Untangling equations involving uncertainty

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Untangling equations involving uncertainty

• Scott Ferson, Applied Biomathematics
• Vladik Kreinovich, University of Texas at El Paso
• W. Troy Tucker, Applied Biomathematics

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Overview

• Three kinds of operations
• Deconvolutions
• Backcalculations
• Updates (oh, my!)
• Very elementary methods of interval analysis
• Low-dimensional
• Simple arithmetic operations
• But combined with probability theory

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Probability box (p-box)

• Bounds on a cumulative distribution function
  (CDF)
• Envelope of a Dempster-Shafer structure
• Used in risk analysis and uncertainty arithmetic
• Generalizes probability distributions and
  intervals

This is an interval, not a uniform distribution
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Probability bounds analysis (PBA)
a T( 0 , 10 , 20) 0, 5 b
N(20,23,1,12) Disagreement between
theoretical and observed variance Disagreement
between theoretical and observed
variance Disagreement between theoretical and
observed variance c a b c a b
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PBA handles common problems

• Imprecisely specified distributions
• Poorly known or unknown dependencies
• Non-negligible measurement error
• Inconsistency in the quality of input data
• Model uncertainty and non-stationarity
• Plus, its much faster than Monte Carlo

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Updating

• Using knowledge of how variables are related to
  tighten their estimates
• Removes internal inconsistency and explicates
  unrecognized knowledge
• Also called constraint updating or editing
• Also called natural extension

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Example

• Suppose
• W 23, 33
• H 112, 150
• A 2000, 3200
• Does knowing W ? H A let us to say any more?

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Answer

• Yes, we can infer that
• W 23, 28.57
• H 112, 139.13
• A 2576, 3200
• The formulas are just W intersect(W, A/H), etc.

To get the largest possible W, for instance, let
A be as large as possible and H as small as
possible, and solve for W A/H.
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Bayesian strategy
Prior
Likelihood
Posterior
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Bayes rule

• Concentrates mass onto the manifold of feasible
  combinations of W, H, and A
• Answers have the same supports as intervals
• Computationally complex
• Needs specification of priors
• Yields distributions that are not justified (come
  from the choice of priors)
• Expresses less uncertainty than is present

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Updating with p-boxes
1
1
1
A
H
W
0
0
0
2000
3000
4000
20
30
40
120
140
160
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Answers
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Calculation with p-boxes

• Agrees with interval analysis whenever inputs are
  intervals
• Relaxes Bayesian strategy when precise priors are
  not warranted
• Produces more reasonable answers when priors not
  well known
• Much easier to compute than Bayes rule

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Backcalculation

• Find constraints on B that ensure C A B
  satisfies specified constraints
• Or, more generally, C f(A1, A2,, Ak, B)
• If A and C are intervals, the answer is called
  the tolerance solution

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Cant just invert the equation

• When conc is put back into the forward equation,
  the dose is wider than planned

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Example

• dose 0, 2 milligram per kilogram
• intake 1, 2.5 liter
• mass 60, 96 kilogram
• conc dose mass / intake
• 0, 192 milligram liter-1
• dose conc intake / mass
• 0, 8 milligram kilogram-1

Doses 4 times larger than tolerable levels!
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Backcalculating probability distributions

• Needed for engineering design problems, e.g.,
  cleanup and remediation planning for
  environmental contamination
• Available analytical algorithms are unstable for
  almost all problems
• Except in a few special cases, Monte Carlo
  simulation cannot compute backcalculations trial
  and error methods are required

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Backcalculation with p-boxes

• Suppose A B C, where
• A normal(5, 1)
• C 0 ? C, median ? 15, 90th ile ? 35, max ?
  50

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Getting the answer

• The backcalculation algorithm basically reverses
  the forward convolution
• Not hard at allbut a little messy to show
• Any distribution totally inside B is
  sure to satisfy the constraint
  its kernel

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Check by plugging back in

• A B C ? C

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When you Know that A B C A B C A ?
B C A / B C A B C 2A C A² C
And you have estimates for A, B A, C B ,C A,
B A, C B ,C A, B A, C B ,C A, B A, C B ,C A, B A,
C B ,C A C A C
Use this formula to find the unknown C A B B
backcalc(A,C) A backcalc (B,C) C A B B
backcalc(A,C) A backcalc (B,C) C A B B
factor(A,C) A factor(B,C) C A / B B
1/factor(A,C) A factor(1/B,C) C A B B
factor(log A, log C) A exp(factor(B, log C)) C
2 A A C / 2 C A 2 A sqrt(C)
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Kernels

• Existence more likely if p-boxes are fat
• Wider if we can also assume independence
• Answers are not unique, even though tolerance
  solutions always are
• Different kernels can emphasize different
  properties
• Envelope of all possible kernels is the shell
  (i.e., the united solution)

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Precise distributions

• Precise distributions cant express the nature of
  the target
• Finding a conc distribution that results in a
  prescribed distribution of doses says we want
  some doses to be high (any distribution to the
  left would be even better)
• We need to express the dose target as a p-box

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Deconvolution

• Uses information about dependence to tighten
  estimates
• Useful, for instance, in correcting an estimated
  distribution for measurement uncertainty
• For instance, suppose Y X ?
• If X and ? are independent, ?Y² ?X² ??²
• Then we do an uncertainty correction

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Example

• Y X ?
• Y, ? normal
• X N(decon(?Y, ?X), sqrt(decon(??², ?Y²))
• Y N(5,9, 2,3) ? N(?1,1, ½,1)
• X N(dcn(?1,1,5,6), sqrt(dcn(¼,1,4,9)))
• X N(6,8, sqrt(3, 63)

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Deconvolutions with p-boxes

• As for backcalculations, computation of
  deconvolutions is troublesome in probability
  theory, but often much simpler with p-boxes
• Deconvolution didnt have an analog in interval
  analysis (until now via p-boxes)

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Relaxing over-determination

• Most constraint problems almost never have
  solutions with probability distributions
• The constraints are too numerous and strict
• P-boxes relax these constraints so that many
  problems can have solutions

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P-boxes in interval analysis

• P-boxes bring probability distributions into the
  realm of intervals
• Express and solve backcalculation problems better
  than is possible in probability theory by itself
• Generalize the notion of tolerance solutions
  (kernels)
• Relax unwarranted assumptions about priors in
  updating problems needed in a Bayesian approach
• Introduce deconvolution into interval analysis

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Acknowledgments

• Janos Hajagos, Stony Brook University
• Lev Ginzburg, Stony Brook University
• David Myers, Applied Biomathematics
• National Institutes of Health SBIR program

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End
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