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1.1 Subproject full title; Graphical Models for Decision Under Risk
1.2 Subproject acronym: GMDUR
1.3 RISKMAN research area; Area 1 Integrated approach to risk management
2.1 Proposing organization: Delft University of Technology
2.2 Contact Person: Prof. Dr. R.M.Cooke
2.3 Address: Mekelweg 4, 2600GA Delft, The Netherlands
2.4 Tel: 31 15278 2548
2.5 Fax: 31 15278 7255
2.6 Email: r.m.cooke@its.tudelft.nl
2.7 Web site http://ssor.twi.tudelft.nl/~risk/
2.8 Participating Organisations:
U. of Strathclyde
United Kingdom
George Washington Univ. USA
National Aeronautics Lab The Netherlands
HKV
The Netherlands
3. Proposal summary: Graphical models are becoming the weapon of choice for modeling
decision problems involving uncertainty and risk. Most popular are Bayesian belief nets
(bbn’s) (Smith, 1988 Lauritzen and Spiegelhalter, 1988, Lauritzen, 1996, Bedford and Cooke,
2001). By capturing ‘influence” relationships between uncertain variables, these models
enable engineers and problem owners to specify their problem in a language that is easy to
understand and communicate. Using available software packages, these models provide a user
interface to drive underlying calculational models. Bbn’s in particular allow the effects of
different decisions to be captured in a perspicuous manner. In spite of their growing
popularity, the current generation of graphical models pose challenging problems: they have
very heavy assessment and calculational burdens. As normally used, they require the
specification of a great many conditional distributions, and require variables to be discretized
drastically. Continuous variables can be used only in very special cases. This research applies
a new graphical modeling tool, called regular vines, to model dependence in high dimensional
continuous distributions. These allow influence to be captured as conditional rank correlation,
and support samplying on the fly, and updating with arbitrary continuous marginal
distributions.
4 Objectives: Regular vines provide a graphical modeling tool for representing the
correlation structure of high dimensional distributions. Specifically, they specify sets of
conditional or partial rank correlations which are algebraically independent and which are
sufficient to determine the rank correlation matrix. The graphical structure also provides a
sampling scheme allowing these distributions to be sampled on the fly. Recent reseach has
shown how a distribution specified by a regular vine can be translated into a belief net or an
independence graph. Using well-known recursive relations for partial correlations, regular
vines can be transformed in such a way as to allow simple conditionalization and updating.
The univariate marginal distributions need not be normal. The goal of this research is to
develop regular vines into a practical modeling tool with software implementation.
5. Deliverables: The research will result in a software implementation. The uncertainty
analysis package UNICORN has recently been supplied with a vine dependence module. This
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module will be extended so as to enable a bbn interface to drive the UNICORN Monte Carlo
sampling engine. Hence, users will specify a set of continuous univariate distributions, and
will specify a bbn. The software will translate this into a regular vine for sampling and
updating. UNICORN’s graphical output tools and report generator will then be available for
analyzing the results.
6. Justification and potential impact: Given the widespread use of bbn’s and the evident
problems which users encounter, the potential impact of this research is great. Bbn’s have
been applied at the TU Delft, for example, in analysing airport safety, underground
construction, and emergency planning. Enhancing the flexibility of modeling problems with
continuous variables, together with tractable computational algorithms would have an
immediate impact in improving decision making whenever substantial uncertainties are
present. The work will result in a continuous bbn front end user interface which drives the
UNICORN engine, and as such will be immediately available to problem owners. However,
the theoretical work can implemented by other uncertainty analysis and belief net codes, of
which Europe has several. In this way this research contributes to the competitiveness and
employment in Europe. Finally, a significant improvement in the modeling of decisionmaking
under uncertainty will contribute in a straightforward way to the health safety and quality of
life.
7. Description of the work:
Introduction
High dimensional probabilistic models are often formulated as Bayesian belief nets (bbn’s),
that is, as directed acyclic graphs with nodes representing random variables and arcs
representing “influence”. bbn’s are conditionalized on incoming information to support
probabilistic inference real-time modelling accident management and emergency planning.
For continuous random variables, an adequate theory of bbn’s exists only for the joint normal
distribution. In general, an arbitrary correlation matrix is not compatible with arbitrary
marginals, and conditionalization is quite intractable. Transforming the marginal distributions
to normals does not help, as the joint distribution is not thereby joint normal. Moreover, the
joint normal cannot realize exactly a specified rank correlation matrix. A continuous belief
net can be represented as a regular vine, where an arc from node i to j is associated with a
(conditional) rank correlation between i and j. Using the elliptical copula and the partial
correlation transformation properties, it is very easy to conditionalize the distribution on the
value of any node, and hence update the bbn.
We associate nodes in a bbn with continuous univariate random variables and to interpret
“influence” in terms of correlation. This should ideally be done in a way that does not impose
intractable constraints, and which supports conditionalization. Simply associating arcs with
correlations is unsuitable for two reasons; (i) the compatibility of marginal distributions and a
specified product moment correlation is not easily determined, and (ii) the correlation matrix
must be positive definite.
One option is to represent influence as rank correlation. Any rank correlation in the interval [1, 1] is compatible with arbitrary continuous invertible marginals. We could transform the
variables to standard normal, induce a product moment correlation structure using well known
methods, and transform back to the original variables. The rank correlation thus induced
would not be exactly equal the specified rank correlation, but would be in the right ball park.
Exact replication of a given rank correlation matrix could be obtained with this method if the
joint normal could realize every rank correlation matrix. This is not the case; indeed the rank
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correlation matrices of joint normal distributions are very sparse in the set of correlation
matrices. Of course, the problem of positive definiteness noted above would still be present.
Vines offer a more promising solution.
Vines
A vine on N variables is a nested set of trees, where the edges of tree j are the nodes of tree
j+1; j = 1,… ,N-2, and each tree has the maximum number of edges (Cooke 1997, Bedford
and Cooke 2001). A regular vine on N variables is a vine in which two edges in tree j are
joined by an edge in tree j+1 only if these edges share a common node, j = 1,… ,N-2. There
are (N-1)+(N-2)+ … +1 =N(N-1)/2 edges in a regular vine on N variables. Figure 1 shows a
regular vine on 5 variables. The four nested trees are distinguished by the line style of the
edges; tree 1 has solid lines, tree 2 has dashed lines, etc. The conditioned ( before |) and
conditioning ( after |) sets associated with each edge are determined as follows: the variables
reachable from a given edge are called the constraint set of that edge. When two edges are
joined by an edge of the next tree, the intersection of the respective constraint sets are the
conditioning variables, and the symmetric difference of the constraint sets are the conditioned
variables. The regularity condition insures that the symmetric difference of the constraint sets
always contains two variables. Note that each pair of variables occurs once as conditioned
variables.
We recall two generic vines, the D-vine D(1,2,…n ) and C-vine C(1,2,…,n), shown on Figures
1 and 2.
12
1
23
34
45
2
3
4
13|2
24|3
35|4
14|23
5
25|34
15|234
Figure 1: The D-vine on 5 variables D(12345) showing conditioned and conditioning sets.
1
34|12
12
13
2
23|1
3
24|1
4
14
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Figure 2: The C-vine on 4 variables C(1,2,3,4) showing conditioned and conditioning sets.
Each edge in a regular vine may be associated with a constant conditional rank correlation
(for j=1 the conditions are vacuous) and, using minimum information copulae, a unique joint
distribution satisfying the vine-copulae specification with minimum information can be
constructed and sampled on the fly (Cooke 1997). Moreover, the (constant conditional) rank
correlations may be chosen arbitrarily in the interval [-1,1].
The edges of a regular vine may also be associated with partial correlations, with values
chosen arbitrarily in the interval (-1,1). Using the well known recursive formulae it can be
shown that each such partial correlation regular vine uniquely determines the correlation
matrix, and every full rank correlation matrix can be obtained in this way (Bedford and Cooke
2002). In other words, a regular vine provides a bijective mapping from (-1,1)N(N-1)/2 into the
set of positive definite matrices with 1's on the diagonal. One verifies that ij can be computed
from the sub-vine generated by the constraint set of the edge whose conditioned set is {i, j}
using the following recursive formulae. When (X,Y) and (X,Z) are joined by the elliptical
copula (Kurowicka et. al. 2000) and the conditional copula (Y,Z|X) does not depend on X,
then the conditional correlation (Y,Z) given X does not depend on X and conditional product
moment correlation of Y,Z given X is equal to partial correlation (Kurowicka and Cooke
2001).
Figure 3: The scatter plot of the elliptical copula with correlation 0.8.
Moreover, there exists very compact functional form of the conditional distribution using the
elliptical copula.
Using the elliptical copula, the vine structure can be uniquely associated with a full rank
correlation matrix and can be converted into an on-the-fly sampling routine.
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For a regular vine-rank correlation specification with elliptical copula updating with
information is very simple. Since for elliptical copulae partial and conditional product
moment correlations are equal then using the recursive formula we can convert any partial
correlation vine into any other partial correlation vine. We convert given vine to the C-vine
with variable which we observe, say 1, as a root. Conditional correlations don't depend on a
value of 1 then we drop the "1"s from all conditions and as a result we obtain C-vine with
variable 2 as a root. We can convert this "updated vine" to any other regular vine, recalculate
conditional rank correlations and sample this "updated" vine if desired.
Belief Nets
A finite valued Belief Net is a directed acyclic, graph, together with an associated set of
probability tables. The graph consists of nodes and arcs. The nodes represent variables, which
can be discrete or continuous. The arcs represent causal/influential or functional relationships
between variables.
1
1
3
2
3
2
b
a
Figure 4: A simple example of bbn’s.
The graph in Figure 4a) tells us that variables 2 and 3 are conditional independent given
variable 1. The message of the graph on Figure 4b) is that 2 and 3 are independent and a
distribution of 1 given 2 and 3 is arbitrary.
If variables 1,2 and 3 in Figure 4b take values “True” or “False” then two 2x2 probability
tables must be specified (example shown in tables below).
1
2
True
True 0.1
False 0.9
1
3
True
True 0.6
False 04
Even for such a simple example, figuring
out the right probabilities in the
probability tables requires some work (e.g.
statistical data or expert’s opinions). For a
large net with many dependences and nodes that can take more values this is extremely
difficult, and often produces superficial quantification.
False
0.5
0.5
False
0.7
0.3
The main use of bbns is in situations that require learning. If we know events that have
actually been observed, we might want to infer the probabilities of other events, which have
not yet been observed. Using Bayes Theorem it is then possible to update the values of all the
other probabilities in the bbn. Updating bbn’s is very complex involving arc reversing and
addition but with the algorithm proposed by (Lauritzen and Spiegelhalter 1988) it is possible
to perform fast updating in large bbns.
It has recently been shown (Kurowicka and Cooke 2002) that bbns can be translated into
regular vines, where influence on the bbn gets interpreted as conditional rank correlation. This
enables a specification of belief nets with arbitrary continuous marginal distributions, and
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arbitrary conditional rank correlations. Such a specification is always consistent. By
translating to regular vines, the joint distribution may be efficiently sampled and updated. The
following figure shows a simple bbn whose edges are associated with conditional rank
correlations. From the theory sketched above, it follows that these correlations uniquely
determine correlation matrix and are consistent with arbitrary continuous univariate
marginals. Further, the regular vine representation leads to an efficient sampling algorithm.
Figure 5, : A Bayesian belief net with conditional rank correlations
(1,3)
1
(1,2)
2
3
(3,4|2)
4
(3,5)
(2,4)
(5,4|3)
5
Anyone who has used bbns will appreciate how dramatically the use of conditional rank
correlations and vines simplifies the assessment and computational burden. The user need
specify only the marginal distributions and the conditional rank correlations as indicated in
Figure 5. Extensive experience with eliciting such correlations from experts has been
accumulated.
The work involes :
RTD: developing the algorithms for bbn-vine conversions, and developing the software to
effect a handshake between a graphical bbn and the UNICORN Monte Carlo engine
DEM: in the last phase of this work, prototype applications involving problems supplied by
GW university and HKV will be used to demonstrate this approach/
Previous work
The TU Delft has developed two commercial codes, EXCALIBUR and UNICORN.
UNICORN is a generic uncertainty analysis program which incorporates the regular vine
method for dependence modeling in high dimensional distributions. It has been sold to
research institutions involved in risk modeling, including RIVM, ABS consulting and GW
University. A previous PhD at the TU Delft (Kurowicka, 2001) contributed substantially to
the theoretical developments underlying the present proposal. Information regarding these
programs, and light versions can be downloaded gratis from the website
http://ssor.twi.tudelft.nl/~risk/. An experienced programmer at the TU Delft is available to
assist the software implementation.
Milestones
The first year will focus on extending the theoretical developments. Algorithms for translating
bbns into regular vines will be extended and optimized. Prof. Bedford (U. Strathclyde) will be
consulted during this phase. The second year will be devoted to building a graphical software
front end to drive the UNICORN engine. In the third year, prototype case studies will be
performed, under supervision of Prof. van Noortwijk and Dr. Kok, (HKV) and Prof.
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Mazzuchi (GWU,. TU Delft). The fourth year will be concerned with write up and
dissemination.
8. Partners involved
University of Strathclyde
George Washington University
HKV consulting
Prof. T.J. Bedford
Prof. T.A. Mazzuchi
Prof. J. van Noortwijk, dr. M. Kok
Bedford has participated in the theoretical developments underlying this proposal, and will
help supervise the implementation of this work. Mazzuchi has extensive experience in real
time risk modelling and will contribute a practical viewpoint, and will provide prototype
applications for testing the software implementation. Similarly, Kok and Van Noortwijk have
extensive experience in risk modelling in water management. They will also oversee
prototype applications.
9. Resources Total resources for this project are:
Item
PhD position
Travel, PhD
Travel Bedford
Personal
computer
Total
180,000 euro
7,000 euro
8,000 euro
2,000 euro
197,000 euro
Travel for Mazzuchi is not anticipated, as he is employed at the TU Delft for 6 months per
year.
10. Duration
The duration of the subproject is four years.
11. Financial Plan
Investment from the TU Delft and partner institutions in this project are:
Item
Project Leader (Cooke,
TUD)
Assisst.Prof
(Kurowicka)(TUD)
Prof. Bedford (U.
Strathclyde)
Programmer
(Kritchallo)(TUD)
Prof.Mazzuchi (GWU)
Prof. van Noortwijk (HKV)
% committed per year
0.1
0.2
0.05
0.3
0.1
0.05
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Total
0.8
12. Other Issues:
none
References
Bedford, T.J. and Cooke R.M. (2001) Probabilistic Risk Analysis, Foundations and Methods,
Cambridge University Press, 2001.
Bedford, T.J., and Cooke. R.M.,2002, Vines - a new graphical model for dependent random
variables, Ann. of Statistics vol. 30 no. 4, 1031-1068.
Bedford, T.J. and Cooke R.M. (2001) "Probability density decomposition for conditionally
dependent random variables modelled by vines", Annals of Mathematics and Artificial
Intelligence 32,245-268.
Cooke, R.M.,1997, Markov and entropy properties of tree and vines-dependent variables,
Proceedings of the ASA Section of Bayesian Statistical Science.
Kurowicka, D. (2001) Techniques for Representing High Dimensional Distributions, PhD
Thesis, Department of Information, Technology and Systems , T.U. Delft.
Kurowicka, D., Misiewicz, J., and Cooke, R.M., 2000, Elliptical copulae, Proc. of
International Conference on Simulations- Monte Carlo, 201-214.
Kurowicka, D., and Cooke, R.M., 2001, Conditional, Partial and Rank Correlation for
Elliptical Copula; Dependence Modeling in Uncertainty Analysis, Proc. of ESREL
2001.
Kurowicka, D. and Cooke, R.M. “The vine copula method for representing high dimensional
dependent distributions: Application to continuous belief nets” Proceedings of the
2002 Winter Simulation Conference E. Yücesan, C.-H. Chen, J. L. Snowdon, and J.
M. Charnes, eds.
Lauritzen, S.L., 1996, Graphical Models, Clarendon Press, Oxford.
Lauritzen S.L.,Spiegelhalter D.J., 1988, Local computations with probabilities on graphical
structures and their application to expert systems, Journal of the Royal Statistical
Society, Series B, vol. 50, pp.157-224.
Smith, J.Q., 1988, Statistical principals on graphs, in Proceedings Influence Diagrams for
Decision Analysis, Inference and Prediction, Smith and Oliver eds, Wiley, Chichester.