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Graphical Interactive Simulation Input Modeling with Bivariate E%zier Distributions MARY ANN FLANIGAN Purdue University

WAGNER

and JAMES R. WILSON North Carolina State University

A graphical interactive technique for modeling bivariate simulation inputs is based on a family of continuous univariate and bivariate probability distributions with bounded support that are described by B6zier curves and surfaces, respectively. This family of distributions has a natural, extensible parameterization so that all parameters have a meaningful interpretation; and the complete family is capable of accurately representing an unlimited variety of shapes for marginal distributions together with many common types of blvariate stochastic dependence. This approach to simulation input modeling is implemented in a Windows-based software system called Pmw-probabilistic Input Modeling Environment. Several examples illustrate the application of PRIME to subjective and data-driven estimation of bivariate distributions representing simulation inputs. Categories and Subject Descriptors: G.3 [Mathematics of Computing]: Probability and StatisI,ics-staz%tical software; 1.6.5 [Simulation aml Modeling]: Model Development—modeling m eth odo[ogtes; 1.6.7 [Simulation and Modeling]: Simulation Support Systems—en Luronmen ts General Terms: Algorithms, Design, Theory Additional Key Words and Phrases: Graphical interactive distribution

fitting

1. INTRO13LKHION One of the central problems in the design and construction of stochastic simulation experiments is the selection of valid input models—that is,

This work was partially supported by a David Ross Grant from the Purdue Research Foundation and by NSF grant DIvE-87 17799. A preliminary version of this work was presented at the 1994 Winter Simulation Conference, which was held December 11–14, 1994, in Ch-lando, Florida, and was sponsored by ASA, ACM, IEEE, IIE, NIST, ORSA, TIMS, and SCS. Authors’ addresses: Mary Ann Flanigan Wagner, Boeing Information Services, 7990 Boeing ecld; James ~. Wilso% Department of Court, Vienna, VA 22183-7000: email: (maf lani@w~. Industrial Engineering, North Carolina State University, Raleigh, NC 27695-7906; email: (]wllsOn@:eOs

.ncsu.

edu).

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164

M A. Flanigan Wagner and J R, Wilson

.

probability distributions input processes driving only to capture random variable between modeling UNOS

that accurately the system. In

mimic the behavior of the random many applications, it is critical not

the shape of the marginal distribution of each major input but also to accurately represent the stochastic dependencies

those variates [Lewis and Orav 1989, p. 291]. For example, in the arrival streams of liver-transplant donors and patients for the Liver

Allocation

the stochastic and ultimately

Model

[Pritsker

et al. 1995],

initially

we had to model

dependence between the age and weight of each new arrival; we had to expand our stochastic model to include the sex and

blood type of each new arrival. Although many practitioners appreciate the need for valid models variate simulation inputs, they lack effective and widely available

of multitook for

building such input models. Stanfield and Wilson [1993] developed a technique for fitting a multivariate distribution when the correlation matrix and the first four moments for each marginal distribution have been specified or estimated by the user. Because the fitted joint distribution is built from univariate

marginals

1949a; Swain Stanfield and joint distribution system [Johnson tional

belonging

et al. Wilson

to the

Johnson

translation

does not belong to the multivariate 1949b; Johnson 1987]; moreover, the

distributions

system

1988], the multivariate input-modeling has substantial flexibility. Unfortunately,

do not

belong

to the

Johnson

[Johnson

technique of the fitted

Johnson translation corresponding condi-

system—and

this

lack

of

“closure” makes it impossible to obtain convenient closed-form expressions for the conditional distributions that naturally arise in many applications. Although DeBrota et al. [1989] developed a graphical interactive software system that enables the user to edit (manipulate) univariate bounded Johnson distributions, it is unclear that a similar tool could be based on the multivariate Other

distribution-fitting approaches

(Transform-Expand-Sample) 1992b; Melamed et al. 1992] cesses [Cario

procedure

to multivariate

and Nelson

1995].

processes and ARTA Both

of Stanfield

input

modeling

and Wilson. can

[Jagerman and (AutoRegressive

methodologies

enable

be based

on TES

Melamed 1992a, To Anything) prothe user to specify

the autocorrelation function out to an arbitrary lag for a univariate stochastic process with a user-specified marginal distribution, but ARTA processes seem to be substantially easier to use. Unfortunately, the conditional distributions associated with TES and ARTA processes do not appear to possess any advantages in analytical or numerical tractability when compared to multivariate processes based on the Johnson translation system. Software packages for fitting TES and ARTA processes are not widely available at the present time. In this article we extend the univariate input-modeling methodology of Wagner and Wilson [1993, 1995] to handle continuous bivariate populations with bounded support, and we present a flexible, interactive, graphical technique for modeling a broad range of bivariate simulation inputs. We employ B6zier surfaces as the parametric form for representing the distribution function of continuous bivariate random vectors that are to be randomly sampled in a simulation experiment, and we show that the corresponding ACM

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on Modeling

and Computer

Simulation,

Vol. 5, No

3, July

1995.

Simulation Input Modeling marginal

and

conditional

variate B6zier Windows-based Environment.

distributions

belong

to the

original

.

family

165 of uni-

distributions. We implemented this methodology in a Microsoft software system called PRIME —l?Robabilistic Input Modeling A public-domain version of the software is available upon

request. The remainder of this article is organized as follows. In Section 2 we summarize the main properties of univariate B6zier distributions that are relevant for our development of bivariate B6zier distributions, and we establish some basic notation that is used throughout the paper. In Section 3 we detail

our methodology

ate B6zier

for constructing,

distributions

as well

manipulating,

as the

associated

and

sampling

marginal

and

bivari-

conditional

univariate B6zier distributions. In Section 4 we describe the implementation of this methodology in PRIME, including techniques for interactively fitting bivariate simulation input models using subjective information (expert opinion) or sample data. In Section 5 we present some examples illustrating the diversity of bivariate distributions that ogy. Finally, in Section 6 we summarize and

we make

recommendations

based on Flanigan and Wilson [1994]. 2. OVERVIEW

In many

research.

this methodolof this work,

Although

were

this

also presented

paper

is

in Wagner

BEZIER DISTRIBUTIONS

of B4zier Curves

applications

a smooth

interval

for future

some of our results

OF UNIVARIATE

2.1 Definition mate

[1993],

can be modeled using the main contributions

of computer

(continuously

by forcing

graphics,

a B&ier

differentiable)

the B6zier

curve

curve

univariate

is used to approxi-

function

to pass in the vicinity

on a bounded

of selected

control

points {p, = (x,, Z,)T: i = O, 1,..., n}. (Throughout this paper, all vectors will be column vectors unless otherwise stated; and the reman superscript T will denote the transpose of a vector or matrix so that each control point is understood points

to be a column

{pO, pi,... P(t)

, P.}

= [Pz(t;

vector.)

is given

n,x),

A B6zier

curve

parametrically

PZ(t;

n,z)lT

of degree

n with

=

~B.,,(t)p,

[0,11,

for t ~

~=o where

x - (X O, Xl,...,

ing function

B .,,(t)

x~)T

and

z = (Z O, ZI, ...,

k the Bernstein

control

by

z~)T,

and

where

the

(1) blend-

polynomial

n! tl(l Bn,l(t)

~

i!(n

– t)n-’

– i)!

,

[ o,

fort

=[0,

l]

and

i=

O,l,...,

n,

otherwise. (2)

B6zier

curves

graphically

have

based

certain

characteristics

approximation

that

of functions

are particularly [Farin

(a) A B6zier curve exactly interpolates its initial and final control means that the curve will pass through these control points. ACM

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and Computer

important

for

points;

this

1990]:

Simulation,

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166

.

M. A. Flanlgan Wagner and J. R. Wilson

(b) A B6zier curve in the location In

the

is edited under global control; this means that any change of a control point affects the shape of the entire curve.

definition

(a) of the

each

t G [0, 1], we have

mial

B.

trials

with

curve

traced

by P(t)

points

{p,:

i = 0,1, ...,

control,

the

greatest

at the

creases

from

decreases

exerted

that

on the

B6zier

P(t)

on the

that

control

Formulation

In this

curve

section

distributions. Wilson

[ 1993,

support

[ x*,

hull

of the under

shape

of the

control global curve

particular,

initial

in n

B&zier

is

as t in-

control

point

PO pn of

location

“magnets”; t E [0, I]}

the

P(t)

and

vicinity” the

the

on the B&zier “magnetic

by the

ith

control

t so that

curve.

attraction” point

p,

is

the corresponding

of p,.

If the

B6zier

curve

weight

(magnetic

is forced

to pass

exactly.

Bezler Probability

we summarize For a detailed x”]

the

B .,.( t) of the final control point weighted average (convex combination)

is 1, then

of Univariate

1995].

the

thus

edited

t. In

for the parameter

is “in

point

on the

B., O(t ) of

{P(t):

point

are

parameter

the current

t = i/n

curve

p,

for

polyno-

of i successes trial;

convex

curves

that

Bernstein

weight

act like

of a control

point the

weight the

determines points

at the value

attraction)

2.2

1, the

for

O to 1 in the overall

control

strongest

t = i/n

in the

B6zier

control

ith

t on each

t G [0, 1] lies

ith

1 to O and

points the

through

O to

from

control

all

the

probability

probability

n}. Although

of the

value

from

increases

point

effect

t = [0, 1]},notice

{P(t):

as the binomial

success

for

curve

1?.,, ( t) = 1 because

,( t ) can be interpreted

independent

Thus

B6zier

E;..

briefly

Distributions

some key properties

of univariate

development

of these

properties,

a continuous

random

variable

Given

and unknown

cumulative

distribution

Fx(. ) arbitrarily closely we can approximate with sufficiently high degree n, where

by a Biizier

B6zier

see Wagner X

with

and

bounded

function

(c.d.f.)

Fx (. ),

curve

of the form

(1)

x(t) = ~B@r, ~=o for all t G [0,1].

(3)

Fy[x(t)] = ~ B~,, (t)z, ~=fJ If Fx(.) is given probability density

x(t)

1

parametrically by Equation (3), then the function (p.d.f.) &-(.) is given parametrically

= i Bn,, (t)x, l=(J

f’x[x(t)l

= z

Bn-l,

~=o ACM

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z(t)Azt /

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Z B..l,, ~=o .%mulatlon,

I

for all t S [0,1],

n–1

n–1

corresponding by

(t) Ax,

Vol

5, No. 3, July

1995

(4)

Simulation

Input Modeling

167

.

with Axl

=Xl+l

‘Xl

Azt

=Zl+l

‘Zl

fori=O,

l,...,

n–l.

)

In Wagner and Wilson [1993, 1995], we presented several applications of the B6zier family of univariate distributions for modeling simulation inputs. This distribution family has a natural, extensible parameterization that allows unlimited flexibility in representing real-world processes. Moreover, because performed

efficiently,

simulation bivariate

this

family

input modeling. B6zier distributions

These that

3. FORMULATION

is

the probabilistic its numerical

well

suited

to

behavior evaluation

graphical

of many can be interactive

considerations motivated the extension is detailed in the next section.

OF BIVARIATE

to

BEZIER DISTRIBUTIONS

We begin the development of bivariate B6zier distributions by considering in Section 3.1 the setup for general two-dimensional B6zier surfaces. In Section 3.2 we specialize this setup to obtain a parametric representation for the c.d.f. FXY(”, ‘ ) corresponding to the B6zier random vector (X, Y )T with prespecified univariate marginal B6zier c.d.f.’s F’x(”) and FY( “); and in Section 3.3 we establish we

the required

derive

the

properties

parametric

Section

3.5 we formulate

Section method

3.6 for

Section

3.7.

a set

of

{qL, J - (~1, j, Y1,J, ZL. J)T: sponding two-dimensional

Q(tx,

for all

distributions.

bivariate

conditional

Bi$zier

c.d.f.’s

3.4

fx y (”, ” ). In

I” ), Fy ,x(”\. ); and in

Fxly(”

between random

In Section

p.d.f.

X and Y. An efficient vectors is presented in

of Bezier Surfaces

from

space that

the

of the

we calculate the covariance generating bivariate B6zier

3.1 Definition Starting

of the marginal

form

is given

control i =

points

0,1,..,

~x;

B6zier

~ =

surface

parametrically

ty) = [Qx(tx,

represented 0, 1)7

in

by

the

ny},

We

column have

three-dimensional

vectors the

Corre-

Euclidean

as

ty; nZ, ny, x),

Qy(tx,

=

~ ; ~=oj=o

Bnt, L(tx)Bny,,(ty)(

=

; t Bni, t(tl)Bn,,,(ty)qL,, 2= OJ=0

ty; nx, nJ, y), QZ(tx,

xL,,, y,,,,

ty; nx, n,y, z)]T

z,,j)T

(5)

tx, ty G [0, 1], where

‘“[XLJ]=FO:l ACM

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.

M. A. Flanigan Wagner and J. R, Wilson

Yo, o

Yo,l

““”

3’0, nv

Yl, o .

Yl,l ,

““” .

Yl, ny .

Yn=,l

““”

Ynz, rz,

[ Y=[yt,

ll=

\ Ynz,o L

>

and ~o, o

2.,1

...

~1, o

21,1

““”

Z=[z,j]=

z

respectively coordinates

denote

the

of the given

n.,

“

i?n

( n ~ + 1) x (n, control

.,

~

‘O, n Y ‘l,

‘-.

1

nY

Zn,, nY

+ 1) matrices

of the

x-,

y-,

and

Extending the discussion at the end Bernstein polynomials (2) in regulating

of Section the shape

2.1 about the role of the of a B6zier curve, we see

that the geometry of a B6zier surface is determined by weights of the B~,,,(t, )BmY,~(tY), where for each (tX, tY)T = [0, 1] X [0, 1], we have

~ ~ B~r,,(tX)B Z=OJ=O Thus hull

the B&zier

as both

tx and

form

.yj(t)=[~OBn,(,)][}oB.,,(y)]=ll=l

surface

of the control

z-

points:

{Q(tX,

points

ty) : (t,,

ty)T

●

{q,, ~: z = O, 1,...,

ty increase

from

[0, 1] X [0, 1]} lies in the convex

nX; j = O, 1,...,

O to 1, the

weight

nY}. In particular

B.,, o(tX)B~Y, 0( t,)

of the

initial control point qo, o decreases from 1 to O and the- weight B ~z2~JtX)B .,, .JtY) of the final control point qnt,,, increases from O to 1 in the overall weighted average (convex combination) of control points that determines the current location Q(tx, ty ) on the B6zier surface. Thus the control points act like magnets; and the magnetic attraction exerted on the B6zier surface {Q(tx, ty) : (tx, ty)T G [0, I] X [0, I]} by the control point qL, J is tx and tY strongest at the values t, = i/n Z and ty = J“/nY for the parameters so that

the

corresponding

q,,,. If the weight surface is forced 3.2

Q( tx,ty) on the

(magnetic attraction) to pass through that

Bezier Distribution

Bivariate

If (X, Y )’

point

is a continuous

surface

is in the vicinity

of a control point is 1, then control point exactly.

of

the B~zier

Functions

random

vector

with

bounded

support

[.x*,

x*]

x

p.d.f. &Y(., . ), then we can [Y., y’], unknown c.d.f. F.~Y(”, “ ), and unknown approximate Fry (., . ) arbitrarily closely with an appropriate B6zier surface of the form (5) that has sufficiently large values of n ~ and n ~ [Farin 1990], where the control points {q,, J: i = O, 1,....nX; j = O, 1,....ny} have been arranged so as to ensure the basic requirements of a joint distribution function: (a) FAY~,( x, y) is monotonically nondecreasing and continuous from the right in x and y; (b) FXY(XX, y) = O for all y and FXY(X, y*) = O for all ACM

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Simulahon x;

(c)

FXY(X*,

y*)

= 1; and

(d)

FXY(ZZ,

Yz) – FXY(XI,

FXY(XI, yl) >0 if xl < X2 and YI < Yz. Given marginal I%zier c.d.f.’s Fx(.) and and

Y, respectively,

(X, Y)T,

where

we seek

(i) Fx(”)

tX

●

Fx[x(tX)]}T

[0, 1]; and (ii) {y(tY),

FY(.)

parametrically

is represented =

~

the

Yl)

+

variables

random

X

vector

by

===~ BJtJ[xz, ~=()

FY[y(ty)]}T

random

Fx Y(., “ ) for

169

.

Y2) – Fxy(xz,

FY(. ) for the

c.d.f.

is represented

{x(tJ, for all

a joint

Input Modeling

(6)

@]T

parametrically

B#y)[yJ,

by (7)

Z;Y’]T

J=()

for all tY

●

[0, 1]. To satisfy

of ( X, Y )T according

Equations

(6) and (7), we formulate

to the parametric

{x(tJ,y(tY),Fxy

=

~

representation

the joint

c.d.f.

(5) such that

[x(tX),y(tJ]}T

Bnl,z(tx)Bn,,,(ty)(x,,,,y,,,, z,>,)T

:

(8)

~=OJ=o

for all nates

tz,

ty

E

[0,

1], where

of the control

points

X.

the matrices {q,, ~} have

X.

““”

~, y, and

z of x-, y-, and

the respective

X.

[ Y=[Y,,

z-coordi-

forms

JI=

,

Y(l

Y1

““”

Yn,

Yo

Y1 . . . Y1

““” .

Yn, . . Yn,

. . Yo

1 .

““”

7

(9)

and

Z=[z,,

00

...

0

.. .

‘Z1, l

)l = [ :

Oz o

‘1,

z~(x)

n>–1

.

:~ ll–

1,1

(Y)

z~

““”

Zn t –1,7L–1

...

Z(Y)

~:x~ ,

~

of the matrix ~ ensures surface, we have

that

in the

(lo)

. ~

‘1

1

nj–l

[ The special form associated B6zier

0

0

1

definition

(5) of the

n,

ACM

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1995

170

.

M. A. Flanigan Wagner and J. R. Wilson a univariate B6zier function of tx alone, which (8). Similarly, denoted by x( t,) as in Equation y ensures that

thus Equation (11) defines simplicity is subsequently special form of the matrix

= f Bny,,(ty)y,;

Qy(t,,ty;n.,n,,y) thus

Equation

the first matrices

paragraph x, y, and

B6zier

is sufficient

YO =Y.

to ensure x

,

requirement

~i

=X*,

of

t,Y alone,

which

for

(8). c.d.f, mentioned

in

the following

z,. j =0

and

ifi=O

conditions

or

on the

j=O

(13)

(b); and the condition

Y., =.v*,

is sufficient to ensure requirement presented in Section 3.4 on bivariate (d) are satisfied

function

by y(tY ) as in Equation (a)–(d) of a bivariate

of this section, we impose z. The condition

x(j =x*,

(12)

~=o

( 12) is a univariate

simplicity is subsequently denoted To satisfy the basic requirements

and

z,,,,.,

(14)

= 1

(c). Finally, it follows from the B6zier p.d.f.’s that requirements

results (a) and

if

ZL+l,J+l —~l, j+l — ~t+l,j+2 ,J>o fori=O,

l,...,

Remark. points

nlandj=O, If

for the

the

l, l,...

individual

are nondecreasing

x-

(that

(15)

,nY–l.

and

y-components

is, if XO < xl


0 for

G [0,

1].

p.d.f.

B6zier Distributions ), the

conditional

If tX, t, G [0, 11 and

c.d.f. of X at the

fY[y(ty)l

>0,

point

x( tx ) is obtained

as

Vol. 5, No 3, July

1995.

then

Fxly[x(tx)Iy(t.)1

ACM Transactions

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174

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M. A. Flanigan Wagner and J. R. Wilson

It follows that B6zier with

the conditional

distribution

of X given

Y = y(tY ) is univariate

provided ~y[ y(tY’)] >0. Notice that the control points {[ x,, z~xlyJ]T: i = 0, 1, . . . . nt} for the B&zier curve representing the conditional c.d.f. of X given Y = y(t~ ) have the same x-coordinates as in Equation (6); and the corresponding z-coordinates are given by

nv—l Z(XIY) [

=

Z

8Z-,,,(~yYZ,,,+I

(26)

nv—l

for z = O, 1,... , n ~. An analogous tional c.d.f. of Y given X.

formulation

ACM

Simulation,

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-z,,,]

,=0

and Computer

yields Vol

Fy ,x( . I ~), the

5, No. 3, July

1995

condi-

Simulation 3.6

Covariance

The

covariance

control

points

between between

Input Modeling

.

175

Bezier Variates X and

{ql, ~; i = O, l,.

Y, Cov( X, Y ), is readily

... nf;

.j = O, l,.

computed

from

the

... nY}. We have

COV(X, Y)

=/”/”[ —. —.

= nxny/l/l[

x – pxl[y

x(u)

00

–pylfxy(~>Y)ci~clY

-

I%yl[y(w)

-

(27)

Pyl

where

fori=O,l,...,

nX and

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176

M. A. Flanigan Wagner and J. R. Wilson

.

E[ Y 1) forj=O, l,. ... rzY. Notice that the expected value E[ X] (respectively, and the variance Var[ X ] (respectively, Var[ Y ]) are readily evaluated using the computational formulas (17) and (18). Thus we can easily compute Corr( X, Y ), the correlation between X and Y.

Generation

3.7

of B6zier Vectors

The random vector (X, Y )T can be generated efficiently using a variant of the method of conditional distributions that we call “ piecewise conditioning.” To explain piecewise conditioning, we first summarize the conventional method of conditional distributions. Given a pair of independent random Method of Conditional Distributions. numbers UI and Uz, we compute Y from UI by inversion of the marginal c.d.f. FY(. ); then given Y, we compute X from Uz by inversion of the Specifically, this involves the following steps. conditional c.d.f. F ~lY(.lY).

1. [Generate

random

numbers

Ul, Uz.]

Generate

Ul, Uz - Uniform[O,

1] indepen-

dently. 2.

[Compute Z, = Y- 1[F= l(U1)I.I Use a root-finding procedure such as bisection [Conte and de Boor 1980] to find the root f, of the equation

search

q U1 within

the interval

=FY[Y(~.y~]

of uncertainty

=

,;O%,,,WT

(28)

[O, 1].

3

(26) with t, = i, to [@mPute control Points of F.YIY[ I Y( i,~l.1 Evaluate Equation compute the z-coordinates {Z~z-lyl. ~ = O, I, . . . . ~ ~} of the control points for the conditional c.d.f. of X given Y = y(~}).

4,

[Compute ~X = x-1 {F’~l~[ Uz IY( i, )]}.] Use a root-finding search to find the root tx of the equation

u, = F.yly[x(il)ly(iy)]

= ~

procedure

such as bisection

Bn,,, (ix)zy)

(29)

Z=()

within 5.

the interval

of uncertalnt

[Return (X, Y )T.] Deliver

(X, Y)T=

y [0, 1].

the vector

[x(i),

y(iJ]T=

fll,lx,l(;.t)x,,;

[ ~=~

T

Bny,,(i,)y,

J=(J

The chief disadvantage of the conventional method of conditional tions is that the root finding operations required to solve Equations (29) are relatively slow.

I

(30)

distribu(28) and

Method of Piecewise Conditioning. The objective of the method of piecewise conditioning is to accelerate the root-finding operations required to solve Equations (28) and (29) by exploiting a precomputed partition of the range of values of the marginal c.d.f. of Y together with the corresponding precomputed partitions of the range of values of the conditional c.d.f. of X given Y. A formal statement of this algorithm is given in the following. ACM

TransactIons

on Modeling

and Computer

Slmulatlon,

Vol

5, No

3, July

1995

Simulation

Piecewise

Conditioning

Input Modeling

177

.

Algorithm

O. [ Initialize—set up partitions of the ranges of F’Y() and F’x,~( I ~).1 a. Compute the partition of the range of the function Fy(. ) on [ y*, Y*1, [y(ty(g)),

FY{y(tv(g))}lT

forg

= 132, . . ..g~ax.

where the cutoff values {tv(g ): g = 1,2,. ... g~..} b. For each y(t,(g))(g = 1,2,..., g~., 1 compute function FxlY[.ly(t,(g))] on [.~.,, X*I, [x(tz(h)),

F’xlY{x(tL(h))l

y(tY(g))}lT

are regularly

the partition

forh

(31) spaced in [0, 11.

of the range of the

= 1,2,...,

h~~k,

(32)

where the cutoff values {tX(h): h = 1,2, ..., h~.x} are regularly spaced in [0, 1]. U], u? - Uniform[O, 11 indepen1, [Generate random numbers UI, Uz. 1 Generate dently. ] 2. [Compute ;Y = y-l[F~l(ul)]. a. Find the subinterval of uncertainty for fY corresponding to the appropriate subinterval of the partition (31); that is, find ~ such that Fy[y(ty(g

– l))]

< UI s

implies

FY[.Y(tY(E))l

ty(g

– 1) < ~, S ty(+?).

(33)

b. Starting

with the initial interval of uncertainty (33) for ~Y, use a modified bisection search to find the root ~y of Equation (28) in the interval (33). 3. [Compute control points of Fxly[.l y(t,)l. 1 Evaluate Equation (26) with t., = f, to compute the z-coordinates {Z$xly ’: i = O, 1, ..., n ~} of the control conditional c.d.f. of X given Y = y(iY). 4. [Compute ZX= x-l{l’j+.[Uz \y(ZY)l}. a. Find a subinterval of uncertainty-for iX corresponding defined by g = ~ – 1; that is, find hl such that

Fxly[x(tJk, -

l))ly(t,(g - 1))]
0

z(x) 1)

\

for all tX G [0,1]

— 20(.x) —o

),

(37)

z(x) = 1 ~. X() < x ~z >

are the order statistics have been incorporated mum LI norm hood estimation, these

fitting

appropriate is

used

to

x(m)

1

for the sample { X~ }. The classical fitting methods that into PRIME include: least squares estimation, mini-

estimation, moment

minimum LX norm estimation, matching, and percentile matching.

schemes

from

distance

function

fit

x(l)

a drop-downAmenu

a univariate

d{ F~(. ), I’x[.; marginal

implicitly

involves

likelione of

selecting

an

nX, x, Z(x ‘]}. The same procedure

B6zier

sample data set {Yh }. Our computational ison to other widely used optimization

maximum Selecting

c.d.f.

experience procedures,

~Y[.;

ny, y, Z(‘)]

to

the

indicates that in comparthe Nelder-Mead simplex

search procedure is faster and more stable in solving the optimization problem (37) for each of the distribution-fitting methods mentioned; see Flanigan [1993, P. 44] and Swain et al. [19:8]. After fitting marginal c.d.f.’s F’x[; rzX, x, z(y)] and fiy[.; nY, y, z(y)] separately to the corresponding components of the random sample {( X~, Yk )T: k = 1,2,..., m}, the user can model the dependencies between these components. with

The dependencies either

the

joint

are modeled B6zier

c.d.f.’s or p.d.f.’s until the next section we illustrate bivariate

B6zier

desired both

distributions ACM

p.d.f.

Transactions

using

by moving the control points associated conditional B&zier fxY (”, . ) or selected

stochastic subjective

dependence is achieved. In the and data-driven estimation of

PRIME.

on Modeling

and Computer

Simulation,

Vol. 5, No 3, July

1995

182

M, A, Flanlgan Wagner and J, R. Wilson

.

Edit

.

NEGPROC.BIV Qptions

3

~le

lJisplay

F~lX=x]

close

Edit

LJpdate

d

b 0.900000 L

fk

show

!ndep

Y] Covariance:

FfT’lX=2.5]

-0.230 1.000 T

Correlation:

1

-0.369

>f

0.9000

!5L-JL

x

0.02.04

.06.08.010.0

F~lY=y] Qlosc

Edit

Llpdate

show

!ndep F~lY=7.0]

1.000 ~

F[x] /

0.800

0.600 0.400 0.200 0.000 L 4$ 0.01.22

1.0001-

/

F[y)

--

0.800

;~( f $L {, (. / ‘ i r.43,64.86.0

Fig 2. A

1.000

bivariate

,/’

0.600

0.400 0.200 ~o.000 L @ distribution

f _.

.J

i ‘~

0.600

Ii

0.400

/

0.200

‘>. >

0.000

times

(X, Y)T

/,/

/

!!, “

F~lX=x]

UNDWNEG.131V

Eile

Cdit

Display

close

Qptions

Edit

~pdate

t 0.1 ki

Covariance:

-5.322

Correlation:

-0.645

show

!ndep

Y]

fk

183

.

F~lX=9.0]

—--....-———

1.000 0.800

0.1000

.,,’

0.600

,,’

0.400

..-,?

/“

0.200

& ;,

H

x

0.02.04

.06.08.010.0

r.... LJpdate

“\.-

- -----

L

0.000

Indep

F~]Y=2.0]

F[x]

1.000 ~ ].800 1.600 +

..

F[y]

1.000

..

0.800 0.600

.,-

1.000

...

0.800

: ,,,’

0.600

i

i

,’”

0.400 0.200

___ .--——

1

~ 0.000 *

0.02.04

@

.06.08.010.0

0.0 2.0 4.0 6.0 8.0 10.0

Fig. 3. A negatively correlated distribution

until

the correlation

of the fitted

joint

distribution

allowed

to move

in the vertical

direction

distributions and the special structure Equation (9). In Figure 2 the displayed

,

t-+--!

I

0.02.04.06,08.0100

1,

with uniform marginals.

to – 0.37. The conditional c.d.f.’s are edited marginal c.d.f.’s, except that the control points only

,’”

was

approximately

equal

in. the same manner as the for the conditional c.d,f.’s are so as to preserve

of the matrices joint distribution

the marginal

x and y defined has a correlation

by of

– 0.369. 5.1.1

Uniform

Marginal

Figure

Distributions.

3 depicts

a PRIME session

in

which each marginal distribution is uniform; that is, X N Uniform[O, 10] and Y - Uniform[O, 10]. Figure 3 displays a bivariate distribution for (X, Y )T with COV(X, Y) = —5.322 and corr(X, Y ) = —0.645. Beneath the window containing the joint p.d.f., there are two windows displaying the marginal c.d.f.’s Fx(.) and FY(.); and these latter windows also display as dashed curves the corresponding marginal p.d.f.’s fx(.) and fy ( .). To the right of the joint

p.d.f.

window

in

Figure

3 are

two

windows

depicting

the

conditional

also display as c.d.f.’s FY ,X(.19.0) and FXIY(”12.0); and these c.d.f. windows dashed curves the corresponding conditional p.d.f.’s. As shown in the joint p.d.f. window of Figure 3, most of the probability mass is concentrated along theline y=–x+lOfor O