From: nuptek@sympatico.ca   
      
   James Andrews wrote:   
   > 1. How do I convert from an intersection rule matrix to the Combs   
   > Method?   
   >   
   > Simple answer: you don't. Don't write fuzzy rules using the old   
   > method and then try to convert them to the Combs Method rules. Look at   
   > each of your inputs, determine how they affect each of your outputs,   
   > and then write your rules accordingly. Construct your rule base this   
   > way even if you already have a fuzzy system written using the old   
   > method. If you try to write the fuzzy rules using the old way and then   
   > rewrite them using the Combs Method, you are probably going to get   
   > bogged down in a big intersection rule matrix. For example, if you have   
   > 6 inputs with 5 ranges each, you are going to need to write 15,625   
   > intersection rules! Start with a Combs Method approach and you will   
   > only need to write 30 rules.   
   >   
   >   
   > 2. I have written an intersection based rule matrix and also a Combs   
   > Method rule base to solve the same problem and the numerical results   
   > are not exactly the same. Is there something wrong with the Combs   
   > Method output?   
   >   
   > No. There can be any number of reasons why the numerical results are   
   > not exactly the same for both methods. Remember, the Combs Method is   
   > based on the logical equivalence between the rule structures, not   
   > numerical equality. Probably the most important reason for a difference   
   > is that the Combs Method does not "plateau." Anyone who has written   
   > more than a few intersection rule matrices has run in to the problem of   
   > trying to decide what result to put in a cell that falls between   
   > adjacent output sets. For example, if you have three cells in a row so   
   > that the first result cell should have a value of low and the third   
   > cell should have a result of medium low, what do you put in the second   
   > (middle) cell? Depending on the design of your fuzzy system (you   
   > don't have an output value between low and medium low), you will end   
   > up putting either low or medium low in the second cell. But that means   
   > that your output will plateau at low or medium low even though the   
   > input has changed. You could try to increase the number of output sets,   
   > but that increases the complexity of your system. The Combs Method does   
   > not plateau and the output values always move smoothly from one set to   
   > the next. A Combs Method fuzzy system will sometimes produce slightly   
   > different numerical results from an intersection based fuzzy system,   
   > but only because the Combs Method is intrinsically more accurate.   
   >   
   > 3. I've got a pair of correlated inputs for which I can't figure   
   > out how to write Combs Method rules. How can I write Combs Method rules   
   > to solve this problem?   
   >   
   > The essence of your problem goes something like this: If both inputs   
   > are high or both inputs are low, then the output is low, but if one   
   > input is high and the other low, then the output is high. What you have   
   > run in to is the old XOR (exclusive or) problem from neural networks.   
   > You could create another (hidden) layer of logic which is also a   
   > solution that works for neural networks, but that adds complexity to   
   > the fuzzy system defeating your purpose for using the Combs Method. The   
   > key to solving this problem is to recognize the difference between an   
   > input to a fuzzy system and an input to the inference step of a fuzzy   
   > system. The two inputs are separate inputs to the fuzzy system, but it   
   > is the relationship between the two inputs that is the input to the   
   > inference step. The two rules for the inference step of the Combs   
   > Method to solve the XOR problem are: "If the difference between the   
   > inputs is low, then the output is low. If the difference between the   
   > inputs is high then the output is high." This solves the problem and   
   > keeps the number of rules down to a reasonable level which is the   
   > essence of the Combs Method. Ironically, this solution carried to an   
   > extreme by an intersection rule matrix, is the main weakness of an   
   > intersection rule matrix: separate rules for every possible   
   > relationship amongst all the inputs.   
   >   
   > 4. Bart Kosko's Fuzzy Approximation Theorem proves that additive   
   > fuzzy systems can approximate any continuous function. Is there a proof   
   > that demonstrates that the Combs Method can also approximate any   
   > continuous function?   
   >   
   > Yes, Bart Kosko's Fuzzy Approximation Theorem. The Combs Method is   
   > an additive fuzzy system and is therefore covered by Dr. Kosko's   
   > proof. The proof does not specify that the input subsets of a fuzzy   
   > system have to be related by intersection. See Bart Kosko's book,   
   > Fuzzy Engineering.   
   >   
      
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