Cumulative Quantities Formulation of the . . . Case Study: . . . Empirical Relation . . . Relationship Between Measurement What We Do in This Talk Results and Expert Estimates Main Idea of Cumulative Quantities, Motivation for Invariance on the Example of Definitions and the . . . Pavement Roughness Proof Home Page Edgar Daniel Rodriguez Velasquez 1 , 2 , Title Page Carlos M. Chang Albitres 2 , and Vladik Kreinovich 3 ◭◭ ◮◮ 1 Universidad de Piura in Peru (UDEP), edgar.rodriguez@udep.pe Departments of 2 Civil Engineering and 3 Computer Science ◭ ◮ University of Texas at El Paso, El Paso, Texas 79968, USA, edrodriguezvelasquez@miners.utep.edu, cchangalbitres2@utep.edu, Page 1 of 22 vladik@utep.edu Go Back Full Screen Close Quit
Cumulative Quantities Formulation of the . . . 1. Cumulative Quantities Case Study: . . . • Many physical quantities can be measured directly: Empirical Relation . . . e.g., we can directly measure mass, acceleration, force. What We Do in This Talk Main Idea • However, we are often interested in cumulative quanti- Motivation for Invariance ties that combine values corresponding to: Definitions and the . . . – different moments of time and/or Proof – different locations. Home Page • For example: Title Page ◭◭ ◮◮ – when we are studying public health or pollution or economic characteristics, ◭ ◮ – we are often interested in characteristics describing Page 2 of 22 the whole city, the whole region, the whole country. Go Back Full Screen Close Quit
Cumulative Quantities Formulation of the . . . 2. Formulation of the Problem Case Study: . . . • Cumulative characteristics are not easy to measure. Empirical Relation . . . What We Do in This Talk • To measure each such characteristic, we need: Main Idea – to perform a large number of measurements, and Motivation for Invariance then Definitions and the . . . – to use an appropriate algorithm to combine these Proof results into a single value. Home Page • Such measurements are complicated. Title Page • So, we often have to supplement the measurement re- ◭◭ ◮◮ sults with expert estimates. ◭ ◮ • To process such data, it is desirable to describe both Page 3 of 22 estimates in the same scale: Go Back – to estimate the actual value of the corresponding Full Screen quantity based on the expert estimate, and – vice versa. Close Quit
Cumulative Quantities Formulation of the . . . 3. Case Study: Estimating Pavement Roughness Case Study: . . . • Estimating road roughness is an important problem. Empirical Relation . . . What We Do in This Talk • Indeed, road pavements need to be maintained and re- Main Idea paired. Motivation for Invariance • Both maintenance and repair are expensive. Definitions and the . . . • So, it is desirable to estimate the pavement roughness Proof Home Page as accurately as possible. Title Page • If we overestimate the road roughness, we will waste money on “repairing” an already good road. ◭◭ ◮◮ • If we underestimate the road roughness, the road seg- ◭ ◮ ment will be left unrepaired and deteriorate further. Page 4 of 22 • As a result, the cost of future repair will skyrocket. Go Back • The standard way to measure the pavement roughness Full Screen is to use the International Roughness Index (IRI). Close Quit
Cumulative Quantities Formulation of the . . . 4. Estimating Pavement Roughness (cont-d) Case Study: . . . • Crudely speaking, IRI describes the effect of the pave- Empirical Relation . . . ment roughness on a standardized model of a vehicle. What We Do in This Talk Main Idea • Measuring IRI is not easy, because the real vehicles Motivation for Invariance differ from this standardized model. Definitions and the . . . • As a result, we measure roughness by some instruments Proof and use these measurements to estimate IRI. Home Page • For example, we can: Title Page – perform measurements by driving an available ve- ◭◭ ◮◮ hicle along this road segment, ◭ ◮ – extract the local roughness characteristics from the Page 5 of 22 effect of the pavement on this vehicle, and then Go Back – estimate the effect of the same pavement on the standardized vehicle. Full Screen Close Quit
Cumulative Quantities Formulation of the . . . 5. Estimating Pavement Roughness (cont-d) Case Study: . . . • In view of this difficulty, in many cases, practitioners Empirical Relation . . . rely on expert estimates of the pavement roughness. What We Do in This Talk Main Idea • The corr. measure – estimated on a scale from 0 to 5 – Motivation for Invariance is known as the Present Serviceability Rating (PSR). Definitions and the . . . Proof Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 22 Go Back Full Screen Close Quit
Cumulative Quantities Formulation of the . . . 6. Empirical Relation Between Measurement Re- Case Study: . . . sults and Expert Estimates Empirical Relation . . . • The empirical relation between PSR and IRI is de- What We Do in This Talk scribed by the 1994 Al-Omari-Darter formula: Main Idea Motivation for Invariance PSR = 5 · exp( − 0 . 0041 · IRI) . Definitions and the . . . • This formula remains actively used in pavement engi- Proof Home Page neering. Title Page • It works much better than many previously proposed alternative formulas, such as ◭◭ ◮◮ √ ◭ ◮ PSR = a + b · IRI . Page 7 of 22 • However, it is not clear why namely this formula works Go Back so well. Full Screen Close Quit
Cumulative Quantities Formulation of the . . . 7. What We Do in This Talk Case Study: . . . • We propose a possible explanation for the above em- Empirical Relation . . . pirical formula. What We Do in This Talk Main Idea • This explanation will be general: it will apply to all Motivation for Invariance possible cases of cumulative quantities. Definitions and the . . . • We will come up with a general formula y = f ( x ) that Proof describes how: Home Page – a subjective estimate y of a cumulative quantity Title Page – depends on the result x of its measurement. ◭◭ ◮◮ • As a case study, we will use gauging road roughness. ◭ ◮ Page 8 of 22 Go Back Full Screen Close Quit
Cumulative Quantities Formulation of the . . . 8. Main Idea Case Study: . . . • In general, the numerical value of a subjective estimate Empirical Relation . . . depends on the scale. What We Do in This Talk Main Idea • In road roughness estimates, we usually use a 0-to-5 Motivation for Invariance scale. Definitions and the . . . • In other applications, it may be more customary to use Proof 0-to-10 or 0-to-1 scales. Home Page • A usual way to transform between the two scales is to Title Page multiply all the values by a corresponding factor. ◭◭ ◮◮ • For example, to transform from 0-to-10 to 0-to-1 scale, ◭ ◮ we multiply all the values by λ = 0 . 1. Page 9 of 22 • In other transitions, we can use transformations y → Go Back λ · y with different re-scaling factors λ . Full Screen • There is no major advantage in selecting a specific scale. Close Quit
Cumulative Quantities Formulation of the . . . 9. Main Idea (cont-d) Case Study: . . . • So, subjective estimates are defined modulo such a re- Empirical Relation . . . scaling transformation y → λ · y. What We Do in This Talk Main Idea • At first glance, the result of measuring a cumulative Motivation for Invariance quantity may look uniquely determined. Definitions and the . . . • However, a detailed analysis shows that there is some Proof non-uniqueness here as well. Home Page • Indeed, the result of a cumulative measurement comes Title Page from combining values measured: ◭◭ ◮◮ – at different moments of time and/or ◭ ◮ – values corresponding to different spatial locations. Page 10 of 22 • For each individual measurement, the probability of a Go Back sensor’s malfunction may be low. Full Screen • However, often, we perform a large number of measure- ments. Close Quit
Cumulative Quantities Formulation of the . . . 10. Main Idea (cont-d) Case Study: . . . • So, some of them bound to be caused by such malfunc- Empirical Relation . . . tions and are, thus, outliers. What We Do in This Talk Main Idea • It is well known that even a single outlier can drasti- Motivation for Invariance cally change the average. Definitions and the . . . • So, to avoid such influence, the usual algorithms first Proof filter out possible outliers. Home Page • This filtering is not an exact science; we can set up: Title Page – slightly different thresholds for detecting an outlier, ◭◭ ◮◮ – slightly different threshold for allowed number of ◭ ◮ remaining outliers, etc. Page 11 of 22 • We may get a computation result that only takes actual Go Back signals into account. Full Screen • With a different setting, we may get a different result, affected by a few outliers. Close Quit
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