An efficient sampling scheme: Updated Latin Hypercube Sampling

Probabilistic Engineering Mechanics 7 (1992) 123-130

An efficient sampling scheme: Updated Latin Hypercube Sampling Ale~, Florian Dept. of Structural Mechanics, Technical University of Brno, Barvicova 85, 662 37 Bi'no, Czechoslovakia

(Received February 1991; accepted May 1991) An efficient sampling scheme called Updated Latin Hypercube Sampling is presented. The proposed method is an improved variant of Latin Hypercube Sampling. It uses specially modified tables of independent random permutations of rank numbers which form the strategy of generating input samples for a simulation procedure. The method is presented in order to obtain these specially modified tables. The aim of this paper is to compare estimates of certain widely used statistical parameters obtained by Updated Latin Hypercube Sampling, Latin Hypercube Sampling and Simple Random Sampling. It is shown that Updated Latin Hypercube Sampling usually results in a substantial decrease of the variance in the estimates of commonly used statistical parameters and that the bias is quite small for a moderate number of simulations. This sampling technique seems to be generally very useful, efficientand superior to other methods especiallyin the case of statistical, sensitivity and probability analyses of complex analytical models with random input variables.

paper presents a basic theoretical background to this imProved variance reduction sampling scheme. This technique uses specially modified tables of random permutations of rank numbers which form the strategy of generating input samples for a simulation procedure. The method is described for obtaining these specially modified tables. The objective of this paper is to compare the estimates of some statistical parameters obtained for a different number of simulations using Updated Latin Hypercube Sampling, Latin Hypercube Sampling and Simple Random Sampling. Based on results of this comprehensive study, it is shown that Updated Latin Hypercube Sampling usually results in a substantial decrease of the variance in the estimates of commonly used statistical parameters and that the bias is quite small for a moderate number of simulations. As a result we suggest utilization of this variance reduction technique for statistical, sensitivity and probability analyses of complex analytical models with random input variables.

INTRODUCTION Analytical models are widely used to predict intricate relationships between input variables and a certain output. In the case of random input variables there exist some sampling schemes for dealing with this problem, see for example Ref. 1. Latin Hypercube Sampling, which is very efficient and practical, is one of them. It was theoretically described by McKay et al. ~ and Iman & Conover. 2 It has been used in many different statistical analyses of complex analytical models since, see Refs. 1-10. It theoretically provides estimates of some statistical parameters mean value, statistical moments, cumulative distribution function (CDF) - - which are unbiased and whose variance is reduced, see Refs. 1 and 2. As proven by numerical study," these features still hold for estimates of other widely used statistical parameters - - standard deviation, coefficient of variation (COV), skewness, excess, minimal or maximal value of sample. For some of these statistical parameters the bias is negligible but for all of them the variance is substantially reduced in comparison with Simple Random Sampling. To reduce the variance of estimates still further, we propose a new variant of Latin Hypercube Sampling - - so called Updated Latin Hypercube Sampling. The -

-

UPDATED LATIN HYPERCUBE SAMPLING The basic theoretical background is the same for both Latin sampling schemes, see for example Ref. 2. Let

Probabilistic Engineering Mechanics 0266-8920/92/$05.00

Y = f({X})

© 1992 Elsevier Science Publishers Ltd. 123

(1)

A]es~ F~orian

t 24

represent the output variable, f ( , ) is a deterministic analytical model (mathematically tractable or a computer code) and iX} = ( ~ , ~ . . . . X~) r repre= sents the vector of input variables (K being the number of input variables) assumed to be r a n d o m ones. Every input variable Xe, k - I, 2 . . . K, is described by its known C D F Fx~ (x) with the appropriate statistical parameters. The sample {x}~ of input vaiables, n = t, 2 . . o N (N being the number of samples equal to the number of simulations) is selected in the following way. The range of the known C D F Fx~ (x) of each input variable X~ is partitioned into N disjunct intervals &.~. Each interval is characterized by the probability p~ defined as ;~o =

P(X~

e &~)

(2)

TaMe 1. The minima| and max{ma~ values ef Cite S ~ a ~ a = eee~eie~t/~'~ among columns of te~ ~a~nm~y ger~erated tunes for different number ef simuI[a~o~s

/~)

Number of simulations 10

13

20

3(}

4(}

Min -0-89i -0.896 -0.663 -0.480 .-~.428 Max 0-806 @8t9 0.629 0 - 5 4 6 [b4~

50 -.04~5 0.465

The brief study reported in this section examines how significant the correlation can ben The Spearrnan rank corre~a~mn coefficient (the Speanman coefficient), ~'~3 can be defined as

and

(3)

Cs =

In the case of intervals o f equal probability it holds that p ~ = tiN. Each interval is represented in the sample by the representative parameter which is takes randomly within the interval or may be taken at the centroid of the interval. In the latter case the representative parameter is obtained as

¢s~<7

E;~

=

~

F~(m~-0"5) y~

k=

1,2,...i<

k = t,2, o..K

(4)

where F~ ~ (-) is the inverse C D F and m~e is the rank number o f the interval used in the nth simulation for input variable X~. The representative parameter is used just once during the simulation procedure and so there are N observations on each o f the K input variables. The selection of interval Se~ to be sampled for a particular simulation is randomized. N observations on each of input variable X~ are associated with a sequence of integers (rank numbers o f intervals) representing a random permutation o f integers 1, 2 . . . N. It is required that these permutations are mutua!Iy independent. They are ordered in the table of r a n d o m permutations o f rank numbers which has N rows and K columns. The ran~ numbers o f intervals used in the sth simulation are represented by the nth row in the table. It means that this table forms the strategy for obtaining an input sample {x}~. For such a sample one can evaluate, using eqn (!), the corresponding value y~ o f the output variable. From N simulations one obtains a set of statistical data { Y} = < Y~, Y2, - ~ . Y~)S- This set is statistically assessed and thus estimates of some statistical parameters are obtained. Tables used in Latin Hypercube Sampling are commonly generated randomly. The possibility does exist that a certain statistical correlation among columns of the table is randomly introduced, which may have a significant influence on the results of simulation. It na> uralty affects the bias and variance of the estimates obtained.

I -

n(n

s

1) (n + i)

(5!

1"0; I'0)

where d, = difference between the ranks o f two sam.p~es, and ~ = samele size, is used to describe statistical co> relation among table columns. The Spearman coeNcie:~?,~ has the following desirable features: 1. It can be used for descrip~.on o f any monotonic statistical correlation between two random vaiables. 2. The CDFs o f correlated random variables can be non-normal and differem 3. ~t is defined with help of the ranks of two samples and thus it is a meaningful measure for defining statistical c o r r d a t i o n among table columns which are f o x e d by rank numbers. For the purpose of our study, ten tabtes for N = i0. 13. 20. 30. 40 and 50 simulations and K = 12 mpu~: variables are randomly generated. The Seearman coefficients among columns of these tables, see eqn (5! are evaluated. The minimal and maximal values of the Spearman. coeNcient are reported for the ~_e-~p~x~v,.,r ..... number o f simulations in Table i. It is obvious tha,: a value of up to -T- 0.8 is not uncommon and the influence on estn~ates can be generally very high. Updated Latin Hypercube Sampling, proposed in this paper, is an improved variant of Latin Hypercube Samp!ingo As mennoned above, its theoretic~ background is the same as described m this section ~'or Latin Hypercube Sampling. Updated Latin Hypercube Sampling uses specially modified ~ables m ~¢hich undesired s-~atistical correlation among their columns is diminished. As a result, the software prepared fbr Latin Hypercube Sampling can be usee without any resmc~ added, to the simu-tions and oniy modified v~abte~< a ~-~ ~ ~.J, h t i o n erocedure. Also the ac)plicabitity is quite wide - " ~ a!"~-~ can cover statistical, sensitivky and re!iabi~ity analyses as we~t.

125

An Efficient Sampling Scheme: UpdatedLatin Hypercube Sampling THE METHOD FOR DIMINISHING UNDESIRED

STATISTICAL CORRELATION

Table 2. Table of random permutations of rank numbers for K = 5 input variables and N = 10 simulations

Simulation The method for diminishing undesired statistical correlation among columns of tables is the same as the one used in Ref. 12 for inducing correlation among input variables. In this section the theoretical background is described briefly; for more details see Refs. 12 and 13. Let R be an N x K matrix whose columns represent K permutations o f integers 1, 2 , . . . N. That is, matrix R is identical to the table of r a n d o m permuations o f rank numbers used in both Latin sampling schemes. Statistical correlation among columns o f this matrix is described by the rank correlation matrix T, where elements T~j, i, j = 1, 2 . . . . . K, are the Spearman coefficients among columns i and j of R. It is obvious that matrix T is symmetrical and in the case o f uncorrelated columns is equal to unit matrix I. Consider only realizations of R for which matrix T is positive definite and let S be a lower triangular matrix such that S-T-S T =

I

(6)

where S =

Q-l

(7)

Q . QT

(8)

and T

=

Because matrix T is positive definite, the Cholesky factorization scheme may be used to find the lower triangular matrix Q. Transformation RB =

R" S T

(9)

results in an N × K matrix RB. Statistical correlation among columns of this matrix is described by the rank correlation matrix TB. As proven in Ref. 12, matrix TB should be close to matrix I. That is, the difference between appropriate elements in matrix TB and matrix I is lower than in the case of matrix T and matrix I. We can now rearrange the values in each column of input matrix R so that they will have the same ordering as the corresponding column of matrix R.. As a result the rank correlation matrix T equals T8 and the statistical correlation among columns of R and also among columns of the table o f random permutations o f rank numbers is reduced. The method presented has the following desirable features: 1. It may be used with an arbitrary C D F describing an appropriate input variable. 2. It does not affect the statistical parameters o f appropriate input variables. 3. It does not affect the structure o f the table, because it only rearranges its columns. 4. It is simple and effective. The following textbook example shows how the rank

Non-modified table

Modified table

Variable 1

2

3

Variable 4

5

1

2

3

4

5

1

1

3

4

1

5

1

3

4

1

3

2 3 4 5 6 7 8 9 10

8 5 9 6 10 2 4 7 3

6 5 4 10 2 1 7 8 9

10 9 1 7 2 5 6 8 3

2 3 10 8 6 9 4 7 5

4 7 3 1 6 10 8 9 2

8 5 9 6 10 2 4 7 3

6 5 4 10 2 1 7 8 9

10 9 1 5 3 7 6 8 2

2 6 9 8 3 10 5 7 4

2 5 4 1 8 6 9 10 7

correlation matrix T looks and how powerful the method presented can be. The table for K = 5 input variables and N = 10 simulations is randomly generated. It is shown in Table 2. Statistical correlation among columns o f this table is described by the rank correlation matrix T shown in Table 3. The extreme value of the Spearman coefficient is equal to - 0.47. The procedure described in this section is used twice and columns of the table are rearranged. This improved table is shown in Table 2. The appropriate rank correlation matrix is shown in Table 3. It is clearly seen that statistical correlation among columns of the table is diminished. The extreme value o f the Spearman coefficient is now equal to 0.07. N U M E R I C A L STUDY General

As has been mentioned above we can expect a decrease of the variance in the estimates of statistical parameters when Updated Latin Hypercube Sampling is used. The aim o f our study is to compare the estimates of mean value ¢, standard deviation Sy, COV %, skewness ay, minimal value in the sample YM~Nand maximal value in the sample YMAX obtained by Latin Hypercube Sampling and Updated Latin Hypercube Sampling. Estimates of a suitable C D F are compared too, see Ref. 14. Estimates are compared for 10 (13), 20, 30, 40, and 50 simulations. Sampling is repeated 10 times for each number of simulations. In the case of Latin Hypercube Sampling the tables are randomly generated in every run. In the case o f Updated Latin Hypercube Sampling the tables are rearranged and statistical correlation of their columns is diminished. Intervals have the same probability 1/N and representative parameters are taken at the centroid o f intervals - - see eqn (4). The 'exact value' of the appropriate statistical parameter is obtained by Simple R a n d o m Sampling with 10000 simulations repeated 10 times. After performing 10 runs we have at our disposal 10

I26

A [e~ Florian Table 3. Rank correlation matrix of table ef random permeations of rank r~ambers fro~,a Table 2

Variable

Non-modified tame

l 1 2 3 4 5

t-00 0"03 0"04 0'3t 0-20

Modified table

2

Variable 3

4

0-03 1'00 0-37 -0-03 0-47

- 0'04 0"37 1'00 -0-41 0-22

0-31 -- 0"03 - 0-4t 1-00 0"01

values of 35, s,, %, £,, YMIN, flMAX and o f a suitable C D F obtained by both methods. F r o m these particular samp]es, m e a n value 2~, standard deviation s , minimal value MtN~ and maximal value MAX~ are evaluated, where z = 35, sy, %, ay, YMINor YMAX-The value 35z is the estimate o f the appropriate statistical p a r a m e t e r z o f the basic set and the values o f %, MIN~ and MAX~ characterize the variance of estimate and the amplitude o f its possible values. I f the statistics obtained by both sampling schemes are compared, one can c o m p a r e the bias and variance of estimates. Examples

[n order to illustrate the behavior of estimates, two examples o f prestressed concrete cross~secfions are considered herein. In Example i (see Fig. l(a)) prestressing steet and mild steel are considered. In example 2 (see Fig. t(b)) only symmetrical prestressing steel is considered. The following input variables are treated as r a n d o m variables: compressive strength of concrete & , strength and modulus of elasticity o f mild steel R, and Es, respectively, strength and modulus o f elasticity of prestressing steel Rp and Ep, respectively, depth and width of crosssection H and B, respectively, position o f layers o f miid steel and prestressing steel in cross-section, au a n d api , respectively, and initial stress in prestressing steel %. The statistical parameters of input variables are summarized in Table 4. They are based on results published in Re£ 15. Each input variable is described by normal C D F . Resuks

5 - 0-20 0-47 0-22 0-0t ~"00

-

1-00

0,03

0-03

1"00

- 0.05 0'05 0"03

0.07 0"02 -- 0'03

- 0'05 0"07 i'00 -0'03 - 0"03

".a

l

0-05 0"02 i 0.03

0-03

-0-03 -- 0-03 0@7 i .00

bOO 0-0~

Analy~ca~ model

F o r t h e purpose of our numerical study the ana!yucai modal for the calculation o f the carrying capacity of a prestressed concrete cross-section in aure bending (ultimate m o m e n t M u ) is treated, see Fig° 2. The follow.° ing assumptions are accepted.: t. The cross-section remains plane a k e r deforma~io:~, and strains are proportionai to the distance from the neutral axis. 2 Yensite sGength o f concrete is neglected. 3. Compressive stress in concrete is taken ~o 50 the =ompressive strength of concrete in a par:.,: which is 0.8 times the vertical distance from the top edge ot ~he cross-section co the neutral axis. 4. Reinforcement (prestressing steel or mild stee0 is subjected to the same variauo~s m strain as adjacent concrete 5. Stress-strain diagrams of both r e m f o r c e m e m are b/linear. 6, Compresswe stress :n prestressing stee~ is no~ permitted. 7. -Ultimate compressive strain of concrete ~s taxe~ ;c be 0.0025° Table 4. Statistical para~eters of i~p~t vaJables Example ~ Mean ,% Rs & tip E~

MPa 52.8 MPa 260 GPa 2!0 MPa 1800 GPa 200 m 0"1837

B

m

,~__884o3 >!

% a~i a~!

MPa m

(b)

%2

m m

ap3 a~4

m m

Fig° L Prestressed cross-section: (a) Example 1; (b) Example 2 (dimensions in ram).

Variable 3

obtained with generally non°norma~ C D F s @4° i o p normal, Weibull, truncated normal) are incorporated fnto this broad numerical study too.

H

(a)

2

Input Units variable

< <_ 85.4

~

0.0854

900 0- i 597 0"0530 0-0730

---

Stand. dev. 8-Q/84 1.? Q5 9-0 1.0 0-0028 0,00i9 ~ u" ".

Example 2 Mean

Stand. dev

52-8 --

7.28&",

!800 200 0"i510

0-0843 900

0'0024 0"00t8 0.00t8

~0 J.<:~ ~-%~ 0"0013 60"0

--0"0200 0.0400

0"0012 @-0012

0-~ tOC 0 1300

0-00:~2 0-0012

An Efficient Sampling Scheme: Updated Latin Hypercube Sampling

XNA

0'8 XNA I

NEUTRAL AXIS

+

+

+

+

_

=N_

B

ac~Rc

~Epi

a ai

127

a'i=E p. £pi +Op o i = E s - Esi

'1 ~J

(a)

(b)

(c)

Fig. 2. Analytical model for the calculation of carrying capacity of a prestressed concrete cross-section in pure bending: (a) typical prestressed cross-section; (b) strain diagram; (c) stress diagram in ultimate state.

1.70-

15-60 •

A

,%

15-59-

,0-2

5

~.

\

%.

1.60.........

Max. Mean

-

15-58

. . . . .

o.1 _

I~

15"57

~

_

~

Min.

1.50-5

----

Max. I>',

15.56

1-40-

Mean .s. s

15.55

\

Min.

--~-~7/-- --

=10 \

1.30-

\ 20

15-54.

I I 10 13

I 20

I 30

[ 40

I 50

/

/ \

.0.1

/

/

/ v

10 13

n

I 20

I 30

I 40

I 50

n

(a)

(a)

-0-00100 0.152

• 0-10

0-014

0-00075

"" .~ --. 0.11

I>, 0 " 0 1 0 " k 0-00050

\

0-07

k

0.006-

\

~.~,

0-050

""

\

\

\ \

0.00025~

%

0.025 0-03.

0.002 -

I I

10 13

I

20

I

30 n

I

40

I

50

(b)

Fig. 3. Estimates of the mean value: (a) mean value, minimal value, maximal value; (b) standard deviation (dimensions in kNm).

I I

10 13

I

20

t

30 n

l

40

I

50

(b)

Fig. 4. Estimates of the standard deviation: (a) mean value, minimal value, maximal value; (b) standard deviation (dimensions in kNm).

AIeY Fiorian

128

0.60 .300 ~ ~ ~ %.

0,20

:200

-

0

¢,

~ -.

. . . . . .

~100_

-O-20 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I>,

I

- 100

"\

/

l

\

\

\

/

-0-60

/

//C~ %

,

-

-~

/7

J

-20

-

Mean

~

~ ~

°300

0"081120 I--I =1@0 }

- + - - - .....

I-

10 13

20

30

{

1

40

50

--b~ 10 I 3

J 30

40

~--.--~ 50

R

{a}

(a)

[ 0.009- -0.0009

O'40

/%

0'007- -0'0007 /

/

/ /

N\

/

\

0-005

2.0

i

"1.5

\

0.30-

/

,.~

f

x

"0"0005

\

k

\

\

-1.O

\

/ * \ \

0.20 "

\

\\ \

\..

X

0'00 3 -0.0003

0 4 0 ' 0-5 0 . 0 01 -0-0001

"~--~_

10 13

20

30

40

50

--}-J .... 10 13

520

-

- E - ~ ........ 30 40

g--50

n

(b) Fig.

5, E s t i m a t e s

of the

coefficient

o£ v a r i a t i o n :

(a) m e a n

v a l u e , m i n i m a l value, m a x i m a l v a l u e ; (b) s t a n d a r d d e v i a t i o n

Fig. 6° Estimates of the skewness: ta) mean value mmm~ai value, maximal value: (b) standard deviation (dimension\ ss)

(dlmensmmess~. 8. Ultimate tensile strain o f mild steel is taken to be 0-010 and that o f prestressing steel 0.015. 9° Carrying capacity in pure bending is accomplished if the ultimate strain is reached in one o f the materials at least. ! 0. Effects of creep, shrinkage, environmental changes and prestress losses are taken into account by means of'initial stress in mild steel and initial stress in prestressing steel. 11. Position of the neutral axis is searched iteratively from compatibility conditions and equilibrium conditions. RESULTS

For the sake of simplicity, only results obtained for Example 2 are presented in Figs 3-8• We have to emphasize here that the results obtained for both examples with

normal and generally non-normal CDFs describing input variables are qualitatively the same ~f the estimates obtained by both sampling techniques are compared. The results obtained by Updated Latin Hypercube Sampling are plotted as a full line, by Latin Hypercube Sampling as a dashed line and by Simple Random Sampling with 10 000 simulations as a thin dashed line m the figures. Each figure consists of two parts - part (a) and part (b). Part (a) g~ves the plots of: l. mean value of an appropriate statistical parameter obtained from l0 runs: 2. minimal value of an appropriate statistical parameter obtained from 10 runs: 3. maximal value of an appropriate stat~st~ca~ parameter obtained from I0 runs" 4. mean value of an appropriate statistical parameter obtained by Simple R a n d o m Sampling with i0 000 simulations and 10 runs labeled MEAN:

An Efficient Sampling Scheme: Updated Latin Hypercube Sampling

14.0,

.................................... 6 0 ~' "~.

12'01

,4o2

21-0

Max.

..................................

Mean

..................................

Min.

20.0- :5

.E

I>~

129

E

10.0- =20

I>,

..................................

Max.

...................................

Mean

. . . . . . . . . . . . .

19"0 - : 1 0

Min.

18"0

"-5

20 6.0

10 13

0.50

20

30

40

I

50

I

I

I

I

20

30

40

50

n

n

(a)

(a)

0-60' 0'06

tx 0-025

0-45

0-40

!

10 13

/

0'50"

/ \

/

\

/

0-05

\

0-020 1/

E

E U')

\

\

>, 0 . 4 0 0,35

0.30.004--->%

0.30

-

o

~v/;, .................................

0-25

o03 / I

I

10 13

~.___

0.20- "0"010

\ l

I

I

l

20

30

40

50

n

(b)

I

I

10 13

l

I

I

I

20

30

40

50

-

n

(b)

Fig. 7. Estimates of the minimal value: (a) mean value, minimal value, maximal value; (b) standard deviation (dimensions in kNm).

Fig. 8. Estimates of the maximal value: (a) mean value, minimal value, maximal value; (b) standard deviation (dimensions in kNm).

5. minimal value of an appropriate statistical parameter obtained by Simple Random Sampling with 10 000 simulations and 10 runs - - labeled MIN; 6. maximal value of an appropriate statistical parameter obtained by Simple Random Sampling with 10000 simulations and 10 r u n s - labeled MAX.

mates of appropriate parameter z (on the outside scale) and the ratio of Sz to the value MEAN (on the inside scale). Standard deviation of 10 estimates obtained by Simple Random Sampling with 10000 simulations has also been plotted. The results of both studies may be summarized in the following way:

On the x-axis (labeled n) the numbers of simulations are plotted. On the y-axis (labeled 35z) the real value of appropriate parameter z (on the outside scale) and deviations in per cent from the value MEAN (on the inside scale) are plotted. In part (a) we can compare the bias of estimates and also the amplitude of these estimates in regard to value MEAN. Last but not least we can mutually compare the results obtained by the two sampling techniques. In part (b) of the figures we have plotted the standard deviation of estimates. On the y-axis (labeled sz) we have plotted the real value of standard deviation of 10 esti-

1. The bias of mean value, standard deviation and COV is negligible for a moderate number of simulations. The variance of estimates is much smaller in the case of Updated Latin Hypercube Sampling. These estimates are comparable with estimates obtained by Simple Random Sampling with 10000 simulations. 2. The estimates of skewness are comparable for both Latin sampling schemes. The estimates obtained by Updated Latin Hypercube Sampling have a smaller variance.

130

Ale~ Fiorian

3. The estimates o f minimal and maximal values are comparable for both sampling schemes - - they are not markedly influenced by table rearrangement° The same may be said about estimates o f a suitable CDF. 4. Updated Latin Hypercube Sampling usually results in substantial decrease of the variance in the estimates of commonly used statistical parameters. 5. The bias of estimates obtained by Updated Latin Hypercube Sampling is quite small for a moderate number o f simulations. It is comparable with Latin Hypercube Sampling and is much smaller than in the case of Simple R a n d o m Sampling for the same number of simulations.

CONCLUSIONS An efficient sampling scheme called Updated Latin Hypercube Sampling is proposed. It is an improved variant o f Latin Hypercube Sampling. The basic theoretical background is the same for both sampling schemes. They differ in one substantial basic feature from each other: namely in how strong the correlation is among columns o f the table o f r a n d o m permutations o f rank numbers which forms the strategy o f generating input samples for the simulation procedure. Updated Latin Hypercube Sampling uses specially modified tables in which the statistical correlation among their columns is diminished° That is, the rank correlation matrix of the table, characterized by the Spearman rank correlation coefficients, is nearly equal to the unit m a t i x . As a resulL the software prepared for Latin Hypercube Sampling can be used without any restrictions and only modified tables are added to the simulation procedure. A m e t h o d is described by means of which tables can be rearranged in such a way that the correlation among columns o f tables is diminished. This method preserves the statistical parameters o f input variables, may be used with an arbitrary CDF, does not affect the structure of the table, and is simple. A broad numerical study o f estimates of commonly used statistical parameters is accomplished° It is found that Updated Latin Hypercube Sampling usually results in a substantial decrease o f the variance in the estimates of commonly used statistical parameters° Also [t is found that the bias of estimates obtained by Updated Latin Hypercube Sampling is quite small for a moderate number o f simulations and is much smaller than in the case of Simple R a n d o m Sampling. The amplitude of possible values is also much smaller and it is distributed better with respect to the exact value of the appropriate statistical parameter.

Updated Latin Hypercube Sampling seems to be gen-.. eraily very useful, efficient and superior to other sampling techniques. As a result we suggest utilizing this variance reduction technique for statistical, sensitivity and pre,bability analyses o f complex analytical models with r a n d o m input variables~

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