Outline Case Study Optimal Sensor Placement in General Case Limitations of the . . . Scale Invariance: . . . Environmental Research: Main Result Proof: Part 1 Designing a Sensor Network under Uncertainty Title Page ◭◭ ◮◮ Aline Jaimes, Craig Tweedy, Tanja Magoc, ◭ ◮ Vladik Kreinovich, and Martine Ceberio Page 1 of 20 Cyber-ShARE Center Go Back University of Texas, El Paso, TX 79968, USA contact email vladik@utep.edu Full Screen Close Quit
1. Introduction Outline Case Study • Challenge: in many remote areas, meteorological sen- General Case sor coverage is sparse. Limitations of the . . . • Desirable: design sensor networks that provide the largest Scale Invariance: . . . amount of useful information within a given budget. Main Result Proof: Part 1 • Difficulty: because of the huge uncertainty, this prob- lem is very difficult even to formulate in precise terms. Title Page • First aspect of the problem: how to best distribute the ◭◭ ◮◮ sensors over the large area. ◭ ◮ • Status: reasonable solutions exist for this aspect. Page 2 of 20 • Second aspect of the problem: what is the best location Go Back of each sensor in the corresponding zone. Full Screen • This talk: will focus on this aspect of the sensor place- Close ment problem. Quit
2. Outline Outline Case Study • Case study: meteorological tower. General Case • This case is an example of multi-criteria optimization, Limitations of the . . . when we need to maximize several objectives x 1 , . . . , x n . Scale Invariance: . . . Main Result • Traditional approach to multi-objective optimization: � n Proof: Part 1 maximize a weighted combination w i · x i . i =1 Title Page • Specifics of our case: constraints x i > x (0) or x i < x (0) i . i ◭◭ ◮◮ def = x i − x (0) or y i = x (0) • Equiv.: y i > 0, where y i − x i . ◭ ◮ i i • Limitations of using the traditional approach under Page 3 of 20 constraints. Go Back • Scale invariance: a brief description. Full Screen • Main result: scale invariance leads to a new approach: � � Close � n � n � � � x i − x (0) maximize w i · ln( y i ) = w i · ln � . i Quit i =1 i =1
3. Case Study Outline Case Study • Objective: select the best location of a sophisticated General Case multi-sensor meteorological tower. Limitations of the . . . • Constraints: we have several criteria to satisfy. Scale Invariance: . . . Main Result • Example: the station should not be located too close Proof: Part 1 to a road. Title Page • Motivation: the gas flux generated by the cars do not influence our measurements of atmospheric fluxes. ◭◭ ◮◮ • Formalization: the distance x 1 to the road should be ◭ ◮ def larger than a threshold t 1 : x 1 > t 1 , or y 1 = x 1 − t 1 > 0 . Page 4 of 20 • Example: the inclination x 2 at the tower’s location Go Back should be smaller than a threshold t 2 : x 2 < t 2 . Full Screen • Motivation: otherwise, the flux determined by this in- Close clination and not by atmospheric processes. Quit
4. General Case Outline Case Study • In general: we have several differences y 1 , . . . , y n all of General Case which have to be non-negative. Limitations of the . . . • For each of the differences y i , the larger its value, the Scale Invariance: . . . better. Main Result Proof: Part 1 • Our problem is a typical setting for multi-criteria op- timization . Title Page • A most widely used approach to multi-criteria opti- ◭◭ ◮◮ mization is weighted average , where ◭ ◮ – we assign weights w 1 , . . . , w n > 0 to different crite- Page 5 of 20 ria y i and Go Back – select an alternative for which the weighted average Full Screen w 1 · y 1 + . . . + w n · y n Close attains the largest possible value. Quit
5. Limitations of the Weighted Average Approach Outline Case Study • In general: the weighted average approach often leads General Case to reasonable solutions of the multi-criteria problem. Limitations of the . . . • In our problem: we have an additional requirement – Scale Invariance: . . . that all the values y i must be positive. So: Main Result Proof: Part 1 – when selecting an alternative with the largest pos- sible value of the weighted average, Title Page – we must only compare solutions with y i > 0. ◭◭ ◮◮ • We will show: under the requirement y i > 0, the ◭ ◮ weighted average approach is not fully satisfactory. Page 6 of 20 • Conclusion: we need to find a more adequate solution. Go Back Full Screen Close Quit
6. Limitations of the Weighted Average Approach: Outline Details Case Study General Case • The values y i come from measurements, and measure- Limitations of the . . . ments are never absolutely accurate. Scale Invariance: . . . • The results � y i of the measurements are not exactly Main Result equal to the actual (unknown) values y i . Proof: Part 1 • If: for some alternative y = ( y 1 , . . . , y n ) Title Page – we measure the values y i with higher and higher ◭◭ ◮◮ accuracy and, ◭ ◮ – based on the measurement results � y i , we conclude Page 7 of 20 that y is better than some other alternative y ′ . Go Back • Then: we expect that the actual alternative y is indeed better than y ′ (or at least of the same quality). Full Screen Close • Otherwise, we will not be able to make any meaningful conclusions based on real-life measurements. Quit
7. The Above Natural Requirement Is Not Always Outline Satisfied for Weighted Average Case Study General Case • Simplest case: two criteria y 1 and y 2 , w/weights w i > 0. Limitations of the . . . • If y 1 , y 2 , y ′ 1 , y ′ 2 > 0, and w 1 · y 1 + w 2 · y 2 > w 1 · y ′ 1 + w 2 · y ′ 2 , Scale Invariance: . . . then y = ( y 1 , y 2 ) ≻ y ′ = ( y ′ 1 , y ′ 2 ) . Main Result • If y 1 > 0, y 2 > 0, and at least one of the values y ′ Proof: Part 1 1 and 2 is non-positive, then y = ( y 1 , y 2 ) ≻ y ′ = ( y ′ y ′ 1 , y ′ 2 ) . Title Page • Let us consider, for every ε > 0, the tuple ◭◭ ◮◮ = ( ε, 1 + w 1 /w 2 ), and y ′ = (1 , 1). def y ( ε ) ◭ ◮ • In this case, for every ε > 0, we have Page 8 of 20 w 1 · y 1 ( ε )+ w 2 · y 2 ( ε ) = w 1 · ε + w 2 + w 2 · w 1 = w 1 · (1+ ε )+ w 2 Go Back w 2 and w 1 · y ′ 1 + w 2 · y ′ 2 = w 1 + w 2 , hence y ( ε ) ≻ y ′ . Full Screen � � 0 , 1 + w 1 Close • However, in the limit ε → 0, we have y (0) = , w 2 Quit with y (0) 1 = 0 and thus, y (0) ≺ y ′ .
8. Towards a Precise Description Outline Case Study • Each alternative is characterized by a tuple of n posi- General Case tive values y = ( y 1 , . . . , y n ). Limitations of the . . . • Thus, the set of all alternatives is the set ( R + ) n of all Scale Invariance: . . . the tuples of positive numbers. Main Result Proof: Part 1 • For each two alternatives y and y ′ , we want to tell whether Title Page – y is better than y ′ (we will denote it by y ≻ y ′ or ◭◭ ◮◮ y ′ ≺ y ), ◭ ◮ – or y ′ is better than y ( y ′ ≻ y ), Page 9 of 20 – or y and y ′ are equally good ( y ′ ∼ y ). Go Back • Natural requirement: if y is better than y ′ and y ′ is Full Screen better than y ′′ , then y is better than y ′′ . Close • The relation ≻ must be transitive. Quit
9. Towards a Precise Description (cont-d) Outline Case Study • Reminder: the relation ≻ must be transitive. General Case • Similarly, the relation ∼ must be transitive, symmetric, Limitations of the . . . and reflexive ( y ∼ y ), i.e., be an equivalence relation . Scale Invariance: . . . Main Result • An alternative description: a transitive pre-ordering Proof: Part 1 relation a � b ⇔ ( a ≻ b ∨ a ∼ b ) s.t. a � b ∨ b � a . Title Page • Then, a ∼ b ⇔ ( a � b ) & ( b � a ), and ◭◭ ◮◮ a ≻ b ⇔ ( a � b ) & ( b �� a ) . ◭ ◮ • Additional requirement: Page 10 of 20 – if each criterion is better, Go Back – then the alternative is better as well. Full Screen • Formalization: if y i > y ′ i for all i , then y ≻ y ′ . Close Quit
10. Scale Invariance: Motivation Outline Case Study • Fact: quantities y i describe completely different phys- General Case ical notions, measured in completely different units. Limitations of the . . . • Examples: wind velocities measured in m/s, km/h, Scale Invariance: . . . mi/h; elevations in m, km, ft. Main Result Proof: Part 1 • Each of these quantities can be described in many dif- ferent units. Title Page • A priori, we do not know which units match each other. ◭◭ ◮◮ • Units used for measuring different quantities may not ◭ ◮ be exactly matched. Page 11 of 20 • It is reasonable to require that: Go Back – if we simply change the units in which we measure Full Screen each of the corresponding n quantities, Close – the relations ≻ and ∼ between the alternatives y = ( y 1 , . . . , y n ) and y ′ = ( y ′ 1 , . . . , y ′ n ) do not change. Quit
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