probabilistic dependency Robert L Mullen Seminar NIST April 3 th 2015 Rafi Muhanna School of Civil and Environmental Engineering Georgia Institute of Technology Atlanta GA 30332 USA ID: 808002
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Slide1
Uncertainty quantification of structures with unknown probabilistic dependencyRobert L. Mullen
Seminar: NIST
April 3
th
2015
Slide2Rafi MuhannaSchool of Civil and Environmental Engineering Georgia Institute of Technology
Atlanta, GA 30332, USA
Dr.M.V.Rama Rao
Department
of Civil Engineering
Vasavi
College of Engineering,
Hyderabad
500
031 INDIA
Slide3ImprimersIvo BabushkaVladik
Kreinovich
Ray Moore
A. NeumierScott Ferson
Slide4Motivation : A toy problem.
A truss structure (extendable to general
FEA
)Loading given in terms of random variables (RV) (extendable to parameters given by RV)p = f (P1, P2, P3) vs p= f1(P1
), p=
f2
(
P2
), p=
f3
(
P3
)
What can one compute about structural response when dependency between
p
1
,
p2
,
p3
is unknown?
Slide5Common Engineering Solution: Assume independence between P1, P2, P3 use Monte Carlo simulation and forget about it.
Slide6Common Engineering Solution: Assume independence between P1, P2,
P3
and forget about it.
Assuming independence can result in significant underestimation of risk (Ferson et al. 2004).
Slide7OutlineIntroductionRepresenting dependency
Parametric models (and associated statistical measures)
CopulasRepresentation of IgnoranceAny dependency boundsNumerical methods for P-box calculationsResults
Conclusion
Slide8Multi-dimensional PDF
Two dimensional
N dimensional
Equation images from Wikipedia
http://en.wikipedia.org/wiki/Multivariate_normal_distribution
Slide9Realizations of Gaussian RVCalculated using Matlab program
copulaa1
Robert L Mullen 2012
Slide10Sum of two standard Gaussian RVCalculated using Matlab program copula1
Robert L Mullen 2012
Slide11Other distributions can be constructed by transforming marginal distributionTheorem 2.1. (from Devroye
, Springer 1986)
Let F be a
continuous distribution function on R with inverse F-1 definedby F -1 ( u ) = inf {a::F(a:)=u, O<U
<
1
}
.
If
U
i
s
a
uniform
[
O,1
]
random
variable, then
F-1(U) has
distribution function F. Also, If X has distribution function F ,
then
F ( X )
i
s uniformly distributed
on [0,1].
Slide12Other distributions can be constructed by transforming marginal distribution
Slide13Correlated uniform distributions
Slide14Sum of two uniform RV
Slide15Examples of dependendtuncorrelated variables
Picture from: Scott
Ferson
, Roger B. Nelsen, Janos Hajagos, Daniel J. Berleant, Jianzhong Zhang, W. Troy Tucker, Lev R. Ginzburg and William L.
Oberkampf
,
SAND2004
-3072.
Slide16CopulasSklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de L’Institut de Statistiques de l'Université de Paris 8: 229-231
.
Copulas define dependency between random variable with any marginal distribution.
The marginal distributions are transformed to uniform distribution between [0,1], theCopula is then the CDF of the transformed (ie. Uniform) variables.
Slide17Fréchet–Hoeffding copula bounds
Figure
by Matteo
Zandi,”Graph of the Fréchet-Hoeffding copula limits” 22 November 2010Wikimedia commons
Slide18Fréchet–Hoeffding copula bounds
Slide19Fréchet–Hoeffding Copulas allow for bounding operation on RV with given marginal distributions with no assumption on dependency
Slide20OutlineIntroductionRepresenting dependency
Representation of Ignorance
Entropy and Information Intervals and P-box RepresentationAny dependency boundsNumerical methods for P-box calculationsResultsConclusion
Slide21Ignorance and InformationFor a
discrete number of events
with probabilities p1, the expression
Is the only form that is
For
equiprobable
events the entropy should increase with the number of outcomes
.
The amount of entropy should be independent of how the process is regarded as being divided into parts
.
Symmetric and continuous
Claude E. Shannon Warren
Weaver 1949, 1971
Slide22Entropy Maximum Entropy results in Equal probabilities for no informationPrinciple of insufficient reason
However, this results in
no difference
between being ignorant of the outcome behavior and a system that has well known behavior of a uniform distributed outcome. Laplace 1812. See Stigler, Stephen M. (1986)
Slide23Entropy IIUniqueness of the entropy expression only holds for a finite set of events. Other expressions have the same four properties of entropy in the countable (less studies) and the non-countable cases.
(see Jos
Uffink 1997 for example).
Slide24Maximum entropy III
Depends on the choice of scale
A solution in terms of
stiffness is incompatible with one based on flexibility even though the information is equivalentWarner North interprets Ed Jaynes as saying that “two states of information that are judged to be equivalent should lead to the same
probability assignments .
Maxent
doesn’t (
Ferson
2014)
Slide25Representation of Ignorance Entropy Probability, Subjective probability
Fuzzy sets, fuzzy-probabilistic methods
(
Möller, B., Graf, W. and Beer, M 2000)Collection of model resultsBounds, Confidence intervals, Intervals
Slide26Only range of information
(tolerance) is available
Represents an uncertain quantity by giving a range of possible values
How to define bounds on the possible ranges of uncertainty?
experimental data, measurements
,
expert knowledge
Interval
Approach
Slide27Presentation – University of South Carolina Sept. 2009Interval number represents a range of possible values within a closed set
Interval Representation
27
Slide28Presentation – University of South Carolina Sept. 2009Interval
Vectors
and Matrices
An interval matrix is such matrix that contains all real matrices whose elements are obtained from all possible values between the lower and upper bounds of its interval components Interval vectors and matrices
28
Slide29Presentation – University of South Carolina Sept. 2009
2
4
1
3
x
y
Interval
Interval Representation - Vectors
29
Slide30Presentation – University of South Carolina Sept. 2009Let
x
= [
a, b] and y = [c, d] be two interval numbers
1. Addition
x
+
y
= [
a, b
] + [
c, d
] = [
a + c
,
b + d
]
2. Subtraction
x
y
= [
a, b
]
[
c, d
] = [
a
d, b
c
]
3. Multiplication
xy
=
[
min
(
ac,ad,bc,bd
)
,
max
(
ac,ad,bc,bd
)]
4. Division
1 / x
= [
1/b, 1/a
]
Interval Operations
30
Slide31Presentation – University of South Carolina Sept. 2009Let
x
,
y and z be interval numbers1. Commutative Lawx + y = y + x
xy = yx
2. Associative Law
x +
(
y + z
)
=
(
x + y
)
+ z
x
(
yz
)
=
(
xy
)
z
3.
Distributive Law does not always hold
,
but
x
(
y + z
)
xy + xz
Properties of Interval Arithmetic
31
Slide32Probability box (Pbox)A set of allowable CDF defined by upper and lower bounding functions.
Ferson
S, Kreinovich
V, Ginzburg L, Myers DS, Sentz K, Constructing probability boxes and Dempster-Shafer structures, Tech. Rep. SAND2002-4015, Sandia National Laboratories, 2003.Williamson R. Probabilistic arithmetic. PhD thesis, University of Queensland, Austrialia, 1989.
Slide33Pbox representation for Imprecise Probability
0
1
1.0
2.0
3.0
0.0
X
Cumulative probability
Figure from Scott
Ferson
and
Vladik
Kreinovich
-REC 2006
Slide34OutlineIntroductionRepresenting dependency
Representation of Ignorance
Entropy and Information Intervals and P-box RepresentationAny dependency boundsNumerical methods for P-box calculationsResultsConclusion
Slide35An even simpler Toy problem posed by Kolmogorov (Frank, Nelsen, Schweizer, Probab. Th. Rel. Fields 74, 199-211 (1987).
“Specifically
, what are the optimal bounds for the probability distribution of the sum of two random variables X and Y whose individual distribution Fx
and Fy are given? Makarov, G.D.: Estimates for the distribution function of a sum of two random variables when the marginal distributions are fixed. Theor. Probab. Appl. 26, 803-806 (1981)
Slide36Independent
Slide37Correlated
Slide38c2=alpha-c1;if (c2<0)
c2
=c2+1;
endX=c1+c2;
Family of dependency that constructs bounds
Slide39OutlineIntroductionRepresenting dependency
Representation of Ignorance
Any
dependency boundsNumerical methods for P-box calculationsResultsConclusion
Slide40Techniques for operating on PboxesMonte Carlo Integration (Zhang 2001)Discrete approximation of Probability bounds
(
Williamson and Downs 1990)Interval extension to Polynomial Chaos
(Redhorse and Benjamin 2004)
Slide41P-box Monte Carlo methodPerform Interval calculation of realization (using Interval Finite Element)Construct interval representation of random input variables from Pbox
Collect Interval Response
Sort Lower and upper bounds and construct Pbox
for Response variable. Zhang, H., Mullen, R. L. and Muhanna, R. L. “Interval Monte Carlo methods for structural reliability”, Structural Safety, 2010
Slide42Generation of random intervals from a probability-box
Slide43Finite Element – Interval Monte Carlo for Pbox
analysis
In steady-state analysis-
variational formulation
Invoking the
y
of
*, that is
* =
0
Slide44HoweverSolution to linear system of interval equationsIs NP (Vladik
Kreinovich
)Compute approximate bounds using fix point theorem.
Slide45Sharp solutions to systems of equationsImprove sharpness of Secondary quantities (stress/strain).Prevent accumulation of errors in iterative correctors
Required Improvements to Linear Pbox
Finite
Element Methods to solve nonlinear problems
Slide46Slide47Error in secondary quantities
Conventional Finite Element
Secondary quantities such as stress/strain
c
alculated from displacement have shown
significant overestimation of interval bounds
Slide48Use constraints to augment original Variational
Slide49Use secant stiffness methods to prevent iterative accumulation of interval overestimation (Rao et. al. 2013)
Slide50Discrete P-box Random SetConstruct Data type for a discrete
Pbox
Define overloaded operators for discrete Pbox
variableApply conventional deterministic (scalar) algorithms using Pbox variablesResult is in the form of a discrete Pbox
Slide51Interval based discrete Pbox Proved interval bounds for a given range of the CDF
Uniform discretization for algorithmic simplicity
Preserve the guaranteed enclosure philosophy from Interval Arithmetic
Extensions to higher order discretization and non-uniform discretization are possible.Extreme case of Any Dependency or Independent
Slide52P-Box defined by bounding lognormal distribution with mean of [2.47, 11.08] and standard deviation of [2.76, 12.38].
Slide53Best fit vs. Guaranteed enclosure
Slide54Polynomial Chaos MethodsExpand upper and lower CDF as a random variable using the same orthogonal polynomial basis. Replace polynomial coefficients by a single interval number
Slide55Polynomial Chaos MethodsCurrent Research IssuesGalerkin projection and truncated series are estimates and not bounds.
Possible for bounds to cross.
Slide56OutlineIntroductionRepresenting dependency
Representation of Ignorance
Any
dependency boundsNumerical methods for P-box calculationsResultsConclusion
Slide57A truss structure
Slide58Property
Value
cross-sectional areas for elements A
1 to A610.32 cm2 cross-sectional areas for elements A7 to A15 6.45 cm2
Elastic modulus
200
GPa
Slide59Statistics
of random loadings acting on the truss
90% confidence interval
99%confidence
interval
Mean
ln
P1
[4.4465, 4.5199]
[4.4258, 4.5407]
Mean
ln
P2
[5.5452, 5.6186]
[5.5244, 5.6393]
Mean ln P3
[4.4465, 4.5199]
[4.4258, 4.5407]
ln Standard dev. P1, P2, P3
0.09975
0.09975
Confidence structures (c-boxes) are imprecise generalizations of confidence distributions.
Ferson
, S.,
O'Rawe
, J., and Balch, M. (2014)
Slide60Slide61Example 2
Slide62Nonlinear ParametersLoading from
Pbox
Lognormal boundsCov
=.05+/- 1.% Ramberg-Osgood (1943) model with parameters of .7, 3, and 600 Mpa for m1, exponent n, and yield stress σ0, respectively.
Slide63Cubic ModelVertical Deflection at Center
Slide64Ramberg-Osgood modelStrain Element 2
Slide65OutlineIntroductionRepresenting dependency
Representation of Ignorance
Any
dependency boundsNumerical methods for P-box calculationsResultsConclusion
Slide66OutlineIntroductionRepresenting dependency Parametric models (and associated statistical measures)
Copulas
Any dependency boundsIntervals and P-box Representation
ResultsConclusion
Slide67Slide68Cubic ModelStrain Element 2
Slide69Ramberg-Osgood modelVertical Deflection at Center
Slide70Square plate with opening
Contour values of maximum
yy
(
kPa
) Contour values of minimum
yy
(
kPa
)
0.063
kPa
44.22
kPa
0.0
kPa
-10.65
kPa
Examples –
Continuum Load Uncertainty
Slide71Efficiency study
N
v
Sens.
Present
246
1.06
0.72
648
64.05
10.17
1210
965.86
59.7
1692
4100
156.3
2254
14450
358.8
2576
32402
528.45
Slide72SummaryBehavior of Linear and Nonlinear of structures with unknown dependency among random parameters and loading can be bounded using concept of Pbox structures
Slide73Slide74Rene
Magritte, Clairvoyance, 1936
Slide75