede the THEN in any rule) in such a way that we have only 5 rules left? Well, as do any question it is easier to ask than to answer. My research consisted in proposing a general framework first to model the above rules and then finding the conditions under which the above kind of reduction is possible that is also inference invariant. |