Why Data Processing . . . From Probabilistic to . . . Need to Process Fuzzy . . . Os Lus´ ıadas Reduction to Interval . . . Need for Type-2 Fuzzy . . . of Computations Interval-Valued Fuzzy . . . Fast Algorithms for . . . under Uncertainty: New Result: Extension . . . from Probabilities Acknowledgments to Intervals to Fuzzy Home Page to Interval-Valued Fuzzy Title Page and Beyond ◭◭ ◮◮ ◭ ◮ Vladik Kreinovich Page 1 of 38 Department of Computer Science University of Texas at El Paso Go Back El Paso, TX 79968, USA Full Screen Email: vladik@utep.edu http://www.cs.utep.edu/vladik Close http://www.cs.utep.edu/interval-comp Quit
Why Data Processing . . . From Probabilistic to . . . 1. Poetic Introduction Need to Process Fuzzy . . . • There is a lot of uncertainty in our knowledge. Reduction to Interval . . . Need for Type-2 Fuzzy . . . • In the glorious past, explorers sailed into the unknown Interval-Valued Fuzzy . . . seas and brought forth new knowledge. Fast Algorithms for . . . • As a result of their efforts, the whole Earth has been New Result: Extension . . . thoroughly mapped. Acknowledgments Home Page • However, there are many areas which are as uncertain as the unknown lands were in the old days. Title Page • For exploring the microworlds of cells and atoms, the ◭◭ ◮◮ macroworlds of galaxies, our “ships” are computers. ◭ ◮ • Data processing under uncertainty – this is how we Page 2 of 38 bring new knowledge about our world. Go Back • We will try to show that data processing under uncer- Full Screen tainty can be as exciting as sea voyages of yore. Close Quit
Why Data Processing . . . From Probabilistic to . . . 2. Why Data Processing and Knowledge Process- Need to Process Fuzzy . . . ing Are Needed in the First Place Reduction to Interval . . . • Problem: some quantities y are difficult (or impossible) Need for Type-2 Fuzzy . . . to measure or estimate directly. Interval-Valued Fuzzy . . . Fast Algorithms for . . . • Solution: indirect measurements or estimates New Result: Extension . . . � x 1 Acknowledgments ✲ x 2 � Home Page � y = f ( � x 1 , . . . , � x n ) f ✲ ✲ · · · Title Page x n � ◭◭ ◮◮ ✲ ◭ ◮ • Fact: estimates � x i are approximate. Page 3 of 38 def Go Back • Question: how approximation errors ∆ x i = � x i − x i affect the resulting error ∆ y = � y − y ? Full Screen Close Quit
Why Data Processing . . . From Probabilistic to . . . 3. From Probabilistic to Interval Uncertainty Need to Process Fuzzy . . . • Manufacturers of MI provide us with bounds ∆ i on Reduction to Interval . . . measurement errors: | ∆ x i | ≤ ∆ i . Need for Type-2 Fuzzy . . . Interval-Valued Fuzzy . . . • Thus, we know that x i ∈ [ � x i − ∆ i , � x i + ∆ i ]. Fast Algorithms for . . . • Often, we also know probabilities, but in 2 cases, we New Result: Extension . . . don’t: Acknowledgments 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 4 of 38 – 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
Why Data Processing . . . From Probabilistic to . . . 4. Main Problem of Interval Computations Need to Process Fuzzy . . . We are given: Reduction to Interval . . . Need for Type-2 Fuzzy . . . • an integer n ; Interval-Valued Fuzzy . . . • n intervals x 1 = [ x 1 , x 1 ], . . . , x n = [ x n , x n ], and Fast Algorithms for . . . • an algorithm f ( x 1 , . . . , x n ) which transforms n real num- New Result: Extension . . . bers into a real number y = f ( x 1 , . . . , x n ). Acknowledgments 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 5 of 38 y f ✲ ✲ . . . Go Back x n ✲ Full Screen Close Quit
Why Data Processing . . . From Probabilistic to . . . 5. Need to Process Fuzzy Uncertainty Need to Process Fuzzy . . . • In many practical situations, we only have expert esti- Reduction to Interval . . . mates for the inputs x i . Need for Type-2 Fuzzy . . . Interval-Valued Fuzzy . . . • Sometimes, experts provide guaranteed bounds on x i , Fast Algorithms for . . . and even the probabilities of different values. New Result: Extension . . . • However, such cases are rare. Acknowledgments 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 6 of 38 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
Why Data Processing . . . From Probabilistic to . . . 6. How to Describe Fuzzy Uncertainty: Reminder Need to Process Fuzzy . . . • In Zadeh’s approach, we assign: Reduction to Interval . . . Need for Type-2 Fuzzy . . . – to each number x i , Interval-Valued Fuzzy . . . – a degree m i ( x i ) ∈ [0 , 1] with which x i is a possible Fast Algorithms for . . . value of the i -th input. New Result: Extension . . . • In most practical situations, the membership function: Acknowledgments 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 7 of 38 • 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
Why Data Processing . . . From Probabilistic to . . . 7. Processing Fuzzy Data: Formulation of the Prob- Need to Process Fuzzy . . . lem Reduction to Interval . . . • We know an algorithm y = f ( x 1 , . . . , x n ) that relates: Need for Type-2 Fuzzy . . . Interval-Valued Fuzzy . . . – the value of the desired difficult-to-estimate quan- Fast Algorithms for . . . tity y with New Result: Extension . . . – the values of easier-to-estimate auxiliary quantities Acknowledgments 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 8 of 38 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
Why Data Processing . . . From Probabilistic to . . . 8. Towards Solving the Problem Need to Process Fuzzy . . . • Intuitively, y is a possible value of the desired quantity Reduction to Interval . . . if for some values x 1 , . . . , x n : Need for Type-2 Fuzzy . . . Interval-Valued Fuzzy . . . – x 1 is a possible value of the 1st input quantity, Fast Algorithms for . . . – and x 2 is a possible value of the 2nd input quantity, New Result: Extension . . . – . . . , Acknowledgments – 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 9 of 38 value of the 2nd input quantity is equal to m 2 ( x 2 ), etc. Go Back • The degree of confidence d ( y, x 1 , . . . , x n ) in an equality Full Screen y = f ( x 1 . . . , x n ) is, of course, 1 or 0. Close Quit
Why Data Processing . . . From Probabilistic to . . . 9. Towards Solving the Problem (cont-d) Need to Process Fuzzy . . . • The simplest way to represent “and” is to use min. Reduction to Interval . . . Need for Type-2 Fuzzy . . . • Thus, for each combination of values x 1 , . . . , x n , the Interval-Valued Fuzzy . . . degree of confidence d in a composite statement Fast Algorithms for . . . “ x 1 is a possible value of the 1st input quantity, and New Result: Extension . . . x 2 is a possible value of the 2nd input quantity, . . . , Acknowledgments and y = f ( x 1 . . . , x n )” Home Page is equal to Title Page d = min( m 1 ( x 1 ) , m 2 ( x 2 ) , . . . , d ( y, x 1 , . . . , x n )) . ◭◭ ◮◮ • We can simplify this expression if we consider two pos- ◭ ◮ sible cases: Page 10 of 38 – when y = f ( x 1 . . . , x n ), we get Go Back d = min( m 1 ( x 1 ) , m 2 ( x 2 ) , . . . , d ( y, x 1 , . . . , x n )); Full Screen – otherwise, we get d = 0. Close Quit
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