Need to Use Words Need for Data Processing Need for Computing . . . From a Words-Related . . . Computing with Words: Resulting Fuzzy-Based . . . Towards a New Tuple-Based A Seemingly Natural . . . Main Idea Formalization Main Proposition: For . . . Examples Olga Kosheleva, Vladik Kreinovich Home Page Ariel Garcia, Felipe Jovel, Luis A. Torres Escobedo University of Texas at El Paso Title Page El Paso, Texas 79968, USA ◭◭ ◮◮ contact email vladik@utep.edu ◭ ◮ Thavatchai Ngamsantivong King Mongkut’s Univ. of Technology North Bangkok, Thailand Page 1 of 16 Go Back Full Screen Close Quit
Need to Use Words Need for Data Processing 1. Need to Use Words Need for Computing . . . • Often, to describe height etc., we use words such as From a Words-Related . . . “small”, “medium”, “high”, etc. Resulting Fuzzy-Based . . . A Seemingly Natural . . . • If we only use the selected words w 1 , . . . , w n , we get a Main Idea rather crude description of the quantity. Main Proposition: For . . . • A more accurate description may include several words, Examples with degrees associated with different words. Home Page • Example: rather short, but closer to medium height. Title Page • We can describe this by specifying degrees d i to which ◭◭ ◮◮ the quantity fits each word w i . ◭ ◮ • Then, our opinion of each value is described by a tuple Page 2 of 16 of degrees d = ( d 1 , . . . , d n ) . Go Back Full Screen Close Quit
Need to Use Words Need for Data Processing 2. Need for Data Processing Need for Computing . . . • Often, we are interested in the value of a physical quan- From a Words-Related . . . tity y which is difficult to measure directly. Resulting Fuzzy-Based . . . A Seemingly Natural . . . • For example, we are interested in tomorrow’s temper- Main Idea ature. Main Proposition: For . . . • In such situations, a usual approach is: Examples Home Page – find easier-to-estimate quantities x 1 , . . . , x m related to y by a known dependence y = f ( x 1 , . . . , x m ), and Title Page – to use the estimates of x i to compute the estimate ◭◭ ◮◮ for y . ◭ ◮ • This computation of y based on x 1 , . . . , x m is known as Page 3 of 16 data processing . Go Back Full Screen Close Quit
Need to Use Words Need for Data Processing 3. Need for Computing With Words Need for Computing . . . • When the estimates for x j are given in the form of From a Words-Related . . . tuples d , we face the following problem: Resulting Fuzzy-Based . . . � � A Seemingly Natural . . . – we know the tuples d ( j ) = d ( j ) 1 , . . . , d ( j ) which n Main Idea describes our knowledge about each input x j ; Main Proposition: For . . . – we want to describe the resulting knowledge about Examples y in a similar tuple form. Home Page • In particular: Title Page – we have quantities x 1 and x 2 characterized by the ◭◭ ◮◮ tuples d (1) and d (1) ; ◭ ◮ – we want to compute tuples corresponding to x 1 + x 2 , Page 4 of 16 x 1 − x 2 , x 1 · x 2 , etc. Go Back • In general, instead of computing with numbers, we Full Screen should be able to compute with words (L. Zadeh). Close Quit
Need to Use Words Need for Data Processing 4. How to Represent the Original Words Need for Computing . . . • A natural way to represent the original words in the From a Words-Related . . . computer-understandable form is to use fuzzy logic. Resulting Fuzzy-Based . . . A Seemingly Natural . . . • Usually, the corresponding membership functions µ i ( x ) Main Idea are triangular, and different functions differ by a shift. Main Proposition: For . . . • In precise terms, for some some starting point s and Examples step h , we have Home Page � 0 , 1 − | x − ( s + i · h ) | � Title Page µ i ( x ) = max : h ◭◭ ◮◮ ◭ ◮ µ i ( x ) ✻ ❅ � ❅ � ❅ � ❅ � Page 5 of 16 ❅ � ❅ � ❅ � ❅ � � ❅ � ❅ � ❅ ❅ � Go Back � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ ✲ x Full Screen Close Quit
Need to Use Words Need for Data Processing 5. From a Words-Related Tuple Representation to Need for Computing . . . a Membership Function From a Words-Related . . . • A tuple d = ( d 1 , . . . , d n ) represents a value x if one of Resulting Fuzzy-Based . . . the following conditions hold: A Seemingly Natural . . . Main Idea – the quantity q is characterized by the word w 1 , and Main Proposition: For . . . x satisfies the property described by this word, . . . Examples – the quantity q is characterized by the word w n , and Home Page x satisfies the property described by this word. Title Page • If we use min for “and” and max for “or”, we get ◭◭ ◮◮ µ d ( x ) = max(min( d 1 , µ 1 ( x )) , . . . , min( d n , µ n ( x ))) . ◭ ◮ Page 6 of 16 min( d 1 , µ 1 ( x )), min( d 2 , µ 2 ( x )) ✻ Go Back � ❅ � Full Screen ❅ � � ❅ ❅ � � ❅ ❅ ✲ Close x Quit
Need to Use Words Need for Data Processing 6. Resulting Fuzzy-Based Formalization of Com- Need for Computing . . . puting with Words From a Words-Related . . . • We know the tuples d (1) and d (2) describing the two Resulting Fuzzy-Based . . . quantities x 1 and x 2 ; then: A Seemingly Natural . . . Main Idea – first, we generate membership functions µ (1) ( x 1 ), µ (2) ( x 2 ) corresponding to the tuples d (1) , d (2) ; Main Proposition: For . . . Examples – then, we use Zadeh’s extension principle to com- Home Page pute the membership f-n µ ( x ) corr. to y = f ( x 1 , x 2 ); Title Page – finally, we generate the tuple d corresponding to the resulting membership function µ ( x ). ◭◭ ◮◮ ◭ ◮ • To implement this idea, we need to generate a tuple corresponding to a given membership function. Page 7 of 16 Go Back Full Screen Close Quit
Need to Use Words Need for Data Processing 7. A Seemingly Natural Idea and Its Limitations Need for Computing . . . • We look for the degree d i to which it’s possible that: From a Words-Related . . . Resulting Fuzzy-Based . . . – a quantity described by a membership function µ ( x ) A Seemingly Natural . . . – is in agreement with w i . Main Idea • This means that some value x is in agreement with the Main Proposition: For . . . membership function and with the word w i . Examples Home Page • If we use min for “and” and max for “or”, we get Title Page d ′ i = max x (min( µ ( x ) , µ i ( x ))) . ◭◭ ◮◮ • Example: we start with the word w i , i.e., with the ◭ ◮ tuple d = (0 , . . . , 0 , 1 , 0 , . . . , 0). Page 8 of 16 • Then, for f ( x ) = x , we would like to get d back. Go Back Full Screen Close Quit
Need to Use Words Need for Data Processing 8. A Seemingly Natural Idea and Its Limitations Need for Computing . . . (cont-d) From a Words-Related . . . • We start with the word w i : d = (0 , . . . , 0 , 1 , 0 , . . . , 0). Resulting Fuzzy-Based . . . A Seemingly Natural . . . • We compute d ′ i = max x (min( µ ( x ) , µ i ( x ))) , and get Main Idea d ′ i = (0 , . . . , 0 , 0 . 5 , 1 , 0 . 5 , 0 . . . , 0) � = d : Main Proposition: For . . . Examples Home Page µ i ( x ), µ i +1 ( x ) ✻ �❅ �❅ Title Page � ❅ � ❅ � � ❅ ❅ ◭◭ ◮◮ � � ❅ ❅ � � ❅ ❅ ✲ x ◭ ◮ min( µ i ( x ) , µ i +1 ( x )) ✻ Page 9 of 16 Go Back d ′ i +1 = 0 . 5 � ❅ � ❅ Full Screen � ❅ ✲ x Close Quit
Need to Use Words Need for Data Processing 9. Main Idea Need for Computing . . . • Problem: membership f-s µ i ( x ) and µ i +1 ( x ) intersect. From a Words-Related . . . Resulting Fuzzy-Based . . . • Solution: remove the intersecting parts, i.e., take A Seemingly Natural . . . µ ′ i ( x ) = max(0 , µ i ( x ) − max( µ i − 1 ( x ) , µ i +1 ( x ))) : Main Idea Main Proposition: For . . . µ ′ i ( x ) Examples ✻ ✁❆ � ❅ � ❅ � ❅ Home Page ✁ ❆ � ❅ � ❅ � ❅ � ❅ � ✁ ❆ � ❅ ❅ Title Page � � ❅ ✁ � ❆ ❅ ❅ � � ✁ ❅ � ❆ ❅ ❅ ✲ x ◭◭ ◮◮ • The reduced functions µ ′ ◭ ◮ i ( x ) no longer overlap: Page 10 of 16 µ ′ 1 ( x ), µ ′ 2 ( x ), . . . ✻ Go Back ❆ ✁❆ ✁❆ ✁❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Full Screen ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆✁ ✲ Close x Quit
Need to Use Words Need for Data Processing 10. Main Idea (cont-d) Need for Computing . . . • Instead of the original functions µ i ( x ), we compute the From a Words-Related . . . reduced functions Resulting Fuzzy-Based . . . A Seemingly Natural . . . µ ′ i ( x ) = max(0 , µ i ( x ) − max( µ i − 1 ( x ) , µ i +1 ( x ))) . Main Idea Main Proposition: For . . . • Similarly, instead of the membership function µ ( x ), we compute the reduced function Examples Home Page µ ′ ( x ) = max(0 , µ ( x ) − max( µ i − 1 ( x ) , µ i +1 ( x ))) . Title Page • Then, we compute the degrees based on these reduced ◭◭ ◮◮ functions, as ◭ ◮ d ′ x (min( µ ′ ( x ) , µ ′ i = max i ( x ))) . Page 11 of 16 Go Back Full Screen Close Quit
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