Symmetry: a . . . Basic Symmetries: . . . Basic Nonlinear . . . Symmetries: Symmetries Explain . . . Symmetries Explain . . . A General Approach to What Else We Do in . . . Integrated Uncertainty First Example: . . . Second Example: . . . Management Towards an Optimal . . . Title Page Vladik Kreinovich 1 , 2 , Hung T. Nguyen 2 , 3 , ◭◭ ◮◮ and Songsak Sriboonchitta 2 ◭ ◮ 1 University of Texas at El Paso, USA, vladik@utep.edu Page 1 of 29 2 Faculty of Economics, Chiang Mai University, Thailand 3 New Mexico State University, Las Cruces, USA Go Back hunguyen@nmsu.edu Full Screen Close Quit
Symmetry: a . . . 1. Symmetry: a Fundamental Property of the Physi- Basic Symmetries: . . . cal World Basic Nonlinear . . . Symmetries Explain . . . • One of the main objectives of science: prediction. Symmetries Explain . . . • Basis for prediction: we observed similar situations in What Else We Do in . . . the past, and we expect similar outcomes. First Example: . . . • In mathematical terms: similarity corresponds to sym- Second Example: . . . metry , and similarity of outcomes – to invariance. Towards an Optimal . . . Title Page • Example: we dropped the ball, it fall down. ◭◭ ◮◮ • Symmetries: shift, rotation, etc. ◭ ◮ • In modern physics: theories are usually formulated in Page 2 of 29 terms of symmetries (not diff. equations). Go Back • Natural idea: let us use symmetry to describe uncer- Full Screen tainty as well. Close Quit
Symmetry: a . . . 2. Basic Symmetries: Scaling and Shift Basic Symmetries: . . . Basic Nonlinear . . . • Typical situation: we deal with the numerical values of Symmetries Explain . . . a physical quantity. Symmetries Explain . . . • Numerical values depend on the measuring unit. What Else We Do in . . . • Scaling: if we use a new unit which is λ times smaller, First Example: . . . numerical values are multiplied by λ : x → λ · x . Second Example: . . . Towards an Optimal . . . • Example: x meters = 100 · x cm. Title Page • Another possibility: change the starting point. ◭◭ ◮◮ • Shift: if we use a new starting point which is s units ◭ ◮ before, then x → x + s (example: time). Page 3 of 29 • Together, scaling and shifts form linear transforma- Go Back tions x → a · x + b . Full Screen • Invariance: physical formulas should not depend on Close the choice of a measuring unit or of a starting point. Quit
Symmetry: a . . . 3. Basic Nonlinear Symmetries Basic Symmetries: . . . Basic Nonlinear . . . • Sometimes, a system also has nonlinear symmetries. Symmetries Explain . . . • If a system is invariant under f and g , then: Symmetries Explain . . . – it is invariant under their composition f ◦ g , and What Else We Do in . . . – it is invariant under the inverse transformation f − 1 . First Example: . . . Second Example: . . . • In mathematical terms, this means that symmetries Towards an Optimal . . . form a group . Title Page • In practice, at any given moment of time, we can only ◭◭ ◮◮ store and describe finitely many parameters. ◭ ◮ • Thus, it is reasonable to restrict ourselves to finite- Page 4 of 29 dimensional groups. Go Back • Question (N. Wiener): describe all finite-dimensional groups that contain all linear transformations. Full Screen • Answer (for real numbers): all elements of this group Close are fractionally-linear x → ( a · x + b ) / ( c · x + d ) . Quit
Symmetry: a . . . 4. Symmetries Explain the Basic Formulas of Differ- Basic Symmetries: . . . ent Uncertainty Formalisms: Neural Networks Basic Nonlinear . . . Symmetries Explain . . . • What needs explaining: formula for the activation func- Symmetries Explain . . . tion f ( x ) = 1 / (1 + e − x ). What Else We Do in . . . • A change in the input starting point: x → x + s . First Example: . . . • Reasonable requirement: the new output f ( x + s ) equiv- Second Example: . . . alent to the f ( x ) mod. appropriate transformation. Towards an Optimal . . . Title Page • Reminder: all appropriate transformations are frac- tionally linear. ◭◭ ◮◮ • Conclusion: f ( x + s ) = a ( s ) · f ( x ) + b ( s ) ◭ ◮ c ( s ) · f ( x ) + d ( s ) . Page 5 of 29 • Differentiating both sides by s and equating s to 0, we Go Back get a differential equation for f ( x ). Full Screen • Its known solution is the above activation function – Close which can thus be explained by symmetries. Quit
Symmetry: a . . . 5. Symmetries Explain the Basic Formulas of Differ- Basic Symmetries: . . . ent Uncertainty Formalisms: Fuzzy Logic Basic Nonlinear . . . Symmetries Explain . . . • Main quantity: certainty degree a = d ( S ). Symmetries Explain . . . • One way to define d ( S ) is by polling n experts and What Else We Do in . . . taking the fraction a = m/n of those who believe in S . First Example: . . . • To make this estimate more accurate, we can go beyond Second Example: . . . top experts and ask n ′ other experts as well. Towards an Optimal . . . Title Page • In the presence of top experts, other experts may ◭◭ ◮◮ – either remain shyly silent – or shyly confirm the majority’s opinion. ◭ ◮ • In the first case, the degree reduces from a = m/n to Page 6 of 29 a ′ = m/ ( n + n ′ ), i.e., to a ′ = λ · a , where λ = n/ ( n + n ′ ). Go Back • In the second case, a changes to a ′ = ( m + m ′ ) / ( n + m ′ ) Full Screen – a linear transformation. Close • In general, we get all linear transformations. Quit
Symmetry: a . . . 6. Symmetries Explain the Basic Formulas of Differ- Basic Symmetries: . . . ent Uncertainty Formalisms: Fuzzy Logic (cont-d) Basic Nonlinear . . . Symmetries Explain . . . • Fact: we can describe the degree of certainty d ( S ) in a Symmetries Explain . . . statement S : What Else We Do in . . . – either by its own degree of certainty, First Example: . . . – or by a degree of certainty in, say, S & S 0 for some Second Example: . . . statement S 0 . Towards an Optimal . . . Title Page • Reasonable to require: the corresponding transforma- tion d ( S ) → d ( S & S 0 ) is appropriate. ◭◭ ◮◮ • Conclusion: the transformation d ( S ) → d ( S & S 0 ) is ◭ ◮ fractionally linear. Page 7 of 29 • Results: this conclusion explains many empirically ef- Go Back ficient t-norms and t-conorms. Full Screen • Comment: many other uncertainty-related formulas Close can also be similarly explained. Quit
Symmetry: a . . . 7. What Else We Do in This Paper Basic Symmetries: . . . Basic Nonlinear . . . • We have shown: basic uncertainty-related formulas can Symmetries Explain . . . be explained in terms of symmetries. Symmetries Explain . . . • We show: many other aspects of uncertainty can be What Else We Do in . . . explained in terms of symmetries: First Example: . . . – heuristic and semi-heuristic approaches can be jus- Second Example: . . . tified by appropriate natural symmetries, and Towards an Optimal . . . Title Page – symmetries can help in designing optimal algorithms. ◭◭ ◮◮ ◭ ◮ Page 8 of 29 Go Back Full Screen Close Quit
Symmetry: a . . . 8. First Example: Practical Need for Uncertainty Prop- Basic Symmetries: . . . agation Basic Nonlinear . . . Symmetries Explain . . . • Practical problem: we are often interested in the quan- Symmetries Explain . . . tity y which is difficult to measure directly. What Else We Do in . . . • Solution: First Example: . . . – estimate easier-to-measure quantities x 1 , . . . , x n which Second Example: . . . are related to y by a known algorithm y = f ( x 1 , . . . , x n ); Towards an Optimal . . . Title Page – compute � y = f ( � x 1 , . . . , � x n ) based on the estimates � x i . ◭◭ ◮◮ • Fact: estimates are never absolutely accurate: � x i � = x i . ◭ ◮ • Consequence: the estimate � y = f ( � x 1 , . . . , � x n ) is differ- Page 9 of 29 ent from the actual value y = f ( x 1 , . . . , x n ). Go Back def • Problem: estimate the uncertainty ∆ y = � y − y . Full Screen Close Quit
Symmetry: a . . . 9. Propagation of Probabilistic Uncertainty Basic Symmetries: . . . Basic Nonlinear . . . • Fact: often, we know the probabilities of different val- Symmetries Explain . . . ues of ∆ x i . Symmetries Explain . . . • Example: ∆ x i are independent normally distributed What Else We Do in . . . with mean 0 and known st. dev. σ i . First Example: . . . • Monte-Carlo approach: Second Example: . . . Towards an Optimal . . . – For k = 1 , . . . , N times, we: Title Page ∗ simulate the values ∆ x ( k ) according to the known i ◭◭ ◮◮ probability distributions for x i ; ∗ find x ( k ) x i − ∆ x ( k ) ◭ ◮ = � i ; i ∗ find y ( k ) = f ( x ( k ) 1 , . . . , x ( k ) n ); Page 10 of 29 ∗ estimate ∆ y ( k ) = y ( k ) − � y . Go Back – Based on the sample ∆ y (1) , . . . , ∆ y ( N ) , we estimate Full Screen the statistical characteristics of ∆ y . Close Quit
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