Formulation of the . . . What If We Allow All . . . 1-D Interval-Based . . . Discussion Voting Aggregation Leads to What If We Require . . . (Interval) Median An Alternative . . . Alternative . . . 1-D Case: Can We . . . Olga Kosheleva 1 , and Vladik Kreinovich 2 Multi-D Interval- . . . Home Page 1 Department of Teacher Education 2 Department of Computer Science Title Page University of Texas at El Paso El Paso, TX 79968, USA ◭◭ ◮◮ olgak@utep.edu, vladik@utep.edu ◭ ◮ Page 1 of 23 Go Back Full Screen Close Quit
Formulation of the . . . What If We Allow All . . . 1. Formulation of the Problem 1-D Interval-Based . . . • For many real-real problems, there are several different Discussion decision making tools. What If We Require . . . An Alternative . . . • Each of these tools has its advantages: otherwise, it Alternative . . . would not be used. 1-D Case: Can We . . . • To combine the advantages of different tools, it there- Multi-D Interval- . . . fore desirable to aggregate their results. Home Page • One of the most widely used methods of aggregating Title Page several results is voting: ◭◭ ◮◮ – if the majority of results satisfy a certain property, ◭ ◮ – then we conclude that the actual value has this Page 2 of 23 property. Go Back • Example: if most classifiers classify the disease as pneu- Full Screen monia, we conclude that it is pneumonia. Close Quit
Formulation of the . . . What If We Allow All . . . 2. What We Do and Why This Is Non-Trivial 1-D Interval-Based . . . • What we do: we analyze how voting can be used to Discussion aggregate several numerical estimates. What If We Require . . . An Alternative . . . • Observation: voting is closely related to the AI notion Alternative . . . of a “typical” object of a class. 1-D Case: Can We . . . • Example: what is an intuitive meaning of a term “typ- Multi-D Interval- . . . ical professor”? Home Page – if most professors are absent-minded, Title Page – then we expect a “typical” professor to be absent- ◭◭ ◮◮ minded as well. ◭ ◮ • Problem: no one is perfectly typical, e.g., due to their Page 3 of 23 specific research area. Go Back • This is why in AI, there is an ongoing discussion on Full Screen how best to describe typical (not abnormal) objects. Close Quit
Formulation of the . . . What If We Allow All . . . 3. Main Definition 1-D Interval-Based . . . • We have: n estimates x i = ( x i 1 , . . . , x iq ) (1 ≤ i ≤ n ) Discussion for the values of q quantities. What If We Require . . . An Alternative . . . • We want: to combine these estimates into a single es- Alternative . . . timate x , so that: 1-D Case: Can We . . . – if the majority of x i satisfies a property P , Multi-D Interval- . . . – then x should satisfy also this property. Home Page R q . • Let q ≥ 1 , let S be a class of subsets of I Title Page R q is a possible S -aggregate of ◭◭ ◮◮ • We say that x ∈ I R q if for every S ∈ S : x 1 , . . . , x n ∈ I ◭ ◮ – if the majority of x i are in this set, Page 4 of 23 – then x should be in this set. Go Back • The set of all possible S -aggregates is called the S - Full Screen aggregate of the elements x 1 , . . . , x n . Close Quit
Formulation of the . . . What If We Allow All . . . 4. What If We Allow All Properties? 1-D Interval-Based . . . R q be the set of all subsets of I def = 2 I R q . • Let U Discussion What If We Require . . . • For every q ≥ 1 and n ≥ 3 , if all n elements x 1 , . . . , x n An Alternative . . . are all different, then their U -aggregate is empty. Alternative . . . • Proof: 1-D Case: Can We . . . Multi-D Interval- . . . – All x i belong to X = { x 1 , . . . , x n } , so x ∈ X hence Home Page x = x i for some i . Title Page – But most elements x j are different from x i , i.e., x j ∈ −{ x i } , thus, x � = x i . ◭◭ ◮◮ – Contradiction shows that no such x is possible, i.e., ◭ ◮ that the U -aggregare is empty. Page 5 of 23 • Thus, the aggregation problem is indeed non-trivial. Go Back Full Screen Close Quit
Formulation of the . . . What If We Allow All . . . 5. When Is the U -Aggregate Non-Empty? 1-D Interval-Based . . . • Let q ≥ 1 . Discussion What If We Require . . . • When n = 1 then the U -aggregate set of x 1 is { x 1 } . An Alternative . . . • When n = 2 , then the U -aggregate set is { x 1 , x 2 } . Alternative . . . • For all odd n ≥ 3 : 1-D Case: Can We . . . Multi-D Interval- . . . – if the majority of x 1 , . . . , x n are equal to each other, Home Page then the U -aggregate set is this common element. Title Page • For all even n ≥ 4 : ◭◭ ◮◮ – if the majority of x 1 , . . . , x n are equal to each other, ◭ ◮ then the U -aggregate set is this common element; Page 6 of 23 – if half of x i are equal to a , and all others ae equal to b � = a , then the U -aggregate is { a, b } . Go Back Full Screen • In all other cases, the U -aggregate set is empty. Close Quit
Formulation of the . . . What If We Allow All . . . 6. 1-D Interval-Based Voting Aggregation Leads 1-D Interval-Based . . . to (Interval) Median Discussion • Let I denote the set of all intervals [ a, b ]. What If We Require . . . An Alternative . . . • For every sequence x 1 , . . . , x n , let x (1) ≤ . . . ≤ x ( n ) Alternative . . . denote the result of sorting the numbers x i in increasing 1-D Case: Can We . . . order. Multi-D Interval- . . . • When n = 2 k +1 for some integer k , then by a median , Home Page we mean the value x ( k +1) . Title Page • When n is even, i.e., when n = 2 k for some integer k , ◭◭ ◮◮ then by a median , we mean the interval [ x ( k ) , x ( k +1) ]. ◭ ◮ • The median will also be called an interval median . Page 7 of 23 • Resyult: For every sequence of numbers x 1 , . . . , x n , Go Back the I -aggregate set is equal to the median. Full Screen Close Quit
Formulation of the . . . What If We Allow All . . . 7. Discussion 1-D Interval-Based . . . • Median is the most robust aggregation, i.e., the least Discussion vulnerable to possible outliers. What If We Require . . . An Alternative . . . • Thus, median is often used in data processing. Alternative . . . • Median is used in econometrics, as a more proper mea- 1-D Case: Can We . . . sure of “average” (“typical”) income than the mean. Multi-D Interval- . . . Home Page • Otherwise, a single billionaire living in a small town: Title Page – increases its mean income ◭◭ ◮◮ – without affecting the living standards of its inhab- itants. ◭ ◮ Page 8 of 23 Go Back Full Screen Close Quit
Formulation of the . . . What If We Allow All . . . 8. What If We Require Strong Majority? 1-D Interval-Based . . . • What if we only require x i ∈ S when the proportion of Discussion values x i ∈ S exceeds a certain threshold t > 0 . 5? What If We Require . . . An Alternative . . . • The set of all possible t- S -aggregates is called the t - S - Alternative . . . aggregate set . 1-D Case: Can We . . . • For every x 1 , . . . , x n , and for every t , the t- I -aggregate Multi-D Interval- . . . set is the interval [ x ( n − k +1) , x ( k ) ] , where k = ⌈ t · n ⌉ . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 23 Go Back Full Screen Close Quit
Formulation of the . . . What If We Allow All . . . 9. An Alternative Derivation of the Interval Me- 1-D Interval-Based . . . dian Discussion • Interval median can be also derived from other natural What If We Require . . . conditions: An Alternative . . . Alternative . . . • That it is a continuous function of x 1 , . . . , x n . 1-D Case: Can We . . . • That it is invariant with respect to arbitrary strictly Multi-D Interval- . . . increasing or strictly decreasing re-scalings: Home Page – such re-scalings which make physical sense; Title Page – e.g., we can measure sound energy in Watts or in ◭◭ ◮◮ decibels – which are logarithmic units. ◭ ◮ • That this is the narrowest such operation: Page 10 of 23 – else we could, e.g., take an operation returning the Go Back � � whole range min x i , max x i . i i Full Screen Close Quit
Formulation of the . . . What If We Allow All . . . 10. Formalizing Scale-Invariance 1-D Interval-Based . . . • We say that a mapping A ( x 1 , . . . , x n ) = Discussion [ a ( x 1 , . . . , x n ) , a ( x 1 , . . . , x n )] is scale-invariant if: What If We Require . . . An Alternative . . . – for each strictly increasing f ( x ): Alternative . . . a ( f ( x 1 ) , . . . , f ( x n )) = f ( a ( x 1 , . . . , x n )) and 1-D Case: Can We . . . Multi-D Interval- . . . a ( f ( x 1 ) , . . . , f ( x n )) = f ( a ( x 1 , . . . , x n )); Home Page – for each strictly decreasing continuous f ( x ): Title Page a ( f ( x 1 ) , . . . , f ( x n )) = f ( a ( x 1 , . . . , x n )) and ◭◭ ◮◮ a ( f ( x 1 ) , . . . , f ( x n )) = f ( a ( x 1 , . . . , x n )); ◭ ◮ Page 11 of 23 Go Back Full Screen Close Quit
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