Fractional Linear . . . Fractional Linear . . . Need to Find . . . Need to Take Interval . . . Fractional Linear What We Do Dependence under Interval Main Idea Main Idea (cont-d) Uncertainty: Main Idea (cont-d) How Good Are the . . . Explicit Bounds Home Page Title Page William Basquez, Elton Villa, and Vladik Kreinovich ◭◭ ◮◮ Computer Science Department ◭ ◮ University of Texas at El Paso El Paso, TX 79968, USA Page 1 of 10 webasquez@miners.utep.edu, euvilla@miners.utep.edu, Go Back vladik@utep.edu Full Screen Close Quit
Fractional Linear . . . Fractional Linear . . . 1. Fractional Linear Dependencies Are Ubiquitous Need to Find . . . • To describe a physical quantity by a numerical value, Need to Take Interval . . . we need to select a measuring procedure. What We Do Main Idea • If we change a measuring unit, then all numerical val- Main Idea (cont-d) ues are multiplied by a constant: x → a · x . Main Idea (cont-d) • If we change a starting point, to all numerical values a How Good Are the . . . constant is added: x → x + b . Home Page • In addition to such linear transformations x → a · x + b , Title Page we may also have nonlinear ones. ◭◭ ◮◮ • The class of all reasonable transformations must be: ◭ ◮ – closed under composition, Page 2 of 10 – contain all linear functions, and Go Back – be described by finitely many parameters. Full Screen • The reason for the last requirement is that in a com- puter, we can only store finitely many values. Close Quit
Fractional Linear . . . Fractional Linear . . . 2. Fractional Linear Dependencies (cont-d) Need to Find . . . • It turns out that all such transformation are fractionally- Need to Take Interval . . . linear. What We Do Main Idea • Indeed, fractional-linear transformations y = a · x + b Main Idea (cont-d) c · x + d are ubiquitous in practice. Main Idea (cont-d) How Good Are the . . . • We can always change the starting point for y so that Home Page x = 0 would correspond to y = 0. Title Page • In this case, we get b = 0, and, dividing both numera- a · x ◭◭ ◮◮ tor and denominator by d , we get y = 1 + c · x . ◭ ◮ • For small x , this is approximately equal to a · x . Page 3 of 10 • So this formula is a reasonable next approximation to Go Back the linear dependence y = a · x . Full Screen Close Quit
Fractional Linear . . . Fractional Linear . . . 3. Need to Find Parameters of the Fractional Lin- Need to Find . . . ear Dependence from Data Need to Take Interval . . . • The parameters a and c need to be determined from What We Do measurement results x k and y k , k = 1 , . . . , n . Main Idea Main Idea (cont-d) • Let us first consider an ideal case, when measurements Main Idea (cont-d) are absolutely accurate. How Good Are the . . . • In this case, we get linear equations y k + c · x k · y k = a · x k Home Page for determining a and c . Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 10 Go Back Full Screen Close Quit
Fractional Linear . . . Fractional Linear . . . 4. Need to Take Interval Uncertainty Into Ac- Need to Find . . . count Need to Take Interval . . . • In practice, we only measure these values with some What We Do uncertainty. Main Idea Main Idea (cont-d) • So, the measurement results � x k and � y k , in general, dif- Main Idea (cont-d) fer from the actual (unknown) values x k and y k . How Good Are the . . . • What do we know about the measurement error, e.g., Home Page def about ∆ x k = � x k − x k ? Title Page • Often, the only information we have is the upper bound ◭◭ ◮◮ ∆ xk on its absolute value: | ∆ x k | ≤ ∆ xk . ◭ ◮ • Then, after the measurement, all we know about x k is Page 5 of 10 that it is between x k = � x k − ∆ k and x k = � x k + ∆ k . Go Back • Under such interval uncertainty, we need to find the Full Screen ranges of possible values of a and c . Close Quit
Fractional Linear . . . Fractional Linear . . . 5. What We Do Need to Find . . . • In this talk, we show how to do it under the assumption Need to Take Interval . . . that x k , y k , and c are all non-negative. What We Do Main Idea • The formulas can be easily modified to the general case. Main Idea (cont-d) Main Idea (cont-d) How Good Are the . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 10 Go Back Full Screen Close Quit
Fractional Linear . . . Fractional Linear . . . 6. Main Idea Need to Find . . . • Based on each measurement result, we can conclude Need to Take Interval . . . � � c + 1 What We Do that a = y k · . x k Main Idea • The largest possible value of this expression is when y k Main Idea (cont-d) is the largest and x k is the smallest, and vice versa. Main Idea (cont-d) How Good Are the . . . • So, we get bounds on a corresponding to each measure- Home Page ment: � � � � c + 1 c + 1 Title Page y k · ≤ a ≤ y k · . x k x k ◭◭ ◮◮ • Such a value a exists if and only if each lower bound ◭ ◮ does not exceed each upper bound: � � � � Page 7 of 10 c + 1 c + 1 y k · ≤ y ℓ · for all k and ℓ. Go Back x k x ℓ • We thus get n 2 linear inequalities, each of which can Full Screen be reformulated as c kℓ ≤ c or c ≤ c kℓ . Close Quit
Fractional Linear . . . Fractional Linear . . . 7. Main Idea (cont-d) Need to Find . . . • Reminder: we have inequalities c kℓ ≤ c or c ≤ c kℓ . Need to Take Interval . . . What We Do • The range of possible values of c can hence be explicitly Main Idea described as Main Idea (cont-d) [max( c kℓ ) , min( c kℓ )] . Main Idea (cont-d) • If we start with inequalities on c , we similarly get ex- How Good Are the . . . plicit bounds on a . Home Page • Namely, we get 1+ c · x k = a · x k , hence c · x k = a · x k − 1 Title Page y k y k − 1 and c = a ◭◭ ◮◮ ; thus: y k x k ◭ ◮ − 1 − 1 a ≤ c ≤ a Page 8 of 10 . y k x k y k x k Go Back Full Screen Close Quit
Fractional Linear . . . Fractional Linear . . . 8. Main Idea (cont-d) Need to Find . . . • Such a c exists if every lower bound is not larger than Need to Take Interval . . . every upper bound: What We Do Main Idea a − 1 ≤ a − 1 for each k and ℓ. Main Idea (cont-d) y k x k y ℓ x ℓ Main Idea (cont-d) • Each such linear inequality can be represented as either How Good Are the . . . a kℓ ≤ a or a ≤ a kℓ . Home Page • So, the range of possible values of a is Title Page ◭◭ ◮◮ [max( a kℓ ) , min( a kℓ )] . ◭ ◮ Page 9 of 10 Go Back Full Screen Close Quit
9. How Good Are the Resulting Formulas Fractional Linear . . . Fractional Linear . . . Need to Find . . . • The resulting formulas require O ( n 2 ) computation steps. Need to Take Interval . . . What We Do Main Idea Main Idea (cont-d) Main Idea (cont-d) How Good Are the . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 10 Go Back Full Screen Close Quit
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