3a-Montecarlo Simulation Concepts 2

8/6/2019 3a-Montecarlo Simulation Concepts 2

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ASST – 8588 (Decision Making Under Risk)

Topic 3a : Montecarlo Simulation (1)



How does Montecarlo Simulation work?

• It relies on the use of probability distributions of input

variables to represent their uncertainty

• Recalculates a deterministic model many times, combiningthe input variables for each iteration by means of random

sampling

• Results for key nominated variables (analytical concerns)are recorded for each pass and reported as an output

probability distribution





Montecarlo Simulation Steps

• Represents the logic of the case that is being modelled

1.- STATIC MODEL (DETERMINISTIC)

• Representation of uncertainty on input data

2.- INPUT DISTRIBUTIONS

• Random sampling of each uncertain input variable

3.- RANDOM VARIABLE GENERATION

• Gathering/storing of results data (histogram) , analysis and insights

4.- OUTPUT GENERATION, ANALYSIS AND SELECTION



Deterministic Model Flow

Model

Variable A

Variable C

Variable BOutput D

Output E

Single Pass



Montecarlo Model Flow

Model

Variable A

Variable C

Variable BOutput D

Output E

Multi Pass





Benefits of Montecarlo over Other Techniques• No meaningful increase in model logic complexity although

complex mathematics can be accommodated if required

• The computer does all the work in calculating outputdistributions

• Montecarlo method is now widely used and understood inthe industry, therefore its results accepted with ease

• It is easy to perform changes to the calculation models

• Correlation and interdependency between variables can be

implemented• A wide variety of professional and public domain software isavailable for users at all levels



What is Random Sampling?

• It is the random selection of hundreds or thousands of

values from a distributed sample of those values• The sampling is performed in such a manner that when

performed a large enough number of times, reproduces the

original distribution’s shape

• The distribution of the values calculated therefore reflects

the probability of the values that could occur





Sampling With Clustering

• Sampling with clustering is a

simple random sampling process

which consists of sampling from

groups or clusters of elements

• Used when it is difficult or

costly to generate a complete list

of population members or

population is dispersed

geographically



Stratified Sampling

• A stratified random sample is

obtained by separating the

population into mutually exclusive

sets or segments and then

drawing simple random samplesfrom each strata

• Latin Hypercube is one of the

most widely used forms of

stratified random sampling



Benefits of Stratified Random Sampling

The key benefits of stratified random sampling are …• Can provide greater precision than a simple random sampleof the same size

• Saves computer time and cost as it requires a smaller

sample for the same level of precision

• A stratified random sampling choice can ensure that arepresentative sample of the whole spectrum is obtained

• Can ensure that enough sample points are obtained to

support a separate analysis of any sub group

• The most well known strata sampling method is the LatinHypercube, used by @Risk and most of similar software



Continuous Random Variable and Histograms

• A random variable is a function X that assigns to each possibleoutcome in an experiment a real number

• If X may assume any value in some given interval, it is called a

continuous random variable

• If it can assume only a number of separated values, it is called a

discrete random variable

• If X is a random variable, we are usually interested in the probability

that X takes on a value in a certain range

• We can use a bar chart, called a (probability distribution) histogram,

to display the probabilities that X lies in selected ranges





Continuous Random Variable – Example

• The histogram for the sample is simply the bar graph of the probability

distribution table…

we can easily answer

questions such as…

- What is the probability of a student being age 20 or older? (78%)

- What is the probability of a student being 25 to 29? (15%)

0.00

0.10

0.200.30

0.40

0.50

15-19 20-24 25-29 30-34 >35

Students




From the histogram we can easily answer some questions…

But other questions can

not easily be answered,

such as…

- What is the probability of a student being age 22 or older? (?%)

0.00

0.10

0.200.30

0.40

0.50

15-19 20-24 25-29 30-34 >35

Students




• The histogram as plotted did allow some partial answers to selected

questions only

• We needed a “smoother” type of histogram

• It could have been achieved by selected finer ranges, for instance the

range could have been divided in steps of 1 year instead, this would havecreated a smoother graph, albeit lower in height

• Nevertheless, it still would have not easily answered questions such as…

What is the probability of a student being 20 ½ years or older?

• The above leads to the need for using continuous distribution graphs

and calculating probabilities by estimating the “area under the curve”



Discrete and Continuous Distributions• We know that a Relative Frequency Diagram is simply a histogram of

actual observations for any given experiment

• The number of total observations is represented by the sum of all

observations in each of the histogram segments

• The histograms can be used to generate probability density

distributions i.e. area under the curve is 1.0 or 100%

• Theory of statistical probability for continuous random variable is

based on continuous probability density distributions

• We know that “smoother” histograms can be generated by finer

definition of the ranges

• Sometimes Continuous Probability Distributions are defined as

functions f(x)



Recommended Reading

1.- “Decision Analysis for Petroleum Exploration”

Paul D. Newendorp

PennWell Books



End of Topic 3a

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3a-Montecarlo Simulation Concepts 2