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Partial specification of risk models Tim Bedford Strathclyde Business School Glasgow, Scotland

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Contents... Modelling considerations Information or KL divergence Vines Copula assessment Conclusions

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Modelling goals Decision Theory* perspective slightly different to Statistics* Modelling aims How do we judge what model is appropriate? –Type –Outputs –Level of detail for cost-effective modelling –Bottom-up or top-down *Caveat: convenient labels for discussion only. Not intended to describe the views of any actual person.

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What is a model? Device for making predictions Creative activity giving understanding A statement of beliefs and assumptions Input dataReal systemOutput Input dataModelOutput

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Nominal versus non-nominal What behaviour are we trying to capture in risk models? Nominal / Non-nominal –EVT is trying to model the extremes of nominal behaviour –Technical risk models try to find non-nominal behaviour Discrete or continuous nature of departures from nominal state is important from modelling perspective

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Model as a statement of beliefs Often many different plausible statistical models consistent with the data May be identifiability problems Non-statistical validation is required Providing finer grain structure to model arising from knowledge of context can be a way of providing validation

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Model dimensions George Mitchell describes 7 dimensions on which models can be compared –Actuality – abstract –Black box – structural –Off the shelf – purpose built –Absolute – relative –Passive – behavioural –Private – public –Subsystem – whole system

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Model dimensions - 2 Black boxStructural FTISRD Statistical Regression PredictiveExplanatory InstrumentalRealistic MacroMicro Back of envelope Large computer

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Role of EJ Decision variables introduced in the model Causal connections differ from correlated connections Black boxStructural model

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Example – common cause Modelling dependent failure in NPPs etc Existing models used for risk assessment use historical data but give no sights –Alpha factor model –Multiple greek letter model –Etc etc

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Common cause model

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Role of experts Structuring – providing framework, specifying important variables, conditional independencies Quantifying – providing assessments on quantities that they can reasonably assess

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Essentially, all models are wrong, but some are useful George Box

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General research theme Develop good ways to determine a decision model when only a partial specification is possible Mainly working in technical/engineering applications

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Expert assessment methods Many methods for expert assessment of distributions - for applications in reliability/risk often non-parametric Experts provide input by –Means, covariances.. –Marginal quantiles, product-moment correlations –Marginal quantiles, rank correlations Here look at methods for building up a subjective joint distribution

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Building a joint distribution Assume experts have given us information on marginals… How do we build a joint distribution with information from experts? –Iman-Conover method –Markov trees –Vines

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Copula Joint distribution on unit square with uniform marginals Copula plus marginals specify joint distribution –If X~F and Y~G then (F(X),G(Y)) is a copula Any (Spearman) rank correlation is possible between –1, 1. But range of PM correlation depends on F and G X Y (F(x),G(y)) G(Y) F(X)

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Information Also known as –Relative entropy –Kullback Leibler divergence Coordinate free measure Requires specification of background distribution –Another role for expert? –“Other things being equal....”

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Minimum information copulae Partially specify the copula, eg by (rank) correlation Find “most independent” copula given information specified Minimize relative information to independent copula= uniform distribution Equivalent to min inf in original space

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Markov trees and Vines (Minimum information) copulae used to couple random variables Marginals specified plus certain (conditional) rank correlations Main advantage is no algebraic restrictions on correlations Disadvantage is difficulty of assessing correlations

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Decomposition Theorem Markov tree example 1

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Vine example

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Information decomposition…

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Information decomposition

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Information calculation

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So… You can build up multivariate distributions from bivariate pieces Minimum information pieces give global minimum information But is it realistic to elicit correlations?

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Observable quantities and expert assessment Experts are best able to judge observable functions of data Distributions of such functions are not free, restrictions depend on –marginals –other functions being assessed REMM project feeds back “contradictory” information real-time to allow experts to reflect on inconsistencies PARFUM method allows probabilistic inversion of random quantities to build joint

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Formulation as minimal information problem Define domain specific functions of the variables to obtain quantiles X Y (F(x),G(y)) G(Y) F(X) h h(x,y) Expectations of regions are fixed by quantiles

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Examples Product Moment correlation –If marginals are known then can just consider range of E(XY) –Equivalent to looking at E(F -1 (X)G -1 (Y)) –All possible values of this can be reached by the min information distributions –Can map out possible values using information function as measure of distance to infeasibility Differences – quantiles for |X-Y|

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Example - exp marginals FR 1, 2 E(XY) as a function of lambda

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Example - exp marginals FR 1, 2 Information as a function of lambda

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Copula density for Lambda=2

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Copula density for Lambda=-2

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Example – constraints on X-Y Marginals as before Expert assesses P(X-Y<0.3)=0.3 P(X-Y<0.9)=0.7

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Sequential – Step 1

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Sequential – Step 2

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Sequential – Step 3

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Conclusions General problem of eliciting dependencies from experts Copulae are a good tool, but how do we select the copula? Can use vines to get simple parameterisation of covariance matrix Can tie together observables and copulae using interactive computer based methods

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Questions Eliciting dependence from experts – some work done but more needed General guidance to use Expert Judgement to add structure and variables to existing models, or to link models Find ways to incorporate “features” specified by experts Methods must recognize limitations of experts –All the usual biases –Poor in the tails –Insight not much deeper than the data

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Contributors/Collaborators Roger Cooke Dorota Kurowicza Anca Hanea Daniel Lewandowski Lesley Walls John Quigley Athena Zitrou

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D 1 AD 2 algorithm - 1 If specify expectations of functions or equivalently of then minimally informative density has form

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D 1 AD 2 algorithm - 2 View discretised density as matrix product D 1 AD 2 where D i are diagonal Iterative algorithm generates D 1 and D 2 normalising by Iteration is contraction in hyperbolic metric

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