Need for Expert Estimates Expert Estimates . . . Traditional Approach . . . Analysis of the . . . How to Take Expert Possible Faults and . . . Uncertainty into Account: What Is the Cost of . . . Combining The Costs: . . . Economic Approach Towards Economically . . . Discussion Illustrated by Pavement Home Page Engineering Applications Title Page ◭◭ ◮◮ Edgar Rodriguez 1 , 2 , Carlos M. Chang 1 , ◭ ◮ Thach Ngoc Nguyen 3 , Olga Kosheleva 1 , Vladik Kreinovich 1 1 University of Texas at El Paso, El Paso, Texas 79968, USA, Page 1 of 19 edrodriguezvelasquez@miners.utep.edu, cchangalbitres2@utep.edu, Go Back olgak@utep.edu, vladik@utep.edu 2 Universidad de Piura in Peru (UDEP), edgar.rodriguez@udep.pe Full Screen 3 Banking University of Ho Chi Minh City, Vietnam, thachnn@buh.edu.vn Close Quit
Need for Expert Estimates Expert Estimates . . . 1. Need for Expert Estimates Traditional Approach . . . • In many practical situations, we use experts to help Analysis of the . . . make decisions. Possible Faults and . . . What Is the Cost of . . . • In medicine, computer-based systems are not yet able Combining The Costs: . . . to always provide a correct diagnosis. Towards Economically . . . • In other case, the corresponding automatic equipment Discussion exists, but it is much cheaper to use human experts. Home Page • For example, in pavement engineering, in principle, we Title Page can use automatic systems: ◭◭ ◮◮ – to gauge the condition of the road surface, ◭ ◮ – to estimate the size of cracks and other faults. Page 2 of 19 • However, the corresponding equipment is still reason- Go Back ably expensive to use. Full Screen • The use of human grades is explicitly mentioned in the corresponding normative documents. Close Quit
Need for Expert Estimates Expert Estimates . . . 2. Expert Estimates Come with Uncertainty Traditional Approach . . . • Expert estimates usually come with uncertainty. Analysis of the . . . Possible Faults and . . . • The experts’ estimates have, at best, the accuracy of What Is the Cost of . . . about 10-15%, up to 20%. Combining The Costs: . . . • This observed accuracy is in the perfect accordance Towards Economically . . . with the well-known “seven plus-minus two law”. Discussion Home Page • According to this law, a person normally divides ev- erything into between 5 and 9 categories. Title Page • Thus, a person has the accuracy between 1 / 9 ≈ 10% ◭◭ ◮◮ and 1 / 5 ≈ 20%. ◭ ◮ Page 3 of 19 Go Back Full Screen Close Quit
Need for Expert Estimates Expert Estimates . . . 3. Traditional Approach to Dealing with This Un- Traditional Approach . . . certainty Analysis of the . . . • Traditionally, we: Possible Faults and . . . What Is the Cost of . . . – come up with the most accurate estimate of the Combining The Costs: . . . desired quantity, and then Towards Economically . . . – if needed, we gauge the economic consequences of Discussion the resulting estimate. Home Page • The main limitation of the traditional approach is that Title Page – while our ultimate objective is economic – how to ◭◭ ◮◮ best maintain the pavement within the budget, – we do not take this objective into account when ◭ ◮ computing the numerical estimate. Page 4 of 19 • In this talk, we show how to take economic factors into Go Back account when producing the estimate. Full Screen • The resulting formulas are in line with the usual way Close how decision makers take risk into account. Quit
Need for Expert Estimates Expert Estimates . . . 4. Analysis of the Problem: Main Idea Traditional Approach . . . • An expert describes his/her opinion by a natural-language Analysis of the . . . word or by a number. Possible Faults and . . . What Is the Cost of . . . • For each such opinion – be it a word or a number – Combining The Costs: . . . – we can find all the cases when this expert expressed Towards Economically . . . this particular opinion, and Discussion – in all these cases, find the actual value of the esti- Home Page mated quantity q . Title Page • As a result, for each opinion, we get a probability dis- ◭◭ ◮◮ tribution on the set of all possible values of q . ◭ ◮ • This distribution can be described: Page 5 of 19 – either in terms of the corresponding probability Go Back density function (pdf) ρ ( x ), – or in the terms of the cumulative distribution func- Full Screen def tion (cdf) F ( x ) = Prob( q ≤ x ). Close Quit
Need for Expert Estimates Expert Estimates . . . 5. Analysis of the Problem (cont-d) Traditional Approach . . . • In many real-life situations, the expert uncertainty is a Analysis of the . . . joint effect of many different small independent factors. Possible Faults and . . . What Is the Cost of . . . • According to the Central Limit Theorem, such distri- Combining The Costs: . . . bution is close to Gaussian (normal). Towards Economically . . . • Thus, it often makes sense to assume that the corre- Discussion sponding probability distribution is normal. Home Page • For the normal distribution with mean µ and standard Title Page � x − µ � deviation σ , we have F ( x ) = F 0 . ◭◭ ◮◮ σ ◭ ◮ • Here, F 0 ( x ) is the cdf of the standard normal distribu- tion – with mean 0 and standard deviation 1. Page 6 of 19 Go Back • Based on the probability distribution, we describe the most accurate numerical estimate. Full Screen Close Quit
Need for Expert Estimates Expert Estimates . . . 6. Analysis of the Problem: Details Traditional Approach . . . • We want to have an estimate which is as close to the Analysis of the . . . actual values of the quantity q as possible. Possible Faults and . . . What Is the Cost of . . . • For the same opinion of an expert, we have, in general, Combining The Costs: . . . different actual values q 1 , . . . , q n . Towards Economically . . . • These values form a point ( q 1 , . . . , q n ) in the corre- Discussion sponding n -dimensional space. Home Page • A natural idea is to select the estimate x 0 for which the Title Page point ( x 0 , . . . , x 0 ) is the closest to the point ( q 1 , . . . , q n ): ◭◭ ◮◮ def ( x 0 − q 1 ) 2 + . . . + ( x 0 − q n ) 2 → min . � d = ◭ ◮ Page 7 of 19 Go Back Full Screen Close Quit
Need for Expert Estimates Expert Estimates . . . 7. Details (cont-d) Traditional Approach . . . • Minimizing the distance d is equivalent to minimizing Analysis of the . . . Possible Faults and . . . d 2 = ( x 0 − q 1 ) 2 + . . . + ( x 0 − q n ) 2 . What Is the Cost of . . . • Differentiating the expression for d 2 with respect to x 0 Combining The Costs: . . . Towards Economically . . . and equating the derivative to 0, we conclude that Discussion 2( x 0 − q 1 ) + . . . + 2( x 0 − q n ) = 0 , thus Home Page = q 1 + . . . + q n def Title Page x 0 = µ . n ◭◭ ◮◮ • This is equivalent to minimizing the mean square value ( x − x 0 ) 2 · ρ ( x ) dx , which leads to ◭ ◮ � Page 8 of 19 � x 0 = µ = x · ρ ( x ) dx. Go Back Full Screen Close Quit
Need for Expert Estimates Expert Estimates . . . 8. Possible Faults and How Much It Costs to Re- Traditional Approach . . . pair Them Analysis of the . . . • In pavement engineering, we are interested in estimat- Possible Faults and . . . ing the pavement fault index x . What Is the Cost of . . . Combining The Costs: . . . • When the pavement is perfect, this index is 0. Towards Economically . . . • The presence of any specific fault increases the value Discussion of this index. Home Page • Repairing a fault takes money; the larger the index, Title Page the more costly it is to repair this road segment. ◭◭ ◮◮ • Let us denote the cost of repairs for a road segment ◭ ◮ with index x by c ( x ). Page 9 of 19 • We are interested in the case when the road is regularly Go Back repaired. Full Screen • In this case, the index x cannot grow too much. Close Quit
Need for Expert Estimates Expert Estimates . . . 9. Possible Faults (cont-d) Traditional Approach . . . • Once there are some faults in the road, these faults are Analysis of the . . . being repaired. Possible Faults and . . . What Is the Cost of . . . • Thus, the values of the index x remain small. Combining The Costs: . . . • So, we can expand the unknown function into Taylor Towards Economically . . . series and keep only the first terms in this expansion. Discussion Home Page • For example, we can keep only linear terms: c ( x ) ≈ c 0 + c 1 · x. Title Page • When the road segment is perfect, i.e., when x = 0, no ◭◭ ◮◮ repairs are needed, so the cost is 0: c (0) = 0. ◭ ◮ • Thus, c 0 = 0, and the cost of repairs linearly depends Page 10 of 19 on the index: c ( x ) ≈ c 1 · x. Go Back Full Screen Close Quit
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