Transcript GoldSim
Sensitivity and Uncertainty Analysis
and Optimization in GoldSim
GoldSim Technology Group LLC, 2006
Slide 1
Overview
Uncertainty Analysis
Sensitivity Analysis with Monte Carlo
simulation
Options to support uncertainty and
sensitivity analysis when doing Monte Carlo
simulation
– Screening realizations
– Saving distributions at multiple timepoints
Sensitivity Analysis with deterministic
simulations
Optimization
GoldSim Technology Group LLC, 2006
Slide 2
Uncertainty and Sensitivity Analysis
Uncertainty analysis answers the question:
– “Which parameters is the uncertainty in the
result most sensitive to”
Sensitivity analysis answers the question:
– “Which parameters is the result most
sensitive to”
GoldSim Technology Group LLC, 2006
Slide 3
Example
A: Uniform(10,20)
B: Uniform(10,20)
C: Uniform(10,20)
E: Uniform(1,5)
D: Constant=10
x = e*d*(a^2 + sin(c/pi))/b
GoldSim Technology Group LLC, 2006
Slide 4
Uncertainty Analysis in GoldSim
GoldSim computes a correlation matrix
n
C rp
(p i m p )(ri m r )
i 1
n
(p i m p )
i 1
2
n
2
(r
m
)
i
r
i 1
Value (Pearson) correlation indicates a linear
relationship
Can capture non-linear relationships with rank
(Spearman) correlation
GoldSim Technology Group LLC, 2006
Slide 5
Example
Run 1000 times
For X, select Final Values | Multivariate
analysis
Select a Stochastic
Add all stochastics
Variable correlations
GoldSim Technology Group LLC, 2006
Slide 6
Sensitivity Analysis in GoldSim Using
Monte Carlo Simulation
Coefficient of determination: This coefficient varies
between 0 and 1, and represents the fraction of the total
variance in the result that can be explained based on a
linear (regression) relationship to the input variables (i.e.,
Result = aX + bY + cZ + …). The closer this value is to 1, the
better that the relationship between the result and the
variables can be explained with a linear model.
SRC (Standardized Regression Coefficient): Standardized
regression coefficients range between -1 and 1 and provide
a normalized measure of the linear relationship between
variables and the result. They are the regression
coefficients found when all of the variables (and the result)
are transformed and expressed in terms of the number of
standard deviations away from their mean. GoldSim’s
formulation is based on Iman et al (1985).
GoldSim Technology Group LLC, 2006
Slide 7
Sensitivity Analysis in GoldSim Using
Monte Carlo Simulation
Partial Correlation Coefficient: Partial correlation
coefficients vary between -1 and 1, and reflect the extent to
which there is a linear relationship between the selected
result and an input variable, after removing the effects of
any linear relationships between the other input variables
and both the result and the input variable in question. For
systems where some of the input variables may be
correlated, the partial correlation coefficients represent the
“unique” contribution of each input to the result. GoldSim’s
formulation is based on Iman et al (1985).
Importance Measure: This measure varies between 0 and 1,
and represents the fraction of the result’s variance that is
explained by the variable. This measure is useful in
identifying nonlinear, non-monotonic relationships between
an input variable and the result (which conventional
correlation coefficients may not reveal). The importance
measure is a normalized version of a measure discussed in
Saltelli and Tarantola (2002).
GoldSim Technology Group LLC, 2006
Slide 8
Example
Run 1000 times
For X, select Final Values | Multivariate
analysis
Select a Stochastic
Add all stochastics
Sensitivity analysis
GoldSim Technology Group LLC, 2006
Slide 9
Example
Run 1000 times
For X, select Final Values | Multivariate
analysis
Select a Stochastic
Add all stochastics
2D Plot
GoldSim Technology Group LLC, 2006
Slide 10
Sensitivity Analysis in GoldSim Using
Deterministic Simulation
Hold all variables at a constant value
(typically mean), and then vary each
parameter over a specified range
X-Y function charts
Tornado charts
– Visually indicates sensitivity
GoldSim Technology Group LLC, 2006
Slide 11
Example
From main menu, select Run | Sensitivity
Analysis…
Add all Stochastics
Add D
Select ranges
Carry out analyses
GoldSim Technology Group LLC, 2006
Slide 12
Optimization
Optimize (minimize or maximize) an
objective function by:
– Adjusting optimization variables
– Meeting a Required Condition
Use’s Box’s Complex
method to2solve 3
2
f ( x, y ) y 2 y 40 x 22 x x x
GoldSim Technology Group LLC, 2006
Slide 13
4
Optimization
Specify Objective Function
Specify Required Condition
Specify Optimization Variables
Example:
f ( x, y ) y 2 y 40 x 22 x x x
2
GoldSim Technology Group LLC, 2006
Slide 14
2
3
4
Optimization
Common application: calibration
Example: the stock market
GoldSim Technology Group LLC, 2006
Slide 15