How Can We Describe . . . Case of Absolute . . . Case of Relative . . . How to Describe Towards a General Case From Measurements . . . Measurement Uncertainty Main Result about f What If There Is No Bias? and Uncertainty of No-Bias Results Expert Estimates? Home Page Title Page Nicolas Madrid and Irina Perfilieva ◭◭ ◮◮ Institute of Research and Applications of Fuzzy Modeling University of Ostrava, Ostrava, Czech Republic ◭ ◮ nicolas.madrid@osu.cz, Irina.Perfilieva@osu.cz Page 1 of 21 Vladik Kreinovich Go Back Department of Computer Science University of Texas at El Paso Full Screen El Paso, TX 79968, USA vladik@utep.edu Close Quit
How Can We Describe . . . 1. How Can We Describe Measurement Uncer- Case of Absolute . . . tainty: Formulation of the Problem Case of Relative . . . Towards a General Case • We want to know the actual values of different quanti- From Measurements . . . ties. Main Result about f • To get these values, we perform measurements. What If There Is No Bias? • Measurements are never absolutely accurate. No-Bias Results Home Page • The actual value A ( u ) of the corr. quantity is, in gen- Title Page eral, different from the measurement result M ( u ). ◭◭ ◮◮ • It is therefore desirable to describe what are the possi- ◭ ◮ ble values of A ( u ). Page 2 of 21 • This will be a perfect way to describe uncertainty: Go Back – for each measurement result M ( u ), Full Screen – we describe the set of all possible values of A ( u ). Close • How can we attain this description? Quit
How Can We Describe . . . 2. In Practice, We Don’t Know Actual Values Case of Absolute . . . Case of Relative . . . • Ideally, for diff. situations u , we should compare the Towards a General Case measurement result M ( u ) with the actual value A ( u ). From Measurements . . . • The problem is that we do not know the actual value. Main Result about f What If There Is No Bias? • A usual approach is to compare No-Bias Results – the measurement result M ( u ) with Home Page – the result S ( u ) of measuring the same quantity by Title Page a much more accurate (“standard”) MI. ◭◭ ◮◮ • From this viewpoint, the above problem can be refor- ◭ ◮ mulated as follows: Page 3 of 21 – we know the measurement result M ( u ) correspond- ing to some situation u , Go Back – we want to find the set of possible values S ( u ) that Full Screen we would have obtained if we apply a standard MI. Close Quit
How Can We Describe . . . 3. Case of Absolute Measurement Error Case of Absolute . . . Case of Relative . . . • In some cases, we know the upper bound ∆ on the Towards a General Case absolute value of the measurement error M ( u ) − A ( u ): From Measurements . . . | M ( u ) − A ( u ) | ≤ ∆ . Main Result about f What If There Is No Bias? • In this case, once we know the measurement result No-Bias Results M ( u ), we can conclude that Home Page M ( u ) − ∆ ≤ A ( u ) ≤ M ( u ) + ∆ . Title Page ◭◭ ◮◮ • In more general terms, we can describe the correspond- ing bounds as f ( M ( u )) ≤ A ( u ) ≤ g ( M ( u )) , where ◭ ◮ Page 4 of 21 def def f ( x ) = x − ∆ and g ( x ) = x + ∆ . Go Back Full Screen Close Quit
How Can We Describe . . . 4. Case of Relative Measurement Error Case of Absolute . . . Case of Relative . . . • In some other cases, we know the upper bound δ on Towards a General Case the relative measurement error: From Measurements . . . | M ( u ) − A ( u ) | ≤ δ. Main Result about f | A ( u ) | What If There Is No Bias? • In this case, for positive values, No-Bias Results Home Page (1 − δ ) · A ( u ) ≤ M ( u ) ≤ (1 + δ ) · A ( u ) . Title Page • Thus, once we know the measurement result M ( u ), we ◭◭ ◮◮ can conclude that ◭ ◮ M ( u ) 1 + δ ≤ A ( u ) ≤ M ( u ) 1 − δ . Page 5 of 21 Go Back • So, we have f ( M ( u )) ≤ A ( u ) ≤ g ( M ( u )) for Full Screen x x def def f ( x ) = 1 + δ and g ( x ) = 1 − δ. Close Quit
How Can We Describe . . . 5. In Some Cases, We Have Both Types of Mea- Case of Absolute . . . surement Errors Case of Relative . . . Towards a General Case • In some cases, we have both additive (absolute) and From Measurements . . . multiplicative (relative) measurement errors: Main Result about f A ( u ) − ∆ − δ · A ( u ) ≤ M ( u ) ≤ A ( u ) + ∆ + δ · A ( u ) . What If There Is No Bias? No-Bias Results • In this case: Home Page M ( u ) − ∆ ≤ A ( u ) ≤ M ( u ) + ∆ Title Page . 1 + δ 1 − δ ◭◭ ◮◮ • So, we have f ( M ( u )) ≤ A ( u ) ≤ g ( M ( u )) , where: ◭ ◮ = x − ∆ = x + ∆ def def f ( x ) 1 + δ and f ( x ) 1 − δ . Page 6 of 21 Go Back Full Screen Close Quit
How Can We Describe . . . 6. Towards a General Case Case of Absolute . . . Case of Relative . . . • The above formulas assume that the measurement ac- Towards a General Case curacy is the same for the whole range. From Measurements . . . • In reality, measuring instruments have different accu- Main Result about f racies ∆ and δ in different ranges. What If There Is No Bias? • Hence, f ( x ) and g ( x ) are non-linear. No-Bias Results Home Page • When M ( u ) is larger, this means that the bounds on possible values of A ( u ) increase (or do not decrease). Title Page ◭◭ ◮◮ • Thus, f ( x ) and g ( x ) are monotonic. ◭ ◮ • To describe the accuracy of a general measuring instru- ment, it is therefore reasonable to use: Page 7 of 21 – the largest of the monotonic functions f ( x ) for Go Back which f ( M ( u )) ≤ A ( u ) and Full Screen – the smallest of the monotonic functions g ( x ) for Close which A ( u ) ≤ g ( M ( u )). Quit
How Can We Describe . . . 7. From Measurements to Expert Estimates Case of Absolute . . . Case of Relative . . . • In areas such as medicine, expert estimates are very Towards a General Case important. From Measurements . . . • Expert estimates often result in “values” from a par- Main Result about f tially ordered set. What If There Is No Bias? No-Bias Results • Examples: “somewhat probable”, “very probable”, Home Page etc. Title Page • Such possibilities are described in different generaliza- tions of the traditional [0 , 1]-based fuzzy logic. ◭◭ ◮◮ ◭ ◮ • In all such extensions, there is order (sometimes par- tial) on the corresponding set of value L : Page 8 of 21 ℓ < ℓ ′ means that ℓ ′ represents a stronger expert’s Go Back degree of confidence than ℓ . Full Screen Close Quit
How Can We Describe . . . 8. Need to Describe Uncertainty of Expert Esti- Case of Absolute . . . mates Case of Relative . . . Towards a General Case • Some experts are very good, in the sense that based on From Measurements . . . their estimates S ( u ), we make very effective decisions. Main Result about f • Other experts may be less accurate. What If There Is No Bias? • It is therefore desirable to gauge the uncertainty of such No-Bias Results experts in relation to the “standard” (very good) ones. Home Page • To make a good decision based on the expert’s estimate Title Page M ( u ), we need to produce bounds on S ( u ): ◭◭ ◮◮ f ( M ( u )) ≤ S ( u ) ≤ g ( M ( u )) . ◭ ◮ • It is thus desirable to find: Page 9 of 21 – the largest of the monotonic functions f ( x ) for Go Back which f ( M ( u )) ≤ S ( u ) and Full Screen – the smallest of the monotonic functions g ( x ) for Close which S ( u ) ≤ g ( M ( u )). Quit
How Can We Describe . . . 9. What Is Known and What We Do in This Talk Case of Absolute . . . Case of Relative . . . • When L = [0 , 1], the existence of the largest f ( x ) and Towards a General Case smallest g ( x ) is already known. From Measurements . . . • We analyze for which partially ordered sets such largest Main Result about f f ( x ) and smallest g ( x ) exist. What If There Is No Bias? No-Bias Results • It turns out that they exist for complete lattices. Home Page • In general, they do not exist for more general partially Title Page ordered sets. ◭◭ ◮◮ • To be more precise, ◭ ◮ – the largest f ( x ) always exists only for complete Page 10 of 21 lower semi-lattices (definitions given later), while – the smallest g ( x ) always exists only for complete Go Back upper semi-lattices. Full Screen Close Quit
How Can We Describe . . . 10. Main Result about f Case of Absolute . . . Case of Relative . . . • By F ( F, G ), we denote the set of all monotonic func- Towards a General Case tions f for which f ( F ( u )) ≤ G ( u ) for all u ∈ U . From Measurements . . . • An ordered set is called a complete lower semi-lattice Main Result about f if for every set S : What If There Is No Bias? No-Bias Results – among all its lower bounds, Home Page – there exists the largest one. Title Page • Theorem. For an ordered set L , the following two ◭◭ ◮◮ conditions are equivalent to each other: ◭ ◮ – L is a complete lower semi-lattice; Page 11 of 21 – for every two functions F, G : U → L , the set F ( F, G ) has the largest element. Go Back Full Screen Close Quit
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