Fractional Linear What We Do Dependence under Interval Main Idea - - PowerPoint PPT Presentation

Fractional Linear . . . Fractional Linear . . . Need to Find . . . 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 1 of 10 Go Back Full Screen Close Quit

Fractional Linear Dependence under Interval Uncertainty: Explicit Bounds

William Basquez, Elton Villa, and Vladik Kreinovich

Computer Science Department University of Texas at El Paso El Paso, TX 79968, USA webasquez@miners.utep.edu, euvilla@miners.utep.edu, vladik@utep.edu

Fractional Linear . . . Fractional Linear . . . Need to Find . . . 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 2 of 10 Go Back Full Screen Close Quit

1. Fractional Linear Dependencies Are Ubiquitous

• To describe a physical quantity by a numerical value,

we need to select a measuring procedure.

• If we change a measuring unit, then all numerical val-

ues are multiplied by a constant: x → a · x.

• If we change a starting point, to all numerical values a

constant is added: x → x + b.

• In addition to such linear transformations x → a·x+b,

we may also have nonlinear ones.

• The class of all reasonable transformations must be:

– closed under composition, – contain all linear functions, and – be described by finitely many parameters.

• The reason for the last requirement is that in a com-

puter, we can only store finitely many values.

Fractional Linear . . . Fractional Linear . . . Need to Find . . . 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 3 of 10 Go Back Full Screen Close Quit

2. Fractional Linear Dependencies (cont-d)

• It turns out that all such transformation are fractionally-

linear.

• Indeed, fractional-linear transformations

y = a · x + b c · x + d are ubiquitous in practice.

• We can always change the starting point for y so that

x = 0 would correspond to y = 0.

• In this case, we get b = 0, and, dividing both numera-

tor and denominator by d, we get y = a · x 1 + c · x.

• For small x, this is approximately equal to a · x.
• So this formula is a reasonable next approximation to

the linear dependence y = a · x.

Fractional Linear . . . Fractional Linear . . . Need to Find . . . 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 4 of 10 Go Back Full Screen Close Quit

3. Need to Find Parameters of the Fractional Lin- ear Dependence from Data

• The parameters a and c need to be determined from

measurement results xk and yk, k = 1, . . . , n.

• Let us first consider an ideal case, when measurements

are absolutely accurate.

• In this case, we get linear equations yk+c·xk·yk = a·xk

for determining a and c.

Fractional Linear . . . Fractional Linear . . . Need to Find . . . 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 5 of 10 Go Back Full Screen Close Quit

4. Need to Take Interval Uncertainty Into Ac- count

• In practice, we only measure these values with some

uncertainty.

• So, the measurement results

xk and yk, in general, dif- fer from the actual (unknown) values xk and yk.

• What do we know about the measurement error, e.g.,

about ∆xk

def

= xk − xk?

• Often, the only information we have is the upper bound

∆xk on its absolute value: |∆xk| ≤ ∆xk.

• Then, after the measurement, all we know about xk is

that it is between xk = xk − ∆k and xk = xk + ∆k.

• Under such interval uncertainty, we need to find the

ranges of possible values of a and c.

Fractional Linear . . . Fractional Linear . . . Need to Find . . . 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 6 of 10 Go Back Full Screen Close Quit

5. What We Do

• In this talk, we show how to do it under the assumption

that xk, yk, and c are all non-negative.

• The formulas can be easily modified to the general case.

Fractional Linear . . . Fractional Linear . . . Need to Find . . . 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 7 of 10 Go Back Full Screen Close Quit

6. Main Idea

• Based on each measurement result, we can conclude

that a = yk ·

• c + 1

xk

• .
• The largest possible value of this expression is when yk

is the largest and xk is the smallest, and vice versa.

• So, we get bounds on a corresponding to each measure-

ment: yk ·

• c + 1

xk

• ≤ a ≤ yk ·
• c + 1

xk

• .
• Such a value a exists if and only if each lower bound

does not exceed each upper bound: yk ·

• c + 1

xk

• ≤ yℓ ·
• c + 1

xℓ

• for all k and ℓ.
• We thus get n2 linear inequalities, each of which can

be reformulated as ckℓ ≤ c or c ≤ ckℓ.

Fractional Linear . . . Fractional Linear . . . Need to Find . . . 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 8 of 10 Go Back Full Screen Close Quit

7. Main Idea (cont-d)

• Reminder: we have inequalities ckℓ ≤ c or c ≤ ckℓ.
• The range of possible values of c can hence be explicitly

described as [max(ckℓ), min(ckℓ)].

• If we start with inequalities on c, we similarly get ex-

plicit bounds on a.

• Namely, we get 1+c·xk = a· xk

yk , hence c·xk = a· xk yk −1 and c = a yk − 1 xk ; thus: a yk − 1 xk ≤ c ≤ a yk − 1 xk .

Fractional Linear . . . Fractional Linear . . . Need to Find . . . 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 9 of 10 Go Back Full Screen Close Quit

8. Main Idea (cont-d)

• Such a c exists if every lower bound is not larger than

every upper bound: a yk − 1 xk ≤ a yℓ − 1 xℓ for each k and ℓ.

• Each such linear inequality can be represented as either

akℓ ≤ a or a ≤ akℓ.

• So, the range of possible values of a is

[max(akℓ), min(akℓ)].

Fractional Linear . . . Fractional Linear . . . Need to Find . . . 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

9. How Good Are the Resulting Formulas

• The resulting formulas require O(n2) computation steps.