Text

Summarizes Response Surface Research Review Group Meeting on 781121.Reviews Preliminary Results of Sandias Relap Blowdown Calculations & Progress Toward Developing a Response Surface for Peak Clad Temp in Blowdown
	ML19269B882
Person / Time
Issue date:	01/10/1979
From:	Clay Johnson; NRC OFFICE OF NUCLEAR REGULATORY RESEARCH (RES)
To:	Tong L; NRC OFFICE OF NUCLEAR REGULATORY RESEARCH (RES)
References
	NUDOCS 7901170329
	Download: ML19269B882 (76)
	v • d • e

SAAfog, Do UNITED STATES

/o NUCLEAR REGULATORY COMMisslON

.E WASHINGTON, D. C. 20555 JAN I 01979 MEMORANDUM FOR:

L. S. Tong, Assistant Director for Water Reactor Safety Research FROM:

Carl E. Johnson, Technical Assistant Office of Water Reactor Safety Research

SUBJECT:

RESPONSE SURFACE RESEARCH REVIEW GROUP MEETING

SUMMARY

On November 21, 1978, the Response Surface Research Review Group met at the Willste Building in Silver Spring.

The purpose of the meeting was to review preliminary results of Sandia's RELAP blowdown calculations and progress toward developing a response surface for peak clad temperature in blowdown.

The agenda and attendance at the meeting are listed in Enclosures 1 and 2.

The discussion is summarized below.

Description of LOCA Statistical Analysis Project The background and status of NRC's statistical analysis project at Sandia was described including objectives, historical background, plan, and selection of input variables and their distribution (Enclosure 3).

During the discussion, individuals offered the following comments on input variables.

- Check the range of U0 conductivity to be sure the study covers the 2

low values expected after irradiation.

- Review carefully the distributions of the variables that are determined to be important.

- The input distributions should be best-estimate, not conservative.

- Consider as additional input variables the following initial conditions:

Coolant inlet temperature in steady state Plugged steam generator tubes RELAP Blowdown Calculationn Preliminary results of Sandia's initial RELAP calculations for blowdown were described by M. Berman and G. Steck, Sandia (Enclosure 4).

79011703M

L. S. Tong J2' 1 c F" During the discussion, individuals raised the following comments:

- How about running a RELAP case with values for each input variable selected to give the highest clad temperature possible within the overall range of input variables?

- The pressure in the fuel / clad gap looks high.

Suggest we check gap pressure vs_ values given in a forthcoming INEL report by J. Kerrigan.

- Recheck whether PCT for fresh fuel is hotter than for once burned fuel at off-nominal values of input variables leading to higher-than-nominal peak clad temperatures.

What reference temperature does RELAP use to calculate stored energy?

Is this the same as FRAP uses?

- The Sandia analysis models only one vendor's plant. Other plants may behave differently.

Refill " Bridge" Calculation A simplified method to estimate cladding temperatures at the beginning of reflood was described by S. Margolis, INEL. The method is based on semiscale data.

In test calculations by C. Johnson,'NRC (Enclosure 5) and by T. Bartel, Sandia (Enclosure 6), the simplified method agreed with other methods for calculating PWR cladding temperatures near the end of refill typically with 100*F.

The discussion brought out the following comments:

- How about incorporating some improvements, such as (1) tie the time reference, T(20), to physical events; and (2) consider differences between UO rods and electric heater rod, for example the large AT 2

from the surface to the centerline for U0 rods.

2

- How about evaluating the bridge calculation method (which is based on semiscale data) for application to full size PWRs?

Is the semiscale suppression-tank pressure typical of the values at containment pressure used in the Sandia study?

- How about a report that describes the rationale, limitations, and uncertainties of using the " bridge" to estimate initial clad temperatures for RELAP best-estimate reflood calculations on PWRs?

--=m-.4---

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L. S. Tong J M 1 0 7979

- How should one estimate initial conditions at beginning of reflood for parameters other than cladding temperature (i.e., U0 temperatures 2

and pressures in primary and secondary systems, etc.)?

Planned Use of Response Surface to Analyze Peak Clad Temperature in LOCA The planned use of Latin Hypercube Sampling was described by G. Steck, Sandia.

The following comments were raised:

- Why use latin hypercube sampling instead of fractional factorial experiment design?

- Why use a relatively new untried method on this study instead of the more conventional factorial approach?

- Response surface methodology implies fractional factorial design.

To use the term " response surface" with latin hypercube sampling implies more credibility than is warranted.

- Fractional Factorial design should give a better first-order response than latin hypercube sampling.

- Latin hypercube sampling allows consideration of more possible terms in the response surface so that the engineer can see which variables ahd cross products the data show to be most important.

(A description of Latin Hypercube sampling is given in enclosure 7, an article by McKay, Conover and Beckman.)

s~E'/ d-Carl E. Jo nson, Technical Assistant Office of Assistant Director for Water Reactor Safety

Enclosures:

as stated cc See next page

cc w/encls 1, 2, & 7: Warren Lyon, NRC Brian Sheron, NRC Robert Van Houten, NRC Lee Abramson, NRC Norman Lauben, NRC H. L. Ornstein, NRC George Marino, NRC Harold Scott, NRC Loren Thompson, NRC B. B. Chu, EPRI William Hunter, U. of Wisconsin Marshall Berman, Sandia George Steck, Sandia Tim Bartel, Sandia R. Byers, Sandia John A.

Dearien,

EG&G Stephen G. Margolis, EG&G Corwin L. Atwood, EG&G P. R. Davis, ITI Agnes Holler, B&W J. H. Holderness, CE Jack A. Marshall, W-Donald Paddleford, W cc w/encls 1, 2, 3, 4, 5, 6, 7 :

PDR (2) "

Carl Johnson (2)

3 RESPONSE SURFACE REVIEW GROUP MEETING Willste Building, Room 150 7315 Eastern Avenue Silver Spring, Maryland November 21, 1978 AGENDA j

8:45 - 9:00 Introduction C. E. Johnson, NRC 9:00 - 10:30 Description of Statistical Analysis C. E. Johnson, NRC Project 10:30 - 10:45 Break 10:45 - 12:00 Preliminary Results of First RELAP M. Berman, Sandia Blowdown Runs and Progress in G. Steck, Sandia Developing Blowdown Response Surface- (Sandia) 12:00 - 1:00 Lunch 1:00 - 1:45 Description of " Refill Bridge" S. Margolis, INEL Correlation for Clad Temperature at End of Refill 1:45 - 2:30 Comparison of Results of " Refill C. E. Johnson, NRC Bridge" Calculations with other Methods 2:30 - 2:45 Break 2:45 -

3:45 Planned use of Response Surface G. Steck, Sandia to Analyze Probability Distribution of Peak Clad Temperature in LOCA 3:45 - 4:45 Discussion 4:45 - 5:00 Summa ry C. E. Johnson, NRC

Attendance Review Group Members Carl Johnson, Chairman Warren Lyon, NRC Brian Sheron, NRC Robert Van Houten, NRC Others Lee Abramson, NRC Norman Lauben, NRC H. L. Ornstein, NRC George Marino, NRC Harold Scott, NRC Loreh Thompson, NRC B. B. Chu, EPRI William Hunter, U. of Wisconsin Marshall Berman, Sandia George Steck, Sandia Tim Bartel, Sandia R. Byers, Sandia John A.

Dearien,

EG&G Stephen G. Margolis, EG&G Corwin L. Atwood, EG&G P. R. Davis, ITI Agnes Holler, B&W J. H. Holderness, CE Jack A. Marshall, W Donald Paddleford, W_

J LOCA STATISTICAL ANALYSIS OBJECTIVE DEVELOP METHOD TO EVALUATE UNCERTAINTY IN BEST-ESTIMATE PREDICTIONS OF PEAK CLAD TEMPERATURE IN LOCA.

S0 THAT EVENTUALLY THE SAFETY MARGIN IN LOCA CAN BE QUANTIFIED.

~

LOCA STATISTICAL ANALYSIS HISTORY e

EPRI/H STUDY PROPOSED RESPONSE SURFACE METHODOLOGY e

NRR/SANDIA/INEL STUDY TESTED FEASIBILITY OF METHODOLOGY FOR CALCULATING SAFETY MARGINS IN LOCA e

INEL DEVELOPED RELAP 11/ MOD 6 e

CURRENT SANDIA STUDY STARTED e

RESPONSE-SURFACE REVIEW GROUP ESTABLISHED

EXPECTED RESULTS:

1.

PCT PROBABILITY DISTRIBUTION 5

I E

I l

l C

E I

u_

E I %

8 o

IN a:

I l

PEAK CLAD TEMPERATURE IN PWR LOCA (OF) e

~

EXPECTED RESULTS:

2.

SENSITIVITY TO IMPORTANT VARIABLES I

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PEAK CLAD TEMPERATURE IN PWR LOCA (OF)

STUDY PLAN STABLISH BASE CA RUN ~ 50 DEVELOP CALCULATE PCT ANGE OF VARIABLES RELAP CALCULATIONS RESPONSE SURFACE PROBABILITY (BLOWDOWN)

(BLOWDOWN)

DISTRIBUTION (BLOWDOWN)

^

(BLOWDOWN) i f PROBABILk RAllGE OF VARIABLES &/

CALCULATE NSITIVITY I

H DISTRIBUTION

)

CALCULATION METHOD "I

OCA

/

(REFILL)

/

RUN-50 DEVELOP CALCULATE PCT RANGE OF VARIABLE,/

RELAP CALCULATIONS RESPONSE SURFACE PROBABILITY 7>

(REFLOOD) blSTRIBUTION (REFLOOD)

(REFLOOD)

9 O

9 INPUT VARIABLES A.

PLANT CONDITIONS B.

FUEL PARAMETERS C.

HYDRAULIC PARAMETERS D.

HEAT-TRANSFER PARAMETERS

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APPLICABILITY

[

BLOWDOWN INPUT VARIABLES

& REFILL REFLOOD NOMINAL VALUE RANGE

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A, PLANT CONDITIONS 1.

TIME IN LIFE X

X 0 -+ 40 YEARS 2.

P]ER X

X 100%

94% -> 106%

3.

PEAKING FACTORS X

X F(T)

(1.47 -> 2.0) 16%

4.

ECC TEMPERATURE X

X 900F 400F -> 1400F O

5.

ACCUMULATOR X

643 PSIA 593 PSI A -> 693 PSI A

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& REFILL REFLOOD NOMINAL VALUE RANGE B.

FUEL PARAMETERS U0 THERMAL X

X MATPRO (0.6 +1.3) x NOMINAL 1.

7 CONDUCTIVITY 2.

FUEL / CLAD DIAMETRAL X

X F(T)

(0.3 MILS ->7.6 MIL 9 3 MILS GAP 3.

DECAY HEAT X

X F(T)x(NEW ANS)

(0.94+1.2) x NOMINAL ZR-H O REACTION X

X CATHCART.PAWEL 0.85(CATHCART PAWEL)-*(BAKER 4.

2 JUST) 5.

FLOW BLOCKAGE X

X RELAP (0.4-+1.6) x NOMINAL

RANGE OF VARIABLES APPLICABILITY BLOWDOWN I INPUT VARIABLES

& REFILL REFLOOD NOMINAL VALUE RANGE C.

HYDRAULIC PARAMETERS 0.9 0.7+].2 SUBC00 LED X

1.

CD 1.0 0.75+2.5 SATURATED X

2.

CD NOMINAL 110 PSIA 3.

CONTAINMENT X

X 44 PSIA PEAK PRESSURE 4.

TWO-PHASE PRESSURE X

X RELAP (0.4+1.6) x NOMINAL DROP RELAP (0.6 +1.0) x NOMINAL 5.

PUMP DEGRADATION X

RELAP

0. 3 +6.0 6.

SLIP X

70%

30%+100%

7.

WATER SWEPT OUT OF X

LOWER PLENUM DURING BLOWDOWN I

RANGE OF VARIABLES APPLICABILITY BLOWDOWN INPUT VARIABLES

& REFILL REFLOUD NOMINAL VALUE RANGE C,

HYDRAULIC PARAMETERS (CONTINUED) 40%

10-7100%

8.

ECC BYPASS X

X RELAP WCI:

0.54>1.0 9.

COUNTER CURRENT WC2:

0. 41"> 0. 65 '

FLOW LIMIT X

RELAP EN 2:

0.34r0.9 10.

CORE ENTRAINMENT X

RELAP 11.

UPPER PLENUM CARRY 0VER 4

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RANGE OF VARIABLES APPLICABILITY BLOWDOWN INPUT VARIABLES

& REFILL REFLOOD NOMINAL VALUE RANGE D.

HEAT TRANSFER PARAMETERS HSU (0.3+3.0) x NOMINAL 1.

DNB CORRELATION X

2.

TRANSITION & FILM X

X CONDIE BENGSTON (0,5+2.0) x NOMINAL BOILING, HIGH G 3.

TRANSITION B0ILING, X

X MODIFIED HSU VARIES WITH DNB LOW G li,

FILM B0ILING, LOW G X

X BROMLEY P0MERANZ (0.5+2,0) x NOMINAL 5,

FORCED CONVECTION X

X DITTUS B0ELTER (0,5+2.0) x NOMINAL 6.

FREE CONVECTION &

X RELAP (0.6+1.5) x NOMINAL RADIATION 7.

STEAM GENERATOR X

RELAP (0,5+2,0) x NOMINAL HEAT TRANSFER 8.

ENERGY PARTITION X

RELAP (0-> 1,0) x NOMINAL

STATISTICAL L0CA G0AL-DEVELOPMETHODSTOESTIMAT$PCTPROBABILITY DISTRIBUTION CALCULATED BY RELAP FROM INPUT DISTRIBUTIONS.

APPLICATIONS:-

1.

QUANTIFY THE CONSERVATISM 0F THE REQUIREMENTS

. OF 10 CFR 50 APPENDIX K.

2.

DETERMINE PCT SEWSITIVITY TO INPUT PARAMETERS AND THEIR VARIATIONS.

3.

APPLY METHODOLOGY TO OTHER SIMILAR PROBLEMS WHERE SAMPLE SIZES ARE SMALL.

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STATISTICAL L0CA FY78/79 G0AL - F3TIMATE. PCT PROBABILITY DISTRIBUTION FOR. BLOWDOWN AND REFLOOD FOR ZION PWR USING RELAP4/ MOD 6, i

PROGRAM:

1.

IMPLEMENT MOD 6/UPD3 ON SANDIA 7600.

2.

CHECK BLOWDOWN DIALS, 3.

SENSITIVITY STUDIES INCLUDING INITIALIZATION, 4.

AUT0i1 ATE BD INPUT, 5.

MAKE RUNS, BUILD BD RESPONSE SURFACE

6.

GENERATE REFLOOD INPUT USING " BRIDGE."

7.

REPEAT 2-5 FOR REFLOOD, 8.

GENERATE PCT PROBABILITY DISTRIBUTION,

4 e

0 9

SENSITIVITY STUDIES 1.

MOD 5 VS MOD 6 HYDRAULICS 2.

MOD 5 VS MOD 6 HEAT TRANSFER 3.'

JUNCTION INERTIAS 4.

ENTHALPY TRANSPORT 5.

STEADY STATE STUDIES 6.

DIAL CHECK 0UT - PART OF-MAIN STUDY 1

EFFECT OF HT SURFACE MOD ON PCT HEAT TRANSFER ACCUMULAT08 RELAP SURFACE ECJ_

IIE ON MOD 5 HTRC 1 2112 5.3 12.3 MOD 6 HTRC 1223 13.3 12.7 MOD 6 HTS 2 1173 9.1 12,7 e

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RELAP4 NODALIZATioN FOR BE/EM STUDY Q [*" PB ADDITIONAL 6S I

PRESSURIZER M

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TilERMAL BALAtlCE RUNS STEADY RUN STATE TilERMAL SLAB 15 SLAB 16 AT NAME POWER PERIOD BALANCE?

CORE M

WH M

MAX TEMP

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6.50 1084 20.01 YES 59.30 80.01 YES 59.30 6.70 1076*

6.51 LO71 BDST-05E 106%

.01 YES 62.87 6.63 1134 6.52 1128 06B

.01 NO 59.30 6.63 1132 6.50 1125 20.01 YES 62.87 05F 80.01 YES 62.87 6.69 1126 6.51 1116 06C N0 59.30 7.56 1116 7.53 1104 07D 94%

.01 YES 55.73 6.61 1040 6.50 1037 08A

.01 N0 59.30 6.64,

1045 6.49 1041 20.01-YES 55.73 07E 80.01 YES 55.73 6.50 1033 6.50 1032 08B NO 59.30 -

5.79 1041 5.47 1040 4

Proprocennor Input Parametera - Summary Nominal Parameter Range Value 1.

DLEURY = subcooled discharge 0.7 + 1.2 0.9 coefficient 2.

DL!lEM = raturated discharge

-0.25 + 1.0 0.

coefficient 3.

SLIP = slip correlation dial

- 1. -* 1.

O.

4.

DLTF = 2-phase form loss dial 0.4 + 1.6 1.0 DLTTFM = 2-phase fanning friction loss dial These dials are assu.acd to be equal, and a singic variable 5.

DCHF = critical heat flux dial 0.3 + 3.0 1.0 6.

DHTC6 = Condic-Dengston dial 0.5 + 2.0 1.0 t

7.

DHTC7 o frce convection and 0.6 + 1.5 1.0 radiation dial 8.

DHTC8 = Dittus-Boelter dial 0.5 + 2.0 1.0 9.

DHTC9 = !!su and Bromlcy-Pomeranz 0.7 + 1.5 1.0 dial 10.

DLBLK = flow blockage dial 0.4 + 1.6 1.0 multiplier 11.

DLMWR = multiplier of Cathcart-0.85 -* 1.15 1.0 Pawel reaction rates 12.

DLPWR = powcr level multiplier 0.94 + 1.06 1.0

13..DLCPR = increment to bw added

-5. + 10. psia 0.

to containment pri.ssure 14.

DLPUMP = dial for 2-phase pump

-1. + 1.

O.

head multiplier 15.

ECCTMP = temperature of accumulator

40. + 140 F 900F 0

and safety injection system water 16.

DLACC = accumulator pressurc 593.2 + 693.2 psia 643.2 psia 17.

TLF = time in fe 0 + 440 months 226 months 18.

PFUNC = peaking factor uncertainty 84 + 1.16 1.0 multiplier 19.

DLECON = thermal conductivity dial

.6 + 1.3 1.0 multiplier 20.

DLGAP = additive. uncertainty in

l.5 mils 0.

radial gap si:c NOB = 0 - fresh fuel

= 1 + once burned fuel 21.

DLDEC = decay heat multiplier

.06 + 1.0 0.

n-

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BD5 SERIES (CONT.)

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REFILL BRIDGE CALCULATION COMPARED WITH OTHER ANALYSES BRIDGE-ESTIMATED BRIDGE TEMPERATURE AT ANALYSI.S ERROR END OF REFILL FRAP-T5 1231 F

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Enclosu.,?,

LA-UR Si Tn'LE: A COMPARISON OF THREE METHODS FOR SELECTING VALUES OF INPUT VARIABLES IN THE ANALYSIS OF OUTPUT FROM A COMPUTER CODE AUTHOR (S): M.D. McKay W.J. Conover R.J., Becks..

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whole or in part said article under any copyright secured by the publisher.

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publisher identify this article as woric performed under the suspices of the USERDA.

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UNITED STATES

,q. s., gm ENERGY RESEARCH AND

,;3 ins DEVE!.OPMENT ADMINISTRATION CONTRACT W.7405.ENG. 36

. s A COMPARISON OF THREE METHODS FOR SEIECTING VALUES OF INPUT VARIABLES IN IHE ANALYSIS OF OUTPUT FROM A COMPUTER CODE M. D. McKay W. J. Conover Los Alamos Scientific Laboratory Texas Tech University Los Alamos, New Mexico Lubbock, Texas

~

R. J. Beckman Lc3 Alamos Scientific Laboratory Los Alamos, New Mexico ABSTRACT Two types-of sampling plans are ernmined as alternatives to simple a

random sampling in Monte Carlo studies. These plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution fune: ion.

KEY WORDS:

LATIN HYPERCUBE SAMPLING SAMPLING TECHNIQUES SIMULATION TECHNIQUES VARIANCE REDUCTION

2 1.

INTRODUCTION Numerical methods have been used for years to provide approximate solutions to fluid flow problems that defy analytical solutions because of their complexity. A mathematical model is constructed to resemble the fluid flow problem, and a computer program (called a " code"), in-corporating methods of obtaining a numerical solution, is written. Then for any selection of input variables X = (X

,...,X ) an output variable 7

g Y = h(X,) is produced by the computer code.

If the code is accurate the output Y resembles what the actual output would be if an experi-ment were performed under the conditions X.

It is of ten impractical or impossible to perform such an experiment. Moreover, the computer codes are somatimes sufficiently complex so that a single set of input variables may require'several hours of time on the fastest computers' presently in

~

existedce~in order to produce one output.

We should mention that a

'~~

single output Y is usually a graph Y(t) of output as a function of time, calculated at discrete time points t,

t0<t<ty.

When modeling real world phenomena with a computer code one is of ten faced with the problem of what values to use for the inputs.

This diffi-culty can arise from within the physical process itself when system par = meters are not constant, but vary in scme manner about nominal values.

We model our uncertainty about the values of the inputs by treating them as random variables. The information desired from the code can be obtained from a study of the probability distribution of the output Y(t).

Con-sequently, we model the " numerical" expertnent by Y(t) as an unknown trans-formationh(X)oftheinputsX,,whichhaveaknownprobabilitydistribution F(x) for x c S.

3 Obviously several values of X,

say X

,...,X

, cust be selected as 7

g successive inputs sets in order to obtain the desired information con-cerning Y(c). When N must be small because of the running tLae of the code, the input variables should be selected with g: eat care.

The next section describes three methods of selecting (sampling) input variables. Sections 3, 4 and 5 are devoted to comparing the three methode with respect to their performance in an actual computer code.

The computer code used in this paper was developed in the Hydro-dynamics Group of the Theoretical Division at the Los Alamos Scientific Laboratory, to study reactor safety (8]. The computer code is named SOLA-PLCOP and is a one-dimensional version of another code SOLA (7].

2 The code was used by us to model the blowdown depressurization of r..

straight pipe filled with water at fixed initial temperature and pressure.

Input variables include:

X ' P ase change rate; X, drag coefficient h

1 2

for drif t velocity; X, number of bubbles per unit volume; and X. Pipe 3

4 roughness.

The input variables are assumed to be uniformly distributed over given ranges.

The output variable is pressure as a function of time, where the initial time t is the time the pipe ruptures and de-0 pressurization initiates, and the final time e is 20 milliseconds later.

y The pressure is recorded at 0.1 millisecond time intervals.

The code was used repeatedly so that the accuracy and precision of the three sampling methods could be compared.

4 2.

A DESCRIPTION OF THE THREE METHODS USED FOR SELECTING THE VALUES OF INPUT VARIABLES From the many different methods of selecting the values of input variables, we have chosen three that have considerable intuitive appeal.

These are called Random Sampling, Stratified Sampling, and Latin Hypercube Sampling.

Random Sampling.

Let the input values g,...,g be a random sample from F(x).

This method of sampling is perhaps the most obvious, and an entire body of statistical literature may be used in making inferences regarding the distribution of Y(t).

Stratified Sampling. Using stratified sampling, all areas of the sample space.of X are represented by input values. Let the sample space 2

S of X be partitioned into I disjoint strata S.

Let p = P(X c S )

g g

represent the size of S. Obtain a random sample X, j=1,...,n from g

S.

Then of course the n sum to N.

If I = 1, we have randem sampling g

over the entire sample space.

Latin Hvoercube Samoling. The same reasoning that led to stratified sampling, ensuring that all portions of S were sampled, could lead further.

If we wish to ensure also that each of the input variables (

has all portions of its distribution represented by input values, we can divide the range of each X int N strata of equal marginal probability k

1/N, and sample once from each stratum.

Let this sample be X, j=1,...,N.

These form the X component, k=1,...,K,in X,

i=1,...,N.

The components k

4 of the various X 's are matched at random. This method of selecting input k

~

'~

~

values is an elttension of Quota Sampling [l3].andcanbeviewedas a K-dimensional extension of Latin square sampling (11).

5 One advantage of the Latin hypercube sample appears when the output Y(t) is dominated by only a few of the components of X.

This method ensures that each of those componen:.s is represented in a fully stratified mnnner, no matter which components might turn out to be important.

Ue mention here that the N intervals on the range of each component

, combine to form [ cells which cover the sample space of of X X.

These cells, which are labeled by coordinates corresponding to the intervals, are e

used when finding the properties of the sampling plan.

2.1 ESTIMATORS In the Appendix, stratified sampling and Latin hypercube sampling are a - hed and compared to random sampling with respect to the class of

.a estimators of Ihe form N

T (Y,...,Y ) " (

8 1

N i'

2.= 1 where g(.) = arbitrary function.

If g(Y) = Y then T represents the sample mean which is used to estimate th E(Y).

If g(Y) = Y" we obtain the r sample moment. By letting g(Y) = 1 for Y s y, 0 otherwise, we obtain the usuri empirical distribution function at the point y.

Our interest is centered around these particular statistics.

Let T denote the expected value of T when the Y 's constitute a random sample from the distribution of Y = h(X).

We show it the Appendix that both stratified sampling and Latin hypercube sampling yield unbiased estimators of t.

6

~

If T is the estimate of T from a random sample of size N, and R

T is the estimate from a stratified sample of size N, then 3

Var (T ) s Var (T ) when the stratified plan uses equal probability 3

R strata with one sample per stratum (all p = 1/N and n

= 1).

No direct q

means of comparing the variance of the corresponding estimator from Latin hypercube sampling, T, to Var (T ) has been found.

However, the g

3 following theorem, proved in the Appendix, relates the variances of T and T

R THEOREM.

If Y = h(Xp...,X ) is monotonic in each of its arguments, g

and g(Y) is a monotonic function of Y, then Var (T ) s Var (TR*

2.2 THE SOLA-PIDOP E.%M'LE The thre'e sampling plans were compared using the SOLA-PIDOP computer code with N = 16. First a random sample consisting of 16 values of X, = (X,X 'X ' 4) was selected, entered as inputs, and 16 graphs of Y(t) 1 2 3 were observed as outputs. These output values were used in the estimators.

For the stratified sampling metSciIhe range of each input variable was divided at the median into two parts of equal probability. The 4

combinations of ranges thus formed produced 2 = 16 strata S.

One t

observation was obtained at~ random from'each S as input, and the re-g sulting outputs were used to obtain the estimates.

To obtain the Latin hypercube sample the range of each input variable I was stratified into 16 intervals of equal probability, and one observa-tion was drawn at random fron each interval. These 16 values for the 4 input variables were matched at random to form 16 inputs, and thus 16 outputs from the code.

7 The entire process of sampling and estimating for the three selection methods was repeated 50 times in order to get some idea of the accuracies and precisions involved. The total computer time spent in running the SOLA-PLOOP code in this study was 7 hours8.101852e-5 days <br />0.00194 hours <br />1.157407e-5 weeks <br />2.6635e-6 months <br /> on a CDC-6600. Some of the standard deviation plots appear to be inconsistent with the theoretical results. These occasional discrepancies are believed to arise from the non-independence of the estimators over time and the small sample sizes.

3.

ESTIMATING THE MEAN The goodness of an unbiased estimator of the mean can be measured by the size of its variance.

For each sampling method, the estimator of E(Y(t)) is of the form Y(t) = ('l/N)

Y (t)

(3.1) i=1 where Y (t) = h(X ),

i=1,...N.

g f

In the' case of the stratified sample, the X comes from stratum t

S,p = 1/N and n = 1.

For the Latin hypercube sample, -he g is g

g obtained in the manner described earlier.

Each of the three estimators Y'

S, and Y is an unbiased estimator of E(Y(t)).

The variances R

g of the estimators are given in (3.2).

Var (Y (t)) = (1/N) Var (Y(t))

g Var (Y (t)) = Var (Y (t)) - (1/N )

(p -p[

g R

i i=1

3 V(V (c)) = V(Y (t)) + ((N-1)/N)

1/(N (N-1) )) E (p - p)(p - p)

R g

R Where p = E(Y(t)),

u = E(Y(t) l X c S ) in the stratified sample, or g

p = E(Y(t) l X c cell 1) in the Latin hypercube sample, and R means the restricted space of all pairs p, y) having no cell' g

coordinates in common.

In the,SOLA-PLOOP computer code the means and standard deviations, based on 50 observations, were computed for the estimators just described.

~

Comparative plots of the means are given in Figure 3.1.

All of the plots of the means are comparable, demonstrating the unbiasedness of the estimators.

Comparative plots of the standard deviations of the estimators are given in Figur's 3.2.

The standard deviation of Y (t) is smaller than that g

of Y (t) as expected.

However, Y (t) clearly demonstrates superiority as R

L an estimator in this example, with a standard deviation roughly one-forth that of the random sampling estimator.

4.

ESTIMATING THE VARIANCE For each sampling method, the form of the estimator of the variance is S (t) = (1/N)

(Y (t) - Y(t))

(4 ~.1) i=1 and its expectation is E(S (t)) = Var (Y(t)) - Var (Y(t)),

(4.2)

L(

where Y(t) is one of Y ( } '

S(

R

9 In the case of the random sample, it is well known that ased estimator of the variance of Y(t).

The NS sa u R

bias in the case of the stratified sample is unknown.

However, because Var (Y (t)) s Var (Y (*

3 R

(1 - 1]NL Var (Y(t)) s E(S, (t)) s Var (Y(t)).

(4.3)

The bias in the Latin hypercube plan is also unknown, but for the SOLA-PLOOP example it was small. ' Variances for these estimators were not found.

_ _ _.. Again using the SOLA-PLOOP example, means and standard deviations (based on 50 observations) were computed. The mean plots are given in Figure 4.1.

They indicate that all three estimators are in relative agreement concerning the quantities they are estimating.

In terms of standard deviations of the estimators, Figure 4.2 shows that, although stratified sampling yields about the same precision as does random sampling, Latin hypercube furnishes a clearly better estimator.

5.

ESTIMATION OF THE DISTRIBUTION FUNCTION The distribution function, D(y,t), of Y(t) = h(X) may be estimated by the empirical distribution function.

The empirical distribution function can be written as N

G(y,t) = (1/N)

E u(y - Y (t)),

(5.1) i=1 where u(z) = 1 for z > 0 and is zero otherwise.

Since equation (5.1) is of the form of the estimators in section 2.1, the expected value of G(y,t) under the three sampling plans is the same, and under random sampling, the expected value of G(y,t) is D(y,t).

10 The variances of the three estimators are given in (5.2).

Dy again refers to either stratum i or cell i, as approximate, and R represents the same restricted space as it did in (3.'2).

Var (G (I'*)) " ( !")

(I' ) ( -

(Y' ))

R N

Var (G (y,t)) = var (G (y,t)) - (1/N )

E (D (y,t) - D(y,t))

(5.2) 3 R

g i=1 var (G (y,t)) - var (G (y,t))

t g

+ (N - 1)/N

1/N (N - 1) ) E (D (y,t) - D(y, t)) * (D (y, t) - D(y, t)

R A6 with the cases of the mean and variance estimators, the distribution function estimators were compared for the three sampling plans. Figures 5.1 and 5.2 give the means and standard deviations of the estimators at t = 1.4 ms.

This time point was chosen to correspond to the time of maximum variance in the distribution of Y(t). Again the estimates obtained from a Latin hypercube sample appear to be more precise in general than the other two types of estimates.

6.

DISCUSSION AND CONCLUSIONS We have presented three sampling plans and associated estimators of the mean, the variance, and the population distribution function of the output of a computer code when the inputs are treated as random variables.

The first method is simple random sampling. The second method involves stratified sanpling and improves upon the first method.

The third method is called here Latin hypercube sampling.

It is an extension of Quota Sampling (13], and it is a first cousin to the " random balance" design discussed by Satterthwaite [12], Budne [2], Youden, et al [15],

11 Anscombe [1], and to the highly fractionalized factorial designs dis-cussed by Ehrenfeld and Zacks (5, 6], Dempster [3, 4], and Zacks [16, 17], and to lattice sampling as discussed by Jensey [9].

This third method improves upon simple random sa=pling when certain monotonicity conditions hold, and it appears to be a good method to use for selecting values of input variables.

7.

ACKNOWLEDGMENTS We extend a special thanks to Ronald K. Lohrding, for his early suggestions related to this work and for his continuing support and encouragement.

We also thank our colleagues Larry Bruckner, Ben Duran, C. Phive, and Tom Boullion for their discussions concerning various aspects of the problem, and Dave Whiteman for assistance with the computer.

This paper was prepared under the support of the Analysis Develop-ment Branch, Divisio,n of Reactor Safety Research, Nuclear Regulatory Commission.

8.

APPENDIX In the sections that follow we present some general results about stratified sampling and Latin hypercube sampling in order to make comparisons with simple random sampling. We move from the general case of stratified sampling to stratified sampling with proportional allocation, and then to proportional allocations with one observation per stratum. We examine Latin hypercube sampling for the equal marginal probability strata case only.

12 8.1 TYPE I ESTIMATORS Let X denote a K variate random variable with probability density function (pdf) f(x) for x C S.

Let Y denote a univariate transfomation of X,

given by Y = h(X).

In the context of this paper we assume X ~ f(x,),

xcS KNOWN pdf Y = h(X)

UNKNOWN but observable transfomation of X,.

The class of estimators to be considered are those of the fom N

~

(#1*****"N) = (

N), E g (u ),

(8.1) g i=1 where g(-) is an arbitrary, known function.

In particular we use g(u) = u to estimate moments, and g(u) = 1 for u.2 0, = 0 elsewhere, tb estimate ~

~

'the distributi'on function.

The sampling schemes described in the following sections will be compared to random sampling with respect to T.

The symbol T denot'es R

T(Y

...,Y w n e arg e nts Y

,...,Y c nstitute a random sample of Y.

y N

ted by I and b N.

The mean and variance of T are e n e statistic T R

given by (8.1) will be evaluated at arguments arising from stratified sampling to fom T, and at arguments arising from Latin hypercube g

sampling to fom T.

The associated means and variances will be com-pared to those for random sampling.

8.2 STRATIFIED SAMPLING Let the range space, S, of X be partitioned into I disjoint subsets I

S of size p = P(X c S ), with E p = 1.

Let X j=1,...,n

, be a g

1 i=1 random sample from stratum S.

That is., let X

~ ild f(x)/p

13 j=1,...,n, for x C S, but with zero density elsewhere. The corresponding g

g values of Y are denoted by Y

= h(X

), and the strata means and variances of g(Y) are denoted by p = E(g(Yg)) =,

g(y)(1/p )f(x)dx g

2f = Var (g(Yg)) =

(g(y)-p )2(1/p )f(x)dx.

o g

g 1

I i

(P /n ) E g(Y

),

It is easy to see that if we use the general form T

=

S 1

i=1 j=1 that T is an unbiased estimator of T with variance given by 3

Var (T ) = E (p /n )c.

(8.2) g g f

.i=1 The following results can be found in Tocher [14].

Stratified Sampling with Procortional Allocation.

If the probability sizes, p, of the strata and the sample sizes, g

n, are chosen so that n = p N, proportional allocation is achieved.

g g

g In this case (8.2) becomes I

2 Var (T } " V""CT ) - (1/N)

I p (p -C.

(8.0 S

R f g i=1 Thus, we see that stratified sampling with proportional allocation offers an improvement over random sampling, and that the variance reduction is a function of the differences between the strata means p and the overall 1

mean T.

14 Proportional Allocation with One Samole per Stratum.

Any stratified plan which employs subsampling, n > 1, can be im-1 proved by further stratification.

When all n = 1, (8.3) becomes Var (T ) = Var C ) ~ (

g R

i i=1 8.3 LATIN HYPERCUBE SAMPLING In stratified sampling the range space S of X can be arbitrarily partitioned to form strata.

In Latin hypercube sampling the partitions are constructed in a specific manner using partitions of the ranges of each component of X.

We will only consider the case where the components of 3 are independent.

Let the ranges of each of the K components of X be partitioned Into N intervals of probability size 1/N.

The cartesian product of these intervals partitions S into [ cells each of probability size N'.

Each cell can be labeled by a set of K cell coordinates m = (mg.mg,...,mg) where mg g

is the interval number of component X) represented in cell 1.

A Latin hypercube sample of size N is obtained from a random selection of N of the cells m,...,g, with the condition that for each j the set g

m is a i= 1 permutation of the integer's 1,...,N.

One random observatica is made in each cell.

The density function of X given X c cell i is [f(x) if x c cell i, zero otherwise. The marginal (bnconditional) distribution of Y (t) is easily 1

seen to be the same as that for a randomly drawn X as fol;ows.

P(Y 5 y) =

E P(Yg $ ylX, c cell q)P(X c cell q) 1 all cells q

15

=E

[f(x)dx(1/N)

~ ~

' cell q h(x)g f(x)dx.

=

h(x)g ~ ~

From this we have T, as an unbiased estimator of T.

t doarriveataformforthevarianceofT we introduce indicator variables w, with g

1 if cell i is in the sample

,v

=

g 0

if not.

The estimator can trow be written.tc

[

T '= (1/N) E v g(Y ),

( 8.S) 7, g

i=1 where Y = h(g) and g e cell 1.

The variance of T is given by g

Var (T ) = (1/N ) E Var (w g(Y ))

g i=1 bN E Cov(w g(Y ),w)g(Y))).

(8. 6)

+ (1/N ) E g

i=1 j=1 JfL The following results about the w are i= mediate:

~

t P(w =1) = (1/[~ ) = E(w ) = E(wf) 1.

g t

Var (w ) = (1/[ )(1 - 1/[~ ).

g 2.

If w and w correspond to cells having no cell coordinates in common, then

16 E(v w)) = E(w t)lw) = 0)P(w) = 0) g g

- E(w w) \\w),= 1)P(w) = 1) t

~

= 1/(N(N-1))

3.

If w and w correspond to cells having at least one comon cell coordinate, then E(w w)) = 0.

g e

Now Var (w g(Y )) = E(w ) Var g(Y ) + E (g(Y )) Var (w )

(8.7) t g

t g

g g

so that

[

E Var (w g(Y )) = N E E(g(Y )

g) + (N

-N

)g (8.8)

~

t g

g i=1 i=1 i=1 where

= E{g(Y)lX e cell i}.

Since f

E(g(Y )-p ) =[

(g(y)-T) f(x)dx + (p -T)

(3.9) g t

we have E Var (w g(Y )) = N Var (Y) - N E(pt-T)

+ (N

-N"

) Ep.

(8.10)

~

y g

i i

i Furthermore

[N

~

E E Cov(w g(Y ),w g(Y )) = E E p E(v v } - N ZEpp (8.11) g f

i=1 j=1 1/j 1/j ifj

17 which combines with (8.10) to give

~

Var (T ) = (1/N) Var (Y) - N E(p -T)

+ (N

-N

)Ep g

t g

i L

K~1

-2K (g*12)

~

E g Yb ~

gg%

+ (N-1)

N R

.. ifj where R means the restricted space N (N-1) pairs (p.p ) corresponding g y to cells having no cell coordinates in con: mon. Af ter some algebra, and with E p

=NT, the final form for Var (T ) becomes Var (T ) = Var (T ) + (N-1)/N[N (N-1)~ E(

-T) (p -T) ].

(8.13) g R

R Note that Var (T ) $ Var (T }

"7 g

R N (N-1)

E(p -T)(

-T) $ 0, (8.14)

~

g which is equivalent to saying that the covariance between cells having no cell coordinates in con: mon is negative. A sufficient condition for (8.14) to hold is given by the following theorem.

THEOREM.

If Y = h(X,...,K ) is monotonic in each of its arguments, t

g and if g(Y) is a monotonic function of Y,

then Var (T ) $ Var (T )

g R

PROOF. The proof employs a theorem by Lehmann [10]. Two functions r(x,...,x ) and s (y,...,y ) are said to be concordant in each argn=ent y

g y

g if r cnd s either increase or decrease together as a function of 1.

Also, y), j/ i held fixed, for each

= y, with all x), jf i and xt g

a pair of-random variables (X,Y) are said to be negatively auadrant dependent

18 if P(X $ x, Y $ y) $ P(X $ x)P(Y $ y) for all x, y.

Lehmann's theorem states that if (i)

(X,Y ),(X 'Y ),...,(X

,Y ) are independent, y g 2 2 g g (ii)

(X,Y ) is negatively quadrant dependent for all 1,

and t g (iii) X = r(X

...,X ) and Y = s(Y,...,Y ) are concordant in each arguent, y

g 7

g then (X,Y) is negatively quadrant dependent.

We earlier described a stage-wise process for selecting cells for a Latin'hypercube sample, where a cell was labeled by cell coordinates m

....,m g

g Two cells (4,...,4 ) and (m,... m ) with no coordinates in common may be 1

g y

g selected as follows. Randomly select two integers (R77,R21) without replace-ment from the first N integers 1,...,N.

Let 47=R77 t=R21 and m Repeat thc procedure to obtain (R12,Rg),(R13' 23)'***'( 1K' 2K)

""U

" ' 'k = Rik Thus two cells are randomly selected and i f g for k=1,...,K.

and g = R2k.

Note that the pairs (Rik,R2k), k=1,...,K, are mutually independent.

I 7) " I*7 ~ *i"(**7)]/("("~1))

, Also note that...because P(Rg $ x, R2k 3

$ P(R

$ x)P(R

$ Y), Where (* } represents the " greatest integer" ik 2k function, each pair (R.k,R2k) i8 "*88'i"*i7 9""d#d"' d

P ""'I""'

Let be the expected value of g(Y) within the cell designated by 1

(1

,...,2 ), and let be similarily defined for (m,...,g).

Then 7

2 y

pt = p(R77,R12,...,R7g) and p2 " E( 21' 22,...,R2K) are c ne rdant in each argument under the assumptions of the theorem.

Lehmann's theorem then yields that pt and p2 are negatively quadrant dependent. Thetafore, P( 1 $ x, p2 $ y) $ P(pt $ x)P(p2 I I)*

~

Using Iloeffding's equation p+=

da Cov(X,Y) = j (P(X $ x, Y $ y) - P(X $ x)P(Y $ y)] dx dy,

. = '

19 (see Lehmann [10] for a proof), we have Ov( y,p ) < 0.

Since 2

Var (T ) = Var (T

(-

(M 'U ), the theorem is prwed.

L R

1 2

~

Since g(t) as used in both Sections 3 and 5 is an increasing function of t,

we can say that if Y = h(X) is a monotonic function of each of its arguments, Latin hypercube sampling is bettsr f.han random sampling for

_ estimating the mean and the population distribution function.

REFERENCES

[1] Anscombe, F. J. (1959). Quick Analysis Methods for Random Balance Screening Experiments. Technometrics, 1, 195-209.

[2] Budne, T. A. (1959). The Application of Randon Balance Designs.

Technometrics, 1, 139-155.

[3] Dempster, A. P. (1960).

Randon Allocation Designs I:.On General Classes of Estimation Methods. Ann. Math. Statist., 31, 885-905.

[4]

Dempster, A. P. (1961). Random Allocation Designs II: Approximate Theory for Simple Random Allocation. Ann. Math. Statist., 32, 387-405.

[5] Ehrenfeld, S., and Zacks, S. (1951). Randomization and Factorial Experiments. Ann. Math. Statist., 32, 270-297.

[6] Ehrenfeld, S., and Zacks, S. (1967). Testing Hypotheses in Randomized Factorial Experiments. Ann. Math. Statist., y, 1494-1507.

[7] Hirt, C. W.,Nichols, B. D., and Romero, N. C. (1975).

SOLA - A Numerical Solution Algorithm for Transient Fluid Flows. Los Alamos Scientific Laboratory Report LA-5852, Los Alamos.

[8] Hirt, C. W., and Romero, N. C. (1975). Application of a Drif t-flux Model to Flashing in Straight Pipes. Los Alamos Scientific Laboratorv Report LA-6005-MS, Los Alamos.

-em

.m.

e

_,w.

20

[9] Jessen, Raymond J. (1975), Square and Cubic Lattice Sampling.

Biometrics, 3J,449-471.

[10] Lehmann, E. L. (1966).

Some Concepts of Dependence. Ann. Math.

Statist., 35,, 1137-1153.

[11] Raj, Des (1968).

Samoling Theory. McGraw-Hill, New York, 206-209.

[12]'Satterthwaite, F. E. (1959).

Randon Balance Experimentation.

T'echnometrics, l_, 111-137.

[13] Steinberg, H. A. (1963), Generalized Quota Sampling. Nuc Sci and Engr., 15, 142-145.

[14] Tocher, K. D. (1963), The Art of Simulation. D. Van Nostrand, Princeton, N.J., 106-107.

[15] Youden, W.'J., Kempthorne, 0.,

Tukey, J. W.,

Box, G. E.

P., and Hunter, J. S. (1959).

Discussion of the Papers of Messrs. Satterth-waite and:Budne. Technometrics, 1., 157-193.

a

[16] Zacks, S. (1963).

On a Complete Class of Linear Unbiased Estimators for Randomized Factorial Experiments. Ann. Math. Statist., 34, 769-779.

[17] Zacks, S. (1964).

Generalized Least Squares Estimators for Randomized Fractional Replication Designs. Ann. Math. Statist., 35, 693-704.

Tig. 3.1.

Escinating the Mean: The Fig. 3.2.

Esci=ating the Mean: The 5_aarple Maan of Y itandarl Deviation _of T (c), and Y,(t)g(t),

Y ("I' I (t, and y c).

S 7

R S

O 4

CNTCa g

W,= :,

haCAse CF C3tmaica 8

. _s

a<

l*

T b.

Me" I/Il I

n!

g Kt I

IiI y

'i 1

l !!

Ii

=

g., if 1-

,'rfD C

n g,

cio ft). 3 (c), w s[,(e3, a

3 Tig. 4,1, g,

na sampi.,,,, 3, {) >

8 a

s$(c).and3(e),

L h

t f

on 2 W CSindATOR

{

& F CSTDdA104 w

t I

a$fl d

i 2-At cI ll sl

}.

llt t11

g' k

.s,.

Fig. 5.2.

Esci:nating the CDF: The Standard Deviacion of C (I'C)' O (y.:), and 2

S Fig. 5.1.

Estimating the CDF: The Sample Mean of C (y,c),

G (y,c), d G,(,t) ac 3

g c = 1.4.

3.0. OF CSTndarca WCAN CF CITlWATOR i

I a

I I

3 e

!-N

.l

'il

!A 4

\\

3. 5 il

=

a 33 ii

/

i:

H:-

3 ii

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ML19269B882

Contents

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SUBJECT:

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ML19269B882

Text

SUBJECT:

SUMMARY

Enclosures:

Dearien,

Dearien,

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