How to Transform Partial From the Idea to an . . . Order Between - PowerPoint PPT Presentation

Why Fuzzy Logic: A . . . Sometimes, the . . . Towards Formulating . . . Main Idea How to Transform Partial From the Idea to an . . . Order Between Degrees into General Case Examples (cont-d) Numerical Values Interval-Valued Degrees Remaining Open . . . Olga Kosheleva, Vladik Kreinovich, Home Page Joe Lorkowski, and Martha Osegueda Escobar Title Page University of Texas at El Paso ◭◭ ◮◮ El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu, ◭ ◮ lorkowski@computer.org, mcoseguedaescobar@miners.utep.edu Page 1 of 22 Go Back Full Screen Close Quit

Why Fuzzy Logic: A . . . Sometimes, the . . . 1. Why Fuzzy Logic: A Brief Reminder Towards Formulating . . . • In many practical situations, there are experts who are Main Idea skilled in performing the corresponding task: From the Idea to an . . . General Case – skilled machine operators successfully operate ma- Examples (cont-d) chinery, Interval-Valued Degrees – skilled medical doctors successfully cure patients, Remaining Open . . . etc. Home Page • It is desirable to design automated systems that would Title Page help less skilled operators and doctors make proper de- cisions. ◭◭ ◮◮ • It is important to incorporate the knowledge of the ◭ ◮ experts into these system. Page 2 of 22 • Some of this expert knowledge can be described in pre- Go Back cise (“crisp”) form. Full Screen • Such knowledge is relative easy to describe in precise Close computer-understandable terms. Quit

Why Fuzzy Logic: A . . . Sometimes, the . . . 2. Why Fuzzy Logic (cont-d) Towards Formulating . . . • However, a significant part of human knowledge is de- Main Idea scribed in imprecise (“fuzzy”) terms like “small”. From the Idea to an . . . General Case • One of the main objectives of fuzzy logic is to translate Examples (cont-d) this knowledge into machine-understandable form. Interval-Valued Degrees • Zadeh proposed to describe, for each imprecise state- Remaining Open . . . ment, a degree to which this statement is true. Home Page • Intuitively, we often describe such degrees by using Title Page words from natural language, such as “very small”. ◭◭ ◮◮ • However, computers are not very good in precessing ◭ ◮ natural-language terms. Page 3 of 22 • Computers are more efficient in processing numbers. Go Back • So, fuzzy techniques first translate the corresponding Full Screen degrees into numbers from the interval [0 , 1]. Close Quit

Why Fuzzy Logic: A . . . Sometimes, the . . . 3. Sometimes, the Corresponding Degrees Are Towards Formulating . . . Difficult to Elicit Main Idea • Some experts can easily describe their degrees in terms From the Idea to an . . . of numbers. General Case Examples (cont-d) • Other experts are more comfortable describing degrees Interval-Valued Degrees in natural-language terms. Remaining Open . . . • In this case, we need to translate the resulting terms Home Page into numbers from the interval [0 , 1]. Title Page • What information can we use for this translation? ◭◭ ◮◮ • For some pairs of degrees, we know which degree cor- ◭ ◮ responds to a larger confidence. Page 4 of 22 • For example, it is clear that “very small” is smaller than “somewhat small”. Go Back Full Screen • It is reasonable to assume that these expert compar- isons are transitive and cycle-free. Close Quit

Why Fuzzy Logic: A . . . Sometimes, the . . . 4. Towards Formulating the Problem Towards Formulating . . . • Thus, we usually have a natural (partial) order relation Main Idea between different degrees. From the Idea to an . . . General Case • This order is not necessarily total (linear): we may Examples (cont-d) have two degrees with no relation between them, e.g., Interval-Valued Degrees • “reasonably small” and Remaining Open . . . • “to some extent small”. Home Page • Thus, in general, this relation is a partial order . Title Page ◭◭ ◮◮ • We would like to assign numbers from the interval [0 , 1] to different elements from a partially ordered set. ◭ ◮ • Of course, there are many such possible assignments. Page 5 of 22 • Our goal is to select the assignment which is, in some Go Back sense, the most reasonable. Full Screen Close Quit

Why Fuzzy Logic: A . . . Sometimes, the . . . 5. Main Idea Towards Formulating . . . • Let us number the elements of the original finite par- Main Idea tially ordered set by numbers 1, 2, . . . , k . From the Idea to an . . . General Case • Then we get the set { 1 , 2 , . . . , k } with some partial or- Examples (cont-d) der ≺ . Interval-Valued Degrees • This order is, in general, different from the natural Remaining Open . . . order < . Home Page • The desired mapping means that we assign, to each of Title Page the numbers i from 1 to k , a real number x i ∈ [0 , 1]. ◭◭ ◮◮ • In other words, we produce a tuple x = ( x 1 , . . . , x k ) of ◭ ◮ real numbers from the interval [0 , 1]. Page 6 of 22 • The only restriction on this tuple is that if i ≺ j , then x i < x j . Go Back Full Screen • Let us denote the set of all the tuples x that satisfy this restriction by S ≺ . Close Quit

Why Fuzzy Logic: A . . . Sometimes, the . . . 6. Main Idea (cont-d) Towards Formulating . . . • Out of many possible tuples from the set S ≺ , we would Main Idea like to select one s = ( s 1 , . . . , s k ). From the Idea to an . . . General Case • Which one should we select? Examples (cont-d) • Selecting a tuple means that we need to select, for each Interval-Valued Degrees i , the corresponding value s i . Remaining Open . . . • The ideally-matching tuple x has, in general, a different Home Page value x i � = s i . Title Page • It usually makes sense to describe the inaccuracy ◭◭ ◮◮ (“loss”) of this selection by the square ( s i − x i ) 2 . ◭ ◮ • We do not know what is the ideal value x i . Page 7 of 22 • We only know that this ideal value is the i -th compo- Go Back nent of some tuple x ∈ S ≺ . Full Screen • We have no reason to believe that some tuples are more probable than the others. Close Quit

Why Fuzzy Logic: A . . . Sometimes, the . . . 7. Main Idea (final) Towards Formulating . . . • We have no reason to believe that some tuples are more Main Idea probable than the others. From the Idea to an . . . General Case • As a result, it makes sense to consider them all equally Examples (cont-d) probable. Interval-Valued Degrees • So, if we select the tuple s , then the expected loss is Remaining Open . . . S ≺ ( x i − s i ) 2 dx. � proportional to Home Page • It is therefore reasonable to select a value s i for which Title Page this loss is the smallest possible: ◭◭ ◮◮ � ( x i − s i ) 2 dx → min . ◭ ◮ s S ≺ Page 8 of 22 Go Back Full Screen Close Quit

Why Fuzzy Logic: A . . . Sometimes, the . . . 8. From the Idea to an Algorithm Towards Formulating . . . • Our objective is to come up with numbers describing Main Idea expert degrees. From the Idea to an . . . General Case • So, we need a simple algorithm transforming a partial Examples (cont-d) order into numerical values. Interval-Valued Degrees • Let us differentiate the objective function with respect Remaining Open . . . to s i and equate the resulting derivative to 0. Home Page � • As a result, we get S ≺ ( s i − x i ) dx = 0 , hence Title Page s i = N � � ◭◭ ◮◮ def def D, where N = = x i dx, D dx. ◭ ◮ S ≺ S ≺ Page 9 of 22 • Since ≺ is a partial order, we may have tuples ( x 1 , . . . , x k ) with different orderings between x i . Go Back • For example, if we know only that 1 ≺ 2 and 1 ≺ 3, Full Screen then we can have 1 ≺ 2 ≺ 3 and 1 ≺ 3 ≺ 2. Close Quit

Why Fuzzy Logic: A . . . Sometimes, the . . . 9. From the Idea to an Algorithm (cont-d) Towards Formulating . . . • In principle, we can also have equalities between x i , Main Idea but such have 0 volume. From the Idea to an . . . General Case • There are k ! possible linear orders ℓ between x i . Examples (cont-d) • Let us denote the set of all the tuples with an order ℓ Interval-Valued Degrees by T ℓ . Remaining Open . . . • Then, each set S ≺ is the union of the sets T ℓ for all Home Page linear orders ℓ extending ≺ : S ≺ = � T ℓ . Title Page ℓ : ℓ ⊇≺ ◭◭ ◮◮ • Thus, each of the integrals N and D over S ≺ can be represented as the sum of integrals over the sets T ℓ : ◭ ◮ � def � Page 10 of 22 D = D ℓ , where D ℓ = x i dx, T ℓ ℓ : ℓ ⊇≺ Go Back � def � Full Screen N = N ℓ , where N ℓ = dx. T ℓ ℓ : ℓ ⊇≺ Close Quit