UQ & Sensitivity analysisfor correlated variables

Anne Eggels

CASA dayApril 19, 2017

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Uncorrelated data

For uncorrelated data,

Independent variables ⇒ stochastic collocation

Number of nodes k grows exponentiallywith number of variables p.

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Wind and wave data

Our data is correlated!

Dependent variables⇒ clustering-based collocation (CBC)

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ResultI The method was tested by integrating test

functions over different data sets.

I Higher correlation leads to smaller error.

Example with 50 nodes and 105

data points

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What now?

Collocation points and weights are known.

Output can be computed.

What to do with the output?

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Goals of sensitivity analysis (1/2)

Increased understanding of the relationships betweeninput and output variables in a system or model

I Which input variables influence the output and howmuch?

I How to detect/quantify correlations between inputvariables?

I Causality

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Goals of sensitivity analysis (2/2)

Improved uncertainty analysis to better estimate theoutput probability distribution function

I More weight to more important variables

I What to do with groups of correlated variables?

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Example - x,y ∼ U[0, 1], f(x,y) = f(x) = x2

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Adaptive CBC

I Apply CBC with a fraction of the computationalbudget

I Apply SA and determine important variables

I Re-apply CBC with the rest of the budget on onlythe important variables

I Iterate?

Remark: if iterations are used, then variables can notbe added again.

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Example for SA - minimum spanning trees

L = 0.0995 L = 3.3695

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Reference distribution

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Method for SA - minimum spanning trees

I Length of the minimum spanning tree is an inversemeasure for correlation

I Include the weights of the clusters

I Compare length to reference length

I Use only “important” variables in the collocationmethod

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In practice, ...

I we would like to compute the MST on the completedataset

I output data is far from complete

Solution:

I Interpolate the output data

I Use the previous method based on CBC forinput/output-relations

I Use a multilevel method (with corrections) forinput/input-relations

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Mutual information

I Measure of mutual dependence between tworandom variables

I “Amount of information” obtained about onerandom variable by knowing the other

I I(X,Y) = HS(X) +HS(Y)−HS(X,Y)

I HS(·) Shannon entropy (measure of information /unpredictability)

I Easy for distributions, hard for datasets

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Minimum spanning trees and mutualdependence

I α-entropy:

Hα(f) =1

1− αlog

(∫Ωfα(ω)dω

)for α ∈ (0, 1)

I Length of MST converges (within a constant) to the1/2-entropy!

log(

LNβ√N

)→ H1/2

I Hα(X) = Hα(Y) for all X,Y =⇒ Hα(X,Y) measure fordependence

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How to handle groups of important variables?

Clique idea (future work)

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Testcase - wind op zee

Dataset: OWEZ (Offshore Windpark Egmond aan Zee)

Model: wake model for Horns Rev

Real input does not match restrictionsof model on input data!

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Current and future work

I How to handle groups of important variables?(clique idea)

I How to determine which variables are importantenough?

I How to allocate simulation budget to first andsecond collocation?

I Real-life tests

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Conclusions

I We developed a new method for sensitivity analysiswhich works for correlated variables.

I The method has a theoretical basis, but detailshave to be worked out.

I When combined with uncertainty analysis, this canlead to more insight in complex problems.

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Questions?

This research is supported by the Dutch Technology Foundation STW, whichis part of the Netherlands Organisation for Scientific Research (NWO), and

which is partly funded by the Ministry of Economic Affairs.