Moment propagation

Scott Ferson, scott@ramas.com11 September 2007, Stony Brook University, MAR 550, Challenger 165

Outline of Moment Propagation

Delta method

Intervals (worst case analysis)

• Easy to understand and calculate with

• Often good enough to make a decision

• Appropriate for use with even the worst data

• Results often too wide to be practically useful

• Don’t say anthing about tail risks

Moments (delta method)

• Easy to compute

• More precise than is justfied

What to do?

• Solution is to marry intervals and moments– Intervals can be tighter if we use moment information– Bounding moments would tell us about tails

What do moments say about risks?

If we know the mean is 10and the variance is 2, theseare best possible bounds on the chance the variableis bigger than any value (Chebyshev inequality).

-10 0 10 20 300

Exc

eeda

nce

risk

1

Moment propagationMean Variance

k + X EX + k VXkX k EX k2 VXexp(X) rowe(exp) rowevar(exp)ln(X) , 0<Xrowe(ln) rowevar(ln)log10(X), 0<X rowe(log10) rowevar(log10)1/X, X¹0 rowe(reciprocal) rowevar(reciprocal)X2 EX2 + VX rowevar(square)sqrt(X), 0X rowe(sqrt) rowevar(sqrt)X + Y EX + EY (VX ± VY)2

X Y EX EY (VX ± VY)2

X Y EX EY ± (VX VY ) Goodman formulaX Y, Y0 E(X (1/Y)) V(X (1/Y))XY, 1X E(exp(ln(X)´Y)) V(exp(ln(X)´Y))where EZ and VZ are the mean and variance of the random variable Z

Range propagation (interval analysis)Least possible value Greatest possible value

k + X k + LX k + GXkX k LX , if 0 k k GX , if k < 0

k GX , if 0 k k LX , if k < 0exp(X) exp(LX) exp(GX)ln(X) , 0<X ln(LX) ln(GX)log10(X) , 0<X log10(LX) log10(GX)1/X, X 0 1/GX 1/LXX2 0, if LX 0 GX max((LX)2,(GX)2)

min((LX)2,(GX)2), elsesqrt(X), 0 X sqrt(LX) sqrt(GX)|X| 0, if LX 0 GX max(|LX|,|GX|)

min(|LX|,|GX|), elseX + Y LX + LY GX + GYX Y LX GY GX LYX Y min(LX LY, LX GY, GX LY, GX GY) max(LXLY,LXGY,GXLY,GXGY)X Y, Y 0 L(X 1/Y) G(X 1/Y)XY, 1 X min(LXLY, GXGY, LXGY, GXLY) max(LXLY, GXGY, LXGY, GXLY)min(X, Y) min(LX, LY) min(GX, GY)max(X, Y) max(LX, LY) max(GX, GY) where LZ and GZ are the leaster and greatest possible values of Z

Intervals about moments

• Even if we can’t say what the distributions and dependencies are, we can project the means and variances through calculations.

• If we know the variables are independent, then the projections will be tighter.

• This can be combined with propagation of the ranges as well.

Range and moments together

EX

VX

GXLX

Pro

babi

lity

(x <

X) 1

0

Interpreting a p-boxP

roba

bilit

y (x

< X

) 1

0

{min = 0, max = 100, mean = 50, stdev = s}

s = 1 s = 5 s = 10 s = 15

s = 20 s = 25 s = 30 s = 35

s = 40 s = 45 s = 49 s = 50

{min = 0, max = 100, mean = 10, stdev = s}

s = 1 s = 2 s = 3 s = 4

s = 5 s = 6 s = 8 s = 10

s = 15 s = 20 s = 25 s = 29

Interval bounds on moments

• What if we don’t know the variance? Mean?

Travel time (Lobascio)

iK

LKocfocBDnT

ParameterL source-receptor distancei hydraulic gradientK hydraulic conductivityn effective soil porosityBD soil bulk densityfoc fraction organic carbonKoc organic partition coefficient

Min800.00033000.215000.00015

Max1200.000830000.3517500.00520

Mean1000.0005510000.2516500.0025510

Stdv11.550.00014437500.051000.0014153

Shapeuniformuniformlognormallognormallognormaluniformnormal

Unitsmm/mm/yr-kg/m3

-m3/kg

Inputs as mmms p-boxes

0 0.002 0.0040

1

foc

1400 1600 18000

1

BD

kg m-3

0.0003 0.0006 0.00090

1

i

0 10 20 300

1

Koc

m3 kg-1

0 2000 40000

1

K

m yr-10.2 0.3 0.4

0

1

n

70 90 110 1300

1

L

m

0 1000000

0.5

1

Tind [yr]

Quantitative results

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000Traveling time (years)

Cu

mu

lati

ve p

rob

abil

ity

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

Traveling time (years)

Cu

mu

lati

ve p

rob

abil

ity

relax independence assumptions

original model

Is independence reasonable?

• Soil porosity and soil bulk density?• Hydraulic conductivity and soil porosity?• Hydraulic gradient and hydraulic conductivity?• Fraction organic carbon and organic partition

coefficient?• You’re the groundwater modelers; you tell us• Remember: independence is a much stronger

assumption than uncorrelatedness

Assumptions no longer needed

• A decade ago, you had to assume all variables were mutually independent

• Software tools now allow us to relax any pesky independence assumption

• No longer necessary to make independence assumptions for mathematical convenience

• But do the assumptions make any difference?

0 1000000

0.5

1

Tdep [yr]

Quantitative results

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000Traveling time (years)

Cu

mu

lati

ve p

rob

abil

ity

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

Traveling time (years)

Cu

mu

lati

ve p

rob

abil

ity

relax independence assumptions

original model

Dependence bounds

• Guaranteed to enclose results no matter what correlation or dependence there may be between the variables

• Best possible (couldn’t be any tighter without saying more about the dependence)

• Can be combined with independence assumptions between other variables

Conclusions

• The model is a cartoon, but it illustrates the use of methods to relax independence and precise distribution assumptions

• Relaxing these assumptions can have a big impact on quantitative conclusions from an assessment

Take-home message

• Whatever assumption about dependencies and the shape of distributions is between you and your spreadsheet

• There are methods now available that don’t force you to make assumptions you’re not comfortable with

Acknowledgments

• Srikanta Mishra• Neil Blandford • William Oberkampf

• Sandia National Laboratories • National Cancer Institute• National Institute of Environmental Health

Sciences

More information

• Website: http://www.ramas.com/riskcalc.htm• Email: scott@ramas.com, troy@ramas.com• Paper: Ferson, S. 1996. What Monte Carlo methods

cannot do. Human and Ecological Risk Assessment 2: 990–1007.

• Software/book: Ferson, S. 2002. RAMAS Risk Calc 4.0 Software: Risk Assessment with Uncertain Numbers. Lewis Publishers, Boca Raton, Florida.

[31.6, 233800] years

• Is ‘6’ the last decimal digit of the lower bound?

• Did you check that the units balance?

• Did you include units in the answer?

How to understand this result

• Highly reliable result, given the assumptions– Can’t get any worse

• Represents parametric uncertainty– Neglects (possibly big) model uncertainty

• Expresses only best and worst cases– How likely is 32 years? 50 years? 100 years?

Lobascio’s original formulation

Kd= foc Koc = [ 0.0005, 0.1] m3 kg-1

R = 1 + BD Kd / n = [ 3.143, 876] V = K i / (n R) = [ 0.000293, 3.82] m yr-1

T = L/V = [ 20.95, 408800] yr

Quickest plume reaches the well = 20.95 yrLongest plume reaches the well = 408,800 yr

What explains the difference? (hint: n is repeated above)

Repeated parameters

a = [1,2]b = [2,3]c = [2, 5]

z = a × (b + c) zz = a × b + a × c

b + c = [0, 8] a × b = [2, 6]z = [0, 16] a × c = [4, 10] zz = [2, 16]

inflated uncertainty

What to do about repeated parameters

• Always rigorous, but maybe not best possible when uncertain parameters are repeated

• Inconsequential if all are non-negative and all operations are increasing (+, ×, but not – or ÷)

• Use cancellation to reduce repetitions, e.g., caia/m + cwiw/m + cdid/m = (caia + cwiw + cdid)/m

• Cancellation not always possible, e.g., (a + b) / (a + c) = ??

If you can’t cancel

• Use tricks with algebrae.g., a² + a = (a +½)² – ¼

• Employ subinterval reconstitutionA brute-force and computationally intensive strategy

Workable if there aren’t too many repeated parameters

• Live with the suboptimalityDecisions may not require perfect precision

TricksBasic identities

u + 0 = u

u – 0 = u

0 – u = –u

u 0 = 0

u 1 = u

u / 1 = u

u0 = 1

u1 = u

u & 1 = u

u | 1 = 1

u & 0 = 0

u | 0 = u

u & u = u

u | u = u

u & not(u) = 0

u | not(u) = 1

(u&a) | (u&b) = u&(a | b)

(u | a)&(u | b) = u | (a&b)

etc.

One repetitionu + u = 2uu – u = 0u u = u2

u / u = 1(1+u) / u = 1/u + 1(1+u)/(1–u) = (1/tan(acos(u)/2))2

au + bu = u(a + b)au – bu = u(a – b)a/u + b/u = (a + b) / ua/u – b/u = (a – b) / uu/a + u/b = u(b + a)/(ab)u/a – u/b = u(b – a)/(ab)aub + cub = (a + c) ub

aub cud = a c u(b + d)

au bu = exp(u (ln(a) + ln(b)))u2 + u = (u + ½)2 – ¼u2 – u = –¼ sin(2 asin(sqrt(u)))2

u2 + au = (u + a/(2))2 – a2/4etc.

u, v, etc. represent the uncertain numbers a, b, etc. represent arbitrary expressions

Two repetitionsu + v – uv = 1 – (1 – u) (1 – v)(u + v) / (1 – uv) = tan(atan(u) + atan(v))(u – v) / (1 + uv) = tan(atan(u) – atan(v))(1 + uv) / (u – v) = 1 / tan(atan(u) – atan(v))(1 – uv) / (u + v) = 1 / tan(atan(u) + atan(v))(uv – 1) / (u + v) = –1 / tan(atan(u) + atan(v))u sqrt(1 – v2) + v sqrt(1 – u2) = sin(asin(u) + asin(v))u sqrt(1 – v2) – v sqrt(1 – u2) = sin(asin(u) – asin(v))u v + sqrt(1 – u2) sqrt(1 – v2) = cos(acos(u) – acos(v))u v – sqrt((1 – u2) (1 – v2)) = cos(acos(u) + acos(v))u v – sqrt(1 – u2 – v2 + u2 v2) = cos(acos(u) + acos(v))sin(u) sqrt(1 – sin(v)2) + sin(v) sqrt(1 – sin(u)2) = sin(u + v)cos(u) cos(v) – sin(u) sin(v) = cos(u + v)sin(u) cos(v) – cos(u) sin(v) = sin(u – v)sqrt((1 + u) / (1 – u)) = 1 / tan(acos(u)/2)etc.

Subinterval reconstitution

• Partition each repeated interval into subintervals• Compute the function for every subinterval• The union of all the results contains the true range

i j k

kji zyxwvufzyxwvuf ),...,,,,...,,(...),...,,,...,,(

i j k

kji wwvvuu ...;;;

where u, v, …, w are repeated intervals and x, y,…, z are other interval and scalar inputs, and

Example: (a + b)a, a = [0.1, 1], b = [0,1]

Partition the repeated uncertain a:ai = [(i 1)w/m + a, iw/m + a] where i = 1,2,…, m, and m is the number of subintervals, w is thewidth of a, and a is its lower bound

m U(ai+b)ai

1 [ 0.1, 2]2 [ 0.282, 2]3 [ 0.398, 2]4 [ 0.473, 2]5 [ 0.525, 2]10 [ 0.624, 2]100 [ 0.686, 2]1,000 [ 0.692, 2]10,000 [ 0.692, 2]a

0 0.5 1

m U(ai+b)ai

1 [ 0.1, 2]2 [ 0.282, 2]3 [ 0.398, 2]4 [ 0.473, 2]5 [ 0.525, 2]10 [ 0.624, 2]100 [ 0.686, 2]1,000 [ 0.692, 2]10,000 [ 0.692, 2]

Cauchy-deviate method

• Propagates intervals through black-box model– Don’t need to know, but have to be able to query it

• “Sample” from around interval – Points not necessarily inside the interval!

• Scale results to get an asymptotically correct estimate of the interval uncertainty of the output

(Trejo and Kreinovich 2001)

Cauchy-deviate method

• Depends on the number of samples, not inputs– Works just as well for 2000 variables as 20– Similar in performance to Monte Carlo

• Need about 200 samples to obtain 20% relative accuracy of half-width of output range– With fewer samples, we’d get lower accuracy, but

we can compensate for this by scaling by N, which works under the linearity assumption

Limitations of the method

• Asymptotically correct, but not rigorous

• Intervals narrow relative to the nonlinearity– Function almost linear OR uncertainties small– Could combine with subinterval reconstitution

• Most efficient when dimensionality is high

• Only handles interval uncertainty

Computing

• Sequence of binary operations– Need to deduce dependencies of intermediate results

with each other and the original inputs– Different calculation order can give different results

(which should be intersected)

• Do all at once in one multivariate calculation – Can be much more difficult computationally– Can produce much better tightening

Specifying input intervals

Interval uncertainty

• Statisticians often ignore this uncertainty

• “Interval uncertainty doesn’t exist in real life”(Tony O’Hagan et

al.)

Hammer salesmen saying screws don’t exist?

When do intervals arise?

• Periodic observationsWhen did the fish in my aquarium die during the night?

• Plus-or-minus measurement uncertaintiesCoarse measurements, measurements from digital readouts

• Non-detects and data censoringChemical detection limits, studies prematurely terminated

• Privacy requirementsEpidemiological or medical information, census data

• Theoretical constraintsConcentrations, solubilities, probabilities, survival rates

• Bounding studies Presumed or hypothetical limits in what-if calculations

Ways to characterize intervals

• Theoretical constraints

• Modeled from other intervals

• Expert assertions

• Discounting (widening) intervals (Shlyakhter)

• Confidence procedures (Grosof)– But 95% confidence isn’t the same as surety– Use in interval calculations requires an assumption

Problems with confidence intervals

• Cannot be combined in arithmetic or logical operations without an assumption

• Don’t measure epistemic belief anyway

Example (Walley): For instance, a 95% confidence interval could have zero chance of containing the value. For example, suppose X ~ normal(, 1), where 0 < . If the sample mean happens by chance to be 21.3, the 95% confidence interval on the mean is the empty set.

Why we have to be careful

• Interval analysis yields contingent results• Results are contingent on assumptions that model inputs are within

their respective intervals• But all analysis results are contingent on similar assumptions that the

models they came from are true

• Naïve elicitation has big problems• Intervals are usually unrealistically narrow• People make incoherent statements

• Can’t mix together different kinds• Not clear how to translate data into intervals

Determining endpoints

• The largest observed may not be the largest possible (and it usually isn’t)

• Sampling theory >> theory of extremes

• Rigor of analysis is contingent on inputs

• If you’re nervous, just widen the bounds

Point sample data

Range (envelope)

Extreme value model

Tolerance interval

Confidence interval

Central value and width

P-box

Level cutcut

Model

Output rangesimulation, etc.Plus-minus interval

Shlyakhter widening

Envelope

Interval function

Distribution

Support

Percentile range

Trejo-KreinovichCauchy deviates

Tolerance solutionbackcalculation

Certain and tenable ranges

Intersection

Credibility interval

Prediction interval

Percentile range

Support

Eliciting dependence

• As hard as getting intervals (maybe a bit worse)

• Theoretical or “physics-based” arguments

• Inference from empirical data – Risk of loss of rigor at this step (just as there is

when we try to infer intervals from data)

Updating

Aggregation (updating)

• How do you combine different sources?

• If you trust them all, take the intersection– [max(x1, y1, z1, …), min(x2, y2, z2, …)]– What if there is no intersection (right<left)?

• If you’re not sure which is right, use the envelope– [min(x1, y1, z1, …), max(x2, y2, z2, …)]– But are you sure this is wide enough?

Example

• Suppose we have two rigorous interval estimates of the same quantity: [1,7] & [4,10]

• Their intersection [4,7] is also a rigorous interval for the quantity

AB

0 5 10

Constraint analysis (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’

Example

• Suppose we know

W = [23, 33] m

H = [112, 150] m

A = [2000, 3200] m2

• Does knowing W H = A let us to say any more?

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.

Updating with p-boxes

2000 3000 40000

1

20 30 400

1

120 140 1600

1 A W H

2000 3000 40000

1

20 30 400

1

120 140 1600

1

Answers

intersect(W, A/H) intersect(H, A/W) intersect(A, WH)

A W H

Bayesian strategy

20003200

])3200,2000[(

112150

])150,112[(

2333

])33,23[(),,Pr(

AIHIWI

AHW

)(),,|( HWAAHWHWAL

),,Pr()()|,,( AHWHWAHWAAHWf

Prior

Likelihood

Posterior

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

(coming from the choice of priors)• Expresses less uncertainty than is present

Backcalculation

Backcalculation

• Needed for cleanup and remediation planning

• Untangles an equation in uncertain numbers when we know all but one of the variables

• For instance, backcalculation finds B such that A+B = C, from estimates for A and C

Can’t just invert the equation

Dose = Concentration × Intake

Concentration = Dose / Intake

When concentration is put back into the forward equation, the resulting dose is wider than planned

prescribed knownunknown

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 four times larger than tolerable levels we planned

Untangling backcalculations

• Solving for B given A + B = C

B = backcalc(A, C) = [C1 A1, C2 A2]

• Solving for B given A B = C

B = factor(A, C) = [C1 / A1, C2 / A2]

• Sometimes called “tolerance solutions”

1

2

3

4

5

6

1 1.5 2A

Bshell

Kernal versus shell

Given A [1,2] C [2,6] C A BThere are two different ways to solve for B

kernel

Shell (united solution)B C A [0,5]

Kernel (tolerance solution)B backcalc(A,C) [1,4]

When you need for

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)

Interval algebra

• Commutativity a+b=b+a, a×b=b×a• Associativity (a+b)+c=a+(b+c), (a×b)×c=a×(b×c)• Neutral elements a+0=0+a=a, a×1=1×a=a

• Subdistributivity a×(b+c) a×b+a×c• Subcancellation a a+bb, a a×b/b• No inverse elements a+( a) 0, a×(1/a) 1

Conclusions

• Interval analysis is a worst case analysis (that also includes the best case)

• Repeated uncertain parameters can cause unnecessary inflation of uncertainy

• Results will always be rigorous, but might not be best possible

• Moving a uncertain parameter to the other side of an equal sign often requires backcalculation

Exercises

1. Do the inputs in the travel time example seem dependent?2. What does subinterval reconstitution with m=100 on the

original Lobascio formulation give for the travel time?3. What contaminant concentrations C in water will lead to

doses D no larger than 6 mg per kg per day if it comes from both drinking and eating fish as D = (Iwater C) / W + (Ifish B C) / W, where

Iwater = [1.5, 2.5] liters per day // water intake

Ifish = [0, 8] g per day // dietary ingestion of fishB = [0.9, 2.1] liters per mg // bioaccumulation factorW = [60, 90] kg // receptor biomass

How do you check the solution?4. Is there a Bayesian analog of backcalculation?

Conclusions

• Easy to compute rigorous bounds

• Mathematical programming may be needed to get answers that are also best possible

• Rigor of analysis is contingent on inputs

• If you’re nervous, just widen the bounds

Exercises

1. Calculate the probability of tank rupture under pumping that assumes the interval inputs and makes no assumption about the dependencies among the events.

2. Develop an fault tree for establishment of snake populations on a Hawaiian island (or a star exploding).

3. Compute the probability of the conjunction of two events having probabilities 0.29 and 0.22, assuming a Pearson correlation of 1.0. Compare the result to the Fréchet range for such probabilities. What’s going on?

4. Derive an algorithm to compute the probability that n of k events occur, given intervals for the probability of each event, assuming they’re independent. Derive an analogous algorithm for the Fréchet case.

Rigorousness

• The computations are guaranteed to enclose the true results (so long as the inputs do)

• “Automatically verified calculations”

• You can still be wrong, but the method won’t be the reason if you are

Conclusions

Why bounding?

• Often sufficient to specify a decision • Possible even when estimates are impossible• Usually easy to compute and simple to combine• Rigorous, rather than an approximation

(after N.C. Rowe 1988)

Reasons to use interval analysis

• Requires very little data• Applicable to all kinds of uncertainty• Can be comprehensive• Fast and easy to compute answers• Conservative when correlations unknown • Can be made “best possible”• Backcalculations easy• Updating relatively easy

Reasons not to use it

• Same thing as worst case analysis

• Doesn't say how likely extreme event is

• Repeated parameters are cumbersome

• Not optimal when there’s a lot of data

• Can't use distribution information

• Can't use correlation information

Interval (worst case ) analysisHow?

– bound inputs, a = [a1, a2], where a1 a2

– addition: [a1, a2] + [b1, b2] = [a1+b1, a2+b2]– subtraction: [a1, a2] – [b1, b2] = [a1–b2, a2–b1]– multiplication, division, etc. are a little more complex

Why?– natural for scientists and easy to explain to others– works no matter where uncertainty comes from

Why not?– paradoxical: can’t give exact value but can give exact bounds– ranges could grow quickly, yielding very wide results– doesn’t give probabilities of extreme outcomes (tail risks)

Interval probabilityHow?

– bound event probabilities, p = [p1, p2], where 0 p1 p2 1

– evaluate event trees as composition of ANDs, ORs, etc.

– standard probabilistic rules if events are independent

– Fréchet rules if their dependence is unknown

– other dependency relations can also be represented

Why?

– can capture incertitude about event probabilities

Why not?

– paradoxical: can’t give exact value but can give exact bounds

– ranges can grow quickly, especially without independence

ReferencesDwyer, P. 1951. Linear Computations. John Wiley & Sons, New York.Ferson, S. 2002. RAMAS Risk Calc 4.0: Risk Assessment with Uncertain Numbers. Lewis Publishers, Boca Raton.Grosof, B.N. 1986. An inequality paradigm for probabilistic knowledge: the logic of conditional probability intervals.

Uncertainty in Artificial Intelligence. L.N. Kanal and J.F. Lemmer (eds.), Elsevier Science Publishers, Amsterdam.Hailperin, T. 1986. Boole’s Logic and Probability. North-Holland, Amsterdam.Kyburg, H.E., Jr. 1998. “Interval Valued Probabilities,” http://ippserv.rug.ac.be/documentation/interval_prob/interval_prob.html,

The Imprecise Probabilities Project, edited by G. de Cooman and P. Walley, http://ippserv.rug.ac.be/home/ipp.html.Lobascio, M.C. 1993. Uncertainty analysis tools for environmental modeling: application of Crystal Ball® to predict

groundwater plume traveling times. ENVIRONews 1: 6-10.Loui, R.P. 1986. Interval based decisions for reasoning systems. Uncertainty in Artificial Intelligence. L.N. Kanal and

J.F. Lemmer (eds.), Elsevier Science Publishers, Amsterdam.Moore, R.E. 1966. Interval Analysis. Prentice-Hall, Englewood Cliffs, New Jersey.Moore, R. 1979. Methods and Applications of Interval Analysis. SIAM, Philadelphia.Rowe, N.C. 1988. Absolute bounds on the mean and standard deviation of transformed data for constant-sign-derivative

transformations. SIAM Journal of Scientific Statistical Computing 9: 1098–1113. Shlyakhter A. 1994. Improved framework for uncertainty analysis: accounting for unsuspected errors. Risk Analysis

14(4):441-447.Tessem, B. 1992. Interval probability propagation. International Journal of Approximate Reasoning 7: 95-120.Trejo, R. and V. Kreinovich. 2001. Error estimations for indirect measurements: randomized vs. deterministic

algorithms for ‘black-box’ programs. Handbook on Randomized Computing, S. Rajasekaran, P. Pardalos, J. Reif, and J. Rolim (eds.), Kluwer, 673–729. http://www.cs.utep.edu/vladik/2000/tr00-17.pdf

Vesely, W.E., F.F. Goldberg, N.H. Roberts, D.F. Haasl. 1981. Fault Tree Handbook. Nuclear Regulatory Commission, Washington, DC.

Vick, S.G. 2002. Degrees of Belief: Subjective Probability and Engineering Judgment. ASCE Press, Reston, Virginia.

End

Software

• RAMAS Risk Calc 4.0 (NIH, commercial)

• GlobSol (Baker Kerfoot)

• WIC (NIH, freeware)

• Interval Solver (<<>>)

Web presentations and documents

Interval computations home page

Uncertainty about distributions

70 90 110 1300

1

0.0003 0.00070

1

0 2000 40000

1

m/yr0.2 0.3 0.4

0

1

1400 1600 18000

1

0 0.002 0.0040

1

0 10 200

1

m

m/kgkg/m3

70 90 110 1300

1

0.0003 0.00070

1

0 40000

1

m/yr0.2 0.3 0.4

0

1

1400 1600 18000

1

0 0.0020

1

0 100

1

L i K n

BD foc Koc

m

3/kgkg/m

Distribution uncertainty

• Could be much bigger

• Could be smaller (could be zero)

• Could be mixed for different variables

• Could be parametric

• Could be uncertainty about the shape

• Could arise from sampling information

Dependence and distribution

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000Traveling time (years)

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000Traveling time (years)

Cu

mu

lati

ve p

rob

abil

ity

original model

relax dependence

original model

relax both dependence and distribution assumptions

Uncertainty about distributions

70 90 110 1300

1

0.0003 0.00070

1

0 2000 40000

1

m/yr0.2 0.3 0.4

0

1

1400 1600 18000

1

0 0.002 0.0040

1

0 10 200

1

m

m/kgkg/m3

70 90 110 1300

1

0.0003 0.00070

1

0 40000

1

m/yr0.2 0.3 0.4

0

1

1400 1600 18000

1

0 0.0020

1

0 100

1

L i K n

BD foc Koc

m

3/kgkg/m

Probability bounds

• Guaranteed to enclose results no matter the distribution (so long as it’s inside the probability box)

• In many cases, the results are best possible (can’t be tighter without more information)

• Can be combined with precise distributions