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

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

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

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Why do we use stochastic approach?

• 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.
• 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)?

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

• Let m appear at successive time intervals
• 0 t0 t1 tm1 ,
• Each set of m events is independent
• Probability of appearing m events in the interval
  0,t could be determined by formula
• Pn(0, t)m exp(-?t) (?t)m /m!

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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.
• Based on observation data, one could estimate
  probability of extreme event appearing later than
  T
• Ptgt T exp(-? T)
• t waiting time for the next extreme event
• ? - stream intensity, i.e., average number
  of extreme events within a unit time interval.

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Guaranteed interval of life without extreme
events

• Let us find the waiting time T? for an extreme
  event such as the probability of surpassing its
  value would be higher than some big threshold
  value ? then
• T? (1/?) ln(1/?)
• The equality corresponds to the threshold
  probability

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

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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.
• Probability ?(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
• ??(t, x)/?t a(x)??(t, x)/?x 1/2b(x)?
  2?(t, x)/?x 2 ,
• where a(x) drift coefficient (rate of
  process variation),
• b(x) diffusion coefficient (rate of
  dispersion variation).
• Initial and boundary conditions ?(0, x) 0,
  x?(q, r) ? ?(t, q) 0 ? ?(t, r)1

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Next complication non-Markov process
Red line - threshold of extreme phenomena

• 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-(t0, t'') PZ
  N(t', t'')

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Own risk and system risk

• 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
• 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.

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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) risk of appearing event X
• Pn(X) damage of the territory by event X,
• Pb(X) probability of appearing the event in
  time
• (activity),
• Cw(X) vulnerability under the event X
• Wy(X) total damage from the event X

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Value at Risk (VaR)

• 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.
• Technology VaR is based on assumptions of risk
  factors distribution, or empirical distribution
  functions.
• Random value X(t), has an increment ?X with a
  distribution function Fx,
• VaR? uP?X(?t)u ?
• VaR? is the maximal loss, which could take place
  during time ?t with a probability ?.
• VaR? F-1 (1 - ?).

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Latent factors for system risk

• Latent risk factors fk (k1,s) could be
  determined based on regional data matrix by means
  of factor analysis.
• VaR? S wk VaR? (fk)
• VaR? regional system risk
• VaR? (fk) risk introduced by the latent factor
  fk
• wk weight of the factor fk, which is
  proportional to its significance,
• ? given level of confidence.

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Pattern of Regional Data Matrix
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Conclusions

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