Title: Untangling equations involving uncertainty
1Untangling equations involving uncertainty
- Scott Ferson, Applied Biomathematics
- Vladik Kreinovich, University of Texas at El Paso
- W. Troy Tucker, Applied Biomathematics
2Overview
- 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
3Probability 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
4Probability 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
5PBA 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
6Updating
- 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
7Example
- Suppose
- W 23, 33
- H 112, 150
- A 2000, 3200
- Does knowing W ? H A let us to say any more?
8Answer
- 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.
9Bayesian strategy
Prior
Likelihood
Posterior
10Bayes 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
11Updating with p-boxes
1
1
1
A
H
W
0
0
0
2000
3000
4000
20
30
40
120
140
160
12Answers
13Calculation 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
14Backcalculation
- 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
15Cant just invert the equation
- When conc is put back into the forward equation,
the dose is wider than planned
16Example
- 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!
17Backcalculating 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
18Backcalculation with p-boxes
- Suppose A B C, where
- A normal(5, 1)
- C 0 ? C, median ? 15, 90th ile ? 35, max ?
50
19Getting 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
20Check by plugging back in
21When 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)
22Kernels
- 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)
23Precise 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
24Deconvolution
- 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
25Example
- 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)
26Deconvolutions 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)
27Relaxing 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
28P-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
29Acknowledgments
- Janos Hajagos, Stony Brook University
- Lev Ginzburg, Stony Brook University
- David Myers, Applied Biomathematics
- National Institutes of Health SBIR program
30End
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