monte carlo simulation - statvision.com carlo simulation.pdf · monte carlo simulation “monte...
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Monte Carlo Simulation
Presented by
Dr. Neil W. Polhemus
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Monte Carlo Simulation
“Monte Carlo simulations are used to model the
probability of different outcomes in a process that
cannot easily be predicted due to the intervention
of random variables.” Investopedia
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Monte Carlo Simulation
“The world … is full of more complicated systems …. the complex interaction of many variables — or the inherently probabilistic nature of certain phenomena — rules out a definitive prediction. So a Monte Carlo simulation uses essentially random inputs (within realistic limits) to model the system and produce probable outcomes.” MIT News
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Statgraphics Centurion XVII
Tools menu has a section for Monte Carlo
Simulation:
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Outline
1. Random number generators
2. Example #1: Estimating the amount of oil
available in an oil reservoir
3. Example #2: Simulating the consumer price
index using an ARIMA time series model
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Random number generators
• Includes:
– 7 discrete distributions
– 39 continuous distributions
– user-specified discrete distribution
• user specifies values for x and p(x)
– user-specified continuous distribution
• user specifies values for x and F(x)
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Example – Gamma Distribution
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Example – Triangular Distribution
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Example – User Specified Discrete
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Example – User Specified Continuous
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Datasheet
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Box-and-Whisker Plot
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General Simulation Models
• Begin with a mathematical function:
Y = f(X1, X2, X3, …, Xp)
• Specify probability distributions for X1 through Xp.
• Generate n sets of random numbers and plug them into the function.
• Calculate statistics from the n values of Y.
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Example: Oil Reserves
Murtha presents the following volumetric model for oil in place:
N = 7,758Ah(1 - Sw) / Bo stock-tank barrels
where
– A = area
– h = net pay
– = porosity
– Sw = water saturation
– B0 = formation volume factor
Question: Given a selected site, how small or large might N reasonably be?
Source: Murtha, J.A. 1994. “Incorporating Historical Data Into Monte Carlo Simulation”. SPE Comp App 6 (2): 11-17. SPE-26245-PA.
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How do get parameter distributions?
• If we have historical data, we can fit a probability distribution using
the Distribution Fitting (Uncensored Data) procedure in Statgraphics.
– Murtha used data from 83 similar reservoirs
• We could specify the minimum value, most likely value, and
maximum value and use a triangular distribution.
• We could specify the mean and standard deviation and use a
normal distribution.
• We could specify 2 or more percentiles and find a matching non-
normal distribution.
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Area
• Characterized by a lognormal distribution with
mean = 788 and standard deviation = 906
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Net pay
• Characterized by a lognormal distribution with
mean = 250 and standard deviation = 227
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Porosity as %
• Characterized by a normal distribution with
mean = 29.8 and standard deviation = 5.36
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Water saturation as %
• Characterized by a gamma distribution with
shape = 21.56 and scale = 0.74
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Formation volume factor
• Characterized by a beta distribution with shape1
= 1.14 and shape2 = 9.04, in the interval [1,2]
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Matching Percentiles
• Water saturation: suppose we are 90% certain
that it is between 20 and 40 and want to use a
gamma distribution.
• I have written an R script that I’ve saved in
matchpercentiles.sgp that will find the
parameters of various distributions for which the
cdf satisfies:
• F(20) = 0.05
• F(40) = 0.95
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R Script
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General Simulation Models
• Step 1: create datasheet with variables
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General Simulation Models
• Step 2: define the simulation
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General Simulation Models
• Step 3: run the simulation
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General Simulation Models
• Step 4: calculate summary statistics
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General Simulation Models
• Step 5: calculate percentiles
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General Simulation Models
• Step 6: create a quantile plot for the response
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Example #2: Consumer Price Index All urban consumers, all items, 1982-2016
Source: https://www.bls.gov/cpi/data.htm
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CPI during last 35 years
Average yearly change: 4.3
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CPI during last 5 years
Average yearly change: 3.2
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Simulating ARIMA Time Series Models
• Steps
– Fit an ARIMA model to the CPI data
– Generate several possible realizations
– Plot the realizations on a single graph
• Useful as input to financial models
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Automatic Forecasting
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Analysis Options
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Estimated Model
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Forecast Plot
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ARIMA Model Simulation
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Simulated Time Series
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Save Data
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Multiple Realizations
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Plot Multiple Realizations
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Plot Multiple Realizations
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Upcoming Enhancement
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Using Historical Data as Starting Values
Time Sequence Plot
12/2016 12/2018 12/2020 12/2022
Time index
240
245
250
255
260
265
270
CP
I
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Resources
• StatFolios and data files are at: www.statgraphics.com/webinars
• Recorded webinar will be posted soon.