Regional Regional probabilistic risk probabilistic risk assessments of assessments of extreme events, extreme events, their magnitude their magnitude and frequencyand frequency

Стохастическое прогнозирование Стохастическое прогнозирование вероятностей и рисков вероятностей и рисков

экстремальных явленийэкстремальных явлений

Dushin V.R., Evlanova V.A., Dushin V.R., Evlanova V.A., Ilyushina E.A., Smirnov N.N. Ilyushina E.A., Smirnov N.N.

Why do we use stochastic Why do we use stochastic approach?approach?

• Processes in hydrosphere and Processes in hydrosphere and atmosphere are stochastic.atmosphere are stochastic.

• Impacts of various external factors Impacts of various external factors and internal mechanisms responsible and internal mechanisms responsible on future system behavior could not on future system behavior could not be evaluated in advance.be evaluated in advance.

• A future state of a system could be A future state of a system could be described only in terms of described only in terms of probability.probability.

• Main problem: how long the system Main problem: how long the system will preserve its current state (with will preserve its current state (with absence of extreme events)?absence of extreme events)?

Data base on Data base on extreme phenomenaextreme phenomena

Data matrixData matrix

Factor analysisFactor analysis

Development of latent Development of latent risk factorsrisk factors ( (VaRVaR))

Time series for geophysical Time series for geophysical observation dataobservation data

Identification of Identification of processes:processes: ARIMAARIMA

Estimate of Estimate of distribution distribution functionsfunctions

Estimates of magnitude and Estimates of magnitude and waiting period for an extreme waiting period for an extreme

eventevent

(It is necessary to introduce(It is necessary to introducedamage codes and scales)damage codes and scales)

Simple stream of eventsSimple stream of events

• LetLet mm appear at successive time appear at successive time intervalsintervals

0 = 0 = tt00 ≤ ≤ tt11≤…≤ ≤…≤ ttm+1m+1 ,,

• Each set of m events is independentEach set of m events is independent• Probability of appearing m events in Probability of appearing m events in

the interval the interval [0,[0,tt]] could be could be determined by formula:determined by formula:

PP{{nn(0, (0, tt)=)=mm} = {} = {expexp(-(-λtλt)})}·· ((λt)λt)m m /m/m! !

Waiting time for next Waiting time for next extreme event distributionextreme event distribution

• We accept the model of a simple stream We accept the model of a simple stream of events for description of extreme of events for description of extreme events because it has a property of events because it has a property of independence of future on past under independence of future on past under given present conditions.given present conditions.

• Based on observation data, one could Based on observation data, one could estimate probability of extreme event estimate probability of extreme event appearing later than appearing later than T T

PP{{ττ>> T T } = } = expexp(-‹(-‹λλ› › TT)) ττ – – waiting time for the next extreme waiting time for the next extreme

eventevent ‹‹λλ›› - - stream intensity, i.e., average stream intensity, i.e., average

number of extreme events within a unit number of extreme events within a unit time interval.time interval.

Guaranteed interval of life Guaranteed interval of life without extreme eventswithout extreme events

• Let us find the waiting time Let us find the waiting time TTγγ for an extreme event such as the for an extreme event such as the probability of surpassing its probability of surpassing its value would be higher than some value would be higher than some big threshold value big threshold value γγ;; thenthen

TTγγ ≤ (1/≤ (1/‹‹λλ››) ) lnln(1/γ) (1/γ)

The equality corresponds to the The equality corresponds to the threshold probabilitythreshold probability

Example:Example:Waiting time for a next Waiting time for a next

stormstorm• It the western and eastern parts of Black Sea It the western and eastern parts of Black Sea

in the periodin the period 1969-1990 1969-1990 years years 17 17 severe severe storms were observedstorms were observed ( (data bydata by Galina Galina VV..SurkovaSurkova, , Alexandre VAlexandre V..KislovKislov))

• With the confidenceWith the confidence 95% 95% the next storm will the next storm will not appear earlier than in 0.1 year (1.5 not appear earlier than in 0.1 year (1.5 months)months)

• For the threshold probability 80% the waiting For the threshold probability 80% the waiting time for the next storm would be not less than time for the next storm would be not less than 5 months5 months

• These understated estimates are the result of These understated estimates are the result of one-parameter model.one-parameter model.

Probability of magnitude and Probability of magnitude and time for an extreme event: time for an extreme event: model with 2 parametersmodel with 2 parameters

• The multiplicity of non-extreme states M of a system The multiplicity of non-extreme states M of a system form a domain limited by a surface of extreme states.form a domain limited by a surface of extreme states.

• ProbabilityProbability ψ( ψ(tt, , xx) ) of first intersecting the boundary by of first intersecting the boundary by the trajectory of system state at definite time for the trajectory of system state at definite time for diffusion Markov processes could be determined after diffusion Markov processes could be determined after solving the differential equation:solving the differential equation:

∂∂ψ(ψ(tt, , xx)/∂)/∂tt = = aa((xx)*∂ψ()*∂ψ(tt, , xx)/∂)/∂xx + 1/2* + 1/2*bb((xx)*∂ )*∂ 22ψ(ψ(tt, , xx)/∂)/∂xx 22 , , wherewhere aa((xx)) – – drift coefficient (rate of process drift coefficient (rate of process

variation),variation), bb((xx)) – – diffusion coefficient (rate of dispersion diffusion coefficient (rate of dispersion

variation)variation).. Initial and boundary conditions: Initial and boundary conditions: ψ(0, ψ(0, xx) = 0, ) = 0, xxє(є(qq, , rr) и ) и

ψ(ψ(tt, , qq) = 0 и ψ() = 0 и ψ(tt, , rr)=1 )=1

Next complication: non-Next complication: non-Markov processMarkov process

Probability of time interval between two extreme events Probability of time interval between two extreme events could be determined calculating mean number of crossing could be determined calculating mean number of crossing threshold by the trajectory of system state Nthreshold by the trajectory of system state N++(t′, t′′) - N(t′, t′′) - N--(t(t00, , t′′) ≤ P{Z} ≤ Nt′′) ≤ P{Z} ≤ N++(t′, t′′)(t′, t′′)

Plot of variable: ветровые волны

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Red line - Red line - threshold of threshold of extreme extreme phenomenaphenomena

Own risk and system riskOwn risk and system risk

• The cause of risks are natural phenomena, The cause of risks are natural phenomena, however consequences are closely however consequences are closely connected with social and economic connected with social and economic components. components.

• System approach makes it possible to System approach makes it possible to distinguish two types of riskdistinguish two types of risk: :

Own risk – the sum or weighted sum of Own risk – the sum or weighted sum of risks of extreme events, which could risks of extreme events, which could happen in the present coastal territory happen in the present coastal territory (storms, runs up, avalanches, etc.),(storms, runs up, avalanches, etc.),

System risk characterizes System risk characterizes maximal maximal possible losses, which could take place possible losses, which could take place during some period of time in the system during some period of time in the system ‘nature-social media-economics’ as a ‘nature-social media-economics’ as a wholewhole. .

Components of own riskComponents of own risk In a general case each component is In a general case each component is

determined as a mean damage using determined as a mean damage using formulasformulas::

•R(X) = Pn(X)R(X) = Pn(X)**Pb(X)*Cw(X)*Wy(XPb(X)*Cw(X)*Wy(X)),,•RR((XX) – ) – risk of appearing eventrisk of appearing event XX •PnPn((XX) – ) – damage of the territory by eventdamage of the territory by event

XX,,•PbPb((XX) – ) – probability of appearing the probability of appearing the

event in timeevent in time ((activityactivity),),•CwCw((XX) – ) – vulnerability under the eventvulnerability under the event XX•WyWy((XX) – ) – total damage from the eventtotal damage from the event XX

Value at Risk (VaRValue at Risk (VaR)) • System risk estimate could be performed System risk estimate could be performed

based on total regional data base of based on total regional data base of extreme eventsextreme events..

• Value at Risk Value at Risk technology characterize technology characterize maximal loss, which could have a region maximal loss, which could have a region with a given probability during a definite with a given probability during a definite time interval.time interval.

• TechnologyTechnology VaR VaR is based on assumptions is based on assumptions of risk factors distribution, or empirical of risk factors distribution, or empirical distribution functions. distribution functions.

• Random valueRandom value XX((tt), ), has an incrementhas an increment ΔX ΔX with a distribution function Fwith a distribution function Fxx,,

VaRVaRάά = {u|P[ΔX(Δt)≤u} = ά = {u|P[ΔX(Δt)≤u} = ά• VaRVaRάά is the maximal loss, which could take is the maximal loss, which could take

place during time Δtplace during time Δt with a probabilitywith a probability ά. ά.VaRVaRάά = = FF-1-1 (1 - ά).(1 - ά).

Latent factors for system Latent factors for system riskrisk

Latent risk factorsLatent risk factors ffkk ((k=1,…s)k=1,…s) could could be determined based on regional data be determined based on regional data matrix by means of factor analysismatrix by means of factor analysis..

VaRVaRάά = Σ = Σ wwkk * VaR* VaRάά ((ffkk))

• VaRVaRάά – – regional system riskregional system risk

• VaRVaRάά ((ffkk)) – – risk introduced by the risk introduced by the latent factorlatent factor ffkk

• wwkk – – weight of the factorweight of the factor ffkk, , which is which is proportional to its significanceproportional to its significance,,

• ά ά – – given level of confidencegiven level of confidence..

Pattern of Regional Data Pattern of Regional Data Matrix Matrix

ConclusionsConclusions• The problem is reduced to estimation The problem is reduced to estimation

of risk factors distribution function on of risk factors distribution function on the basis of empirical data.the basis of empirical data.

• Annual statistical re-analysis of data Annual statistical re-analysis of data matrix makes it possible to determine matrix makes it possible to determine empirical distributions which are empirical distributions which are necessary for system risk.necessary for system risk.

• The corresponding formulas and The corresponding formulas and mathematical model are developed.mathematical model are developed.