Partial specification of risk models Tim Bedford Strathclyde Business School Glasgow, Scotland.
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Partial specification of risk models
Tim Bedford
Strathclyde Business School
Glasgow, Scotland
Contents...
Modelling considerations Information or KL divergence Vines Copula assessment Conclusions
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.
What is a model?
Device for making predictions Creative activity giving understanding A statement of beliefs and assumptions
Input data Real system Output
Input data Model Output
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
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
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
Model dimensions - 2
Black box Structural
FT ISRDStatisticalRegression
Predictive Explanatory
Instrumental Realistic
Macro Micro
Back of envelope
Large computer
Role of EJ
•Decision variables introduced in the model•Causal connections differ from correlated connections
Black box Structuralmodel
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
Common cause model
Role of experts
Structuring – providing framework, specifying important variables, conditional independencies
Quantifying – providing assessments on quantities that they can reasonably assess
Essentially, all models are wrong, but some are useful
George Box
General research theme
Develop good ways to determine a decision model when only a partial specification is possible
Mainly working in technical/engineering applications
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
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
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)
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....”
dQ
dQ
dP
dQ
dPQPI log)|(
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
dudvvufvuffI )),(log(),()(
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
Decomposition Theorem
Markov tree example
1
Vine example
Vine example
Vine example
Vine example
Vine example
Information decomposition…
Information decomposition
Information calculation
So…
You can build up multivariate distributions from bivariate pieces
Minimum information pieces give global minimum information
But is it realistic to elicit correlations?
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
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
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|
Example - exp marginals FR 1, 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-5 -4 -3 -2 -1 0 1 2 3 4 5
E(XY) as a function of lambda
Example - exp marginals FR 1, 2
-20
0
20
40
60
80
100
120
140
160
180
200
-5 -4 -3 -2 -1 0 1 2 3 4 5
Information as a function of lambda
Copula density for Lambda=2
1 3 5 7 9
11 13 15 17 19S1
S6
S11
S16
0
2
4
6
8
10
12
14
Copula density for Lambda=-2
1 3 5 7 9
11 13 15 17 19
S1
S6
S11
S16
0
0.5
1
1.5
2
2.5
3
3.5
Example – constraints on X-Y
1 3 5 7 9
11 13 15 17 19
S1
S6
S11
S16
0
2
4
6
8
10
12
14
Marginals as before
Expert assesses
P(X-Y<0.3)=0.3
P(X-Y<0.9)=0.7
Sequential – Step 1
Sequential – Step 2
Sequential – Step 3
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
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
Contributors/Collaborators
Roger Cooke Dorota Kurowicza Anca Hanea Daniel Lewandowski Lesley Walls John Quigley Athena Zitrou
D1AD2 algorithm - 1
If specify expectations of functions
or equivalently of
then minimally informative density has form
),('...,),,('),,(' 21 yxhyxhyxh k
),(...,),,(),,( 21 vuhvuhvuh k
),()()( 21 vuavdud
)),(...),(),(exp(),( 2211 vuhvuhvuhvua kk
D1AD2 algorithm - 2
View discretised density as matrix product D1AD2 where Di are diagonal
Iterative algorithm generates D1 and D2
normalising by
Iteration is contraction in hyperbolic metric
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