propagation of uncertainties in deterministic systems

004s7949187 L3.00 + 0.w Perg;rmon Journals Lti.

PROPAGATION OF UNCERTAINTIES IN DETERMINISTIC SYSTEMS

W. M. DONG?, W.-I_. Cwmcit and F. S. WONG$

tThe John A. Blume Earthquake Engineering Center, Civil Engineering Department, Stanford University, Stanford CA 94305, U.S.A.

~WeidIinger Associates, 620 Hansen Way, Suite 100, Palo Alto, CA 94304, U.S.A.

(Received 14 March 1986)

Abstract-This paper addresses the propagation of uncertainties in deterministic systems. i.e. the system definition is known but the system parameters and input to the system contain uncertain information. The effect of the uncertain information on the system response is to be assessed, Three models of uncertainties corresponding to differing degrees of knowledge about the uncertainty are considered: interval, fuzzy and random. A method to propagate uncertainties expressed as intervals is described; the method, called the Vertex method, is based on a generalization of combinatorial interval analysis techniques. It is shown how the Vertex method can be extended naturally to treat the propagation of uncertainties modeled as fuzzy sets. Finally, propagation of random uncertainties is described using the classical probabilistic technique of derived distribution functions. The computational implications of the three models of uncertainties and the corresponding methods of propagation are contrasted. It is suggested that when the available info~ation is too crude to support a random definition, the interval or fury model should be used to take advantage of the expediency with which interval and fuzzy un~~ainties can be propagated and processed.

INTRODUCTION or randomness. The result of an experiment is uncer-

An engineering system can be modeled by an tain implies that there is randomness in the test. This

input-output relation. For simple systems or systems understanding is based on the so-called objective

which are well understood, engineers have developed probability point of view. Uncertainty is considered

certain functionals to describe their input-output in a frequency sense; chance can be measured by

relations. Consider a simply-supported beam of repeated experiments. It exists objectively and can be

length L with a concentrated load P acting at the verified in some way. In this interpretation, informa-

mid-point. The relation between load P and tion can be modeled as random variables, using the

deflection y at mid-span is given by the well-known well-develo~d probability theory. The dist~bu~~n

relation for the output of a system can be derived from distributions of the input random variables.

Y =$$ =.f(P),

However, there are other kinds of uncertainty

(1) which may not be interpreted as randomness. For instance in evaluating the bearing capacity of a

where EI is flexural rigidity and L is the span. The specific soil, an engineer may give an interval estimate

deflection y is then a function of P: P is the input, y of, say, 16-18 kips. What does this estimate mean? Is the be

is the output and J(P) models the beam system. EI aring capacity of the specific soil random? The

and t are the parameters of the system. When the answer to the second question is obviously no. The be

system parameters are variables, the system model is aring capacity has a certain value, but is unknown

a multi-variate function to the engineer. The interval that the engineer gave is only a subjective’ belief or guess. Althou~ the max-

Y = f(p, E, 1, L). (2) imum entropy principal or betting strategies can be used to “extend” the interval information into a

A general system is illustrated symbolically in Fig. 1. density distribution, the distribution does not exist The system is deterministic when the relation (func- tional) between the inputs and the output is fixed or certain. That is, when the inputs are certain, the output is also certain. When the inputs are uncertain, they can be propagated through the system without amplification of uncertainty. This paper investigates how various kinds of uncertain information related to the system input and parameters can be propa-

I; I/_

:

gated in a deterministic system. 47

By un~~ainty, most of people will refer to chance, Fig. 1. System ~p~ntation.

415

416 W. M. DOKG CI al

and is not supported by data. In this cm. it is cquall!

if not more reasonable to use the interval number as the model of the uncertain information, instead of interjecting extraneous assumptions on the nature of the information.

Engineers use not only numbers but also words to convey certain concepts and communicate with one another. For example, “about I7 kips” is a concept. If the denotation of this concept is quantified by the interval 16-18 kips, any value in this interval will belong to “about I7 kips” but a value such as 15.9999 will not. Clearly, the interval representation implies that there is an abrupt jump from belongingness to not belongingness when a certain crisp threshold is crossed. That is, of course, unrealistic. To circumvent such difficulties, Zadeh [I] established the theory of fuzzy sets which allows graduate change of belongingness using the membership function. The membership value for a certain element (or number) indicates the degree of belongingness to a specific concept, or the degree of qualification to be a member of this fuzzy set. Note that the membership value may have nothing to do with randomness or chance.

Depending on the quality and nature of the available data, a piece of uncertain information may be modeled as an interval, a fuzzy number, or a random number. Given that the system parameters and inputs are uncertainties of these types, it is of interest to know what the corresponding uncertainty of the system response is. This problem of uncertainty propagation is the subject of this paper, and will be discussed in more detail in subsequent sections. Emphasis of the discussion is on the computational aspects of the processing of uncertain information and, in particular, the implications of a particular type of representation (interval, fuzzy, or random) on the processing. Mathematical details are kept at a minimum in order not to obscure the basic ideas involved; details are deferred to the cited references. For the same reason, the examples used are designed to illustrate the concepts. They are not intended to be representative of the types of problems which can be addressed using the methods described in the paper.

UNCERTAINTY AS INTERVAL NUMBERS [2]

An interval representation of uncertainty states that it is possible to be any number within the interval but impossible to be without. In such a representation, a portion of the subjective belief is committed to a set (the interval) without allocating the belief to individual elements of the set. Hence, outside of the interval, the belief distribution is known; the possibility is zero (the minimum). Within the interval, the exact belief distribution is not known. The possibility for any element of the interval could be maximum (one) since no evidence exists which would refute such a possibility. Figure 2 shows the possibility distribution for the interval 16-18 kips.

;I’----r_l I5 I6 17 I8

P (klpsl

Fig. 2. Possibility distribution of an interval number.

Representation

Define an ordered pair of numbers [a,61 as an interval number I where a is the lower bound and b the upper bound, i.e., a I b. When a = b, the interval number [a,b] degenerates to a real number. An interval number can be referred to as a set of real number. By the set notation, interval number I = [a,b] can be written as

I=[a,b]={xlalxlb}. (3)

Hence, all the symbols for sets such as E, 3, U, fl can he used for interval numbers. In paticular,

x E [a,b] x belongs to [a,b]

[a, b] c [c,d] [a, b] is contained in [c, d].

Furthermore, the union and intersection of two intervals [a,b] and [c,d] depends on six possible cases as listed in Table 1.

Use the general operation * to denote any of the symbols, +, -;, /. The arithmetic operations on interval numbers are defined by

[a,b]*[c,d]={x*ylaIxIb,cI~Id}, (4)

with the restriction when * is /, 0 e [c,d].

Propagation

Just as the value of a variable x can be a real number such as 3, a, etc. we can define an interval variable, denoted by X, whose value is an interval number such as I, [a,b], [4,7], etc. Suppose that all input values of the function

)’ =/(x,, x2, .3x,) (5)

are interval estimates, X, = [a,,b,], i = I, n. Define the output Y as function of interval variables X,,

Y=fW,,...,X,)

=f{(x,, . ..I XJlX,EX I,... X,EX”}, (6) whose value usually would be an interval number.

Table I. Union and intersection of two intervals Lb] and [c,d]

Cases Union Intersection

1. a>d

2. c>b 3. a>c,b<d

lc,dl u [a,61 0 WI u lc,dl 0 k. dl

4. c>a,d<b 5. acccb<d

6. cca<d<b la.dl

It would be very convenient if one could use ordinary arithmetic operations on interval variables to compute the function value Y. For instance, set

y =x,.x*-x,*x,

Propagation of un~rtainties in deterministic systems

Given that the interval estimates are

X, = [6,8], C, = [600, 1500]

x2 = [lo, 121, c, = [500, 12001,

X, = [1,2], X,= [2,3], X, = [1,4]. (7)

Then, define the arithmetic expression

Y = F(X,,X,,X,) = x,-x, - X,*X, (8)

which leads to Y = [1,2].[2,3] -[1,2],[1,4]

= [2,6] - [I,81 = [-6,5].

Unfortunately, this is not the correct answer. The correct answer is Y = [ -4,4] which is a smaller interval than [ -6,5]. To check this, change the form of the function to

Hence,

Y =X,*(X, - X,). (9)

Y = [1,2]*([2,3] - [1,4])= [1,2]-[-2,2] = I-4,4].

In general, the following relation holds:

0X, t * . * 9 X”) = _w,, f . * f u, fW

which means that the interval by the sequential arithmetic operations always contains the actual function value. The reason for this excess is due to multioccurrence of any interval variable in the expression; when we process the variables sequentially, we treat two occurrences of a variable as two independent variables.

In order to reduce this excess, we can refine the intervals into small subdivisions

then

= (j F(X ,.,,, . . .,X,,,,.) c F(X,, . 0.1 U (11) ,t= 1

and when N -) co

lim Ftm(X,, . . . , X,)=/(X,, . , . , X,). (12) N-1

Example

Suppose that there are two brands of cement in a warehouse. There are 6-8 tons of Brand A at a cost of $600-1500 per ton, and IO-12 tons of Brand B at a cost of $500-1200 per ton. Determine the average cost per ton of cement.

Set X,-the weight of brand i cement C,-the cost of brand i cement.

The average cost per ton is

y_w,+w2

x,+x, ’ . -

From the above exampie, one can see that interval types of info~ation are very easy to process using (13) interval operations and the Vertex method.

417

y = F(X,, x,, c,, C,) = c,.x, + c>.xz

4 + x2

= [3600,12000] f [SOOO, MOO]

[16,201

= ‘86;10; ;y = [450,1650]. 9

Obviously, this is not correct since intuition dictates that the average cost should be contained in the interval C, U C, = [500,1500].

By dividing each interval into two subinte~als, one gets

= [479,1482] c [450,1650]. ( 14)

While the refinement does improve the result, it also involves much computation effort. For an n-variate function, and each interval variable is divided into N subdivisions, the interval expression must be computed N” times.

To facilitate the computation, we introduce the Vertex method [3,4]. For an interval function such as

Y=f(X,,...,X,) (15)

all interval variables form an n-dimensional rectangle x, x . . . X, with 2” vertices. The ordinates of al1 vertices are actually the combination of n pairs of end points of interval numbers. We use v, to denote the jth combination (or the jth vertex) j = 1, 2”.

When y=f(~~,..., x,) is continuous in the n- dimensional rectangular region and there are no extreme points inside the rectangular region, the value of the interval function can be obtained by

Y=f(X,,...,X,)

= min (T@,B, max WV,)) 1 . (16) i I

For the above example, the list in Table 2 is obtained which can be used to compute the value of Y:

Y = [533,1333] c [479,1482] c [450,1650].

418 W. M. Doso PI ol

Table 2. Example computations usinp the vertex method

i Y, ./(r,)

I (6,600, 10, 500) 537 2 (6,600, 10. 1200) 975 3 (6,600, 12,500) 533 4 (6,600, 12, 1200) 1000 5 (6,1500, 10,500 860 6 (6. 1500, 10, 1200) 1312 7 (6, 1500, 12,500) 833 8 (6, 1500, 12, 1200) I300 9 (8,6@tO, 500) 544

10 (8,600, 10, 1200) 933 11 (‘3,600,12,500) 540 12 (8,600, 12, 1200) 960 13 (8, 1500, IO, 500) 944 14 (8, 1500, 10, 1200) 1333 15 (8,1500,12,500) 900 16 (8, 1500, 12, 1200) 1320

UNCERTAINTIES AS FUZZY NUMBERS

If the information is of the interval type, it is appropriate to model them by interval numbers and use the method developed in the previous section to process them. However, as mentioned before, there is a jump between “possible” and “impossible”, and between member and nonmember at the boundaries of the interval representation. Most people would prefer a gradual change in this regard.

Representation

The abrupt jump associated with an interval number is related to the concept of a crisp set. An element is either in the set or not in the set. This is sometimes denoted by a characteristic function such as that

(Cl)

d -_-_-.- . s

,*. - ‘,‘\ ,#,>

, \\ 8%

I’ 0:

::

:

:: A 11

:: ‘#’

:: ,+

,‘a’ ‘:::di’ .x

(b)

illustrated in Fig. 3a. To relax the binary (yes and no) nature of the belongingness to a set, Zadeh proposed using the membership function

P”(X) l P, 11. (17)

Note that the value of pA can be 0, 1 or any value in between (see Fig. 3b). A load of “about 17 kips” can be modeled as a fuzzy set as shown in Fig. 4.

Note that the membership function can also be interpreted as the possibility distribution discussed in previous section. The membership function has a little bit more knowledge about possibility. For instance, in Fig 4, value 17 has possibility 1, value 16 has 0.5 and so on. Fuzzy set is also a special case of Dempster and Shafer’s belief function (so called consonant belief function) [5].

If we set a threshold level, say 0.5, and assume that all elements x with p,(x) 2 0.5 are members of the fuzzy set I, then we will have an interval I,, = [ 16,181, which are collection of members with threshold level 0.5. In general, we will have I, = {x [p,(x) z a}, called a-cut as member set with a degree of qualification. We shall exploit this extended interval representation of fuzzy numbers in the following discussion.

Propagation

Given that

Y=f(x,,x,,...,%) (18)

and the inputs x,, .,x, are fuzzy numbers, X,, . ,X,, evaluate Y:

Y=f(X I,..., X,). (19)

^ 1 __a-___

* -I n 2

,I 0 b

X

Fig. 3. Extending crisp intervals to fuzzy sets. (a) Crisp set. (b) Fuzzy set.

Propagation of uncertainties in deterministic systems 419

I5 16 17 I8

/J 1 klps 1

Fig. 4. Possibility distribution of a fuzzy number.

Note that this is an extension of a function of ordinary variables to fuzzy variables. There are many methods to implement the computation of this function [69]. What we suggest in this paper is an approach based upon the combination of a-cut concept and interval analysis (Vertex method).

Take any value a E [0, 1). The a-cuts of all variables X, are intervals X,.,, which can bc intcrprctcd as members of X, with a or above degree of qualification. Then we have

y, = fGf,,,Y . > X”,,), (20)

which is the set of members with at least a degree of qualification as members of Y. This can be easily done using the interval operations described in the previous section. With differing a values, we obtain the fuzzy number Y as

y=$&Y,+Y,+..., (21) .a aI 4

where Z and + stand for union.

Example

Change the condition for the cement cost problem as follows:

X, = about 7 C, = about IO50

X2 = about 11 C2 = about 850

with their membership function as shown in Fig. 5.

Select a = 0' ,

x,,cl+ = [6,81,

c ,,o+ = [600,1500],

x 2.0+ = (10,121,

c 2,fJ+ = [SOO, 12001,

which are the same as the original example,

Y o* = [533,1333].

Select a = 0.5,

X ,.o.J = [6.5,7.51,

C 1.0.) = [825,12751,

X 2.0.) = [10.5,11.51,

c 2.0 5 = [675,10251,

Y O,S = (729,1129].

Select a = 1 .O, all a-cuts degenerate to numbers,

XI.1 = [71,

C,,, = [10501,

x2.1 = [111,

Cz., = [8501,

Y, = [928].

Figure 6 gives the membership function for the average cost Y.

- I

0’ 1” 0 5 6 7 8 603 1500

x2 C2

Fig. 5. Fuzzy quantities and fuzzy costs for two brands of cement.

420 W. M. DOS Ed al

7 - 05 . zi

0 533 723 928 1129 ‘333

Y

Fig. 6. Fuzzy average cost. ?’

UNCERTAINTIES AS RANDOM VARIABLES

A brief discussion on calculating the output distribution through a memory-less transformation is presented in this section. For the most general case, set

)’ = f(x,, s*, , x,),

where X,, i = 1, n, are random variables with

P(Xl, x27 . . ,x,) as their joint probability density function. Then the probability density function of Y is:

where

h,(x,, . ” 1 x, _ , , y) is the i th solution for x, from the equation Y =/(x,. . . , 3,)

J, =

Cl,. is the integration domain of variables

100. 0 0 0 10. 0 0

(x,, ,x,_ ,), which, in general, depends on Y. It should be noted that equation (22) can be found, in different forms, in most of the statistics text books [ 10, 111.

A simple example is solved to show how equation (22) can be used. Let

Y = x, + 2x,

P(X,) = f, 15x,53

p(x2) = 4, 1 5 x1 I 6.

There is only one solution of x1 from the equation

Hence.

It is necessary to divide the integration domain n, into three intervals, i.e. 3 5 Y _< 5. 5 I Y < 13, 13 I Y I 15. Application of equation (22) gives

Finally,

MY) =’ +f+dx,

)-I

b fdx,=$+3) 31.~~5

P,(Y) = i

3 &dx,=$ 511’1 13

It should be noted that the difficulty in utilizing equation (22) to calculate the derived distribution arises from the y-dependent nature of the integration domain R.

To investigate the cement cost estimation problem in the probabilistic setting, assume the costs and weights of each brand of cements are uniformly distributed independent random variables. The problem can now be stated as follows. Given

p(xl)=f 61x,58

/7(X?)=; 105X,I 12

p(c,) = & 600 s c, I 1500

p(c*) = f 500 I c* I 1200

and

X,‘C, +x,.c, )‘=

x,+x,

What is the pdf of Y? Unfortunately, the integration domain zl, for this

problem is so complicated that a close form solution of equation (22) is impossible. It is rather surprising that an analytic solution of the derived distribution of y is not achievable for such a simple input-output relation. In view of the analytical difficulties, a sequential operation is used instead. The procedure is as follows.

Step 1

Y = x, + 2x,, y--t i.e. xt = - 2

Calculate the pdf for Z, = c,x,.


step 2

Calculate the pdf for z9 = :I + r2.

Step 3

Calculate the pdf for y = 2. 23

Schematic presentations shown in Figs 7 and 8 explain how the integration domain a,, is divided into subdomains. Figure 7 shows the subdomains for evaluating the pdf of z, = xt *cl. The subdomains are between the adjacent elliptical isocurves which pass through the nodes. There are three such domains. Figure 8 shows the subdomains for evaluating the pdfofz,= z, + 2,. The subdomains are between pairs of adjacent 45” straight line which pass through the nodes. There are 15 such domains.

Finally, to calculate y = Z./Z,, the integration domain must be divided into 4 x 16 - 1 = 63 subdomains. The pdf of y must be evaluated in 63 intervals, which is really exhausting.

Several points need to be made at this time. First, it is, in general, not possible to get the analytic solution for the derived distribution of output y except when there are only a few random variables or the input random variables have special property (e.g. y =x,x2.. . x,, and the X,S are independentIy and lognormally distributed). Second, the derived distri- tion of the output y using the sequential approach have a wider spread than the “exact” derived di. bution of y because multi-occurrences of the same random variable are treated as independent random variables. We note also that the use of the ~quential approach is not supported by mathematical theory. it is presented in the above example to emphasize the di%cuity of getting a closed-form derived distribution.

Since an analytic solution is not available in most cases, numerical methods must be used to solve the problem. A brief account of the numerical study based on discretization and Monte-Carlo simulation is given in the following together with some numerical results for the cement cost estimation problem.

For an input-output relation y =1(x,, . . . ,x,X the state space of the input random xi, which is assumed continuous, is discretized into Nj intervals. The n-dimensional state space is then divided into

cells. Each cell is represented by its centroid and has a probability mass in proportional to its volume. Numerical calculation in terms of the centroid is carried through all the ceils to evaluate the derived distribution of the output. For the cement cost

estimation example, 50 subintervals are used for each input random variable, resulting in SO’ cells and the histogram shown in Fig. 9.

The accuracy and smoothness of the resulting histogram depend naturally on how fine the state space is divided. For the present example, only four random variables are involved and the state space is bounded. Hence, an accurate result can be obtained. without much effort. However, as the number of input random variables increases, the computing effort will increase significantly and can make the numerical calculation intractable.

Monte-Carlo simulation

Monte-Carlo simulatio~l is a simple but time- consuming method. A large number of samples is necessary to achieve a statistically meaningful result, We apply this method to study the cement estimation problem, and ten thousand points are simulated for each input variables separately. The output histogram is shown in Fig. 10.

CONCLUSON

Three different methodologies to handle various types of info~ation have been described. Which method is used depends on the re~nement of information and its interpretation. Generally speaking, the possibility measure which has ignorance imbedding in it, is much more flexible than the probability

Fig. 7. Subdomains for evaluating the pdf of 2, = x,‘c,.

Fig. 8. Su~omains for evaluating the pdf of z, = 2, -I- z2.

422

0324 t

ooie 0017 0016 0015 0014

? 0013

cl 0012 001 I

001 0009 0008 QC.07 OK6 0005 0004 0033 0002 0031

0 500 loo0 1500

Cost

Fig. 9. Histogram of the average cost of cement computed by numerical discretization

Fig. IO. Histogram of the average cost of cement computed by Monte-Carlo simulation.

measure. For instance, the possibility of I for one member does not exclude the possibility of some value for another member. The probability measures, on the other hand, must sum up to 1. If one element has probability 1.0, then all others must have zero probability measure.

The cement cost estimation problem is carried through using all three methods so that the different approaches can be contrasted. It is not suggested that probability theory and fuzzy theory are equivalent. The selection of methodology should conform with the “nature” and quality of information, i.e. fuzzy theory deals with fuzzy information and probability theory handles random information. Furthermore, the interpretation and usage of the results also de-

pend on the choice of methods. The interval evalu- ation gives a range within which any value is possible. The inferred fuzzy number also gives a range within which any value is possible, but with different degrees of belief or confidence. For instance, in the cement problem, $729 is the weighted average cost with possibility 0.5. Finally, the derived distribution contains frequency information, i.e. what is the probability that the cost would be in a specific interval. Hence, in evaluating the results from the different methods towards decision making, different criteria may be necessary. This problem will be addressed in another paper which describes how random numbers, fuzzy numbers and interval numbers can be ordered in a decision context.


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propagation of uncertainties in deterministic systems

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