ASK-BATAN_AnalisiSensitifitas&Ketidakpastian

8/14/2019 ASK-BATAN_AnalisiSensitifitas&Ketidakpastian

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Paper presented at CCCNS 2008 LIPI (to be publish on the related proceeding)

LATIN HYPERCUBE SAMPLING FOR UNCERTAINTY ANALYSIS

Nursinta A.Wahanani, Anik Purwaningsih, Topan Setiadipura

Computational Division, PPIN-BATANemail: [email protected]

Abstract

The assessment and presentation of the effects of uncertainty are now widely recognized as

important parts of anlysis for complex system. Thus, uncertainty analysis is an important tools

for a wide area of application, from the nuclear reactor waste analysis to economic calculation.

Sampling-based method is one of the powerful method for the uncertainty analysis. There are

several steps needed to run an uncertainty analysis,(i) construction of distribution to characterize

subjective uncertainty, (ii) sampling procedures, (iii) propagation of uncertainty through model,

(iv) display of uncertainty in model prediction. In this paper, development of the Latin

Hypercube Sampling (LHS) module at PPIN-BATAN as the sampling procedure in the

uncertainty analysis is reported. The LHS module is able to demonstrate the stability of the LHS

method in estimating the cumulative density function. This development is a platform for further

advance sampling analysis method, and a step to build an in-house complete tool of uncertainty

analysis. Review of the LHS method and its superior to standard random sampling and

importance sampling is also discussed.

Keywords : sampling-based uncertainty analysis, Latin Hypercube Sampling.

INTRODUCTION

The assessment and presentation of the effects of uncertainty are now widely recognized as

important parts of anlysis for complex system1. Thus, uncertainty analysis is an important tools

for a wide area of application, from the nuclear reactor waste analysis to economic calculation.

Specifically, uncertainty analysis refers to the determination of the uncertainty in analysis resultsthat derives from uncertainty in analysis inputs, slightliy different from sensitivity analysis

which is refers to the determination of the contributions of individual uncertain inputs to the

uncertainty in analysis results. The uncertainty under consideration here is often reffered to as

epistemic uncertainty which derives from a lack of knowledge about the appropriate value to use

for a quantity that is assumed to hace a fixed value in the context of a particular analysis. This

uncertainty, in the conceptual and computational organization of an analysis, is generally

considered to be distinct from aleatory uncertainty which arises from an inherent randomness in

the behaviour of the system under study.2

The underlying idea is that analysis results y(x) = [ y1(x), y2(x), ., ynY (x)] are functions of uncer-

tain analysis inputs x = [ x1, x2, ., xnX ]. In turn, uncertainty in x results in a corresponding uncer-

tainty in y(x). This leads to two questions: (i) What is the uncertainty in y(x) given the uncer-tainty in x?, and (ii) How important are the individual elements of x with respect to the uncer-

tainty in y(x)? The goal of uncertainty analysis is to answer the first question, and the goal of

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sensitivity analysis is to answer the second questions. In practice, the implementation of an un-

certainty analysis and the implementation of a sensitivity analysis are very closely connected on

both a conceptual and a computational level3. The illustration of the above problem is depicted

in the figure.1

Figure1. Illustration of uncertainty analysis of the output which arises from uncertainty of the

input4.

There are several steps for the sampling-based uncertainty analysis, as will be mentioned in the

next section, this paper focus on the sampling phase. The objective of the research is to develop

a sampling module using latin hypercube method as part of the uncertainty analysis tools, and to

demonstrate the stability of the latin hypercube sampling estimate compare to standard random

sampling. This development is a platform for further advance sampling analysis method, and a

step to build an in-house complete tool of uncertainty analysis. Review of the LHS method and

its superior to standard random sampling and importance sampling is also discussed.

Sampling-based uncertainty Analysis

Sampling-based methods are based on the use of a probabilistic procedure to select model input

and result in a mapping between analysis inputs and analysis outcomes that is then used to pro-

duce uncertainty and sensitivity analysis results. Desirable features of Monte Carlo analysis in-

clude (i) extensive sampling from the ranges of the uncertain variables, (ii) uncertainty results

that are obtained without the use of surrogate models (e.g., Taylor series in differential analysis

and response surfaces in RSM), (iii) extensive modifications of, or manipulations with, the ori-

ginal model are not required (Le., as is the case for the other techniques), (iv) the extensive

sampling from the individual variables facilitates the identification of nonlinearities, thresholds

and discontinuities, (v) a variety of sensitivity analysis procedures are available, and (vi) the ap-

proach is conceptually simple, widely used, and easy to explain. The major drawback is compu-

tational cost. This is especially the case if long-running models are under consideration or prob-

abilities very close to zero or one must be estimated1.

There are five basic components that underlie the implementation of a sampling-based uncer-tainty and sensitivity analysis: (i) Definition of distributions D1, D2, ., DnX that characterize the

uncertainty in the components x1, x2, ., xnX of x, (ii) Generation of a sample x1, x2, ., xnS fro the

x.s in consistency with the distributions D1, D2, ., DnX , (iii) Propagation of the sample through

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the analysis to produce a mapping [xi, y(xi)], i = 1, 2, ., nS , from analysis inputs to analysis res-

ults, (iv) Presentation of uncertainty analysis results (i.e.,approximations to the distributions of

the elements of y constructed from the corresponding elements of y(xi), i = 1, 2, ., nS ), and (v)

Determination of sensitivity analysis results (i.e., exploration of the mapping [xi, y(xi)], i = 1, 2,

., nS )3. This paper is limited to the second phase, generation of sample, particularly using latin

hypercube sampling.

Some type of sampling procedure are used to generate the sample in Monte Carlo analysis, such

as Simple Random sampling , Stratified sampling and Latin Hypercube sampling. With random

sampling, there is no assurance that a sample element will be generated from any particular

subset of the sample space. In particular, important subsets of sample space with low probability

but high consequences are likely to be missed. Stratified sampling has the advantage of forcing

the inclusion of specified subsets of sample space while maintaining the probabilistic character

of random sampling but there is a problem with stratified sampling, the necessity of defining

the strata and calculating their probabilities. when the dimensionality of sample space is high,the determination of strata and strata probabilities become a major undertaking. The

determinations are further complicated when many analysis outcomes are under consideration, in

particular, strata definitions that are appropriate for one analysis outcome may be inappropriate

for other analysis outcomes. Latin hypercube sampling is based on a combination of simple

random sampling and stratified sampling techniques that leads to statistically significant results

with substantially fewer realizations1. Latin hypercube sampling (LHS) displays properties

between simple random sampling, which involves no stratification, and stratified sampling,

which stratifies on sample space.

The basis of LHS is a full stratification of the sampled distribution with a random selection

inside each stratum, sample values are randomly shuffled among different variables. To generate

probability distributions, the LHS was performed using the following steps: (i) Assign an inverse

cumulative distribution function (cdf) for each input variable, (ii) choose the number of

simulations (N) to be performed, (iii) divide the cdf for each variable into N equi-probable

intervals, (iv) for each interval, choose a random sample, x, from the inverse cdf and develop a

data set for each parameter, (v) randomly select from each parameter input data set to create N

model input sets, (vi) use an analytical or numerical model to determine a realization for each

model input set. The current development accomodate uniform, triangular, and normaldistribution which are commonly used in practice5. The particular different of the LHS and

standard random sampling is in the third step of the above steps.

Development Results and Analysis

A software of the LHS and standard random sampling based on the above mentioned procedure

is developed as a stand alone software using JAVA programming language. Result of the

sampling is tested to produce an estimate of a monoton simple function which also demonstrate

the powerfull of LHS method to SRS method. For the normal distribution the inputs needed is

the sample number, mean, and standard deviation. The triangular distribution using four

parameter which are the sample number, minimum value, modus, and maximum value. Theuniform distribution need three parameter the sample number,minimum, dan maximum value.

For the purpose of comparing LHS and SRS sampling, simple monotonic function y(x1 ,x2 ) = x1

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+ x2, where x1 and x2 assumed to be uncorrelated. The repeated estimate for the cumulative

density function of the y(x1 ,x2 ) from both sampling method are compared.

Figure.2. Comparation of SRS (left) and LHS (right) where x1(100,3,0.1) and x2(100,5,0.2) arenormal distribution.

Figure.3. Comparation of SRS (left) and LHS (right) where x1 and x2 are triangular distribution.

Figure.4. Comparation of SRS (left) and LHS (right) where x1 and x2 are uniform distribution.

To check the combination of different input distribution another simple monotonic function y(x1 , x2, x3 ) = x1+ x2+ x3 to accommodate the three distribution involved. The result of the cdf of the y

is depicted in figure 5.

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Figure.5. Comparation of SRS (left) and LHS (right) where x1 ,x2 , and x3 are Triangular

(100,2.5,6), Uniform (100,2,6), and Normal (100,7,0.1) distribusion, respectively.

The above results show that the LHS method produce more stable estimate for the cdf than is

being produced by SRS method. The stability of the LHS estimate over the SRS estimate is

produce from the normal, triangular, and uniform distribution of the inpur parameter. Even if the

function is involving all the distribution, the LHS method still more stable than the SRS. This

well behaviour of the LHS method might come from the full stratification of the method which

assure that the sample element is generated from all the subset sample space.

Conclusion

A sampling module based on the latin hypercube sampling method is already developed (beside

the standard random sampling) at PPIN-BATAN. The module able to demonstrate the powerfull

behaviour of the LHS where the estimate for the cumulative density function (cdf) is more stable

than is being produced by standard random sampling. The module is a platform for further

advance sampling analysis method, and a step to build an in-house complete tool of uncertainty

analysis.

Aknowledgement

Authors are very gratefull to Dr.Syahril at BATAN for trigerring the uncertainty problem at

computational division, particularly for the techno-economic problem. Also to Mrs.Khairina,

chair of the Computational Division for its support in the development procceses.

Refference

1. Helton J.C, Davis F.J, 2002, “ Latin Hypercube Sampling and the Propagation of

Uncertainty in Analyses of Complex System”, Sandia Report SAND2001-0417.

2. Helton J.C, Johnson J.D, Sallaberry C.J, Storlie C.B, 2006, “ Survey of Sampling Based

Method for Uncertainty and Sensitivity Analysis”, Sandia Report SAND2006-2901.

3. Helton J.C, 2005, “ Sampling-Based Methods for Uncertainty and Sensitivity Analysis”,

Sensitivity Analysis of Model Output, Kenneth M.H, Hernez F.M (eds), Los Alamos Na-

tional Laboratory.

4. Tarantula S, 2008, “Global Sensitivity Analysis”, JRC-EC

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5. Swiller L.P, Wyss G.D, 2004, “ A User’s Guide to Sandia’s Latin Hypercube Sampling

Software : LHS Unix Library/Stand Alone Version”, Sandia Report SAND2004-2439.

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