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. Overview. Three kinds of operations Deconvolutions Backcalculations Updates (oh, my!) - PowerPoint PPT PresentationTRANSCRIPT
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Untangling equations involving uncertainty
Scott Ferson, Applied BiomathematicsVladik Kreinovich, University of Texas at El PasoW. 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, nota uniform distribution
Cum
ulat
ive
prob
abili
ty
0 10 20 30 400
0.5
1
10 20 30 400
0.5
1
10 20 300
0.5
1
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Probability bounds analysis (PBA)
assumingindependence
0 800
1
assumingnothing
1
0 800
0 400
1
0 200
1
CD
F
assumingindependence
0 800
1
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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, it’s 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
• SupposeW = [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 thatW = [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
20003200])3200,2000[(
112150])150,112[(
2333])33,23[(),,Pr(
AIHIWIAHW
)(),,|( HWAAHWHWAL
),,Pr()()|,,( AHWHWAHWAAHWf
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
2000 3000 40000
1
A20 30 40
0
1
W120 140 160
0
1
H
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2000 3000 40000
1
A20 30 40
0
1
W120 140 160
0
1
H
Answers
intersect(W, A/H) intersect(H, A/W) intersect(A, WH)
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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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Can’t just invert the equation
When conc is put back into the forward equation, the dose is wider than planned
dose body mass intakeconc =
conc intake body massdose =
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Exampledose = [0, 2] milligram per kilogramintake = [1, 2.5] litermass = [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-boxesSuppose A + B = C, where
A = normal(5, 1)C = {0 C, median 15, 90th %ile 35, max 50}
2 3 4 5 6 7 80
1 A
-10 0 10 20 30 40 50 600
1 C
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Getting the answer
• The backcalculation algorithm basically reverses the forward convolution
• Not hard at all…but a little messy to show• Any distribution
totally inside B is sure to satisfy the constraint … it’s “kernel”
-10 0 10 20 30 40 500
1 B
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Check by plugging back in
A + B = C* C
-10 0 10 20 30 40 50 600
1
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, BA, CB ,CA, BA, CB ,CA, BA, CB ,CA, BA, CB ,CA, BA, CB ,C
ACAC
Use this formulato find the unknownC = A + BB = backcalc(A,C)A = backcalc (B,C)C = A – BB = –backcalc(A,C)A = backcalc (–B,C)C = A * BB = factor(A,C)A = factor(B,C)C = A / BB = 1/factor(A,C)A = factor(1/B,C)C = A ^ BB = factor(log A, log C)A = exp(factor(B, log C))C = 2 * AA = C / 2C = A ^ 2A = 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 can’t 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 didn’t 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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20000
1
110 120 130 140 150 1600
1
H
20 30 400
1
W