How to Compare Different Localness Property Range Estimations: - PowerPoint PPT Presentation

Need for Data Processing Need to Take . . . Dynamic Case How Preferences Are . . . How to Compare Different Localness Property Range Estimations: Other Common-Sense . . . Main Result A Symmetry-Based How to Compare . . . Auxiliary Result: How . . . Approach Home Page Title Page Olga Kosheleva and Vladik Kreinovich ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu Page 1 of 27 Go Back Full Screen Close Quit

Need for Data Processing Need to Take . . . 1. Outline Dynamic Case • How to compare different range estimators for multi- How Preferences Are . . . variate functions under uncertainty? Localness Property Other Common-Sense . . . • To answer this question, we analyze which utility func- Main Result tions can be used for this task. Specifically, we: How to Compare . . . – introduce various invariance assumptions, Auxiliary Result: How . . . – describe the class of all utility functions which sat- Home Page isfy these assumptions, and Title Page – show how the resulting utility functions can be used ◭◭ ◮◮ to compare different range estimators. ◭ ◮ Page 2 of 27 Go Back Full Screen Close Quit

Need for Data Processing Need to Take . . . 2. Need for Data Processing Dynamic Case • In many practical situations, we are interested: How Preferences Are . . . Localness Property – in the future value of a quantity y Other Common-Sense . . . – or in its current value which is not easy to measure Main Result directly. How to Compare . . . • In such situations: Auxiliary Result: How . . . Home Page – we find easier-to-measure quantities x 1 , . . . , x n which are related to y in a known way y = f ( x 1 , . . . , x n ), Title Page – and we use the measured values � x i of the auxiliary ◭◭ ◮◮ quantities x i to estimate y as � y = f ( � x n ). x 1 , . . . , � ◭ ◮ • This is usually called data processing. Page 3 of 27 Go Back Full Screen Close Quit

Need for Data Processing Need to Take . . . 3. Need to Take Measurement Uncertainty Into Dynamic Case Account How Preferences Are . . . • Measurements are never absolutely accurate. Localness Property Other Common-Sense . . . • Thus, the measured values � x i are somewhat different Main Result from the actual (unknown) values x i . How to Compare . . . • Therefore, the estimate � y = f ( � x n ) is, in general, x 1 , . . . , � Auxiliary Result: How . . . different from the desired value y = f ( x 1 , . . . , x n ). Home Page • Often, the only information that we have about Title Page def ∆ x i = � x i − x i is the upper bound ∆ i : | ∆ x i | ≤ ∆ i . ◭◭ ◮◮ • In this case, the only information that we know about ◭ ◮ each x i is that x i ∈ x i = [ � x i − ∆ i , � x i + ∆ i ]. Page 4 of 27 • We therefore need to find the range y of f ( x 1 , . . . , x n ) Go Back when x i ∈ x i ; this is called interval computations. Full Screen • The exact computation of this range is known to be NP-hard, so we find an enclosure Y ⊇ y . Close Quit

Need for Data Processing Need to Take . . . 4. Dynamic Case Dynamic Case • Often, we are interested in the values y ( t ) correspond- How Preferences Are . . . ing to different t (time, location, etc.). Localness Property Other Common-Sense . . . • Different interval techniques lead to different enclo- Main Result sures Y ( t ), Y ′ ( t ), . . . for y ( t ). How to Compare . . . • How can we compare these results? Auxiliary Result: How . . . Home Page • Of course, if a technique leads to narrower bounds for all t , it is better. Title Page • However, often, one technique is better for some t while ◭◭ ◮◮ another technique is better for other t . ◭ ◮ • In this talk, we propose a symmetry-based approach to Page 5 of 27 comparing two methods. Go Back Full Screen Close Quit

Need for Data Processing Need to Take . . . 5. How Preferences Are Described in Decision The- Dynamic Case ory: The Notion of Utility How Preferences Are . . . • To describe preferences in numerical terms, we select a Localness Property very bad alternative A 0 and a very good one A 1 . Other Common-Sense . . . Main Result • Then, for each p ∈ [0 , 1], we form a lottery L ( p ) in How to Compare . . . which we get A 1 with probability p else A 0 . Auxiliary Result: How . . . • When p ≈ 0, we have L ( p ) < A ; when p ≈ 1, we have Home Page A < L ( p ). Title Page • There exists a threshold value ◭◭ ◮◮ u ( A ) = sup { p : L ( p ) < A } for which: ◭ ◮ – for all p < u ( A ), we have L ( p ) < A , and Page 6 of 27 – for all p > u ( A ), we have L ( p ) > A . Go Back • This threshold value is called the utility of A . Full Screen • Here, A is better than A ′ if and only if u ( A ) > u ( A ′ ). Close Quit

Need for Data Processing Need to Take . . . 6. Application to Our Problem Dynamic Case • The numerical value of the utility depends on the se- How Preferences Are . . . lection of the alternatives A 0 and A 1 . Localness Property Other Common-Sense . . . • One can show that: Main Result – if we select a different pair of alternatives A ′ 0 < A ′ 1 , How to Compare . . . – then the new utility values u ′ ( A ) is linearly related Auxiliary Result: How . . . to u ( A ): u ′ ( A ) = a · u ( A ) + b. Home Page • Let us apply the notion of utility to our problem. Title Page ◭◭ ◮◮ • Usually, we only describe the bounds Y ( t ) = [ Y ( t ) , Y ( t )] for the values t from a grid: t i = t 0 + i · ∆ t. ◭ ◮ • In this case, the utility describing the quality of each Page 7 of 27 estimation has the form u ( Y ( t 0 ) , Y ( t 1 ) , . . . ). Go Back • Let us describe common sense properties of preference Full Screen in terms of utility functions. Close Quit

Need for Data Processing Need to Take . . . 7. Localness Property Dynamic Case • Let us assume that: How Preferences Are . . . Localness Property – the two enclosures Y ( t ) and Y ′ ( t ) differ only for Other Common-Sense . . . moments t ∈ [ t, t ], and Main Result – that Y is preferred to Y ′ ( Y ≥ Y ′ ). How to Compare . . . • This means that, for the user, on the interval [ t, t ], the Auxiliary Result: How . . . bounds Y ( t ) are better than the bounds Y ′ ( t ). Home Page • This preference should not depend on whatever com- Title Page mon bounds Y ( t ) = Y ′ ( t ) we have for t �∈ [ t, t ]. ◭◭ ◮◮ • Formally: ◭ ◮ – if we have bounds Z ( t ) and Z ′ ( t ) for which: Page 8 of 27 ∗ Z ( t ) = Z ′ ( t ) for all t �∈ [ t, t ], and Go Back ∗ Z ( t ) = Y ( t ) and Z ′ ( t ) = Y ′ ( t ) for all t ∈ [ t, t ], Full Screen – then Z should be preferable to Z ′ : Z ≥ Z ′ . Close Quit

Need for Data Processing Need to Take . . . 8. Localness in Terms of Utility Dynamic Case • It is known that preferences are local if and only if How Preferences Are . . . u ( x 1 , . . . , x n ) is: Localness Property Other Common-Sense . . . – additive u ( x 1 , . . . , x n ) = u 1 ( x 1 ) + . . . + u n ( x n ) for Main Result some u i ( x i ); or How to Compare . . . – multiplicative u ( x 1 , . . . , x n ) = u 1 ( x 1 ) · . . . · u n ( x n ). • In our case, we maximize � u ( Y ( t i ) , t i ) or � Auxiliary Result: How . . . u ( Y ( t i ) , t i ), Home Page i i def where u i ( Y ( t i )) = u ( Y ( t i ) , t i ). Title Page • Maximizing the product is equivalent to maximizing ◭◭ ◮◮ its logarithm – which is a sum. ◭ ◮ • So, we always maximize the sum. Page 9 of 27 • When ∆ t → 0, the sum tends to an integral, so we � Go Back maximize the integral u ( Y ( t ) , t ) dt . Full Screen • So, to specify preferences, we must find an appropriate Close function u ( Y , t ). Quit

Need for Data Processing Need to Take . . . 9. Other Common-Sense Properties Dynamic Case • Small changes in time t and enclosure Y ( t ) should lead How Preferences Are . . . to small changes in utility. Localness Property Other Common-Sense . . . • It is therefore reasonable to require that the function Main Result u ( Y , t ) is smooth. How to Compare . . . • Specifically, we require that this function is at least Auxiliary Result: How . . . twice differentiable. Home Page • Preferences should not change: Title Page – if we simply change the unit for measuring time ◭◭ ◮◮ t → λ · t (e.g., from minutes to seconds), ◭ ◮ – or change the starting point t → t + t 0 . Page 10 of 27 • Similarly, we can use different starting points and dif- Go Back ferent units for describing the quantity y . Full Screen • Thus, the preference relation should not change if we replace y with y + y 0 or with λ · y . Close Quit