Functional . . . Need for Polynomial . . . How to Represent . . . Bernstein . . . Why Fuzzy Interpretation of . . . Bernstein Polynomials Fuzzy Interpretation . . . How Can We . . . Are Better: Fuzzy Analysis (cont-d) Why Bernstein . . . Fuzzy-Inspired Home Page Justification Title Page ◭◭ ◮◮ Jaime Nava, Olga Kosheleva, and Vladik Kreinovich ◭ ◮ Page 1 of 18 University of Texas at El Paso El Paso, TX 79968, USA Go Back nava.jaime@gmail.com, olgak@utep.edu, vladik@utep.edu Full Screen Close Quit
Functional . . . Need for Polynomial . . . 1. Functional Dependencies are Ubiquitous How to Represent . . . • In practice, we are often interested in a quantity y Bernstein . . . which is difficult to measure directly. Fuzzy Interpretation of . . . Fuzzy Interpretation . . . • Examples: distance to a faraway star, amount of oil in How Can We . . . a well. Fuzzy Analysis (cont-d) • To estimate such quantities, we: Why Bernstein . . . Home Page – measure related quantities x 1 , . . . , x n , and – use the known dependence y = f ( x 1 , . . . , x n ) to es- Title Page timate y . ◭◭ ◮◮ • To predict a future value y of a quantity, we use the ◭ ◮ known relation between y and current values x i . Page 2 of 18 • In all these cases, we need, given x i , to compute Go Back y = f ( x 1 , . . . , x n ) . Full Screen Close Quit
Functional . . . Need for Polynomial . . . 2. Need for Polynomial (and Piece-Wise Polyno- How to Represent . . . mial) Approximations Bernstein . . . • In a computer, only addition, subtraction, and multi- Fuzzy Interpretation of . . . plication are hardware supported. Fuzzy Interpretation . . . How Can We . . . • All other operations with real numbers, including divi- Fuzzy Analysis (cont-d) sion, are implemented as a sequence of +, − , · . Why Bernstein . . . • A composition of +, − , · is a polynomial. Home Page • Also hardware supported are logical operations; so, we Title Page can have different expressions in different sub-domains. ◭◭ ◮◮ • Thus, whatever we want to compute, we are computing ◭ ◮ a (piece-wise) polynomial function (spline). Page 3 of 18 • In other words, every computation means that we (lo- Go Back cally) approximate functions by polynomials. Full Screen Close Quit
Functional . . . Need for Polynomial . . . 3. Possibility of a Polynomial Approximation How to Represent . . . • The possibility to approximate functions by polynomi- Bernstein . . . als was first proven by Weierstrass in 19 cent. Fuzzy Interpretation of . . . Fuzzy Interpretation . . . • Specifically, Weierstrass showed that: How Can We . . . – for every continuous function f ( x 1 , . . . , x n ), Fuzzy Analysis (cont-d) – for every box [ a 1 , b 1 ] × . . . × [ a n , b n ] , Why Bernstein . . . – for every real number ε > 0, Home Page – there exists a polynomial P ( x 1 , . . . , x n ) which is, on Title Page this box, ε -close to the original function f ( x 1 , . . . , x n ): ◭◭ ◮◮ | P ( x 1 , . . . , x n ) − f ( x 1 , . . . , x n ) | ≤ ε for all x i ∈ [ a i , b i ] . ◭ ◮ • Polynomial approximations are ubiquitous in physics : Page 4 of 18 – many dependencies are analytical, i.e., representable Go Back as Taylor series, Full Screen – so, to get a good approximation, we keep a few first terms in the Taylor series. Close Quit
Functional . . . Need for Polynomial . . . 4. How to Represent Polynomials in a Computer: How to Represent . . . Traditional Approach Bernstein . . . • In the physics case, it is natural to write Fuzzy Interpretation of . . . f ( x ) = c 0 + c 1 · x + c 2 · x 2 + . . . + c d · x d . Fuzzy Interpretation . . . How Can We . . . Fuzzy Analysis (cont-d) • So, we can represent this polynomial by its coefficients Why Bernstein . . . c 0 , c 1 , c 2 , . . . , c d . Home Page Title Page • A general polynomial has the form � ◭◭ ◮◮ c d 1 ...d n · x d 1 1 · . . . · x d n f ( x 1 , . . . , x n ) = n . ◭ ◮ d 1 ,...,d n Page 5 of 18 • It is natural to represent it as a corresponding multi-D array of coefficients c d 1 ...d n . Go Back • This is how polynomials are traditionally represented. Full Screen Close Quit
Functional . . . Need for Polynomial . . . 5. Bernstein Polynomials: an Alternative Com- How to Represent . . . puter Representation Bernstein . . . • In practice, alternative computer representations often Fuzzy Interpretation of . . . work better. Fuzzy Interpretation . . . How Can We . . . • For example, Bernstein proposed to represent a poly- Fuzzy Analysis (cont-d) nomial as � Why Bernstein . . . f ( x 1 , . . . , x n ) = c k 1 ...k n · p k 1 1 ( x 1 ) · . . . · p k n n ( x n ) , Home Page k 1 ...k n Title Page def = ( x i − a i ) k i · ( b i − x i ) d − k i . where p k i i ( x i ) ◭◭ ◮◮ • In this representation, in the computer, we store the ◭ ◮ coefficients c k 1 ...k n . Page 6 of 18 • Bernstein polynomials are often successful, but they Go Back are the least studied. Full Screen • Our objective is to study them. Close Quit
Functional . . . Need for Polynomial . . . 6. Bernstein Polynomials Are Useful in Interval How to Represent . . . and Fuzzy Computations Bernstein . . . • Bernstein polynomials are useful in interval computa- Fuzzy Interpretation of . . . tions , when we compute the range Fuzzy Interpretation . . . def How Can We . . . f ([ a 1 , b 1 ] , . . . , [ a n , b n ]) = Fuzzy Analysis (cont-d) { f ( x 1 , . . . , x n ) : x 1 ∈ [ a 1 , b 1 ] , . . . , x n ∈ [ a n , b n ] } . Why Bernstein . . . Home Page • Interval computations are used in fuzzy computations : Title Page – we know fuzzy values µ i ( x i ) of the inputs x 1 , . . . , x n , – we want to find the fuzzy value of y = f ( x 1 , . . . , x n ). ◭◭ ◮◮ ◭ ◮ • It is known that for every α , the α -cut for y is the range of f ( x 1 , . . . , x n ) over α -cuts x i ( α ): Page 7 of 18 y ( α ) = f ( x 1 ( α ) , . . . , x n ( α )) . Go Back Full Screen • This is how fuzzy computations are usually performed: we perform interval computations over the α -cuts. Close Quit
Functional . . . Need for Polynomial . . . 7. Bernstein Polynomials: Explaining Success How to Represent . . . • Fact: Bernstein polynomials are successful. Bernstein . . . Fuzzy Interpretation of . . . • Problem: there is no clear explanation for this success. Fuzzy Interpretation . . . • Idea: use fuzzy logic. How Can We . . . • Preliminary step: to make fuzzy interpretation possi- Fuzzy Analysis (cont-d) ble, let us reduce each interval [ a, b ] to [0 , 1]: Why Bernstein . . . Home Page – when x ∈ [ a, b ], = x − a Title Page def – we take t b − a ∈ [0 , 1]. ◭◭ ◮◮ • For each f ( x 1 , . . . , x n ), we take ◭ ◮ F ( t 1 , . . . , t n ) = f ( a 1 + t 1 · ( b 1 − a 1 ) , . . . , a n + t n · ( b n − a n )) . Page 8 of 18 • Then, once we get an approximation � F for F , we can Go Back approximate f ( x 1 , . . . , x n ) by � x 1 − a 1 � Full Screen , . . . , x n − a n f ( x 1 , . . . , x n ) = � � F . Close b 1 − a 1 b n − a n Quit
Functional . . . Need for Polynomial . . . 8. Fuzzy Approximations: Reminder How to Represent . . . • Fuzzy logic has a natural approach for approximation Bernstein . . . – rules: Fuzzy Interpretation of . . . Fuzzy Interpretation . . . “if x 1 is P 1 , x 2 is P 2 , . . . , and x n is P n , then y = c .” How Can We . . . Fuzzy Analysis (cont-d) • For each rule k , the degree to which this rule is acti- vated is: Why Bernstein . . . Home Page d k = f & ( µ 1 ( x 1 ) , . . . , µ n ( x n )) . Title Page • In the simplest case f & ( a, b ) = a · b , we have: ◭◭ ◮◮ d k = µ 1 ( x 1 ) · . . . · µ n ( x n ) . ◭ ◮ • The conclusions c k of several rules can be combined as Page 9 of 18 a weighted average Go Back c 1 · d 1 + . . . + c r · d r . Full Screen Close Quit
Functional . . . Need for Polynomial . . . 9. Fuzzy Interpretation of the Usual Polynomial Representation c 0 + c 1 · x + c 2 · x 2 + . . . + c m · x m How to Represent . . . Bernstein . . . • This representation corresponds to fuzzy rules: Fuzzy Interpretation of . . . Fuzzy Interpretation . . . • c 0 (with no condition); How Can We . . . • if x , then c 1 ; Fuzzy Analysis (cont-d) • if x 2 , then c 2 ; . . . Why Bernstein . . . • If we take x as the degree to which x ∈ [0 , 1] is large, Home Page then x 2 is usually interpreted as “very large”, etc. Title Page • Thus, the above rules have the form: ◭◭ ◮◮ • c 0 ; ◭ ◮ • if x is large, then c 1 ; Page 10 of 18 • if x is very large, then c 2 , etc. Go Back • Similarly, c 012 · x 2 · x 2 3 means the following rule: Full Screen “if x 2 is large and x 3 is very large, then c 012 .” Close Quit
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