Why Boxes for Multi-D Our Result Proof Uncertainty? Home Page - - PowerPoint PPT Presentation

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Why Boxes for Multi-D Uncertainty?

Christian Servin1, Erick Duarte2, Francisco Rodriguez2, Olga Kosheleva2, and Vladik Kreinovich2

1El Paso Community College, 919 Hunter

El Paso, TX 79915 USA, cservin@gmail.com

2University of Texas at El Paso, El Paso, TX 79968 USA

{ejduarte2,farodriguezcabral}@miners.utep.edu

• lgak@utep.edu, vladik@utep.edu

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1. Formulation of the Problem

• In practice, we often do not have a complete informa-

tion about the values of several quantities x1, . . . , xn.

• Thus, several possible tuples x = (x1, . . . , xn) are con-

sistent with our knowledge.

• How can we describe the set of possible values of x?
• In many practical applications,we get reasonable re-

sults if we approximate this set by a box [x1, x1] × . . . × [xn, xn].

• Why boxes? Why not other families of sets?

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2. What are Reasonable Properties of the Corre- sponding Family F of Approximating Sets?

• Usually, we know some bounds on all xi, so each set S

should be bounded.

• It is reasonable to require that each set S ∈ F is closed:

indeed, – if a = lim an for an ∈ S, – then for any measurement accuracy, a is indistin- guishable from some some possible an ∈ S, – thus, we will never be able to tell that a is not possible.

• Similarly, the family F should be closed.

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3. Reasonable Properties (cont-d)

• Also, often, the observed tuple consists of two indepen-

dent components xi = yi + zi; so: – if the set Y of all y’s and the set Z of all z’s are possible, – then the set Y + Z

def

= {y + z : y ∈ Y, z ∈ Z} is also possible; – this set is called the Minkowski sum.

• So, F should be closed under Minkowski sum.
• Finally, the numerical values of each quantity xi change

if we change the starting point and/or measuring unit: xi → ai + bi · xi.

• These changes do not change what is possible.
• So, F should be closed under these transformations.

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4. Our Result

• Every family F that satisfies these properties contains

all the boxes.

• Thus, the boxes form the simplest possible approxi-

mating family.

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5. Proof

• For each set S, we can take:

– first S1 = S, – then Sk+1 = 0.5 · (Sk + Sk) for k = 0, 1, 2, . . .

• In the limit, we get a convex hull of S.
• By appropriate re-scaling xi → bi · xi, we can shrink

this set S in all directions except one i0.

• In the limit, we get an interval parallel to the i0-th axis.
• By shifting and re-scaling, we get all possible intervals

parallel to this axis.

• We can thus have n intervals [xi, xi] each of which is:

– parallel to the i-th axis and – has all other coordinates 0s.

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6. Proof (cont-d)

• We can have n intervals

[0, 0] × . . . × [0, 0] × [xi, xi] × [0, 0] × . . . × [0, 0].

• The Minkowski sum of these intervals is the box

[x1, x1] × . . . × [xn, xn].

• Thus, each family F satisfying the above properties

contains all the boxes.

• (Also, the class of all boxes satisfies all the above prop-

erties.)