Sensitivity and Uncertainty Analysis and Optimization in GoldSim

GoldSim Technology Group LLC, 2006

Slide 1

Sensitivity and Uncertainty Analysis and Optimization in GoldSim


Slide 2

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


Slide 3

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”


Slide 4

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


Slide 5

Uncertainty Analysis in GoldSim GoldSim computes a correlation matrix

Value (Pearson) correlation indicates a linear relationship

Can capture non-linear relationships with rank (Spearman) correlation

)m(r)m(p

)mr)(m(pC

n

1i

2ri

n

1i

2pi

n

1 iripi

rp


Slide 6

Example

Run 1000 times

For X, select Final Values | Multivariate analysis

Select a Stochastic Add all stochastics Variable correlations


Slide 7

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).


Slide 8

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).


Slide 9

Example

Run 1000 times


Select a Stochastic Add all stochastics Sensitivity analysis


Slide 10

Example

Run 1000 times


Select a Stochastic Add all stochastics 2D Plot


Slide 11

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


Slide 12

Example

From main menu, select Run | Sensitivity Analysis…

Add all Stochastics Add D Select ranges Carry out analyses


Slide 13

Optimization

Optimize (minimize or maximize) an objective function by:– Adjusting optimization variables– Meeting a Required Condition

Use’s Box’s Complex method to solve

4322 22402),( xxxxyyyxf


Slide 14

Optimization

Specify Objective Function Specify Required Condition Specify Optimization Variables

Example:

4322 22402),( xxxxyyyxf


Slide 15

Optimization

Common application: calibration

Example: the stock market

Documents

Sensitivity and Uncertainty Analysis and Optimization in GoldSim