Traditional . . . Need for Heavy-Tailed . . . What We Do Normalization-Invariant Fuzzy Logic Need for Normalization Operations Explain Empirical Success of How to Combine Degrees Student Distributions in Describing Deriving Student . . . Measurement Uncertainty Acknowledgments Home Page Hamza Alkhatib 1 , Boris Kargoll 1 , Title Page Ingo Neumann 1 , and Vladik Kreinovich 2 ◭◭ ◮◮ 1 Geod¨ atisches Institut, Leibniz Universit¨ at Hannover ◭ ◮ Nienburger Strasse 1, 30167 Hannover, Germany alkhatib@gih.uni-hannover.de, kargoll@gih.uni-hannover.de Page 1 of 13 neumann@gih.uni-hannover.de 2 Department of Computer Science, University of Texas at El Paso Go Back El Paso, TX 79968, USA, vladik@utep.edu Full Screen Close Quit
Traditional . . . 1. Traditional Engineering Approach to Measure- Need for Heavy-Tailed . . . ment Uncertainty What We Do • Traditionally, in engineering applications, it is assumed Need for Normalization that the measurement error is normally distributed. How to Combine Degrees Deriving Student . . . • This assumption makes perfect sense from the practical Acknowledgments viewpoint. Home Page • For the majority of measuring instruments, the mea- Title Page surement error is indeed normally distributed. ◭◭ ◮◮ • It also makes sense from the theoretical viewpoint: ◭ ◮ – the measurement error often comes from a joint Page 2 of 13 effect of many independent small components, – so, according to the Central Limit Theorem, the Go Back resulting distribution is indeed close to Gaussian. Full Screen Close Quit
Traditional . . . 2. Traditional Engineering Approach (cont-d) Need for Heavy-Tailed . . . What We Do • Another explanation: we only have partial information about the distribution. Need for Normalization How to Combine Degrees • Often, we only know the first and the second moments. Deriving Student . . . • The first moment – mean – represents a bias. Acknowledgments • If we know the bias, we can always subtract it from the Home Page measurement result. Title Page • Thus re-calibrated measuring instrument will have 0 ◭◭ ◮◮ mean. ◭ ◮ • Thus, we can always safely assume that the mean is 0. Page 3 of 13 • Then, the 2nd moment is simply the variance V = σ 2 . Go Back Full Screen Close Quit
Traditional . . . 3. Traditional Engineering Approach (cont-d) Need for Heavy-Tailed . . . What We Do • There are many distributions w/0 mean and given σ . Need for Normalization • For example, we can have a distribution in which we How to Combine Degrees have σ and − σ with probability 1/2 each. Deriving Student . . . • However, such a distribution creates a false certainty – Acknowledgments that no other values of x are possible. Home Page • Out of all such distributions, it makes sense to select Title Page the one which maximally preserves the uncertainty. ◭◭ ◮◮ • Uncertainty can be gauged by average number of bi- ◭ ◮ nary questions needed to determine x with accuracy ε . Page 4 of 13 � • It is described by entropy S = − ρ ( x ) · log 2 ( ρ ( x )) dx . Go Back • Out of all distributions ρ ( x ) with mean 0 and given σ , Full Screen the entropy is the largest for normal ρ ( x ). Close Quit
Traditional . . . 4. Need for Heavy-Tailed Distributions Need for Heavy-Tailed . . . • For the normal distribution, What We Do Need for Normalization − x 2 1 � � ρ ( x ) = √ 2 π · σ · exp . How to Combine Degrees 2 σ 2 Deriving Student . . . • The “tails” – values corresponding to large | x | – are Acknowledgments very light, practically negligible. Home Page • Often, ρ ( x ) decreases much slower, as ρ ( x ) ∼ c · x − α . Title Page � ∞ 0 x − α dx = + ∞ , • We cannot have ρ ( x ) = c · x − α , since ◭◭ ◮◮ � and we want ρ ( x ) dx = 1. ◭ ◮ • Often, the measurement error is well-represented by a Student distribution ρ S ( x ) = ( a + b · x 2 ) − ν . Page 5 of 13 Go Back • Our experience is from geodesy, but the Student dis- tributions is effective in other applications as well. Full Screen • This distribution is even recommended by the Interna- Close tional Organization for Standardization (ISO). Quit
Traditional . . . 5. What We Do Need for Heavy-Tailed . . . • How to explain the empirical success of Student’s dis- What We Do tribution ρ S ( x )? Need for Normalization How to Combine Degrees • We show that a fuzzy formalization of commonsense Deriving Student . . . requirements leads to ρ S ( x ). Acknowledgments • Our idea: uncertainty means that the first value is pos- Home Page sible, and the second value is possible, etc. Title Page • Let’s select ρ ( x ) with the largest degree to which all ◭◭ ◮◮ the values are possible. ◭ ◮ • It is reasonable to use fuzzy logic to describe degrees Page 6 of 13 of possibility. Go Back • An expert marks his/her degree by selecting a number from the interval [0 , 1]. Full Screen Close Quit
Traditional . . . 6. Need for Normalization Need for Heavy-Tailed . . . • For “small”, we are absolutely sure that 0 is small: What We Do µ small (0) = 1 and max µ small ( x ) = 1. Need for Normalization x How to Combine Degrees • For “medium”, there is no x with µ med ( x ) = 1, so Deriving Student . . . max µ med ( x ) < 1. Acknowledgments x • A usual way to deal with such situations is to normalize Home Page µ ( x ) µ ( x ) into µ ′ ( x ) = µ ( y ) . Title Page max y ◭◭ ◮◮ • Normalization is also needed performed when we get ◭ ◮ additional information. Page 7 of 13 • Example: we knew that x is small, we learn that x ≥ 5. Go Back • Then, µ new ( x ) = µ small ( x ) for x ≥ 5 and µ new ( x ) = 0 Full Screen for x < 5, and max µ new ( x ) < 1. x Close Quit
Traditional . . . 7. Need for Normalization (cont-d) Need for Heavy-Tailed . . . What We Do • Normalization is also needed when experts use proba- bilities to come up with the degrees. Need for Normalization How to Combine Degrees • Indeed, the larger ρ ( x ), the more probable it is to ob- Deriving Student . . . serve a value close to x . Acknowledgments • Thus, it is reasonable to take the degrees µ ( x ) propor- Home Page tional to ρ ( x ): µ ( x ) = c · ρ ( x ). Title Page ρ ( x ) • Normalization leads to µ ( x ) = ρ ( y ) . ◭◭ ◮◮ max y ◭ ◮ • Vice versa, if we have the result µ ( x ) of normalizing a Page 8 of 13 µ ( x ) pdf, we can reconstruct ρ ( x ) as ρ ( x ) = µ ( y ) dy. � Go Back Full Screen Close Quit
Traditional . . . 8. How to Combine Degrees Need for Heavy-Tailed . . . • For each x , we thus get a degree to which x is possible. What We Do Need for Normalization • We want to compute the degree to which x 1 is possible How to Combine Degrees and x 2 is possible, etc. Deriving Student . . . • So, we need to apply an “and”-operation (t-norm) to Acknowledgments the corresponding degrees. Home Page • Natural idea: use normalization-invariant t-norms. Title Page • We can compute the normalized degree of confidence ◭◭ ◮◮ in a statement A & B in two different ways: ◭ ◮ – we can normalize f & ( a, b ) to λ · f & ( a, b ); Page 9 of 13 – or, we can first normalize a and b and then apply Go Back an “and”-operation: f & ( λ · a, λ · b ). Full Screen • It’s reasonable to require that we get the same esti- Close mate: f & ( λ · a, λ · b ) = λ · f & ( a, b ) . Quit
Traditional . . . 9. How to Combine Degrees (cont-d) Need for Heavy-Tailed . . . What We Do • It is known that Archimedean t-norms f & ( a, b ) = f − 1 ( f ( a ) + f ( b )) are universal approximators. Need for Normalization How to Combine Degrees • So, we can safely assume that f & is Archimedean: Deriving Student . . . c = f & ( a, b ) ⇔ f ( c ) = f ( a ) + f ( b ) . Acknowledgments Home Page • Thus, invariance means that f ( c ) = f ( a )+ f ( b ) implies f ( λ · c ) = f ( λ · a ) + f ( λ · b ). Title Page • So, for every λ , the transformation T : f ( a ) → f ( λ · a ) ◭◭ ◮◮ is additive: T ( A + B ) = T ( A ) + T ( B ). ◭ ◮ • Known: every monotonic additive function is linear. Page 10 of 13 • Thus, f ( λ · a ) = c ( λ ) · f ( a ) for all a and λ . Go Back • For monotonic f ( a ), this implies f ( a ) = C · a − α . Full Screen • So, f ( c ) = f ( a )+ f ( b ) implies C · c − α = C · a − α + C · b − α , Close and c = f & ( a, b ) = ( a − α + b − α ) − 1 /α . Quit
Traditional . . . 10. Deriving Student Distribution Need for Heavy-Tailed . . . • We want to maximize the degree What We Do Need for Normalization f & ( µ ( x 1 ) , µ ( x 2 ) , . . . ) = (( µ ( x 1 )) − α +( µ ( x 2 )) − α + . . . ) − 1 /α . How to Combine Degrees Deriving Student . . . • The function f ( a ) is decreasing. Acknowledgments • So, maximizing f & ( µ ( x 1 ) , . . . ) is equivalent to minimiz- Home Page ing the sum ( µ ( x 1 )) − α + ( µ ( x 2 )) − α + . . . Title Page def ( µ ( x )) − α dx . � • In the limit, this sum tends to I = ◭◭ ◮◮ � • So, we minimize I under constrains x · ρ ( x ) dx = 0 ◭ ◮ µ ( x ) x 2 · ρ ( x ) dx = σ 2 , where ρ ( x ) = � and µ ( y ) dy. Page 11 of 13 � ( µ ( x )) − α dx under constraints Go Back � • Thus, we minimize Full Screen � � � x 2 · µ ( x ) dx − σ 2 · x · µ ( x ) dx = 0 and µ ( x ) dx = 0 . Close Quit
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