Outline Fuzzy Data Processing Interval Computations Reduction to Interval . . . Towards Fast Algorithms Measurement and . . . for Fuzzy Data Processing: Further Speed-Up Type 1, Type 2, and Beyond Type-2 Fuzzy Case or Beyond min t-Norm How Interval Ideas Travelled from Reduction to Informal . . . Home Page Warszawa, Tokyo, and California Title Page Back to Warszawa ◭◭ ◮◮ Vladik Kreinovich ◭ ◮ Department of Computer Science Page 1 of 45 University of Texas at El Paso, 500 W. University El Paso, Texas 79968, USA, vladik@utep.edu Go Back Many of these results come from joint papers Full Screen with Andrzej Pownuk Close Quit
Outline Fuzzy Data Processing 1. Outline Interval Computations • From the mathematical viewpoint, we can use Zadeh’s Reduction to Interval . . . extension principle to process fuzzy data. Measurement and . . . Further Speed-Up • However, a direct implementation of Zadeh’s extension Type-2 Fuzzy Case principle often requires too many computational steps. Beyond min t-Norm • A known way to speed up computations is to use in- Reduction to Informal . . . terval computations on α -cuts. Home Page • We show that in many cases, we can further reduce Title Page computation time. ◭◭ ◮◮ • The need to decrease computation time is even more ◭ ◮ important for type-2 fuzzy sets. Page 2 of 45 • They require even more computations; we show that Go Back for type-2, a significant speed up is also possible. Full Screen • We also extend the speed-up beyond the min t-norm. Close Quit
Outline Fuzzy Data Processing Part I Interval Computations Fuzzy Data Processing: What Is Reduction to Interval . . . the Problem, and Which Measurement and . . . Algorithms Are Available for Further Speed-Up Type-2 Fuzzy Case Solving This Problem Beyond min t-Norm Reduction to Informal . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 45 Go Back Full Screen Close Quit
Outline Fuzzy Data Processing 2. Why Data Processing and Knowledge Process- Interval Computations ing Are Needed in the First Place Reduction to Interval . . . • Problem: some quantities y are difficult (or impossible) Measurement and . . . to measure or estimate directly. Further Speed-Up Type-2 Fuzzy Case • Solution: indirect measurements or estimates Beyond min t-Norm � x 1 Reduction to Informal . . . ✲ x 2 � Home Page � y = f ( � x 1 , . . . , � x n ) f ✲ ✲ · · · Title Page x n � ◭◭ ◮◮ ✲ ◭ ◮ • Fact: estimates � x i are approximate. Page 4 of 45 def Go Back • Question: how approximation errors ∆ x i = � x i − x i affect the resulting error ∆ y = � y − y ? Full Screen Close Quit
Outline Fuzzy Data Processing 3. From Probabilistic to Interval Uncertainty Interval Computations • Manufacturers of MI provide us with bounds ∆ i on Reduction to Interval . . . measurement errors: | ∆ x i | ≤ ∆ i . Measurement and . . . Further Speed-Up • Thus, we know that x i ∈ [ � x i − ∆ i , � x i + ∆ i ]. Type-2 Fuzzy Case • Often, we also know probabilities, but in 2 cases, we Beyond min t-Norm don’t: Reduction to Informal . . . Home Page – cutting-edge measurements; – cutting-cost manufacturing. Title Page ◭◭ ◮◮ • In such situations: ◭ ◮ – we know the intervals [ x i , x i ] = [ � x i − ∆ i , � x i + ∆ i ] of possible values of x i , and Page 5 of 45 – we want to find the range of possible values of y : Go Back y = [ y, y ] = { f ( x 1 , . . . , x n ) : x 1 ∈ [ x 1 , x 1 ] , . . . , [ x n , x n ] } . Full Screen Close Quit
Outline Fuzzy Data Processing 4. Main Problem of Interval Computations Interval Computations We are given: Reduction to Interval . . . Measurement and . . . • an integer n ; Further Speed-Up • n intervals x 1 = [ x 1 , x 1 ], . . . , x n = [ x n , x n ], and Type-2 Fuzzy Case • an algorithm f ( x 1 , . . . , x n ) which transforms n real Beyond min t-Norm numbers into a real number y = f ( x 1 , . . . , x n ). Reduction to Informal . . . Home Page We need to compute the endpoints y and y of the interval Title Page y = [ y, y ] = { f ( x 1 , . . . , x n ) : x 1 ∈ [ x 1 , x 1 ] , . . . , [ x n , x n ] } . ◭◭ ◮◮ x 1 ◭ ◮ ✲ x 2 Page 6 of 45 y f ✲ ✲ . . . Go Back x n ✲ Full Screen Close Quit
Outline Fuzzy Data Processing 5. Interval Computations: A Brief History Interval Computations • Origins: Archimedes (Ancient Greece) Reduction to Interval . . . Measurement and . . . • Pioneers: Mieczyslaw Warmus (Poland), Teruo Further Speed-Up Sunaga (Japan), Raymond Moore (USA), 1956–59 Type-2 Fuzzy Case • First boom: early 1960s. Beyond min t-Norm • First challenge: taking interval uncertainty into ac- Reduction to Informal . . . Home Page count when planning spaceflights to the Moon. Title Page • Current applications (sample): ◭◭ ◮◮ – design of elementary particle colliders: Martin Berz, Kyoko Makino (USA) ◭ ◮ – will a comet hit the Earth: Berz, Moore (USA) Page 7 of 45 – robotics: Jaulin (France), Neumaier (Austria) Go Back – chemical engineering: Mark Stadtherr (USA) Full Screen Close Quit
Outline Fuzzy Data Processing 6. Need to Process Fuzzy Uncertainty Interval Computations • In many practical situations, we only have expert esti- Reduction to Interval . . . mates for the inputs x i . Measurement and . . . Further Speed-Up • Sometimes, experts provide guaranteed bounds on x i , Type-2 Fuzzy Case and even the probabilities of different values. Beyond min t-Norm • However, such cases are rare. Reduction to Informal . . . Home Page • Usually, the experts’ opinion is described by (impre- cise, “fuzzy”) words from natural language. Title Page • Example: the value x i of the i -th quantity is approxi- ◭◭ ◮◮ mately 1.0, with an accuracy most probably about 0.1. ◭ ◮ • Based on such “fuzzy” information, what can we say Page 8 of 45 about y = f ( x 1 , . . . , x n )? Go Back • The need to process such “fuzzy” information was first Full Screen emphasized in the early 1960s by L. Zadeh. Close Quit
Outline Fuzzy Data Processing 7. How to Describe Fuzzy Uncertainty: Reminder Interval Computations • In Zadeh’s approach, we assign: Reduction to Interval . . . Measurement and . . . – to each number x i , Further Speed-Up – a degree m i ( x i ) ∈ [0 , 1] with which x i is a possible Type-2 Fuzzy Case value of the i -th input. Beyond min t-Norm • In most practical situations, the membership function: Reduction to Informal . . . Home Page – starts with 0, Title Page – continuously ↑ until a certain value, – and then continuously ↓ to 0. ◭◭ ◮◮ ◭ ◮ • Such membership function describe usual expert’s ex- pressions such as “small”, “ ≈ a with an error ≈ σ ”. Page 9 of 45 • Membership functions of this type are actively used in Go Back expert estimates of number-valued quantities. Full Screen • They are thus called fuzzy numbers . Close Quit
Outline Fuzzy Data Processing 8. Processing Fuzzy Data: Formulation of the Interval Computations Problem Reduction to Interval . . . • We know an algorithm y = f ( x 1 , . . . , x n ) that relates: Measurement and . . . Further Speed-Up – the value of the desired difficult-to-estimate quan- Type-2 Fuzzy Case tity y with Beyond min t-Norm – the values of easier-to-estimate auxiliary quantities Reduction to Informal . . . x 1 , . . . , x n . Home Page • We also have expert knowledge about each of the quan- Title Page tities x i . ◭◭ ◮◮ • For each i , this knowledge is described in terms of the corresponding membership function m i ( x i ). ◭ ◮ • Based on this information, we want to find the mem- Page 10 of 45 bership function m ( y ) which describes: Go Back – for each real number y , Full Screen – the degree of confidence that this number is a pos- Close sible value of the desired quantity. Quit
Outline Fuzzy Data Processing 9. Towards Solving the Problem Interval Computations • Intuitively, y is a possible value of the desired quantity Reduction to Interval . . . if for some values x 1 , . . . , x n : Measurement and . . . Further Speed-Up – x 1 is a possible value of the 1st input quantity, Type-2 Fuzzy Case – and x 2 is a possible value of the 2nd input quantity, Beyond min t-Norm – . . . , Reduction to Informal . . . – and y = f ( x 1 . . . , x n ). Home Page • We know: Title Page – that the degree of confidence that x 1 is a possible ◭◭ ◮◮ value of the 1st input quantity is equal to m 1 ( x 1 ), ◭ ◮ – that the degree of confidence that x 2 is a possible Page 11 of 45 value of the 2nd input quantity is equal to m 2 ( x 2 ), – etc. Go Back Full Screen • The degree of confidence d ( y, x 1 , . . . , x n ) in an equality y = f ( x 1 . . . , x n ) is, of course, 1 or 0. Close Quit
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