uncertainty and sensitivity analysis...introduction: sensitivity analysis framework step a...
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1. Introduction & framework
2. Local approaches:
โช One-factor-at-a-time sensitivities on the nominal range
o Tornado diagram to spider plot
o Elementary effect & elasticity measures
โช Morrisโ method
3. Global/probabilistic approaches:
โช Discrete:
o Event and probability tree
o Method of discrete probabilities
โช Continuous:
o Method of moments
o Regression-based method
โข Input/output correlation coefficients
โข SRC/SRRC
o Distribution-based method
โข Variance decomposition method
โข Moment-independent method
4. Model structure uncertainty
Outline
Introduction: sensitivity analysis framework
Step ASimulation
Step BQuantification of sources of uncertainty
Step CUncertainty propagation
Step DSensitivity Analysis
Description of phenomenaโ Adoption of math models
(which are for example turned into operative computer
codes for simulations)
In practice the system under
analysis can not be
characterized exactly and the
knowledge of the undergoing
phenomena is incomplete.
โข Model parameter
โข Hypothesis
โข Model structure
Uncertainty propagates within the model and causes
uncertainty of the model outputs
โข Uncertainty analysis and propagation:
propagates the uncertainties of the input parameters and the model structure to the
model output
โข Parameter prioritization:
identify the most influential inputs to the output uncertainty
โmore information on those parameters would lead to the largest reduction in the
output uncertainty
โข Parameter fixing:
determine non-influential inputs that can be fixed at certain values
โmodel simplification
โข Direction of change (trend):
whether an increase in an input causes an increase (decrease) in the model output
โข Insights about model structure
interactions, etc.
Introduction: objectives of sensitivity analysis
Introduction: how uncertainty propagates
๐ = ๐(๐)
๐ = (๐ฅ1, ๐ฅ2, โฆ , ๐ฅ๐) : vector of ๐ input parameters
๐ : output vector correspondent to ๐ and ๐(๐)
๐ ๐ : model structure
โข Deterministic (e.g., advection-dispersion equation, Newton law)
โข Stochastic (e.g., Poisson model, exponential model)
๐(๐)
๐ = (๐ฅ1, ๐ฅ2, โฆ , ๐ฅ๐) ๐
๐ ๐
LOCAL APPROACHES
โข Tornado diagram to spider plots
โข Elementary effect & elasticity measures
โข Morrisโ method
โข Local approaches generally focus on nominal values of the inputs, called
reference/base value
๐0 = ๐ฅ10, โฆ , ๐ฅ๐
0
observe what happens to ๐ for small variations around ๐0
โข The measure of the contribution of input ๐ฅ๐ is
the variation of ๐ฆ when ๐ฅ๐ is varied by โ๐ฅ๐?
โข One-Factor-At-a-Time (OFAT)
Typically one single parameter ๐ฅ๐ is varied one at a time around its nominal
value, while maintaining the others fixed at their nominal values
Local approaches
โข A graphical representation of two series of OFAT sensitivities
โข For each parameter ๐ฅ๐, a low level ๐ฅ๐โ and high level ๐ฅ๐
+are defined
โข When no information on the distributions of ๐ฅ๐ โs are available, these are the best
estimation of the bounds on the parameters
Tornado diagrams, Howard (1988)
Sensitivity Analysis: An Introduction for the Management Scientist (Borgonovo, 2017)
๐ฆ = ๐ ๐ฅ1 , ๐ฅ2, ๐ฅ3 = ๐ฅ1 + ๐ฅ2 + 5๐ฅ3
๐ฅ10 = ๐ฅ2
0 = ๐ฅ30 = 1, ๐ฅ๐ โ [0.5,1.5]
Spider plot of one-way sensitivity functions,
Eschenbach (1992)
โ๐ฆ1+, โ๐ฆ2
+
Relation between spider plot and tornado diagram
โ๐ฆ1โ, โ๐ฆ2
โ
โ๐ฆ3โ
โ๐ฆ3+
เตฏโ(๐ฅ๐) = ๐(๐ฅ๐ , ๐ฑ~๐0
Tornado diagram & spider plot
The graphs only reflects the dependence of the model on ๐ฅ3
๐ฆ = (๐ฅ1 โ1
2)2(๐ฅ2 โ
1
2)2 + (๐ฅ3 โ
1
2)2
๐ฑ0 = (1
2,1
2,1
2), ๐ฑ+ = (1,1,1), ๐ฑโ = (0,0,0)
โข Computed at a finite change requires only two model runs
โข Its limit tend to the partial derivative of ๐(๐ฅ) at ๐ฅ0
limโ๐ฅ๐โ0
EE๐(๐ฑ0, โ๐ฅ๐) =
๐๐ฆ
๐๐ฅ๐|๐ฑ0
Elementary Effect & Elasticity
Problem: Inputs have different unit?
The unit of EE depends on that of ๐ฅ๐, since it is the percentage variations
around the base case of each parameter on the horizontal axis.
โข Elasticityโ normalizing the EE
๐ ๐ =โ๐ฆ
๐ฆ/โ๐ฅ๐๐ฅ๐
=โ๐ฆ
โ๐ฅ๐ยท๐ฅ๐๐ฆ= EE๐ ยท
๐ฅ๐๐ฆ
โข Results depend on the base point ๐ฅ0
Elementary Effect (EE)
EE๐(๐ฑ0, โ๐ฅ๐): =
โ๐ฆ
โ๐ฅ๐=
เตฏ๐(๐ฅ10, โฆ , ๐ฅ๐
0 + โ๐ฅ๐ , โฆ , ๐ฅ๐0) โ ๐(๐ฑ0
โ๐ฅ๐
Idea of Morris method:
Generalize EE approach by varying the base point over the support of X and
averaging the results
EE & Morrisโ method
Assume that the inputs are independent uniformly distributed in
0,1 ๐, i.e. ๐๐ ~๐.๐.๐
๐ 0,1
1. Define a grid of equally spaced (๐ + 1) points in each
dimension:
grid๐ = {0,1
๐,2
๐, โฆ , 1}, ๐ = 1โฆ๐,Grid = grid1 โโฏโ grid๐
2. Select randomly ๐ different points {๐ฑ1, โฆ , ๐ฑ๐ , โฆ , ๐ฑ๐}
in the grid, and compute the EEijin each point and each
dimension.
๐ฑ๐
The mean (๐๐) and the standard deviation (๐๐) of these ๐ ๐ธ๐ธ๐๐
are used as
sensitivity measures of ๐๐.
๐๐ =1
๐
๐=1
๐
EE๐๐, ๐๐
2 =1
๐ โ 1
๐=1
๐
EE๐๐โ ๐๐
2
โข ๐๐ close to 0 means ๐๐ is non-influential
โข a small ๐๐ means ๐๐ has negligible non-linear effects
โข a large ๐๐ means that the parameter has non-linear effects and/or strong
interactions with other inputs
EE & Morris
EE๐๐(๐ฑ๐ , โ) =
๐(๐ฅ1๐, โฆ , ๐ฅ๐
๐+ โ,โฆ , ๐ฅ๐
๐) โ ๐(๐ฑ๐)
โ
โข Local approaches aim at detecting the important input parameters of a model
based on the knowledge of the range of the inputs (no probabilistic setting)
โข Relative low computational cost (model runs):
2๐ + 1 for Tornado diagram
๐๐ for Spider plot
๐(๐ + 1) for Morrisโ method (can be reduced)
โข Morrisโ method (also regarded as semi local,) also provides an idea about
parameters with nonlinear effect and/or interactions with others
Notes
GLOBAL APPROACHES
โข Discrete:
Event and probability tree
Method of Discrete Probabilities
โข Continuous:
Method of moments
Regression-based method
Distribution-based method
โข Inputs and outputs are random variables: ๐ฟ, ๐
โข Focus on determining the influence of inputs to the output uncertainty
distribution ๐๐(๐)
โข ๐๐(๐) contains all relevant information on the uncertainty of the model
response regardless the values of the input
Global approaches
Global vs Local
โข Global approaches account for the whole input variability range
Local approaches account for small variations around nominal values
โข Global approaches account for interactions between the inputs
Local approaches consider variations of one input at a time
Better results at a higher computational expense
Discrete global approaches:
Event & probability tree (1)
The input variability ranges are discretized in levels
Different scenarios are organized in a tree
โข node = uncertain parameter
โข branch = one possible level of the parameter
โข consider all possible combinations
โข scenario = value of the model output obtained in correspondence of the
values of the โbranchesโ
๐10.3 1.71.0
๐20.5 1.51.0
โข Assign discrete probabilities to the branches
o if discrete probability distributions โ straightforward
o if continuous probability distributions โ discretize
o probabilities of the branches are conditional on the values of the
โpreviousโ branches
โข Evaluate the probability of the scenarios y (product of conditional
probabilities)
Uncertainty
Propagation
Discrete global approaches:
Event & probability tree (2)
Discrete global approaches:
Event & probability tree (3)
โข Extract sensitivity information (see sensitivity on the nominal range)
Sensitivity
Analysis
โข It considers interactions between the inputs
โข Problem: combinatorial explosion!
(๐ inputs, ๐ levels โ ๐๐ combinations)
โข To reduce computational cost โ discrete probability method
๐ฆ = ๐(๐ฅ1, ๐ฅ2)
๐ฅ1
๐ฅ2 = 0.5
๐ฅ2 = 1
๐ฅ2 = 1.5
0.3 1.71.0
Discrete global approaches:
Discrete probability method (1)
Consider the simple model ๐ = ๐(๐, ๐)
1. The pdfโs ๐๐ (๐ฅ) and ๐๐(๐ง) are discretized in ๐ and ๐ intervals of amplitude ๐and ๐โ two sets of pairs < ๐ฅ๐ , ๐๐ >,< ๐ง๐, ๐๐ >, ๐ = 1, 2, โฆ , ๐, ๐ = 1, 2, โฆ , ๐
๐๐ = เถฑ๐ฅ๐
๐๐(๐ฅ)๐๐ฅ ๐ฅ๐ =1
๐๐เถฑ๐ฅ๐
๐ฅ๐๐๐๐ฅ ๐ = 1,2, . . . ๐
๐๐ = เถฑ๐ฅ๐
๐๐(๐ง)๐๐ง ๐ง๐ =1
๐๐เถฑ๐ฅ๐
๐ง๐๐๐๐ง ๐ = 1,2, . . . ๐
Exp(ฮป) = exp(1)
0.63 0.23 0.09 0.03 0.02
2. Build the matrix of the ๐ ร ๐ output values:
๐ฆ๐๐ = ๐(๐ฅ๐ , ๐ง๐), ๐ = 1, 2, โฆ , ๐, ๐ = 1, 2, โฆ , ๐ with probabilities ๐๐๐ = ๐๐๐๐
3. Condense the discrete distribution of ๐ฆ as:
๐๐ =๐ฆ๐๐โ๐
๐๐๐ ๐ฆ๐ =1
๐๐
๐ฆ๐๐โ๐๐๐๐๐ฆ๐๐ ๐ = 1,2, . . . , ๐ก โช ๐๐
Discrete global approaches:
Discrete probability method (2)
independence of inputs!
fX
fZ
rij
yijxi
zj
pi
qj
fY
rl yl
(25 values)
(5 values)
Discrete global approaches:
Discrete probability method (3)
โข Combinatorial explosion avoided
โข Info about dependences is lost in condensation process
๐ฆ = ๐ฅ โ ๐ง where ๐ฅ = ๐ โ ๐ and ๐ง = ๐ + ๐
fX
fZ
fW
fX,Z fX,Z,W
(25 values)
(5 values)
(5 values)
(25 values) (5 values)
5 values instead of
25*5 = 125
๐
๐
๐
๐ฅ
๐ง
๐ฅ (condensed)
๐ง (condensed)
๐ฆ
indep OK indep?
GLOBAL APPROACHES
โข Discrete:
Event and probability tree
Method of Discrete Probabilities
โข Continuous:
Method of moments
Regression-based method
Distribution-based method
The procedure consists in drawing from the assigned pdfโs ๐๐ฟ(๐) a sequence of ๐ samples of each of the ๐ input parameters
๐ฑ๐ = (๐ฅ1๐, ๐ฅ2
๐, โฆ , ๐ฅ๐
๐), ๐ = 1โฆ๐ .
In correspondence of each of the ๐ generated vectors of input values, we
evaluate the model and thus obtain a sequence of output values
๐ฆ๐ = ๐(๐ฑ๐), ๐ = 1โฆ๐ .
The sequences ๐๐ , {๐ฆ๐} can be analysed using classical statistical techniques
for uncertainty and sensitivity analysis
Continuous global approaches:
Random sampling (Monte Carlo-based methods) (1)
๐ฆ
๐๐ ๐ฆ :pdf of ๐
โข It covers the space of the input parameters by random sampling from the
corresponding probability distributions
โข It provides a direct estimation of the output uncertainty via the corresponding
distribution ๐๐(๐ฆ)
โข It is an approximated method, but its accuracy can be assessed and improved
by increasing the sample size s.
e.g. the RMSE of approximating mean using MC method is O sโ1/2
Continuous global approaches:
Random sampling (Monte Carlo-based methods) (2)
Global approaches:
The method of moments (1)
Consider the first-order approximation of the two moments of the distribution of
๐: ๐ผ[๐] and ๐[๐] , with
เต๐ฅ๐0 = ๐ผ ๐๐
๐ฆ0 = ๐ ๐๐
Idea of the method of moments:
If the variations of the ๐๐ โs around their mean value are small, the model
function ๐ may be approximated by a Taylor series expansion.
From this approximation, ๐ผ[๐] and ๐[๐] may be estimated.
Taylor expansion of ๐ at ๐๐
๐ฆ = ๐(๐0) +
๐=1
๐
(๐ฅ๐ โ ๐ฅ๐0)
๐๐ฆ
๐๐ฅ๐ ๐0+1
2
๐=1
๐
๐=1
๐
(๐ฅ๐ โ ๐ฅ๐0)( ๐ฅ๐ โ ๐ฅ๐
0)๐2๐ฆ
๐๐ฅ๐๐๐ฅ๐ ๐0+โฏ
Global approaches:
The method of moments (2)
Compute two moments of the distribution of ๐: ๐ผ[๐] and ๐ ๐ , with เต๐ฅ๐0 = ๐ผ ๐๐
๐ฆ0 = ๐ ๐๐
๐ผ[๐] โ ๐ผ ๐ ๐0 +๐=1
๐
๐ผ ๐๐ โ ๐ฅ๐0 ๐๐ฆ
๐๐ฅ๐ ๐0
+1
2
๐=1
๐
๐=1
๐
๐ผ ๐๐ โ ๐ฅ๐0 ๐๐ โ ๐ฅ๐
0 ๐2๐ฆ
๐๐ฅ๐๐๐ฅ๐ ๐0
0 ๐๐๐ฃ[ ๐ฅ๐ , ๐ฅ๐]
โ ๐(๐0) +1
2
๐=1
๐
๐=1
๐
แ๐๐๐ฃ แ๐๐ , ๐๐๐2๐ฆ
๐๐ฅ๐๐๐ฅ๐ ๐0
only if the model is linear, we have ๐ผ[๐] โ ๐ฆ0 = ๐(๐0)
Global approaches:
The method of moments (3)
๐[๐] = ๐ผ[(๐ โ ๐ฆ0)2] โ ๐ผ
๐=1
๐
๐๐ โ ๐ฅ๐0 ๐๐ฆ
๐๐ฅ๐ ๐0
2
Since ฯ๐=1๐ ๐ง๐
2= ฯ๐=1
๐ ๐ง๐2 + 2ฯ๐=1
๐ ฯ๐=๐+1๐ ๐ง๐๐ง๐, we have
๐[๐] โ
๐=1
๐
๐ผ ๐๐ โ ๐ฅ๐0 2 ๐๐ฆ
๐๐ฅ๐ ๐0
2
+ 2
๐=1
๐
๐=๐+1
๐
๐ผ ๐๐ โ ๐ฅ๐0 ๐๐ โ ๐ฅ๐
0 ๐๐ฆ
๐๐ฅ๐ ๐0
๐๐ฆ
๐๐ฅ๐ ๐0
๐[๐๐] ๐๐๐ฃ[ ๐๐ , ๐๐]
So that
๐ ๐ = ๐ผ[ ๐ โ ๐ฆ02]
โ ๐=1
๐
๐ ๐๐๐๐ฆ
๐๐ฅ๐ ๐ฅ0
2
+ 2๐=1
๐
๐=๐+1
๐
๐๐๐ฃ[ ๐ฅ๐ , ๐ฅ๐]๐๐ฆ
๐๐ฅ๐ ๐ฅ0
๐๐ฆ
๐๐ฅ๐ ๐ฅ0
Input independence
Global approaches:
The method of moments (4)
โข Approximation
โข Partial derivative computation
That is
๐ ๐ = ๐ผ[ ๐ โ ๐ฆ02] โ
๐=1
๐
๐ ๐๐๐๐ฆ
๐๐ฅ๐ ๐ฅ0
2
The sensitivity indices based on Taylor expansion is given by
๐๐ ๐๐ = ๐๐ก๐ ๐๐๐๐ฆ
๐๐ฅ๐ ๐ฅ0
Or
๐๐(๐๐) =
๐๐ฆ๐๐ฅ๐
|๐ฑ=๐๐
2
๐๐๐2
๐[๐]
Continuous global approaches:
Pearsonโs Input/output correlation coefficients, 1880s
Idea of correlation coefficients:
If a parameter ๐๐ strongly influences the output, there must be some correlation
between ๐ and ๐๐
๐๐๐,๐ =cov[๐๐ , ๐]
๐๐๐๐๐.
Advantages:
โข the existence of a linear trend can be visually checked via scatterplot
โข ๐๐๐,๐ โ โ1,1 โpositive / negative values indicate the increasing / decreasing trend of
the input/output mapping.
Drawbacks:
โข only measures a linear relationship between ๐๐ and ๐,
โข in case of nonlinearity (or even non monotonic), the correlation coefficient may be
close to zero even if the parameter ๐๐ is influential.
Idea of the regression-based method:
Fitting a linear approximation around the mean values ๐๐ฟ, look at the
proportion of the variance of ๐ induced by ๐๐.
Continuous global approaches:
Regression-based method (1)
๐ = ๐ฝ0 + ๐ฝ1๐1 +โฏ+ ๐ฝ๐๐๐ + ๐
Principle of least squares:
๐ = argmin๐ผ ๐2 = argmin๐ผ ๐ โ
๐=1
๐
๐ฝ๐๐๐ โ ๐ฝ0
2
The Ordinary Least Squares (OLS) solution of ๐
๐ = (๐ฝ0, โฆ , ๐ฝ๐)โค = (๐โค๐)โ1๐โค๐ฒ
where
๐ =1 ๐ฅ1
1 โฏ ๐ฅ๐1
โฎ โฑ โฎ1 ๐ฅ1
๐ โฏ ๐ฅ๐๐
, ๐ฒ =๐ฆ1
โฎ๐ฆ๐
The Standard Regression Coefficients SRC๐ associated with ๐๐ is given as:
SRC๐: = ๐ฝ๐๐๐๐๐๐
Continuous global approaches:
Regression-based method (2)
Under assumptions
โข the input variables are independent,
โข ฯต is independent of the inputs,
the variance decomposition of ๐ is given by
๐[๐] =
๐=1
๐
๐ฝ๐2๐[๐๐] + ๐๐
2.
Thus, ๐ฝ๐2๐[๐๐] is the proportion of the variance of ๐ induced by ๐๐.
โข The sum of SRC๐2 is the proportion of variance explained by the linear
regression. It is linked to the coefficient of determination ๐ 2:
๐ 2: = 1 โ๐๐2
๐๐2 =
๐=1
๐
SRC๐2
โข SRC๐ โ โ1,1 : a value close to 1 indicates that ๐๐ has a major contribution to
the variance of Y , and a value close to 0 implies little influence.
โข When inputs are independent, ๐๐๐,๐ coincides with the SRC๐ .
Continuous global approaches:
Regression-based method (3)
Notes
โข When the model is linear or accurately approximated by a linear model:
โช Importance measures based on Taylor series expansion
โช Input/output correlation coefficients
โช Standard regression coefficients
โข In case of non linear, yet monotonic models the rank transform shall be used:
โข Input/output rank correlation coefficients
โข Standard Rank Regression Coefficients (SRRC)
โข For more general situations, one may consider distribution-based methods.
GLOBAL APPROACHES
โข Discrete:
Event and probability tree
Method of Discrete Probabilities
โข Continuous:
Method of moments
Regression-based method
Distribution-based method
Let us consider the following simple model (2 inputs):
๐ = ๐ ๐1, ๐2To what extent does the variance of ๐ decreases when we fix ๐1 = ๐ฅ1
โ?
๐๐2 ๐เธซ๐1 = ๐ฅ1โ
In general, ๐1 has a distribution of values! โ Expectation
๐ผ๐1 ๐๐2 ๐เธซ๐1
Variance of the output distribution (Variance Decomposition)
๐ ๐ = ๐๐1 ๐ผ๐2 ๐|๐1 + ๐ผ๐1 ๐๐2 ๐เธซ๐1
The lower, the more
important x1 is!
๐1 =๐๐1 ๐ผ๐2 ๐|๐1
๐ ๐
Variance-based Sensitivity index
Global approaches:
Variance decomposition method (1)
โข What is ๐ผ๐2 ๐|๐1 ?
The expected value of the output y, computed with respect to ๐2, when ๐1 is given
๐ผ๐2 ๐|๐1 = ๐ณ2๐ ๐ฅ1, ๐ฅ2 ๐ ๐ฅ2|๐ฅ1 ๐๐ฅ2 โ function of x1
๐ผ๐2 ๐|๐1
x1
โข What is ๐๐1 ๐ผ๐2 ๐|๐1 ?
๐๐1 ๐ผ๐2 ๐|๐1 = เถฑ
๐ณ1
๐ผ๐2 ๐|๐ฅ1 โ ๐ผ[๐]2๐ ๐ฅ1 ๐๐ฅ1
The โexpectedโ (part of) variability of the output ๐ which is due to ๐1 alone
Global approaches:
Variance decomposition method (2)
๐1 =๐๐1 ๐ผ๐2 ๐|๐1
๐ ๐
1. Sample ๐ random values of ๐1 from ๐๐1๐ฅ1 : ๐ฅ1
1, ๐ฅ12, . . . , ๐ฅ1
๐, . . . , ๐ฅ1
๐
2. For each ๐ฅ1๐, ๐ = 1, 2, โฆ , ๐
1) Sample ๐ values of ๐2 from ๐๐2|๐1 ๐ฅ2|๐ฅ1 = ๐ฅ1๐: ๐ฅ2
1, ๐ฅ22, . . . , ๐ฅ2
๐ , . . . , ๐ฅ2๐
2) Uncertainty propagation:
evaluate the ๐ output values ๐ฆ๐๐ = ๐ ๐ฅ1๐, ๐ฅ2
๐ , ๐ = 1,2, . . . , ๐
3) Estimate the expected value:
๐ผ๐2 ๐|๐1 = ๐ฅ1๐โ เท๐ฆโ ๐ฅ1
๐=1
๐โ
๐=1
๐
๐ฆ๐๐
End For ( j )
The double-loop Monte Carlo estimation for ๐1 =๐๐1 ๐ผ๐2 ๐|๐1
๐ ๐
Global approaches:
Variance decomposition method (3)
๐๐1 ๐ฅ1 = เถฑ
๐ฅ2
๐๐1,๐2 ๐ฅ1, ๐ฅ2 ๐๐ฅ2
๐๐2|๐1 ๐ฅ2|๐ฅ1 = ฮค๐๐1,๐2 ๐ฅ1, ๐ฅ2 ๐๐1 ๐ฅ1
๐ผ๐2 ๐|๐ฅ1 = ๐ฅ11 , ๐ผ๐2 ๐|๐ฅ1 = ๐ฅ1
2 , . . . , ๐ผ๐2 ๐|๐ฅ1 = ๐ฅ1๐
3. Evaluate the following quantities:
๐ผ ๐ = ๐ผ๐1 ๐ผ๐2 ๐|๐1 โ 1
๐
๐=1
๐
๐ผ๐2 ๐|๐1 = ๐ฅ1๐=1
๐
๐=1
๐
เท๐ฆโ ๐ฅ1๐= ว๐ฆ
๐๐1 ๐ผ๐2 ๐|๐1 โ 1
๐ โ 1โ
๐=1
๐
เท๐ฆโ ๐ฅ1๐โ ว๐ฆ
2
๐ ๐ฆ โ 1
๐ โ ๐ โ 1โ
๐=1
๐
๐=1
๐
๐ฆ๐๐ โ ว๐ฆ2
โข no hypotheses on the model
โข Problem: computational cost!
(๐ โ ๐ model evaluations โ around 106!)
โข possible interactions considered
โ Sobol indices
Global approaches:
Variance decomposition method (4)
เท๐1 =๐๐1 ๐ผ๐2 ๐|๐1
๐ ๐
Contribution of ๐1 and ๐2(alone and interacting)
๐1 alone ๐2 alone
= only interaction!
Global approaches:
Variance decomposition method (5)
๐ = ๐ ๐1, ๐2, ๐3๐[๐] = ๐1 + ๐2 + ๐3 + ๐12 + ๐13 + ๐23 + ๐123
๐๐= contribution of input ๐๐ alone
๐๐๐= contribution exclusively due to the interaction of ๐๐ and ๐๐
๐123 = contribution exclusively due to the interaction of ๐1 and ๐2 and ๐3
Note that:
๐1 = ๐๐1 ๐ธ๐2,๐3 ๐ฆ|๐ฅ1๐12 = ๐๐1,๐2 ๐ธ๐3 ๐ฆ|๐ฅ1, ๐ฅ2 โ ๐1 โ ๐2
Example:
๐ฆ = ๐ฅ1 + ๐ฅ22 + ๐ฅ1 โ ๐ฅ3
2
๐12 = 0, but notice that ๐๐1,๐2 ๐ผ๐3 ๐|๐ฅ1, ๐ฅ2 โ 0
๐23 = 0๐123 = 0
โข The index ๐๐ corresponds to the fraction of ๐[๐] associated with the individual
contribution of ๐๐.
Global approaches:
Functional ANOVA decomposition (3)
The first-order Sobolโ indices defined as
๐๐ =๐๐๐[๐]
โข The overall contribution of ๐๐ accountting also for its interactions with the
remaining inputs.
โข The index ๐๐๐ is, then, the total fractional contribution of ๐๐ to ๐[๐].
The total-order Sobolโ indices
๐๐๐ =
๐๐ +๐โ ๐
๐
๐๐,๐ +โฏ+ ๐1,2,โฆ,๐
๐[๐]
โข ฯ๐๐ฎ = 1 and ฯ๐=1๐ ๐๐๐ โฅ 1
โข ๐๐๐ = ๐๐ if ๐๐ is NOT involved in interactions with other inputs.
If this occurs for all inputs, then we have that ฯ๐=1๐ ๐๐ = 1
The model response is additive.
โข The above decomposition requires independence assumption among inputs.
For correlated input, the decomposition is not unique any more.
โthe generalized ANOVA decomposition.
โข To study the trend, one may look at the main effect function ๐๐.
โข To study the interactions, one may look at the higher-order Sobol indices
๐๐,๐,โฆ๐.
โข The definition of ๐๐ in is equivalent to the first-order Sobolโ indices ๐๐ when
inputs are mutually independent.
๐๐ =๐[๐ผ[๐|๐๐]]
๐[๐]โก ๐๐ =
๐๐๐[๐]
Notes
Global approaches:
Moment-independent sensitivity measures (1)
Sensitivity measures that consider the outputโs entire distribution function are
called moment-independent measures.
In general, we can consider a discrepancy operator between probability
distributions over the support of ๐, which is evaluated at the marginal-
conditional pair โ๐ and โ๐|๐๐=๐ฅ๐.
For example,
๐(โ๐, โ๐|๐๐=๐ฅ๐) =1
2เถฑ๐ด
|๐๐|๐๐=๐ฅ๐(๐ฆ|๐๐ = ๐ฅ๐) โ ๐๐(๐ฆ)| d๐ฆ.
๐ = ๐1 + ๐2 + 5๐3, ๐๐ ~๐.๐.๐
๐ฉ(0,1)
๐(๐3 = 0) =1
2เถฑ๐ด
|๐๐|๐3=0(๐ฆ) โ ๐๐(๐ฆ)| d๐ฆ = 0.022
The moment-independent ๐ฟ๐ sensitivity index of ๐๐ with respect to output
Y is defined as:
๐ฟ๐: =1
2๐ผ๐๐[เถฑ
๐ด
|๐๐|๐๐(๐ฆ|๐๐) โ ๐๐(๐ฆ)| d๐ฆ]
Global approaches:
Moment-independent sensitivity measures (2)
Double-loop MC Estimation
๐ฟ๐ represents the normalized expected shift in the distribution of ๐ provoked by
๐๐ . ๐ฟ๐ is, historically, the first moment-independent sensitivity measure.
Properties of the ๐ฟ๐ importance measure:
โข Normalization: 0 โค ๐ฟ๐ โค 1.
โข ๐ฟ1,2,โฆ,๐ = 1
โข Nullity implies independence: ๐ฟ๐ = 0 if and only if ๐ is independent of ๐๐. (Note that ๐๐ do not preserve this property)
โข ๐ฟ๐๐ = ๐ฟ๐ if and only if ๐ is independent of ๐๐
โข Invariant to transformation of ๐: ๐ฟ๐ ๐ = ๐ฟ๐ ๐ ๐
โข Sobolโ indices do not require particular assumptions on the model structure
โข require independence among inputs
โข not transformation-invariant to the output
โข a null value of the first-order sensitivity measure does not implies that ๐ is
independent of ๐๐โข Moment-independent measures solve the above problem
e.g. the moment-independent measure based on Kolmogorov-Smirnov
distance
๐ฝ๐พ๐๐= ๐ผ sup
๐ฆโ๐ด|๐น๐|๐๐(๐ฆ|๐๐) โ ๐น๐(๐ฆ)|
โข Better results at a higher computational expense
Computational cost of double-loop MC estimation: ๐ โ ๐ โ ๐ ,
which can be improved
Notes
MODEL (STRUCTURE)
UNCERTAINTY
โข The alternative model approach
โข The perturbative treatment of the reference model
Model (structure) uncertainty:
The alternative model approach
Consider a set of ๐ plausible models ๐๐, ๐ = 1,2, โฆ ,๐ , each characterized by a
set of uncertain parameters ๐๐ with distribution (๐๐| ๐๐)
๐ฆ๐ ๐ฑ, ๐๐ = ๐๐ ๐ฑ, ๐๐
๐ฆ๐ ๐ฑ = เถฑ
๐๐
๐๐ ๐ฑ, ๐๐ ๐ ๐๐|๐๐ ๐๐๐
Model uncertainty quantified by a discrete probability distribution
๐(๐๐), ๐ = 1, 2, โฆ ,๐
Output ๐ฆ estimation โ total probability theorem
๐ฆ ๐ฑ =
๐=1
๐
๐ ๐๐ เถฑ
๐๐
๐๐ ๐ฑ, ๐๐ ๐ ๐๐|๐๐ ๐๐๐
Model (structure) uncertainty:
The perturbative treatment of the reference model
โข One single model ๐โ is considered, typically the most plausible, and a
perturbation is directly introduced in the model output
โข The perturbation is a sort of error term with which one accounts for the
structural uncertainties due to the incomplete knowledge of the
phenomenon
โข In practice, one adopts additional or multiplicative perturbative terms
๐ฆ = ๐โ(๐, ๐โ) + ๐ท๐โ ๐ฆ = ๐โ ๐, ๐โ ๐ท๐
โ
โข The perturbations ๐ท๐โ and ๐ท๐
โ are not exactly known, but they may be
described in terms of opportune probability distributions
Summary
1. Local approaches focus on sensitivities at nominal range
EE โ Morris
2. Global/probabilistic approaches studies the entire input support
โช Linear/ approximately linear
Importance factors based on Taylor series expansions
Input/output correlation coefficients
Standard regression coefficients
โช Non linear, yet monotonic modelsโ rank transform
โช More general assumptions on the model
variance-based, moment-independent based sensitivities
3. Model (structure) uncertainty