Analysis and Management of Uncertainties

Bali, Indonesia, 9 – 13 September 2013

Rodolfo AvilaRodolfo Avila

Outline

• Part 1. Basic concepts: Hazard, impact, probability, risk

• Part 2. Types of uncertainties

• Part 3. Management of uncertainties

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PART 1. BASIC CONCEPTS

SADRWMS Methodology Report 3

Hazard

• A hazard is a situation or agent that poses a threat to humans, biota or the environment.

The higher the potential to cause harm or damage, the higher is the hazard.

Hazards cannot be avoided, but they can be controlled to avoid or reduce exposures and impacts

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Types of hazards

• Chemical, radiological, biological, physical, …

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Hazard – exposure – effect - impact

• Exposures to a hazard resulting in adverse effects is called “a negative impact”

• Low hazards can sometimes lead to very high negative impacts, whereas high hazards may not lead to impacts at all or to very low impacts

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Hazards – impacts - risk

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Hazards may or may not lead to impacts, or lead to high or low impacts depending on the exposure and effects

Something is missing !

Definition of Risk

The word risk derives from the early Italian “risicare” , which means to dare

Risk is the "possibility of loss or injury: peril.” (Webster Dictionary, 1999)

Antoine Arnauld 1662

Fear of harm ought to be proportional not merely to the gravity of harm, but also to the probability of the event(”La logique, ou l´art de penser” a bestseller at the time)

The concept of risk has (two) elements: the likelihood of something happening and the consequences if it happens

Representation of risk

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obabilityesConsequencRisk Pr

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What is probability?What is probability?

The ancient Greek word meant plausible or probable. Socrates defined it as ”likeness to truth”

Probability is a measure of our confidence that something is going to happen. Probable means to be expected with some degree of certainty

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Conditional Conditional probabilitiesprobabilities

Any probability P(E) of an event E is conditional to some stipulated model or assumption (A1), it should strictly be written P(E|A1), i.e. the probability of E, given A1.

There may be other assumptions (A2, A3, A4, … Ai) in addition to the assumption A1 believed to be the most likely.

How can this fact be taken into account?

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Example 1Example 1

Will a person survive the next twelve months?

From the age and sex we might get an answer from vital statistics, that will apply to the average person.

We may know the habits and conditions, which will make us to believe that the a higher or lower probability is more likely.

Someone with better insight could always assess a better value.

As uncertainties are eradicated the value would approach zero or unity

All estimates of probabilities are subjective and depend of knowledge and experience

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””A priori” and ”a posteriori”A priori” and ”a posteriori”

A priori probability- estimated before the fact. Can be reliable ”only for the most part”

A posteriori probability- estimated after the fact. By taking a sufficient large sample, you can increase your confidence in the estimated probability to whatever degree you wish.

But, how reliable this probability is for predicting future events?

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Example 2Example 2

A person undergoes a medical test for a relatively rare cancer. The cancer has an incidence of 1 % among the general population.

Extensive trials have shown that the test does not fail to detect the cancer when it is present.

The test gives a positive result in 21 % of the cases in which no cancer is present.

When she was tested, the test produced a positive result.

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Example 2 (cont)Example 2 (cont)

What is the probability that the person actually has the cancer?

1 % 4,6 % 79 %

What happens if she repeats the test several times?

Sources of uncertainty

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Future (System) Uncertainty

• Uncertainty due to our inability to make exact predictions of the future evolution of the system, the environment and of future human actions

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Model Uncertainty

• Uncertainty in conceptual, mathematical and

computer models used to simulate the system behavior and evolution

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Data/Parameter Uncertainty

• Uncertainty/variability in data and parameters used as inputs in modeling and dose calculations

• Can be represented using probability distributions

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Management of uncertainties

• Uncertainty always exists in modelling any physical system

• Not possible to remove such uncertainties totally

• Instead need to manage them and reduce their effects

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Approaches for Uncertainty Management

• Awareness – be aware of all major locations of uncertainty

• Importance – determine relative importance of various sources of uncertainty using sensitivity analysis

• Reduction – reduce uncertainties, e.g. through further data collation

• Quantification – quantify effects of uncertainties on model output using sensitivity analysis

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Management of system uncertainty

• System uncertainty is usually managed by performing analysis for a set of scenarios of future evolution of the system

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Management of Model Uncertainty

• Conceptual model: consider alternative conceptual models and collect further data

• Mathematical/computer model: use model verification, calibration and validation, and range of models

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Management of Data/Parameter Uncertainty

• Five approaches can be used:• conservative/worse case approach

• best estimate and what if

• sensitivity analysis

• Probabilistic

• Can also be used to address model uncertainties

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Conservative/worse case approach

• Use pessimistic parameter values to overestimate impact

• Danger of being so pessimistic as to be worthless and misleading

• Difficult to define the worst value, and prove that this is the worst one.

• Not always obvious what is conservative for a particular combination of parameters, exposure pathways and radionuclides

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Magnification of errorsMagnification of errors

RQ=X1*X2 / X3

X1,X2 Lognormal(5,5)

X3

Uniform (10,20)

RQ Percentile

Pessimistic 15 From 99

to +infinity

Probabilistic 6 95

Probabilistic

with

correlation

1 95

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Probabilistic methods - Random Probabilistic methods - Random variablesvariables

Random variables are used to describe aleatory events (from the Latin word aleatorius, which means games of chance), i.e. events whose outcome is uncertain

A discrete random variable can only take specific values from a set of values

A continuous variable can take any value within a defined range of values

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Probability Density FunctionProbability Density Function

0

0,5

1

1,5

2

2,5

3

0 0,625 1,25 1,875 2,5

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Propagation of UncertaintiesPropagation of Uncertainties

00.5

11.5

22.5

33.5

44.5

0.05 0.1625 0.275 0.3875 0.5

00.020.040.060.08

0.10.120.140.16

85 90 95 100 105 110

0

1

2

3

4

5

6

7

0 125 250 375 500

Input Parameter

Endpoint = F(Input, Parameter)

Endpoint

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Main problems of the probabilistic approachMain problems of the probabilistic approach

1. Getting the probability distributions

2. Avoiding impossible combinations

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Maximum entropy distribution selectionMaximum entropy distribution selection

H=∑pi*ln(pi)

Constraint PDF

Upper bound, lower bound

Uniform

Min, Max, Mode Triangular (Beta)

Mean, SD NormalRange, Mean Beta

Mean occurrence rate

Poisson

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Bayesian updatingBayesian updating

Based on the Bayes´theorem

Posterior ≈Prior*Likelihood

Observations Prior Posterior

Binomial Beta Beta

Poisson Gamma Gamma

Negative binomial

Normal Normal

Normal Normal Normal

Normal Gamma Gamma

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Sensitivity analysisSensitivity analysis

“Sensitivity analysis is the study of how the variation in the output of a model (numerical or otherwise) can be apportioned, qualitatively or quantitatively, to different sources of variation, and how the given model depends upon the information fed into it.”

(Andrea Saltelli, 2000)

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Input – Output dependenciesInput – Output dependencies

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Tornado PlotsTornado Plots