1. Regional probabilistic risk assessments of extreme events, their magnitude and frequency Стохастическое прогнозирование вероятностей и рисков экстремальных явлений Dushin V.R., Evlanova V.A., Ilyushina E.A., Smirnov N.N.

2. Why do we use stochastic approach? Processes in hydrosphere and atmosphere are stochastic.Processes in hydrosphere and atmosphere are stochastic. Impacts of various external factors and internal mechanisms responsible on future system behavior could not be evaluated in advance.Impacts of various external factors and internal mechanisms responsible on future system behavior could not be evaluated in advance. A future state of a system could be described only in terms of probability.A future state of a system could be described only in terms of probability. Main problem: how long the system will preserve its current state (with absence of extreme events)?Main problem: how long the system will preserve its current state (with absence of extreme events)?

3. Data base on extreme phenomena Data matrix Factor analysis Development of latent risk factors (VaR) Time series for geophysical observation data Identification of processes: ARIMA Estimate of distribution functions Estimates of magnitude and waiting period for an extreme event (It is necessary to introduce damage codes and scales)

4. Simple stream of events Let m appear at successive time intervalsLet m appear at successive time intervals 0 = t 0 ≤ t 1 ≤…≤ t m+1, 0 = t 0 ≤ t 1 ≤…≤ t m+1, Each set of m events is independentEach set of m events is independent Probability of appearing m events in the interval [0,t] could be determined by formula:Probability of appearing m events in the interval [0,t] could be determined by formula: P{n(0, t)=m} = {exp(-λt)}· (λt) m /m! P{n(0, t)=m} = {exp(-λt)}· (λt) m /m!

5. Waiting time for next extreme event distribution We accept the model of a simple stream of events for description of extreme events because it has a property of independence of future on past under given present conditions.We accept the model of a simple stream of events for description of extreme events because it has a property of independence of future on past under given present conditions. Based on observation data, one could estimate probability of extreme event appearing later than TBased on observation data, one could estimate probability of extreme event appearing later than T P{τ> T } = exp(-‹λ› T) P{τ> T } = exp(-‹λ› T) τ – waiting time for the next extreme event τ – waiting time for the next extreme event ‹λ› - stream intensity, i.e., average number of extreme events within a unit time interval. ‹λ› - stream intensity, i.e., average number of extreme events within a unit time interval.

6. Guaranteed interval of life without extreme events Let us find the waiting time T γ for an extreme event such as the probability of surpassing its value would be higher than some big threshold value γ; thenLet us find the waiting time T γ for an extreme event such as the probability of surpassing its value would be higher than some big threshold value γ; then T γ ≤ (1/ ‹λ› ) ln(1/γ) T γ ≤ (1/ ‹λ› ) ln(1/γ) The equality corresponds to the threshold probability The equality corresponds to the threshold probability

7. Example: Waiting time for a next storm It the western and eastern parts of Black Sea in the period 1969-1990 years 17 severe storms were observed (data by Galina V.Surkova, Alexandre V.Kislov)It the western and eastern parts of Black Sea in the period 1969-1990 years 17 severe storms were observed (data by Galina V.Surkova, Alexandre V.Kislov) With the confidence 95% the next storm will not appear earlier than in 0.1 year (1.5 months) With the confidence 95% the next storm will not appear earlier than in 0.1 year (1.5 months) For the threshold probability 80% the waiting time for the next storm would be not less than 5 monthsFor the threshold probability 80% the waiting time for the next storm would be not less than 5 months These understated estimates are the result of one- parameter model.These understated estimates are the result of one- parameter model.

8. Probability of magnitude and time for an extreme event: model with 2 parameters The multiplicity of non-extreme states M of a system form a domain limited by a surface of extreme states.The multiplicity of non-extreme states M of a system form a domain limited by a surface of extreme states. Probability ψ(t, x) of first intersecting the boundary by the trajectory of system state at definite time for diffusion Markov processes could be determined after solving the differential equation:Probability ψ(t, x) of first intersecting the boundary by the trajectory of system state at definite time for diffusion Markov processes could be determined after solving the differential equation: ∂ψ(t, x)/∂t = a(x)*∂ψ(t, x)/∂x + 1/2*b(x)*∂ 2 ψ(t, x)/∂x 2, ∂ψ(t, x)/∂t = a(x)*∂ψ(t, x)/∂x + 1/2*b(x)*∂ 2 ψ(t, x)/∂x 2, where a(x) – drift coefficient (rate of process variation), where a(x) – drift coefficient (rate of process variation), b(x) – diffusion coefficient (rate of dispersion variation). b(x) – diffusion coefficient (rate of dispersion variation). Initial and boundary conditions: ψ(0, x) = 0, xє(q, r) и ψ(t, q) = 0 и ψ(t, r)=1 Initial and boundary conditions: ψ(0, x) = 0, xє(q, r) и ψ(t, q) = 0 и ψ(t, r)=1

9. Next complication: non-Markov process Probability of time interval between two extreme events could be determined calculating mean number of crossing threshold by the trajectory of system state N + (t′, t′′) - N - (t 0, t′′) ≤ P{Z} ≤ N + (t′, t′′) Red line - threshold of extreme phenomena

10. Own risk and system risk The cause of risks are natural phenomena, however consequences are closely connected with social and economic components.The cause of risks are natural phenomena, however consequences are closely connected with social and economic components. System approach makes it possible to distinguish two types of risk:System approach makes it possible to distinguish two types of risk:  Own risk – the sum or weighted sum of risks of extreme events, which could happen in the present coastal territory (storms, runs up, avalanches, etc.),  System risk characterizes maximal possible losses, which could take place during some period of time in the system ‘nature-social media-economics’ as a whole.

11. Components of own risk In a general case each component is determined as a mean damage using formulas: R(X) = Pn(X)*Pb(X)*Cw(X)*Wy(X),R(X) = Pn(X)*Pb(X)*Cw(X)*Wy(X), R(X) – risk of appearing event XR(X) – risk of appearing event X Pn(X) – damage of the territory by event X,Pn(X) – damage of the territory by event X, Pb(X) – probability of appearing the event in timePb(X) – probability of appearing the event in time (activity), (activity), Cw(X) – vulnerability under the event XCw(X) – vulnerability under the event X Wy(X) – total damage from the event XWy(X) – total damage from the event X

12. Value at Risk (VaR) System risk estimate could be performed based on total regional data base of extreme events.System risk estimate could be performed based on total regional data base of extreme events. Value at Risk technology characterize maximal loss, which could have a region with a given probability during a definite time interval.Value at Risk technology characterize maximal loss, which could have a region with a given probability during a definite time interval. Technology VaR is based on assumptions of risk factors distribution, or empirical distribution functions.Technology VaR is based on assumptions of risk factors distribution, or empirical distribution functions. Random value X(t), has an increment ΔX with a distribution function F x,Random value X(t), has an increment ΔX with a distribution function F x, VaR ά = {u|P[ΔX(Δt)≤u} = ά VaR ά is the maximal loss, which could take place during time Δt with a probability ά.VaR ά is the maximal loss, which could take place during time Δt with a probability ά. VaR ά = F -1 (1 - ά).

13. Latent factors for system risk Latent risk factors f k (k=1,…s) could be determined based on regional data matrix by means of factor analysis. Latent risk factors f k (k=1,…s) could be determined based on regional data matrix by means of factor analysis. VaR ά = Σ w k * VaR ά (f k ) VaR ά – regional system riskVaR ά – regional system risk VaR ά (f k ) – risk introduced by the latent factor f kVaR ά (f k ) – risk introduced by the latent factor f k w k – weight of the factor f k, which is proportional to its significance,w k – weight of the factor f k, which is proportional to its significance, ά – given level of confidence.ά – given level of confidence.

14. Pattern of Regional Data Matrix

15. Conclusions The problem is reduced to estimation of risk factors distribution function on the basis of empirical data.The problem is reduced to estimation of risk factors distribution function on the basis of empirical data. Annual statistical re-analysis of data matrix makes it possible to determine empirical distributions which are necessary for system risk.Annual statistical re-analysis of data matrix makes it possible to determine empirical distributions which are necessary for system risk. The corresponding formulas and mathematical model are developed.The corresponding formulas and mathematical model are developed.