1GS314 RISK ANALYSIS

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• 1GS314 RISK ANALYSIS ASSESSMENT

SCHOOL OF EARTH ENVIRONMENTAL SCIENCES Part
7 Probabilistic Risk Assessment Tools
Procedures Monte Carlo Simulation
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Random Variables

• Parameters such as
• The angle of friction of rock joints
• UCS of rock specimens
• Porosity of a sandstone
• Inclination orientation of discontinuities in
  a rock mass

do not have a single fixed value - they may
assume any number of values
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Variability Uncertainty
Key Qusetion of QRA How do we model the
variability of a given parameter?
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Random Variables
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Probability Density Function (PDF) describes the
relative likelihood that a random variable will
assume a particular value
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Frequency / Probability
Area under a PDF is always unity
The PDF is alternatively referred to in the
literature as the probability function or the
frequency function
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Parameter Value
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Random Variables
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Cumulative Distribution Function (CDF) gives the
probability that the variable will have a value
less than or equal to the selected value
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Probability of Value lt X Axis Value
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CDF is the integral of the corresponding PDF
The CDF is alternatively referred to in the
literature as the distribution function, cumulativ
e frequency function, or the cumulative
probability function.
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Parameter Value
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Representing Uncertainty
Risk analysis relies on the appropriate use of
probability distributions to accurately
represent the uncertainties of the problem
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Probability Distributions

• Risk analysis relies on the appropriate use of
  probability distributions to accurately represent
  the uncertainties of the problem
• Distributions can be
• Discrete or Continuous
• Bounded or Unbounded
• Parametric or Non-parametric
• Inappropriate use of probability distributions
  is a very common failing of risk analysis models

Important to understand theory history behind
probability distribution functions
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Discrete Distributions
May take one of a set of identifiable values
Each of which has a calculable probability of
occurrence Probability of occurrence sometime
called probability mass
Probability Mass Function
Sum of all these values must add up to 1

• Examples
• Binomial Negative Binomial Distributions
• Poisson Distribution
• Geometric Hypergeometric Distributions
• Generalised Discrete Distribution

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Discrete Distributions
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CDF
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Parameter Value
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Examples of Discrete Variables

• Number of bridges in a road scheme
• Number of key personnel to be employed
• Number of seismic events of a given magnitude
• i.e. Not possible to have 1.5 bridges or 2.6
  earthquakes of magnitude 5
• Variables take on specific values
• A 'count' rather than a 'measure'
• e.g. 0, 2, 25, 36 as opposed to 0 - 36
• Descriptors
• Mean, variance, standard deviation,coefficient
  of variation

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Continuous Distributions
Used to represent a variable that can take any
value within a defined range
Probability Density Function

• Examples
• Normal, Lognormal
• Beta, Triangular
• Weibull

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Continuous Distributions
CDF
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Continuous Distributions

• Measurement could be infinitely divisible
• e.g. Time, mass distance
• Also for 'discrete' variables where the gap or
  increment between them is insignificant e.g.
  project costs
• Descriptors
• Mean, variance, standard deviation, coefficient
  of variation, confidence interval

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Bounded Unbounded Distributions

• A bounded distribution is confined to lie
  between two determined values
• Uniform - bounded between a minimum maximum
• Triangular - bounded between a minimum maximum
• Beta - bounded between 0 1
• Binomial - bounded between 0 n
• An unbounded distribution extends from minus
  infinity to plus infinity
• Truncated Normal Distribution
• Partially bounded
• Removes nonsensical tails

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Parametric Non-Parametric

• Theoretically derived parametric distributions
• General non-parametric distributions
• Distribution shape borne of the mathematics
  describing a theoretical problem
• e.g. Exponential Distribution
• Distribution whose mathematics is defined by the
  shape that is required
• e.g. Triangular Distribution
• Parametric Distributions - need to fully
  understand the theory behind them before using in
  practice

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Probability Density Functions

• Beta, BetaPERT
• Binomial
• Cauchy
• Chi Squared
• Cumulative
• Discrete
• Exponential
• Extreme Value
• Gamma

• Geometric
• Histogram
• Hypergeometric
• Inverse Gaussian
• Logistic
• Log-Logistic
• Lognormal
• Negative Binomial
• Normal
• Pareto

• Poisson
• Student's t
• Triangular
• Uniform
• Weibull
• General

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Performing a Risk Assessment

• Probabilistic Risk Assessment
• QRA
• Monte Carlo Simulation
• Probability Density Functions
• Cumulative Density Functions
• Discrete Continuous Variables
• Developing a Risk Analysis Model

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Quantitative Risk Analysis

• Considering Risk Uncertainty
• One of Several Tools for Risk Assessment
• Traditionally single point or 'deterministic'
  models
• Stochastic or probabilistic models
• Number of possible scenarios considered
  generated rather than simply one outcome
• Simulation

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Deterministic
Factor of Safety Bolt Capacity
----------------------

Thickness c Bolt Spacing2
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Stochastic
Factor of Safety Bolt Capacity
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Thickness c Bolt Spacing2
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Monte Carlo Simulation

• Each uncertain variable modelled by a probability
  distribution rather than a single value
• Structure of a quantitative risk assessment model
  similar to a deterministic model i.e. with all
  the multiplication's, additions etc.
• Exception is that probability distributions are
  used to describe the variables rather than single
  values
• Objective is to calculate the combined impact of
  the model's various uncertainties in order to
  determine a probability distribution of the
  possible outcomes

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Monte Carlo Simulation

• Random sampling of each probability distribution
  within the model
• Produces 1000's or even 10,000's of scenarios
• i.e. Iterations or trials
• Each probability distribution is sampled in a
  manner that reproduces the distribution's shape
• Distribution of the values calculated for the
  model outcome therefore reflects the PROBABILITY
  of the values that could occur
• Developed in World War II from Atomic Bomb
  research
• Monte Carlo or Latin Hypercube Sampling

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Performing a Risk Assessment

• Monte Carlo Simulation

Combination of Parameter Distributions
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Methodology

• Use data to inform the choice of input
  distributions for model parameters
• The choice of input distribution should always be
  based on all information (both qualitative and
  quantitative) available for a parameter

EXPERT KNOWLEDGE JUDGEMENT
DATA
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Methodology

• In selecting a distributional form, the risk
  assessor should consider the quality of the
  information in the database ask a series of
  questions including (but not limited to)
• Is there any mechanistic basis for choosing a
  distributional family?
• Is the shape of the distribution likely to be
  dictated by physical or biological properties or
  other mechanisms?
• Is the variable discrete or continuous?
• What are the bounds of the variable?
• Is the distribution skewed or symmetric?
• If the distribution is thought to be skewed, in
  which direction?
• What other aspects of the shape of the
  distribution are known?

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Methodology
Random Number Generated
Process Repeated Many Times
Distributions Sampled
Output Value Calculated
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Monte Carlo Sampling
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Random Number Generated
Curve Sampled
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Parameter Value
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Monte Carlo Sampling
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Clustering - unless a large number of samples
taken
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Monte Carlo Sampling

• Random sampling from input distribution
• Random number generated between zero one
• Used to sample the probability distribution
• Entirely random sampling technique
• Sample may fall anywhere within the range of the
  input distribution
• Randomness of sampling may mean that it will over
  under sample from various parts of the
  distribution
• Cannot be relied upon to replicate the input
  distribution's shape unless a very large number
  of iterations are performed
• Pure randomness of Monte Carlo sampling not
  really relevant

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Latin Hypercube Sampling
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Curve stratified into layers
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Each layer only sampled once
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Parameter Value
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Latin Hypercube Sampling

• Sampling method that appears random
• BUT guarantees to reproduce the input
  distribution with much greater efficiency than
  Monte Carlo sampling
• Uses 'stratified sampling without replacement'
• Sampling is forced to recreate the original input
  probability distribution
• Overcomes clustering problems associated with
  Monte Carlo sampling

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Latin Hypercube Sampling

• Probability distribution split into n levels of
  equal probability where n is the number of
  iterations to be performed on that model
• Stratification of the input probability
  distribution
• Sample taken from each interval or stratification
• This stratification is not sampled from again as
  its value is already represented in the sampled
  set
• Distribution will have been reproduced with
  predictable uniformity over the data range

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Random Number Generator Seeds

• Many algorithms developed to generate a random
  number between 0 1 with equal probability
  density for all possible values
• Start or initial seed value
• Possible to select this seed value
• If model is not changed the same simulation
  results can be repeated exactly if same seed
  value set
• Possible to consider the effects of changing a
  distribution on the effects on the model's
  output
• Hence results due to changes in the model not
  as a result of the randomness of the sampling

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Advantages of Monte Carlo Method

• Complex mathematics can be included with no
  extra difficulty
• Widely accepted technique
• Model behaviour can be investigated with ease
• Changes can be made quickly compared with
  previous models

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Advantages of Monte Carlo Method

• Distributions of the model's variables can be
  precisely defined
• Correlations other inter - dependencies can be
  modelled
• Computer does all of the work required
• Commercially available software
• _at_Risk, CrystalBall
• Level of mathematics required to perform a Monte
  Carlo Simulation is quite basic
• Greater levels of precision achieved by
  increasing number of iterations

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References

• Guiding Principles for Monte Carlo Analysis
  (EPA/630/R-97/001)
• http//www.epa.gov/ncea/monteabs.htm
• Guidance on Assigning Values to Uncertain
  Parameters in Subsurface Contaminant Fate and
  Transport ModellingA McMahon, J Heathcote, M
  Carey, A Erskine J Barker June 2001
• www.environment-agency.gov.uk/commondata/105385/nc
  _99_38_3.pdf
• Vose, D. (2000) Risk analysis A Quantitative
  Guide. John Wiley Sons.