Outline Situations in Which . . . Situations in Which . . . From Gauging Accuracy of Quantity What If a Model Is . . . Estimates to Gauging Accuracy and Situations in Which . . . Example of Asymmetry Resolution of Field Measurements: Situations with No . . . Geophysical Case Study Towards Optimal . . . Towards Optimal . . . Irina Perfilieva 1 , Roumen Anguelov 2 , Acknowledgments Vladik Kreinovich 3 , and Matt Averill 4 Home Page 1 Inst. for Research and Applications Title Page of Fuzzy Modeling University of Ostrava, Czech Republic ◭◭ ◮◮ 2 Department of Math. and Applied Math. ◭ ◮ Univ. of Pretoria, South Africa 3 Department of Computer Science Page 1 of 14 4 Department of Geological Sciences University of Texas at El Paso Go Back El Paso, Texas 79968, USA vladik@utep.edu Full Screen Close
Outline 1. Traditional Applications of Interval Computa- Situations in Which . . . tions: Reminder Situations in Which . . . What If a Model Is . . . • Objective: estimate a difficult-to-measure quantity y . Situations in Which . . . • Approach: measure quantities x 1 , . . . , x n related to x i Example of Asymmetry by a known dependence y = f ( x 1 , . . . , x n ). Situations with No . . . • Fact: measurements are never absolutely accurate. Towards Optimal . . . Towards Optimal . . . • Conclusion: the measurement results � x i are, in general, Acknowledgments different from the actual (unknown) values x i . Home Page • Conclusion: the result � y = f ( � x n ) of data pro- x 1 , . . . , � Title Page cessing differs from y = f ( x 1 , . . . , x n ). ◭◭ ◮◮ • Frequent situation: we only know the upper bound ∆ i def ◭ ◮ on the measurement errors ∆ x i = � x i − x i : | ∆ x i | ≤ ∆ i . Page 2 of 14 def • So: we only know that x i ∈ x i = [ � x i − ∆ i , � x i + ∆ i ]. Go Back • Interval computations: find the corresponding range Full Screen y = { f ( x 1 , . . . , x n ) : x i ∈ x i } of y . Close
Outline 2. In Practice, the Situation is Often More Com- Situations in Which . . . plex Situations in Which . . . What If a Model Is . . . • Dynamics: we measure the values v ( t ) of a quantity v Situations in Which . . . at a certain moment of time t . Example of Asymmetry • Spatial dependence: we measure the value v ( x, t ) at a Situations with No . . . certain location x . Towards Optimal . . . • Geophysical example: we are interested in the values of Towards Optimal . . . the density at different locations and at different depth. Acknowledgments Home Page • Traditional: uncertainty in the measured value, � v ≈ v . Title Page • New: uncertainty in location x , � x ≈ x . ◭◭ ◮◮ • Additional uncertainty: the sensor picks up the “aver- ◭ ◮ age” value of v at locations close to � x . Page 3 of 14 • Question: how to describe and process the new uncer- Go Back tainty ( resolution )? Full Screen Close
Outline 3. Outline Situations in Which . . . Situations in Which . . . • Question (reminder): how to describe and process un- What If a Model Is . . . certainty both Situations in Which . . . – in the measured value � v and Example of Asymmetry – in the spatial resolution � x ? Situations with No . . . Towards Optimal . . . • Natural idea: the answer depends on what we know about the spatial resolution. Towards Optimal . . . Acknowledgments • Possible situations: Home Page – we know exactly how the measured values � v i are � Title Page related to v ( x ), i.e., � v i = w i ( x ) · v ( x ) dx + ∆ v i ; ◭◭ ◮◮ – we only know the upper bound δ on the location error � x − x (this is similar to the interval case); ◭ ◮ – we do not even know δ . Page 4 of 14 • What we do: describe how to process all these types of Go Back uncertainty. Full Screen Close
Outline 4. Situations in Which We Have Detailed Knowl- Situations in Which . . . edge Situations in Which . . . What If a Model Is . . . • Fact: all our information about v ( x ) is contained in Situations in Which . . . the measured values � v i . Example of Asymmetry • Linearity assumption: � v i = v i + ∆ v i , where: Situations with No . . . � def – we have v i = w i ( x ) · v ( x ) dx ; and Towards Optimal . . . Towards Optimal . . . – ∆ v i is the measurement error; e.g., | ∆ v i | ≤ ∆ i . Acknowledgments • Comment: v i can be viewed as the value of v ( x ) at a Home Page “fuzzy” point characterized by uncertainty w i ( x ). Title Page • Description of the situation: we know the weights w i ( x ). ◭◭ ◮◮ � def • Find: range [ y, y ] for y = w ( x ) · v ( x ) dx . ◭ ◮ � • LP solution: minimize (maximize) w ( x ) · v ( x ) dx un- Page 5 of 14 der the constraints � Go Back def def = � v i − ∆ i ≤ w i ( x ) · v ( x ) dx ≤ v i = � v i + ∆ i . v i Full Screen Close
Outline 5. Situations With Detailed Knowledge (cont-d) Situations in Which . . . � Situations in Which . . . • Reminder: find the range of w ( x ) · v ( x ) dx when � What If a Model Is . . . v i ≤ w i ( x ) · v ( x ) dx ≤ v i . Situations in Which . . . • General case: when no bounds on v ( x ), bounds are Example of Asymmetry infinite – unless w ( x ) is a linear combination of w i ( x ). Situations with No . . . • In practice (e.g., in geophysics): v ( x ) ≥ 0. Towards Optimal . . . Towards Optimal . . . • Similar: imprecise probabilities (Kuznetsov, Walley). Acknowledgments • Solution: dual LP problem provides guaranteed bounds Home Page �� � � v = sup y i · v i : y i · w i ( x ) ≤ w ( x ) ; Title Page �� � � ◭◭ ◮◮ v = inf y i · v i : w ( x ) ≤ y i · w i ( x ) . ◭ ◮ • Easier than in IP: w i ( x ) are localized, and we often have ≤ 2 non-zero w i ( x ) at each x . Page 6 of 14 • Piece-wise linear w i ( x ) and w ( x ) – sufficient to check Go Back inequality at endpoints. Full Screen Close
Outline 6. Situations in Which We Only Know Upper Bounds Situations in Which . . . Situations in Which . . . • Situation: we only know; What If a Model Is . . . • the upper bound ∆ on the measurement inaccuracy Situations in Which . . . def ∆ v = � v − v : | ∆ v | ≤ ∆, and Example of Asymmetry • the upper bound δ on the location error Situations with No . . . def ∆ x = � x − x : | ∆ v | ≤ δ . Towards Optimal . . . Towards Optimal . . . • Consequence: the measured value � v is ∆-close to a con- Acknowledgments vex combination of values v ( x ) for x s.t. � x − � x � ≤ ∆ x . Home Page • Conclusion: v δ ( � x ) − ∆ ≤ � v ≤ v δ ( � x ) + ∆, where: Title Page def • v δ ( � x ) = inf { v ( x ) : � x − � x � ≤ δ } , and ◭◭ ◮◮ def • v δ ( � x ) = sup { v ( x ) : � x − � x � ≤ δ } . ◭ ◮ • Fact: measurement errors are random. Page 7 of 14 • So: it makes sense to only consider essential ess inf Go Back and ess sup (i.e., inf and sup modulo measure 0 sets). Full Screen Close
Outline 7. What If a Model Is Only Known With Interval Situations in Which . . . Uncertainty Situations in Which . . . What If a Model Is . . . • Reminder: we can tell when an observation ( � v, � x ) is Situations in Which . . . consistent with a model v ( x ): Example of Asymmetry v δ ( � x ) − ∆ ≤ � v ≤ v δ ( � x ) + ∆ . Situations with No . . . Towards Optimal . . . • Fact: in real life, we rarely have an exact model v ( x ). Towards Optimal . . . • Usually: we have bounds on v ( x ), i.e., an interval- Acknowledgments valued model v ( x ) = [ v − ( x ) , v + ( x )]. Home Page • Question: when is an observation ( � x ) consistent with v, � Title Page an interval-valued model? ◭◭ ◮◮ • General answer: when the observation ( � v, � x ) is consis- ◭ ◮ tent with one of the models v ( x ) ∈ v ( x ). Page 8 of 14 • A checkable answer: an observation ( � x ) is consistent v, � with an interval-valued model [ v − ( x ) , v + ( x )] when Go Back v − v ≤ v + δ ( � x ) − ∆ ≤ � δ ( � x ) + ∆ . Full Screen Close
Outline 8. Situations in Which We Only Know Upper Bounds Situations in Which . . . (cont-d) Situations in Which . . . What If a Model Is . . . • Fact: the actual v ( x ) is often continuous. Situations in Which . . . • Case of continuous v ( x ) : we can simplify the above Example of Asymmetry criterion. Situations with No . . . Towards Optimal . . . • Simplification: the set � m of all measurement results ( � x, � x ) is consistent with the model v ( x ) iff Towards Optimal . . . Acknowledgments ∀ ( � v, � x ) ∈ � m ∃ ( v ( x ) , x ) ∈ v (( � v, � x ) is (∆ , δ )-close to ( v ( x ) , x )) , Home Page i.e., | � v − v | ≤ ∆ and � x − � x � ≤ δ . Title Page • Hausdorff metric: d H ( A, B ) ≤ ε means that: ◭◭ ◮◮ ∀ a ∈ A ∃ b ∈ B ( d ( a, b ) ≤ ε ) and ∀ b ∈ B ∃ a ∈ A ( d ( a, b ) ≤ ε ) . ◭ ◮ • Conclusion: we have an asymmetric version of Haus- Page 9 of 14 dorff metric (“‘quasi-metric”). Go Back Full Screen Close
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