Fundamentals of Statistics with Fuzzy Data

Hung T. Nguyen, Berlin Wu Fundamentals of Statistics with Fuzzy Data

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Hung T. Nguyen Berlin Wu

Fundamentals of Statistics with Fuzzy Data

ABC

Professor Hung T. Nguyen

Professor Berlin Wu

Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003-8001 USA E-mail: [email protected]

Department of Mathematical Sciences National Chengchi University Taipei 11605, Taiwan E-mail: [email protected]

Library of Congress Control Number: 2005938944

ISSN print edition: 1434-9922 ISSN electronic edition: 1860-0808 ISBN-10 3-540-31695-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-31695-4 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006
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To Pattama and Ching-Min

Preface This monograph aims at laying down a rigorous framework for statistical analysis of fuzzy data. By fuzzy data we mean imprecise data which are recorded linguistically, i.e. expressed in some natural language as opposed to precise numerical measurements. Clearly, this type of data is more complex and general than set-valued observations (in, say, coarse data) which generalize data in statistical multivariate analysis. Fuzzy data need to be modeled mathematically before they can be subject to analysis. In this monograph, we will model fuzzy data as fuzzy sets in the sense of Zadeh. Postulating that those data are generated by random mechanisms, we will proceed to formulate the basic concept of random fuzzy sets as bona fide random elements in appropriate metric spaces. With this mathematical model for populations, we are entirely in the framework of standard statistical analysis. With this goal, this monograph is in fact more oriented towards probabilistic foundations than statistical procedures. We formulate, however, basic problems in statistics such as estimation, testing and prediction. We will also illustrate statistical methods for analyzing fuzzy data through some studies. It is our hope that this monograph will serve as a solid starting point for analyzing fuzzy data and thus enlarging the domain of applicability of the field of statistics. We thank Janusz Kacprzyk, editor of the series, for encouraging us to write this monograph. Special thanks are due to Ding Feng for allowing us to use his recent thesis on convergence of random sets. We thank our families for their love and support.

VIII Preface

Hung T. Nguyen December 2005

Berlin Wu

Contents Chapter 1. Introduction .................................................................... 1.1 What is Fuzzy Statistics?.......................................................... 1.2 Why Do We Need Fuzzy Statistics? ........................................ 1.3 How to Carry Out Statistical Inference with Fuzzy Data? ....... Chapter 2. Set-valued Data............................................................... 2.1 Random Closed Sets................................................................ 2.2 Coarse Data ............................................................................. Chapter 3. Modeling of Fuzzy Data ................................................. 3.1 Fuzzy Sets................................................................................ 3.2 Fuzzy Logics ........................................................................... 3.3 Fuzzy Relations ....................................................................... 3.4 D -level Sets of Fuzzy Sets...................................................... 3.5 Fusion of fuzzy Data ............................................................... Chapter 4. Random Fuzzy Sets ........................................................ 4.1 Back to Sampling Surveys....................................................... 4.2 Fuzzy Numbers........................................................................ 4.3 Fuzzy Set-Valued Random Elements ...................................... Chapter 5. Aspect of Statistical Inference ....................................... 5.1 Fuzzy Data in Sampling Surveys ............................................ 5.2 Coarse Data ............................................................................. 5.3 Large Sample Statistics with Coarse Data............................... 5.4 Random Set Data on Finite Spaces.......................................... 5.5 Random Set Data on Metric, Compact Space ......................... Chapter 6. Convergence of Random Fuzzy Sets............................. 6.1 Stochastic Convergence of Random Sets ................................ 6.2 The Choquet Integral............................................................... 6.3 A Variational Calculus of Set Functions ................................. 6.4 Choquet Weak Convergnece ................................................... Chapter 7. Fuzzy Statistical Analysis and Estimation ................... 7.1 The Nature of Fuzzy Samples ................................................. 7.2 Fuzzy Sample Mean for Fuzzy Data ....................................... 7.3 Fuzzy Sample Median for Fuzzy Data ....................................

1 1 2 3 5 5 9 13 14 19 27 28 31 35 35 38 39 45 45 46 50 53 67 71 71 73 79 87 109 109 113 115

X Contents

7.4 Fuzzy Sample Mode ................................................................ 7.5 Heuristic Properties Related to the Fuzzy Statistics ................ 7.6 Miscellaneous Applications ..................................................... 7.7 Concluding Remarks................................................................ Chapter 8. Tests of Hypothesis: Means............................................ 8.1 Introduction.............................................................................. 8.2 Soft Distance with Fuzzy Samples........................................... 8.3 Some Properties of Fuzzy Data................................................ 8.4 Hypothesis Testing with Fuzzy Sample................................... 8.5 Fuzzy F 2 -test of Homogeneity ............................................... Chapter 9 Fuzzy Time Series Analysis and Forecasting ................ 9.1 Introduction.............................................................................. 9.2 Fuzzy Time Series.................................................................... 9.3 Multivariate Fuzzy Time Series Modeling and Forecasting 9.4 Measuring Beliefs in the Forecasting Process ......................... 9.5 Empirical Studies ..................................................................... 9.6 Concluding Remarks................................................................ References........................................................................................... Index....................................................................................................

117 119 121 127 129 129 132 134 137 140 145 145 148 153 162 169 181 183 193

Chapter 1 Introduction First like fuzzy logics are logics with fuzzy concepts, by fuzzy statistics we mean statistics with fuzzy data. Data are fuzzy when they are expressed in our natural language. For example, the linguistic value “young” for the age of Tony in “Tony is young” is a fuzzy concept. We use natural language to describe phenomena when precise measurements are not available. The fuzzy concept “young” is intended to describe Tony’s age from our perception. As such it should “contain” the true age of Tony. However, the boundary of a set which supposes to “contain” Tony’s true age is not sharply defined. We will call this vagueness in the meaning of “young” fuzziness. We take fuzzy concepts in our nature language as primitives. We use our natural language to impart knowledge and information. This monograph is about the topic of fuzzy statistics. Thus let us address three basic questions. What is fuzzy statistics? Why do we need fuzzy statistics? How do we intend to carry out statistical analysis of fuzzy data?

1.1 What is Fuzzy Statistics? As stated above , fuzzy statistics is statistics of fuzzy data , i.e. making inference from data which contain fuzzy concepts such as estimation , testing of hypotheses and prediction, just like in standard statistical science. The closest situation to fuzzy data is set-valued observations in which data consist of sets rather than points of some domain of interest. Set-valued observations can arise in different contexts. For example, realizations of a point process are sets (which are locally finite). Here, we are interested in the observed sets themselves, i.e. the underlying population is a point process. In coarse data analysis, such as in biostatistics, the situation is this. The variable of interest X (population) is point-valued. However, X is unobservable, and instead we observe some coarsening of X, i.e. a random set S containing X with probability one. Here, the population of interest is H.T. Nguyen and B. Wu: Fundamentals of Statistics with Fuzzy Data, StudFuzz 198, 1–4 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com


2 Chapter 1 Introduction

X, and we need to use the imprecise data form the coarsening population S to carry out valid inductive logic about X. Set-valued observations are the most general form of imprecise data in standard statistics, as they generalize imprecise data such as censored data, missing data, grouped data. In this simple form of imprecise data, data are subsets of some set. Subsets have clearly defined boundaries. Fuzzy concepts are a step beyond concepts which can be described with ordinary set theory.

1.2 Why Do We Need Fuzzy Statistics? We rely on data, either by performing experiments or by observing phenomena, to discover knowledge. This is the essence of empirical science. The forms of data obtained depend clearly on problems of interest. When data are imprecise to the extend that they can be only described linguistically, we are facing fuzzy data. Risk assessments, or interest rates, say, by experts, are classified as “low”, “moderate”, and “high”; interest rates as perception-based information or behavior of dynamical systems are, in general, described in natural language; concept in economics and marketing such as “fidelity of costumers”, “effectiveness of promotions”, are intrinsically fuzzy in nature. Of course, it is possible to reduce fuzzy concepts which can be described by set theory, say, by putting thresholds. However, in one hand, whether such procedure will result in some significant loss of information, or, on the other hand, it is not appropriate to do so in view of subjectivity. This last point is similar to the concept of p-value in testing of statistical hypotheses as opposed to significant tests at pre-determined levels of rejection. In any case, there exist important cases in real-world applications in which we have to deal with fuzzy data. As such, it seems obvious that we need to extend standard statistical science to cope with this new type of data. In doing so, we enlarge the domain of applicability of statistical science. In the foreword to the book of Matheron [51], which lays down the mathematical foundations for studying set-valued observations (as a new type of data), G. Watson wrote “Modern statistics might be defined as the applications of computers and

1.3 How to Carry Out Statistical Inference with Fuzzy Data?

3

mathematics to data analysis. It must grow as new types of data are considered and as computing technology advances”. Due to the general form of coarse data and artificial intelligent technologies, we are facing fuzzy data as a new type of data, and as such, within Watson’s vision of statistics, the task of theoretical statistics is to propose appropriate frameworks to carry out statistical inference with this new type of data.

1.3 How to Carry Out Statistical Inference with Fuzzy Data? As pointed out by Frechet [25], nature and technology present various forms of random data. As such, a most general and appropriate framework for statistical analysis is needed. Traditional statistical data are vectors in Euclidean spaces or curves in function spaces. The corresponding sample spaces are metrizable, and metric spaces are appropriate mathematical structures for sample spaces. For set-valued observations (see e.g. [81], [24], and [61]), the sample space of closed sets in a Hausdorff locally compact, and second countable topological space is metrizable, when equipped with the hit-or-miss topology (see [51]). Data as sets are viewed as realizations of random elements with values in the space of sets. The situation for fuzzy data is similar. To arrive at a framework, we need first two basic steps: (i) Modeling fuzzy data (ii) Specifying metric spaces of fuzzy data We are going to use Zadeh’s fuzzy sets [117], see also [62] to model mathematically fuzzy concepts. The essentials of fuzzy sets will be presented in Chapter 3. Chapter 2 will be devoted to a simple form of imprecise data, namely random sets whose generalization to random fuzzy sets will be the topic of Chapter 4. Basic aspects of statistical inference with fuzzy data will be discussed in Chapter 5 where statistics with set-valued observations will provide the foundations for extensions. The Chapter 6

4 Chapter 1 Introduction

provides some asymptotical analysis and aspects of decision theory related to fuzzy data. The second part of the monograph, namely Chapters 7-9, is devoted to illustrations of applications with fuzzy data.

Chapter 2 Set-valued Data

Since fuzzy sets are generalizations of ordinary sets, we present in this Chapter the essentials of random set theory for statistics. This material is useful for considering random fuzzy sets later.

2.1 Random Closed Sets Set-valued mappings are formulated as random elements as follows. Let ʻ : ʿʳ Aʿʳ P) be a probability space. Let (V, v) be a measurable space. Then a random element with values in V is a map X : : o V which is A-Ȟ-measurable, i.e. X 1 (Q ) Ž A. The probability law of X is the probability measure P X on v defined by PX = PX 1 . We are interested in the case where V is a collection of subsets of some set U. In this case, the random element X is called a random set. Thus, a random set is a random element whose values are sets. When U is a finite set, V is its power set 2U , and v is the ı-field on 2U consisting of all subsets of 2U , i.e. the power set of 2U . Its probability law PX is then determined simply from its distribution function F: 2U o >0,1@ , where F(A) = P(X Ž A), or equivalently, by its capacity functional T: 2U o >0,1@ , where T(A) = P( X ˆ A z I ) = 1-F( A c ), where A c denotes the set complement of A, i.e. A c =U-A .

H.T. Nguyen and B. Wu: Fundamentals of Statistics with Fuzzy Data, StudFuzz 198, 5–12 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com


6 Chapter 2 Set-valued Data

Viewing U as a topological space with its discrete topology, any finite random set X is a random closed set, i.e. taking closed subsets of U as values. Random intervals [X,Y] on the real line R, where X, Y are real-valued random variables, are examples of random closed sets on Euclidean spaces. Point processes on R d are also random closed sets. For example, for Poisson process on R  , the collection ( Tn n t 1) of successive arrival times is a (almost sure) random set on R  . Indeed, for any t  R  , EN t = rt 0, we can find an f * such that || f  f * ||d H , where || ˜ || denote a distance between f and f * . Just like neural networks, it turns out that the approximation capability can be proved by calling upon the Stone-Weierstrass Theorem which provides design requirements for obtaining an appropriate dense subset of functions. Specifically, the Stone-Weierstrass Theorem is this. Let ( (U , G ) be a compact metric space. Let C(U) denote the space of real–valued continuous functions on U, and let H Ž C (U ) . Then H is dense in C(U) if H satisfies (i) H is a subalgebra of C(X), i.e. for a  R , and f, g in H, we have that af, f+g, fg belong to H.

34 Chapter 3 Modeling of Fuzzy Data

(ii) H vanishes at no point of U, i.e. for u  U , there is an h  H, such that h(u) z 0. (iii) H separates points. i.e. if u , v  U , then there is an h  H such that h( x ) z h( y ) . Now, the design methodology described above clearly depends on the membership functions A ji , Bi ; the t-norm ' and t-conorm ’ ; and the defuzzification procedure leading to the final output y*. Thus a designed methodology is a triple (M,L,D), where M denote a class of membership functions, L a class of logical connectives, and D a defuzzification procedure. Such a triple produces the input-output map y*= f*(x). The approximate function f* depends on the sample size, i.e. the number of rules in the rule vase, and is related as f n . The rationale or approximation capability of fusion of fuzzy data is that, under suitable conditions, on (M,L,D), the class of function { f n ,n t 1} is dense in the space of continuous functions C(K) defined on a compact subset K of Rd or R with respect to the supnorm. For example, let M consists of Gaussian type membership functions, i.e. A ji ( x)

D ji e

 ( x  a ji ) 2 / k ji

;

' = product t-norm, ’ = its dual t-conorm; and centroid defuzzification

procedure. Then, for any compact set K of Rd, the corresponding (M,L,D) approximates f to any degree of accuracy, this is only an existence theorem, not algorithmic.

Chapter 4 Random Fuzzy Sets

Statistical models for observations which are numbers or vectors are random variables and random vectors, respectively. Similarly, if the observations are subsets of some set, then appropriate statistical models are random sets of that set. Motivated by the desire to generalize numbers to fuzzy numbers and random closed sets to fuzzy sets, we establish here a reasonable concept of random fuzzy sets which can be used in statistical inference with fuzzy data.

4.1 Back to Sampling Surveys Since fuzzy sets generalize ordinary (crisp) sets, random fuzzy sets will generalize random sets. For the most advanced material on random sets, see [54]. Although we have recalled some basics of random closed sets in Chapter 2, we make here a return to the origin of statistics to re-emphasize the crucial role of making models in scientific studies, as well as connecting random sets to random fuzzy sets. Suppose we wish to select a part of a finite population U with N elements (| U | N ) to estimate, say the population total ¦ T (u) , where uU

T (u ) is the annual income of the individual u. To make this inductive logic valid, we need to be able to select a “good sample” as well as specifying the error of the estimator ¦ T (u) , where S is the selected sample. uS

To gain public acceptance, we should select the sample S at random. Thus, we create a random mechanism, just like a game of chance, to select samples. A sample is a subset of the population U. The random process leading to selections of samples is formally a random set. Samples are selected at random. For example, if we decide to select a sample of a fixed

H.T. Nguyen and B. Wu: Fundamentals of Statistics with Fuzzy Data, StudFuzz 198, 35–43 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com


36 Chapter 4 Random Fuzzy Sets

size n, then a probability sampling design could be a random set who probability density function on 2U (the power set of U) is f : 2U o >0,1@

f ( A)

­ §N· °1 / ¨¨ ¸¸ if | A | n ® ©n¹ ° 0 otherwise ¯

Here a man-made randomization is introduced to make inductive logic valid. Of course, depending upon problems at hand, different probability sampling designs (i.e. different random set distributions) can be considered, see e.g. [32]. The probability of observing x T fT ( x )

A

 X for some A Ž U is

fT (T A ) = f ( A)

Thus, if we let ­ f ( A) if x T A , ® otherwise ¯ 0

fT (x)

then fT (x) is a probability density function X with finite suppose. Let PT denote the probability measure on X (or some V  field , V ( X ) of X), i.e. PT (.) ¦ f ( A)G (T A ) (.) , where G (T A ) (˜) is the Dirac measure on X at AŽU

the point (T A ) , i.e. for B Ž V ( X ) ,

G (T

­1 if T A  B

A

) ( B) = ®

¯0 otherwise

We are led to a family of probability spaces indexed by 4 , namely, {( X , V ( X ), PT ),T  4} .

4.1 Back to Sampling Surveys

37

By analogy with games of chances in which outcomes are subsets of some set, such as in sampling from finite populations, we consider uncertain subsets which arise in observing phenomena as outcomes of random sets. Thus, random elements taking subsets as values appear in the very beginning of statistical science! It is interesting to observe that the random set formulation in sampling surveys sets up the general framework for statistical inference. Indeed, let U be a finite population, and let T 0 : U o R be our variable of interest, where, e.g., T 0 (u ) denotes the annual income of the individual u. Without being able to conduct a census T 0 is unknown, but T 0 is located in the parameter space 4 RU {T : U o R} . Let f denote a probability sampling design (i.e. a probability density function on 2U ). Our random experiment consists of ”drawing” a subset A of U according to f. Upon selecting A, we are interested in the values of T 0 on A, i.e. the restriction T 0 A of T 0 to A. Thus, we can view that the outcome is T 0

A

, and not A per se.

The sample space of our random experiment is X

A *R .

AŽU

The statistical interpretation of this statistical model is this. The true but unknown parameter of interest is T 0 . We make an observation x  X which is drawn according to PT 0 . The problem is how to make an “educated guess” about T 0 ? Clearly, the above formulation is sampling surveys suggests the same structure for other statistical problems, namely a statistical model is a family of probability space {( X ,V ( X ), PT ,T  4)} . In standard statistical analysis, X is R or R d , and 4 is some finitely dimensional space (parametric statistical problems). Random variables or vectors are special cases of random closed sets. Intervals are the simplest form of imprecision about point observations (measurements). In statistics, confidence intervals (or regions in higher dimensions) are random intervals. Specifically, consider the parametric statistical model { f ( x T ) : x  X Ž R m ,T  4 Ž R d }. Suppose our parameter of interest is M T . Give a random sample x1 , x2 ,..., xn from the population X (with true, but unknown f ( x T 0 ) , the

38 Chapter 4 Random Fuzzy Sets

essence of confidence region estimator is to find a random set S ( x1 ,..., xn ) containing M (T 0 ) with high probability, e.g., T  4 , PT {M (T )  S ( x1 ,..., xn )} t 1  D for some D  0,1 , and where PT (d x ) f ( x T )dx . The situation for fuzzy data is somewhat similar as we proceed now.

4.2 Fuzzy Numbers The use of fuzzy sets, as mathematical models for fuzzy data, and their associated logics spreads out to many fields in science and technology, especially in computational intelligence. Here we only have statistics in mind. As such, we are going to present basic background on structures of fuzzy sets for a rigorous framework to extend mathematical statistics to the fuzzy setting. Imprecision in the observation of the values of random variables (or vectors) results in replacing numbers by intervals or more generally fuzzy intervals. Let us formulate this coarsening scheme in specific terms. Let R be the set of real numbers. Elements of the space of all fuzzy ~ subsets of R, denoted as P ( R) , are called fuzzy quantities. Operations on ~ R are extended to P ( R) by the extension principle. For example, ~ for A, B  P ( R) , ( A  B )( z ) sup{ A( x) š B( y ) : x, y : x  y ( A ˜ B )( z )

sup{ A( x) š B ( y ) : x, y : xy

z}

z}

Note that these operations generalize interval arithmetic as well as Minkowski’s operations on sets. Fuzzy numbers are special fuzzy quantities. Intuitively, “around 0” is a fuzzy number. Thus a reasonable definition is this. Definition 4.1. A fuzzy number is a fuzzy subset A of R such that: (i) A( x)

1 for exactly one x

4.3 Fuzzy Set-valued Random Elements

39

(ii) The support {x : A( x) ! 0} of A is bounded. (iii) The D -level set AD of A are closed intervals From the above definition, it is not hard to show that? (a) Real numbers are fuzzy numbers. (b) Fuzzy numbers A are upper semi-continuous (u.s.c.), i.e. AD is closed,

D  >0,1@ . (c) Fuzzy numbers are convex (A fuzzy subset A of R is said to be convex if its AD are convex, i.e. intervals).

As stated above, the use of intervals and their arithmetic is appropriate in some situations involving imprecision. When the intervals themselves are not sharply defined, we arrive at the concept of fuzzy intervals. Generalizing fuzzy numbers, we define a fuzzy interval A as a fuzzy subset of R satisfying (i) A is a normal fuzzy set ( i.e. x  R such that A( x)

1 ).

(ii) The support {x  R : A( x) ! 0} of A is bounded. (iii) The AD ' s are closed intervals.

Thus a direct generalization of real numbers to fuzzy data could be a class of fuzzy subsets of R (or of Rd), denoted as L(R) (or L(Rd)), where elements are normal, convex and u.s.c. fuzzy subsets with bounded supports.

4.3 Fuzzy Set-valued Random Elements As stated earlier, in order to analyze fuzzy data, we need to define rigorously the concept of random fuzzy sets, i.e. random elements taking fuzzy

40 Chapter 4 Random Fuzzy Sets

subsets of some set U as values. That boils down to specify a class of fuzzy subsets of U and a V  field on that class. Since fuzzy sets are generalizations of ordinary (crisp) sets, and fuzzy numbers closed sets of R, we will proceed to formulate the concept of random fuzzy sets generalizing the well-establish concept of random closed sets of Mathͼron [51] on Rd. Recall that the relation between crisp sets and fuzzy sets is expressed in terms of D  levels sets. Let A be a fuzzy subset of a set U. For D  >0,1@ , to D  levels set of A is the crisp subset AD of U, where AD

{x U : A( x) t D } .

The membership function A(.) is recovered from its AD ' s via: 1

x U , A( x) ³0 AD ( x)dx . ~ Let ( :, A, P) be a probability space. Suppose S : : o P (U ) . Then for

each D  >0,1@ , we have a set-valued map SD : : o 2U , where SD (Z ) = [ S (Z )]D = {x U : S (Z )( x) t D } . For each SD (Z ) to be a closed set, say when U = Rd, we need precisely that each fuzzy set S (Z ) is upper semi-continuous (u.s.c.). Thus, the range of S should consist of u.s.c. membership functions. The space of fuzzy numbers L(R) (or L(Rd)) is metrizable, and random fuzzy sets can be defined as measurable maps with values in . See e.g. [17]. We specify here the concept of random fuzzy sets on Euclidean spaces R d . For aspects of random fuzzy sets on general metric spaces with Hausdorff distance, see e.g. [51], [40]. Now observe that on a compact, metric space (e.g. a compact metric subspace of Rd), the topology defined by the Hausdorff metric on K \ {I} (space of non-empty compact subsets of R d ) is equivalent to the relative (hit-or-miss) topology (Chapter 2), the measurability of random closed sets is expressed simply as follows.

4.3 Fuzzy Set-valued Random Elements

41

Let ( :, A, P) be a probability space. Let B(F) denote the Borel V  field of the space of closed subsets F(Rd) of Rd. Then S : : o F(Rd) is called a random closed set if for any 3  B(F), S 1 (F)  A. Similar to the situation of random vectors, it can be checked that the above measurability of S is equivalent to: for any open subset G of Rd, S 1 (FG)  A, where FG = {F  F : F ˆ G z I} . In view of all the above,

we take the following approach (for more details, see [45]). ~ Let PK (Rd ) denote the space of fuzzy subsets of Rd which are u.s.c., ~ normal (i.e. if f  PK (Rd ), then {x  R d : f ( x) 1} z I ), and which have compact supports (i.e. the closure of {x  Rd : f ( x) ! 0} ) is a non-empty concept subset of Rd). Note that we consider non-empty random (closed) sets on Rd for practical purposes. For example, in the context of coarse data analysis, a random set S is a coarsening of some random variable X. As such, S takes non-empty subsets as values (since X is an almost selector of S). ~ On PK (Rd ), a natural metric is the following. Let H denote the Haus~ dorff metric on F(Rd) \ {I} . Then for A, B  PK (Rd ), ~ H ( A, B )

sup H ( AD , BD )

0D d1

is a metric (see [34]), where AD is the D -level set of A. ~ Definition 4.2 Let ( :, A, P) be a probability space. Then S : : o PK (Rd) is called a random fuzzy set if for any D  [0,1] , the map SD : : o F(Rd) SD (Z ) [ S (Z )]D

{x  Rd : S (Z )( x) t D }

is a random closed set on Rd. The concept of random fuzzy set considered here is a direct generalization of random closed sets in the sense of Matheron [51]. As such, statistical inference with random fuzzy sets could benefit from the foundations

42 Chapter 4 Random Fuzzy Sets

of statistical inference for random closed sets (on Hausdorff locally compact and second countable topological spaces, such as Rd), see [61]. Random fuzzy sets are population models for observations which are fuzzy sets obtained at random. Various basic properties of fuzzy data can be formulated in terms of their associated D  levels random sets. Recall that if S is a random closed set (defined on ( :, A,P) with values in F(Rd), then its probability law PS is a probability measure on B(F), i.e. for A  B(F), PS ( A)

P( S 1 ( A)) , and the V  field V (S ) generated

by S is {S 1 ( A) : A  B(F)}, V (S ) is a sub V  field of A. Recall also that an (infinite) family { Ai : i  I } of events (i.e. elements of a) is said to be (mutually) independent if for any finite J Ž I , P(  A j ) jJ

– P( A j ) . jJ

A family of sub V  fields {ai : i  I } of a is said to be independent if any family of events { Ai  ai : i  I } is independent. A sequence of random closed sets {S n , n t 1} is said to be independent if its associated sequence {V ( S n ), n t 1} of sub V  fields is independent in the above sense. Of course {S n , n t 1} is said to be identically distributed when all PSn are the same. Thus, i.i.d. samples of random closed sets are defined just like the case of random vectors. ~ Viewing random fuzzy sets as random elements on PK (Rd) with the ~ Borel V  field B( PK (Rd), generated by the metric H f , the concept of i.i.d. samples of random fuzzy sets are defined similarly. While random fuzzy sets are random elements in well-defined measurable spaces, the concept of expected values of random fuzzy sets is delicate. In fact, this situation arises for random sets as well. This is essentially due to the technical problem of integration of set-valued functions. Let S be a random closed set on Rd, then the core of S is non-empty, where the core of S is ] ( S ) ={X: X is an a.s. selector of S} i.e. X : : o Rd, measurable and P ( X  S ) 1 . Suppose EX exists for X  C ( S ) , then one way to define ES is the Aumann expectation of S, namely

4.3 Fuzzy Set-valued Random Elements

ES

43

{EX : X  ] ( S )} .

Thus, if S is a random fuzzy set, its expectation can be defined as a ~ fuzzy subset of PK (Rd), denoted by ES , such that D  [0,1] , (ES )D is the closure of {EX : X  ] ( SD )} .

Chapter 5 Aspects of Statistical Inference With the background in previous chapters, problems of statistical inference with fuzzy data should be somewhat straightforward in principle! By that we mean replacing random vectors by random fuzzy sets in all aspects of statistical inference. Of course, as in any generalization problem, this is just a guideline. Due to the nature of fuzzy data, as observations from random fuzzy sets, technical difficulties are expected in developing the theory. In fact, actual research is aiming at investigating, say, limit theorems for random fuzzy sets in order to provide rationale for large sample samples statistics with fuzzy data. Here we choose to discuss some basic aspects of statistical inference with emphasis on “where do fuzzy data come from”

5.1 Fuzzy Data in Sampling Surveys As primitive as sampling surveys in applied statistics, fuzzy data appear naturally in opinion surveys, say, questionnaires. In an opinion surveys questionnaires are sent to a random sample of individuals in a population. The data collected consists of a sample of responses to the questionnaires. As usual, the questionnaires in a sampling survey contain explicitly or implicitly variables which are either quantitative (numerical) or qualitative variables, the answers are required to be just “yes” or “no”. This practice is not just rooted in two-values logic (true or false), but perhaps, even in obvious cases such as “Is the candidate A suitable for the specified job?”, allowing a response scale larger than just “yes” or “no”, might introduce difficulties in final analysis! Realistic scales reflecting truly respondents’ opinion should be either in linguistic answer or degrees to which the candidate A is suitable for the job. Although, applied statisticians do recognize this phenomenon, the practice is that one could avoid “fuzziness” by using binary logic! Of course, people are entitled to do so, but the question is “how much H.T. Nguyen and B. Wu: Fundamentals of Statistics with Fuzzy Data, StudFuzz 198, 45–70 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com


46 Chapter 5 Aspects of Statistical Inference

information is lost in the process?” This should be answered based on the importance of the issue involved. For example, in the risk analysis for adopting construction designs, or risk assessment of potentially critical situations, security issue is at stake. A rough analysis based on two-values logic might be too naïve. Risk is a linguistic variable with values such as “low”, “medium”, “high”. It is here that fuzzy set theory enters statistics to provide representation and manipulation for realistic data. Thus we can enlarge the statistical analysis of sampling surveys in which questionnaires are better designed and conclusions are more realistic. For example, a fuzzy proportion of people favoring the candidate A (say in an election) is definitely more realistic to use in predication than a crisp proportion based on two-valued logic. See later chapters for concrete examples and applications in social sciences. Another typical example is in computer science: the extension of relational databases to fuzzy relational data bases allows queries to be stated in a more flexible way. For example, a standard database consisting of the variable “name of employee” and relation “height” can be extended so that a query of the form “select tall employees” can be retrieved. These examples illustrate the current tendency of data analysis in extracting more information from data, as opposed to the situation where by nature, problems are fuzzy, but crisp mathematics are used to avoid the natural phenomena of fuzziness! As stated ealier, fuzzy data are not conventional in statistical practices. Perhaps this is due to the lack of a mathematical theory to analyze fuzzy data. While the statistical analysis of random set data is starting to take shape, we believe that it will not take much longer for the statistics community to embrace fuzzy statistics. Fuzzy data in sampling surveys statisticians should consider when dealing with real-world problems.

5.2 Coarse Data Incomplete data are natural in a variety of problems and have been received attention of statisticians for quite some time (e.g. [72]) The type of

5.2 Coarse Data

47

incomplete data that has been studied thoroughly is missing data, in which each data value is either known or entirely unknown. But this “yes” or “no” situation is only a special case of general patterns of incomplete data. Indeed, collected data could be neither entirely missing nor perfectly present. This kind of incomplete data is termed “coarse data” and has received much attention more recently in statistics (e.g. [34]). A typical example of this type of “not yes or no” is the following situation of imprecise observations. In performing an experiment or in observing natural phenomena, we might not be able to record correctly the values of locate them with some degree of accuracy, or more generally, locate them in some regions (subsets) of the sample space. Thus the observations are present but not known! For example, suppose our random variable X of interest takes values in U. Suppose X cannot be observed directly but its outcomes can be located in a finite partition { A1 , A2 ,..., An } of U, i.e. each X(w) is known to be in some A. Such a partition is referred to as a coarsening of X. Providing a model for observing X. If we consider the set-valued map S(w) = the element of the partition containing X(w), then S is formally a random set with probability density P( S

Ai ) = P ( X  Ai ) , i=1,…,k

Thus, coarse data are actually observations of random sets which are special cases of random fuzzy sets. The following is an example of statistical estimation in the context of coarse data. For details, see [38]. Suppose the random variable X admits a probability density function f ( x | T ) , where T  4 Ž R . Let S be the above coarsening scheme of X. Let S1 , S 2 ,..., S n be a random sample drawn from S. Then the likelihood of T when observing S1 , S 2 ,..., S n is

48 Chapter 5 Aspects of Statistical Inference

L(T | S1 , S 2 ,..., S n )

n

– ³Si f ( x | T )dx i 1

Suppose the parameter space 4 is finite. Then the maximum likelihood estimator Tˆn of T , based on the random set observations S1 , S 2 ,..., S n is unique with probability tending to one, and is weakly consistent. This can be shown as follows: Since 4 is finite, an Tˆn which maximizes L(T | S1 , S 2 ,..., S n ) over 4 exist. Let 4 {T 0 ,T1 ,...,T n } with T 0 being the true (unknown) parameter. And let P4 j ( si ) = ³S f ( x | T j )dx i

(i) Observe that P4 ( si ) , i=1,…,n is an i.i.d. (independent, identically distributed) sample of a random variable Z which is a function of the random set S. Similarly,

log

PT ( S i ) , i=1,…,n. PT 0 ( S i )

Obviously, for these samples, the weak law of large numbers holds, e.g. P (S ) P (S ) 1 n P ¦ log T i o ET0 [log T i ] . ni 1 PT 0 ( S i ) PT0 ( S i )

(ii) Since L(T 0 | S1 , S 2 ,..., S n ) > L(T | S1 , S 2 ,..., S n ) , is equivalent to P (S ) 1 n ¦ log T i 0,1@ . Also, let P denote the class of all probability measure on U. We identify P with a compact, convex subset of >0,1@ . Suppose W 0 is known to k

belong to some “model” ) Ž P, where ) is assumed to be closed (therefore compact). The parameter space 4 will be ) ˆ Core(T ) . The explicit description of Core(T ) is useful for statistical inference about W 0 as well as for the extension to the case of a compact, metric space U in Section 5.5. While the fact that Core(T ) contains precisely all probability laws of the a.s. selectors of T, in a fairly general setting, say when U is a Hausdorff, locally compact, second countable topological space, e.g. Rd, is a consequence of a general theorem of Noberg [12] on ordered coupling S of random sets, we will elaborate on this fundamental fact in the finite case. Let S be a non-empty random set on a finite set U. Its probability density function is a map f : 2U o [0,1] such that f (I ) 0 , ¦ f ( A) 1 AŽU

An allocation of f is a function

D : U u (2U  {I}) o [0,1] such that A Ž U , ¦ D (u , A)

f ( A)

aA

For example, let P : U o [0,1] be a probability density (of some random variable taking value in U ), such that u  U , P (u ) z 0 Let

D (u , A) where

f ( A) P(u ) P ( A)

5.4 Random Set Data on Finite Spaces

P ( A)

55

¦ P(u )

uA

then D (˜,˜) is an allocation of f. In particular, if P(u )

1 , let u  U , |U |

i.e. P is the uniform density on U, then

D (u , A)

f ( A) ,AzI | A|

Each allocation D (.,.) gives rise to a probability measure on U via the probability density function gD : U o [0,1] where gD (u )

¦ D (u, A)

uA

the summation is of course voer all A Ž U which contain u. The associated probability measure is

W D ( A)

¦ g D (u )

uA

Note that, equivalently, we can characterize the probability law of a random set S, either by its probability density f on 2U , or by its capacity functional T, where T ( A)

P(S ˆ A z I )

or by its distribution function F ( A)

P ( S Ž A) noting that T ( A) 1  F ( AC )

Clearing if X is an a.s. selector of a nonempty random set S, then PX  Core(T )

The converse holds. In fact, the following result is sufficiently general.

56 Chapter 5 Aspects of Statistical Inference

Theorem 5.1 (Noberg, [12]) Let W be a probability measure on B( R d ) and T be a capacity functional on R( R d ) . Then the following are equivalent: (i) W  Core(T ) (ii) There exists a common probability space ( :, A,P) on which are defined a ramdom closed set S : : o F ( R d ) , and a random variable

X : : o R d such that P( X  S ) 1 , and S and X have T and W as their distributions, respectively. It turns out that elements of Core(T) all come from allocations. Specifically, Theorem 5.2 Let T be a capacity functional on a finite set U. Let f be the Mobius inverse of the dual (distribution) F of T (i.e. f is the associated probability density on 2U ). Then a probability measure W on U is in Core(T) if and only if W comes from an allocation of f. Proof (a) Necessity. Suppose W  Core(T ). By the above Noberg’s theorem, there is a common probability space ( :, A,P) on which are defined S : : o 2U \ {I} , X : : o U

such that P ( S U ) 1, with S and X having T and W as distributions, respectively. Define

D (u , A) P( S

A, X

u)

5.4 Random Set Data on Finite Spaces

Clearly, D (u , A) 0 for u  A , and A Ž U , ¦ D (u , A)

57

f ( A) i.e. D is

uA

an allocation with

W ({u})

P( X

u)

¦ D (u, A),

uA

i.e. W comes from the above allocation D (b) Sufficiency. Consider the following probability space : U u (2U \ {I}) , AЈpower set of : , P({w})= P((u,A)) = D (u , A) ,

where D is an allocation of f. Define S : : o 2U \ {I} X : : o U by S (u , A)

A, X (u , A) u , for any (u , A)  :.

We haveΚ P( S

A)

P( A u U )

¦ P((u , A))

f ( A)

uU

so that f is the density of S (i.e. T is the capacity functional of S ). On the other hand, the probability measure M D arising from the allocation D is precisely the probability law of X. Indeed, P( X

u)

P (({u} u (2U \ {I}))

¦ P ((u , A))

I z AŽU

¦ D (u , A)

I z AŽU

Moreover, X is an P-a.s. selector of S. Indeed, P( X  S )

P((u , A) : u  A

¦ ¦ D (u , A)

AŽU uA

Thus, W D  Core(T )

¦ f ( A) 1

AŽU

58 Chapter 5 Aspects of Statistical Inference

Remarks (i) As we will see, the above Theorem 5.2 for U finite is useful for the extension of statistical inference to more general cases. (ii) Traditionally the above Theorem 5.2 can be proved by using Shapley’s Theorem [83] in game theory. This can be seen as follows. Let U with U

k , and f by the probability density function of a non-

empty random set S. Let U= {u1 , u2 ,...uk } be an ordering of U. Set D (u , A)

u

u j  A where u j Let F ( A)

f ( A) if

max{i : ui  A} , and zero otherwise.

¦ f ( B) be the associated distribution function. Then the

BŽ A

above special allocation D gives rise to the special probability density function gD on U, namelyΚ

g D (ui )

¦ D (u , A)

ui  A

¦ f ( A)

ui AŽ{u1,u2 ,...ui }

F ({u1 , u 2 ,..., ui })  F ({u1 , u 2 ,...., ui 1})

for i=1,2,…,k (if i=1,then F ({u1 , u 2 ,...., ui 1})

F (I ) 0 ).

Let P denote the set of all probability measures on U. Since each probability measure W  P is uniquely determined by its density function on U, we identify W with a vector in the unit simple S k of R k where k

Sk

{p

( p1, p 2 ,..., p k )  [0,1]k , ¦ pi

1}

i 1

By identification, Core(T) is a compact convex subset of S K , where T ( A) 1  F ( Ac )

5.4 Random Set Data on Finite Spaces

59

There are k! different orderings of the elements of the set U. Let ¦ denote the set of all permutations of {1,2,…,k}. For each V  ¦ , the elements of U are indexed as {uD (1) , uD ( 2) ,...., uD ( k ) } , and g V (uV (1) )

F ({uV (1) }) , and for i t 2 ,

gV (uV (i ) )

F ({uV (1) , uV ( 2) ,..., uV (i ) })  F ({uV (1) , uV ( 2) ,..., uV (i 1) })

The associated probability measure W (having gV as density on U) is in Core(T). Indeed, for simplicity, take V (i ) i, for i=1,2,…,k. For W ( A) ¦ g (u ) (here we drop V from our writing, but not from our uA

mind !), we have F ( A) d W ( A) when A=U. For AU, let

j

min{i : ui  Ac } ,

noting that Ac z I .

Let

B {u1 , u2 ,..., u j } , then A ˆ B {u1 , u 2 ,..., u j 1} , A ‰ B A ‰ {u j }. Since F is monotone of order 2, i.e. F ( A ‰ B) t F ( A)  F ( B)  F ( A ˆ B) , we have F ( A ‰ {u j }) t F ( A)  F ({u1 , u 2 ,..., u j })  F ({u1 , u 2 ,..., u j 1})

as g (u j ) d F ( A ‰ {u j })  F ( A) . But g (u j ) W ({u j }) W ( A ‰ {u j })  W ( A) So that F ( A)  W ( A) d F ( A ‰ {u j })  W ( A ‰ {u j })

On the right hand side of the above inequality, viewing A ‰ {u j } as another set A’, and using the same reasoning, we have F ( A ‰ {u j })  W ( A{u j} )

F ( A' )  W ( A' ) d F ( A ‰ {u i })  W ( A ‰ {u i })

where ui z u j . Continuing this process, we arrive at

60 Chapter 5 Aspects of Statistical Inference

F ( A)  W ( A) d F ( A ‰ {u j })  W ( A ‰ {u j }) d F ( A ‰ {u j , u i })  W ( A ‰ {u j , u i }) d ... d F (U )  W (U ) 1  1 0

Thus, F d W , i.e. W  Core(T ) . Now, for each V  6 , W V  Core(T ) . There are k! (not necessarily distinct ) W V .

Shapley Theorem 5.3 [43]. Let T be the Capacity functional of some nonempty random set as U with U k . Then Core(T) is a compact, convex polyhedron with at most k! extreme points which are precisely the W V ' s . The above results imply that Core(T) consists precisely of probability measures coming from allocations of f (the Mobius inverse of F ( A) 1  T ( Ac ) . Indeed , let A denote the subset of P consisting of all probability measures coming from f. Let D be an allocation of f. Then, for any A Ž U F ( A)

¦ f ( B)

BŽ A

¦ ¦ D (u, B)

B Ž A uB

d ¦ ¦ D (u , B) W D ( A) uAuB

i.e. A Ž Core(T ) Conversely, by Shapley’s theorem, Core(T) has the W V ' s as extreme points and hence Core(T) is the set of convex combinations of these extreme points. Since PV ’s are elements of the convex set A, A contains all convex combinations of its elements, in particular, convex combinations of these W V ' s which is Core(T).

5.4 Random Set Data on Finite Spaces

61

We are now ready to present Schreiber’s work [81] With the previous notation, the essence of statistical inference about W 0 is to establish a framework in which we can assert that, based on the observations of S1 , S 2 ,..., S n , estimates of W 0 can be constructed in a consistent manner. For this, first we need to define concepts of “distance”. Recall that P denotes the class of all probability measures on the finite set U. A discrepancy measure on P is a map

' ΚP u P o R  such that (1) ' ( W , X )=0 œ W = X (2) ' is lower semi-continuous (P is identified with compact convex of [0,1]k (3) ' is strictly convex on P u P (4) for each X  P, the map W  P o ' ( W , X ) is continuous.

For example, the following are discrepancy measuresΚ (a) 'f ( W , X )=max{ W (u i )  X (u i ) : i 1,2,..., k} (i.e. the l f -distance). (b) For p t 1 k

1 p

' p ( W , X )= [¦ W (u i )  X (u i ) ] p ] i 1

Let ' be a given discrepancy measure on P. From axioms (2) and (3), it follows that, for each X  P., there is a unique probability measure in P, denoted as X (T ) such that ' (X (T ),X )

inf{'(W ,X ) : W  Core(T )

Thus, let us extend ' to * u P (recall that * is the class of all capacity functional on U ) asΚ

62 Chapter 5 Aspects of Statistical Inference

' (T ,X )

where ' (T ,X )

inf{W (W ,X ) : W  Core(T )}

' (W (T ),X ) ,  X  P . This means that X (T ) is the prob-

ability law of the a.s. selector (of the random set S having T as its capacity functional) which is the best approximation of U in the ' sense. In particular, if V is itself a probability law of some a.s. selector of S, thus by (1), ' (T ,X ) 0 and X (T ) X . The extension ' has nice topological properties. Specifically, ' : * u P o R  is lower semi-continuous and strictly convex. Furthermore for each fixed X  P, the map T  * o ' (T ,X ) is continuous. (See [3] for details). Now, as in [3], let ) be a statistical model for W 0 . Define for each T * , 'ˆ (T | ) )

inf{'* (T ,V ) : V  )}

'ˆ (T | ) )

inf{'* (W ,X ) : X  )}

inf{'(W ,X ) : W  Core(T ),X  )}

It follows from topological properties of '* and the compactness of ) that there is (not necessarily unique) an X  ) such that ' (T ,V )

'ˆ (T | ) )

For a such X , we have ' (X (T ),X )

'ˆ (T | ) ) , and hence X (T) is

viewed as the most likely probability law governing the unobserved data leading the deserved random set data. Thus, consider ) * Ž ) , where ) * {X  ) : '* (T ,X ) 'ˆ (T | ) )} . Note that ) * is compact. The interpretation of ) * is that A is the class of probability laws on V which fit best the observations S1 , S 2 ,..., S n . Thus, it boils down to identity the elements of the “optimal” set )* .

5.4 Random Set Data on Finite Spaces

63

Recall that the empirical capacity functional T (n ) , based on S1 , S 2 ,......, S n is. T ( n ) ( A)

1 n ¦ (Si ˆ A z I ) ni 1

Let )*n {X  ) : '* (T ( n ) ,X ) 'ˆ (T ( n ) | ) )} . It can be checked that ) *n is a.s. non-empty and compact. For I z A Ž P , and X  P , define d (X , A)

inf{'( 2) (X ,W ) : W  A}

On the metric space (P, '( 2 ) ) we have d (X , A) 0 œ X  A , and the map X o d (X , A) , for each A z I , A compact in P, is continuous. The following facts are essential. Let (Tn , n t 1) Ž * such that Tn o T  * , and ) (n )

{X  ) : '* (Tn ,X ) 'ˆ (Tn | ) )} for n t 1

Then, with probability one, we haveΚ (i) ) (n ) Ž ) is non-empty and compact (ii) lim 'ˆ (Tn | ) ) n of

'ˆ (T | ) )

(iii) lim sup{d (X , ) ) : X  ) ( n ) } 0 n of

These facts imply Schreiber’s basic resultΚ Theorem 5.4 (Schreiber [3]) With probability one, we have lim 'ˆ (T ( n ) | ) )

n of

'ˆ (T | ) ) , lim sup{d (X , ) ) : X  ) ( n ) } 0 , n of

where {T ( n ) , n t 1) is the sequence o empirical capacity functionals based on S1 , S 2 ,..., S n .

64 Chapter 5 Aspects of Statistical Inference

In addition, the rate of convergence of 'ˆ (T ( n ) | ) ) to 'ˆ (T | ) ) can be specified. Specifically, for each İЇ0, there is an LH ! 0 such that, for n sufficiently large, P (| 'ˆ (T ( n ) | ) )  'ˆ (T | ) ) |t H ) d e  nLH

Let us elaborate a little bit on this result of Schreiber which is based on large deviations principle ( see e.g. [15] ). Let X 1 , X 2 ,..., X n be an i.i.d. sample form the Bernouilli random variable X taking value in {0,1} with P(X=1)=p. The sample mean X 1  X 2  ......  X n Xn is a weakly consistent estimator of EX = p, n since H ! 0 , P(| X n  p |t H ) d

p (1  p ) o 0 as n ĺ ’. nH 2

The rate of which X n o p can be made explicit as follows. The events { X n  p t H } are very rare events in the sense that then probabilities decay to zero exponentially fast. They represent large deviations of X n from the mean EX =P since X n avoids some neighborhood of EX,

whereas the event {| X n  p | H } represent small deviations of X n from P, i.e. X n is in some neighborhood of P. Let F be the distribution function of X, and M (t ) = E (e tx ) tx ³R e dF ( x) , t  R be the moment generating function of X (or the Laplace transform of F). Here, for the Bernoulli case, M (t ) 1  p  pe t . Let / ( x) sup{xt  log M (t ) : t  R}, x  R , be the so-called Cramer

transform of F. The function / is called a rate function. The following are well-known. Let Pn denote the probability law of X n . ThenΚ (a) For every closed set A of R, 1 lim sup log Pn ( A) d  inf{/ ( x) : x  A}. n

n of

5.4 Random Set Data on Finite Spaces

65

(b) For every open set B of R, lim inf

n of

Let I ( A)

1 log Pn ( B) t  inf{/ ( x) : x  B}. n

inf{/ ( x) : x  A} . Then, A  B(R) , Pn ( A) decays exponen-

tially fast with rate I(A), i.e. 1 lim log Pn ( A) n

n of

 I ( A).

On finite dimension spaces like R d , the exponential rate of convergence is typically given by the Cramer transform which is defined in terms of the Lebesgue measure (as a reference measure on B B ( R d )) . Large deviations of sample paths of stochastic processes such as Brownian motions, require infinite dimensional spaces, like C[0,1], the space of continuous functions defined on the unit interval [0,1]. In order to cover all cases, a general large deviations principle is formulated without referring to any reference measure on the space. To this end, the Cramer transform is abstracted from its basic properties. The general large deviations principle is as follows principle is as follows. Let U be a polish (i.e. a complete, separable, metric space) and Ȍ its Borel ı -field. A rate function is a function I : U o [0, f) , which is lower semi-continuous. A family ( Pİ , İ ! 0 ) of probability measures on Ȍ is said to satisfy the large deviations principle with rate function I if for any A Ȍ ,  inf I d lim inf İ log Pİ ( A) o A

İ o0

d lim sup İ log Pİ ( A) d  inf I İo0

A

where A o , A denote the interior and closure of A, respectively. This is equivalent to

66 Chapter 5 Aspects of Statistical Inference

lim sup İ log Pİ ( A) d  inf I , A closed, and İo0

A

lim sup İ log Pİ ( A) t  inf I , A open, İo0

where I ( A)

A

inf{I ( x) : x  A} .

For example, Cramer large deviations principle on R is extended to R where the Fenchel-Legender transform takes the role of rate functions. Specially, let F be a distribution function on R d . The Laplace transform of dF is d

M (t )

Ee t , x ! d

= ³R d exp[ ¦ t j x j ]dF ( x) j 1

where t

(t1 , t 2 ,..., t d )  R d ,

x

( x1 , x 2 ,...x d )  R d . The Fenchel-

Lebesgue transform of dF is / (t )

Sup{ t , x !  log M (t ) : x  R d } .

For example, let S be a non-empty random set on a finite set U with capacity functional T. Since T ( A) P( S  A z I ) , we can consider the associate random vector X with values in R d with |U| = d, namely

X

§ 1( S  A1 zI ) · ¨ ¸ # ¨ ¸ ¨¨ ¸¸ ©1( S  Ad zI ) ¹

where A j ' s , j = 1,2,…,d, are all subsets of U. If we identify each map h : 2U o R with a vector in R d , namely

5.5 Random Set Data on Metric, Compact Spaces

§ h( A1 ) · ¸ ¨ ¨ # ¸ ¨ h( A ) ¸ d ¹ ©

67

§ t1 · ¨ ¸ ¨ # ¸ = t, ¨t ¸ © d¹

then we can consider the Laplace transform of X as a function defined on maps h:

M S (h) M X (t ) = Ee t , x ! d

= E exp[ ¦ h( A j )1( S  A j zI ) ] j 1

= E exp[ ¦ h( A)] S  A j zI

Also by identifying the capacity functionals T with the vector (T ( A), A Ž U ) in R d , the Fenchel –Legendre transform of X is written as: / S (T ) = Sup{ ¦ h( A)T ( A)  log M S (h) : h : 2U o R d } AŽU

with these identifications, Schreiber’s result is stated as follows: The sequence { 'ˆ (T ( n ) | ) ), n t 1 } satisfies the large deviations principle with rate function given by h o inf{/ (T ) : T  *, 'ˆ (T | ) ) h} . S

5.5 Random Set Data on Metric, Compact Spaces As stated earlier statistics with set-values observations are not only new but also essential in developing framework for statistics with fuzzy set observations, via, say techniques based on D -level sets of fuzzy sets. As such, it is useful for researchers who wish to investigate statistical methods for fuzzy data to be aware of current developments in framework for statistical inference with random set data. This section is devoted to the extension of finite random set models in 5.4 to the more general case of metric,

68 Chapter 5 Aspects of Statistical Inference

compact spaces. The material presented here is drawn from the recent work [23]. Let (U, d) be a metric, compact space, such as [0,1], and < its Borel V -field. All random (closed) sets and variables are assumed to be defined on a probability space ( : , A, P). As in section 5.4, we look at the situation of coarse data in which we observe a random sample S1 , S 2 ,..., S n from a non-empty random set model S which is a random closed (in fact here compact) set, i.e. taking values in the class of closed subsets F (=K, the class of compact subsets) of U. The unknown capacity functional of S is denoted as T. Let P denote the class of all probability measures on < . Since the core of T, i.e. Core(T ) {W  P, : W d T }

is non-empty, there exists, according to Noberg’s theorem (section 5.4), a random variable X : : o U which is an a.s. selector of S, i.e P ( X  S ) =1. We can view X as the random variable of interest but unobservable. The most general statistical problem is the inference about the unknown probability law W 0 of X (a probability measure on < ) from the observations S1 , S 2 ,..., S n . We follow the set–up in Section 5.4. The estimation of W 0 is essentially based upon the convergence of the sequence of cores of empirical functionals T (n ) to Core (T) which is a compact subset of P, where P is a topological space with the weak-star topology (see e.g. [41]). Specially, a metric on P is defined as:

' (W ,X )

| Sf n dW  ³ f n dX | 2 n || f || 1

f

¦

i

where { f n , n t 1} is a sequence of continuous, real-valued functions dense in the space C(U) of continuous, real-valued functions defined on U (endowed with the uniform topology), and

5.5 Random Set Data on Metric, Compact Spaces

69

|| f || max{| f ( x) |: x  U }

In fact, {P, ' } is a compact space. To generalize Schreiber’s result in the finite case, we proceed as follows. We define a pseudo-metric on the space * of all capacity functionals on K, and show that Core (T) depends continuously on T. These mathematical results lead to the existence of strong consistent estimators (W n , n t 1) of W 0 in the weak star convergence sense. Specially, let W 0  Core (T ) and T (n ) be the empirical capacity functionals based on S1 , S 2 ,..., S n . Then a.s., there exists W n  Core (T (n ) ) , n t 1 , such that W n o W 0 in the weak star convergence sense as n o f . Moreover, let e' denote the Hausdorff metric on non-empty, compact subsets of P, i.e. e' ( A, B )

inf{H t 0 : N H ( B ) ‹ A, N H ( A) ‹ B}

where N H ( A) is the neighborhood of A in P. i.e. N H ( A) ={ W  P : X  A, such that ' (W ,X )  H }

Then for any H ! 0 , there exist nH and Le > 0 such that P (e' (Core(T ), Core(T ( n ) )) ! H ) d e  nLH

for all n t nH . Here is the sketch of the proof. On the metric space (U,d), a set A Ž U is called a H -spanning set of U if N H ( A) =U, for H ! 0 . Since U is compact, for each n t 1, there is a finite 1/n-spanning set of U, denoted as H n . Let T n

{N1 / 2 ( A) : A Ž H n } , n t 1. Then a pseudo-metric on *

(space of all capacity functionals on K(U) is defined as:

70 Chapter 5 Aspects of Statistical Inference

'* (T , T ' )

f

¦2

i 1

 n | H n |

¦ | T ( A)  T ' ( A) | .

AQn

It can be proved that if Tn o T in ' * sense, then Core (T ) o Core (T ) in e' -sense. As an application, if Tn T (n ) , i.e. Tn is the sequence of empirical capacity functionals, then for any W 0 ( w) o W is ' - sense.

Remark: Using the characterization of the core of a capacity functional on a finite set (in terms of allocations) and the “finite approximation” of capacity functionals on compact, metric spaces by capacity functionals on finite set, the above estimators W n ( w) are in fact constructive. For details, see [4]. As in the finite case, the exponential rate of convergence of Core (T ( n ) ) to Core (T ) in e' -sense, is obtained by using classical large deviations principle for i.i.d random vectors.

Chapter 6 Convergence of Random Fuzzy Sets As in Chapter 5, we view random fuzzy sets as generalizations of random closed sets on Rd, or more generally on Hausdorff, locally compact, second countable topological spaces. Thus, technically speaking, the study of limit theorems for random fuzzy sets can be carried out by using D -level sets and results from limit theorems for random closed sets. For this reason, we focus in this chapter on convergence of random closed sets.

6.1 Stochastic Convergence of Random Sets Again let ( : ,A, P) be a probability space, and F(Rd) or simply F, be the set of closed subsets of Rd. Let W denote the hit-or miss topology on F, and B(F) its associated Borel V  field . Equipped with W , F is a compact, metrizable space. A random closed set is a map from : to F which is A-B(F) measurable. Let S1 , S 2 ,..., S1 , be an i.i.d sample from a random closed set S. Large sample statistics relies on limit behavior of functions of the S i' s . For example, the laws of large numbers deal with the limits, in probability and almost surely, of “sample means” ( S1  S 2  ...  S n ) / n in the sense of Minkowski. Recall that, the Minkoski’s operations for subsets of Rd, on more generally of a linear space, are defined as follows: For A, B Ž Rd, and D  R, A  B {x  y : x  A, y  B}

DA {Dx : x  A} Let Z n ( S1  S 2  ...  S n ) / n . We are interested in formulating stochastic convergence concepts for set-valued random elements ( Z n , n t 1) .

H.T. Nguyen and B. Wu: Fundamentals of Statistics with Fuzzy Data, StudFuzz 198, 71–108 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com


72 Chapter 6 Convergence of Random Fuzzy Sets

For simplicity, we restrict ourselves to sequences of random sets (convergence of sets can be formulated similarly) a.s Let S, S n be random closed sets on Rd. Then by S n o S , as n o f , we mean P( w : S n ( w) o S ( w)) 1 ,

(6.1)

where S n ( w) o S ( w) in the sense of the topology W . Note that the convergence in the topology W is the same as the Painleve-Kuratowski convergence (see e.g. Salinetti and Wets [76], Molchanov [54]), namely S ( w)

lim sup S n ( w) n

lim inf S n ( w) n

(6.2)

where lim sup S n ( w) ={ x  R d : xn ( k ) o x : for x n ( k )  S n ( k ) , n(k ), k t 1 being n

a subsequence} lim inf S n ( w) = { x : xn  S n ( w), n t 1, xn o x }. n

(6.3) (6.4)

Note that set-theoretic concepts of limits of sequences of sets are different than the above concepts of limsup and liminf . See also Klein and Thompson [40]. Remark: In the context of Banach spaces, random fuzzy sets can be embedded into Banach spaces, where the concept of Mosco convergence can also be need. See Li et al [45]. Since Rd is a metric space, with metric denoted as U , the concept of convergence in probability of sequences of random closed sets can be formulated For H > 0, and A Ž Rd, x  Rd, U ( x, A) H

d

and A = {x  R : U ( x, A)  H }

inf{U ( x, y ) : y  A}

6.2 The Choquet Integral

73

A sequence S n of random closed sets on Rd is said to converge in probability to the random closed set S if for every H ! 0 , and compact set K(Rd), lim P{w : (( S n ( w) \ S H ( w)) * ( S ( w) \ S nH ( w))  K z I} =0.

n of

For a flavor of research on laws of large numbers for random sets and random fuzzy sets, see e.g. Artstein and Vitale [46] and Dozzi et al. [48]. A central limit theorem for random sets was first investigated by Cressie [14]. We discuss in same details next the convergence in distribution of random closed sets.

6.2 The Choquet Integral The convergence in distribution of S n to S is the usual weak convergence of probability measures on metric spaces. Specially, let PS 1 and PS n1 be D S, probability laws of S and S n , respectively, on B(F). Then S n o

when PS n1 o PS 1 in the weak topology of the space of probability measures on (F,B(F), see e.g. Billingsley [5]. While there are equivalent ways to study the convergence in distribution for random closed sets, we choose to formulate this type of convergence as a natural extension of the process based on Lebesque-Stietltje’s theorem for random vectors, via the Chaquet theorem for random closed sets. This program requires the concept of Choque integral which we elaborate in this section. First. Note that, as we will see, not only the concept of Choquet integral is useful for defining the convergence in distribution of random sets but it provides also an interesting tool for general decision–making. Let X be a non-negative random variables, defined on ( : , A, P) Then its expected value can be written as: E P ( x)

³: X ( w)dP( w)

74 Chapter 6 Convergence of Random Fuzzy Sets

= ³R  xdF ( x) f

= ³0 P ( w : X ( w) ! t )dt

(6.5)

For X : : o R, we can write EP ( X ) f

³0

Ep (X  )  Ep (X  ) 0

P ( X ! t )dt  ³f [ P( X ! t )  1]dt ,

(6.6)

where F is the distribution function of X, and X  , X  are positive and negative parts of X, respectively. f If we look at the term ³0 P( X ! t )dt , we recognize that the function t o P ( X ! t ) is measurable since it is a non-increasing function on R  , and this last fact is due to the monotonicity of the set function P. Thus if P is replaced by any set function W on A which is monotone, i.e. A, B 

A. A Ž B Ÿ W ( A) d W ( B ) , then formally the integral f

³0 W ( w : X ( w) ! t )dt is well-define. Moreover if W (:) 1 , then the quantity: CW ( X )

0

f ³0 W ( x ! t )dt + ³f [W ( x ! t )  1]dt

(6.7)

is also well-defined. CW ( X ) is termed the Choquet integral of the measurable function X : : o R with respect to the monotone set function W . This concept of “integral”, extending Lebesgue integral with respect to V -additive set

functions (probability measures), is due to Choquet [12]. While this type of integral is widely used in mathematical economics, see e.g. Marinacci and

6.2 The Choquet Integral

75

Montruccchio [50], it remains somewhat unfamiliar to statistics, see also e.g. Huber [38], Huber and Strassen [37]. As we will see, when replacing distribution functions F of random variables by capacity functionals T of random closed sets (via Choquet theorem characterizing of probability laws of random closed set in terms of capacity functionals which are not V - additive set functions) Choquet integral with respect to capacity functionals is necessary. Here is a situation in coarse data analysis. Suppose that for each random experiment w, we cannot observe the exact outcome X (w) , but we can only locate X (w) in some closed subset of R. Let S be a coarsening of X. Let g: R o R  , measurable. Suppose we are interested in computing Eg ( X ) in this situation. The random variable g ( X ) is bounded by inf{g ( x) : x  S ( w)} d g ( X ( w))

g * ( w)

d sup{g ( x) : x  S ( w)} = g * ( w)

(6.8)

For g * and g * should be such that, for any B  B(R), B*

{w : S ( B ) Ž B}  A

B*

{w : S ( B)  B z I}  A

and

In such a case, let B ( R) o [0,1] be defined as

F*

F* ( B)

P{w : S ( w) Ž B}

P ( B* )

Then E* g ( X )

=

f

³0

Eg * = ³: g * ( w)dP( w)

P{w : g * ( w) ! t}dt

(6.9)

76 Chapter 6 Convergence of Random Fuzzy Sets f

= ³0 P( g *1 (t , f))dt f

= ³0 P( g 1 (t , f))* dt f

= ³0 P{w : S ( w) Ž g 1 (t , f) ! t}dt f

= ³0 F* ( g 1 (t , f))dt f

= ³0 F* ( x : g ( x) ! t )dt

(6.10)

Similarly, for F * ( B)

P(S  B z I )

P( B * ) ,

(6.11)

we have E*g( X )

f

Eg * = ³0 F * ( x : g ( x) ! t )dt .

(6.12)

*

Thus, both E g * and E g are Choquet integrals of the measurable function g with respect to monotone set functions F* , F * , respectively, and E * g d Eg ( X ) d Eg * .

The above situation with imprecise information is considered in Dempester [16]. Here is another simple problem in decision making with incomplete information. Let X be a random variable with values in 4 {T1 ,T 2 ,T 3 ,T 4 } Ž R  , with probability density function f 0 partially specified as f 0 (T1 ) t 0.4 , f 0 (T 2 ) t 0.2 , f 0 (T 3 ) t 0.1 . To make decisions based upon expected value of X, we can only rely on inf{E f ( X ) : f  P} , where E f ( X ) denotes the expected value of X with respect to the density f , and P denotes the class of all probability density functions on 4 satisfying the above partial specification.

6.2 The Choquet Integral

77

Renaming the element of 4 if necessary, suppose T1  T 2  T 3  T 4 . Let F : 2 4 o [0,1] F ( A) inf{Pf ( A) : f  P} ,

where Pf ( A)

(6.13)

¦ f (T ) .

T A

Define g : 4 o [0,1] by g (T1 )

F ({T1 ,T 2 ,T 3 ,T 4 })  F ({T 2 ,T 3 ,T 4 })

g (T 2 )

F ({T 2 ,T 3 ,T 4 })  F ({T 3 ,T 4 })

g (T 3 )

F ({T 3 ,T 4 })  F ({T 4 })

g (T 4 )

F ({T 4 })

Clearly g  P. Now, E g ( X ) = ³4 X (T )dPg (T ) f

= ³0 Pg ({T i : T i ! t})dt

t2

³0

F ({T i : T i ! t})dt ,

(6.14)

i.e. E g ( X ) is a Choquet integral. We are going to show that inf {E f ( X ) : f  P} is attained precisely at g. Since E f (X )

t2

³0

Pf ({T i : T i ! t})dt ,

(6.15)

78 Chapter 6 Convergence of Random Fuzzy Sets

it suffices to show that, for each t  R, 2

E f (X )

t ³0 Pf ({T i : T i ! t})dt , for all f  P .

(6.16)

If {T : T ! t} {T1  T 2  T 3  T 4 } , then 4

Pg (T ! t ) = ¦ g (T i ) =1= Pf (T ! t ) .

(6.17)

i 1

If {T : T ! t} = {T1 ,T 2 ,T 3 ,T 4 } , then 4

4

i 1

i 2

Pg (T ! t ) = ¦ g (T i ) d ¦ f (T i ) = Pf (T ! t ) ,

(6.18)

Since, by construction of g, we have g (T i ) d f (T i ) , for f  P , and i

2, 3, 4 .

(6.19)

Thus, the Choquet integral of X (T ) T with respect to the monotone set function F is the lower bound for expected values in the model. It is interesting to note the concept of Choquet integral is popular in the field of multicriteria decision-making (see e.g. Grabisch et al. [30]). Specifically, ranking of alternatives with respect to a set of criteria is a common and important process in almost every day activities. When the criteria are non-interactive (independent), then weighted a averages seems to be appropriate aggregation operators for ranking. Even in more complex situations, such as those involving fuzzy relations, linear (additive) aggregation operators are still in use, such as the popular Analytic Hierarchy Process (see Saaty [75]). Recently, non-linear aggregation operators for ranking alternatives are shown to be more appropriate for interactive criteria, and the Choquet integral seems to be the most useful one, as it is a generalization of linear

6.3 A Variational Calculus of Set Functions

79

aggregation operators based on probability measures. In this context, the finite version of Choquet integral is used. Specifically, let C {1,2,..., k} be a set of criteria. Let X : C o R be an “evaluation”. The degrees of importance of subsets of C is modeled by a set function A : 2 C o [0,1] such that A (I ) =0, A(C)=1, and A is monotone, i.e. A Ž B Ÿ W (A) d W (B). Such a general set function is called a fuzzy measure. Then, k

CW ( X )

where x(i )

¦ x(i ) [ W ( A(i ) )- W ( A(i 1) ) ],

(6.20)

i 1

xi , x(1) d x( 2) d ... d x( n ) , A(i ) {(i), (i  1),..., (k )} , A( k 1) I .

The discrete version CW ( X ) of the Choquet integral of X with respect to W is used as a map from R n o R to compare various evaluations: the evaluation X is preferred to the evaluation Y if CW (Y ) d CW ( X ) . In the next sections, we are going to use Choquet integral to investigate the convergence in distribution of random sets. Statisticians who wish to know more about the mathematics of Choquet integral could see e.g. Grabisch et al [30], Nguyen and Walker [62], Marinacci and Martrucchio [50].

6.3 A Variational Calculus of Set Functions While coarse data analysis seems to be an important and general view that coarsening schemes are random sets (or more generally random fuzzy sets) has not been sufficiently emphasized. Moreover, distributional aspects of large sample statistics based on random sets, let alone random fuzzy sets, have not yet entered statistical analysis. A typical situation is the estimation of probability density function using the excess mass approach (see e.g. Polonik [28]). Recall that, like membership functions of fuzzy sets, a probability density function f : R d o R  can be recovered from its D  level sets

80 Chapter 6 Convergence of Random Fuzzy Sets

AD ( f ) {x  R d : f ( x) t D }, D  R 

(6.21)

via f ( x)

f d ³0 AD ( f )( x)dD , x  R ,

(6.22)

where we also write AD ( f )( x) as the indicator function of set AD ( f ) . Thus, the pointwise estimation of f (x) can be achieve by estimating each D  level set of f, say by some random set AD ,n (based upon an i.i.d. sample X 1 , X 2 ,..., X n ), and using a plug-in estimator of the form f n ( x, X 1 , X 2 ,..., X n )

f ³0 AD ,n ( x)dD .

(6.23)

Of course, it remains to examine whether such an estimator possesses desirable properties. The method of “excess mass” provides a rationale for proposing random set estimators (see e.g. Hartigan [33], Polonik [70] ). Specifically, let F denote the unknown distribution function of X. We write F ( dx) for associated Stieltjes probability measure, i.e. the probability laws of X on B(Rd). Let L( dx) denote the Lebesgue measure on Rd. Then for each A  B(Rd), and D  R  , H D ( A) ( F  DL)( A) is the excess mass of the set A at level D . By writing A ( A ˆ AD ) ‰ ( A ˆ ADc ) , we see that

H D ( A) d H D ( AD ) ,  A  B(Rd), i.e. the level set AD has the largest excess mass at level D among all Borel sets. This suggest a way to estimate AD from empirical counter-part of the signed measure F  DL , just like the maximum likelihood principle in statistics. Let Fn (dx) denote the empirical measure associated with the sample X 1 , X 2 ,..., X n , i.e. Fn (dx)

1 n ¦G X , nj1 j

(6.24)

6.3 A Variational Calculus of Set Functions

81

where G X being the Dirac measure as X  Rd. The empirical excess mass of A  B(Rd) is then

H D ,n ( A) ( Fn  DL)( A) .

(6.25)

Then, formally, an “excess mass” estimator for AD could be taken as

AD ,n which is the set maximizing H D ,n ( A) over some specified class C of subsets of B(Rd). While the program seems plausible, much more work is needed to justify its use in statistical practice, e.g. limit distributions of the random set estimator AD ,n . This requires, among other things, tractable criteria for convergence in distribution of sequences of random sets. Moreover, the above optimization problem needs special attention: the variable A is neither a vector, nor a function, but a set. How to maximize H D ,n ( A) over A C ?

From a general view point, this boils down to develop a variational calculus of set functions. While the above type of optimization of set functions occurs in many areas of applied mathematics, such as shape optimization, we are not aware of set functions for solving it. We propose below a variational calculus (see Nguyen and Kreinovich [59]). But first, here is some familiar situations in statistics where optimization of set functions arises. (i) Neyman-Pearson lemma Let X be a random vector on R d . Consider simple hypotheses H 0 : The probability law of X is the probability measure P0 , H a : The probability law of X is Pa .

Suppose P0 , Pa can be represented in terms of densities ( f 0 , f a ) with respect to a common measure L on B(Rd). The type-I error is

82 Chapter 6 Convergence of Random Fuzzy Sets

D

P( x  B H a )

³B f o ( x)dL( x) ,

(6.26)

³Bc f a ( x)dL( x) .

(6.27)

where B is the critical region of a test. The type-II error is

E

P( x  B H a )

For fixed D , the problem of finding B to minimize E is this. Find B* in W ={B  B(Rd): ³B f o ( x)dL( x) d D } so that

³B* f 0 ( x)dL( x) D

(6.28)

and

³B*c f a ( x)dL( x) d ³Bc f a ( x)dL( x) for all B  W . (6.29) Neyman and Pearson derived the solution of this optimization problem of set functions using the calculus of variations. The solution is B*

{x  R d : f a ! kf o ( x)}

(6.30)

where k is such that ³B f 0 ( x)dL( x) D . * (ii) Bayes tests between capacities (see Huber and Strassen [37]) Consider the following situation in robust statistics. Let M denote the set of all probability measures on the complete, separable and metrizalbe space : . Consider testing thee null hypothesis H 0 against the alternative H a , where H0

{P  M : P d To }

Ha

{P  M : P d Ta } ,

where T0 , Ta are two capacity functionals.

6.3 A Variational Calculus of Set Functions

83

Let B be a critical region for the above test, i.e. rejecting H 0 if x  B. is observed. Then the upper probability of falsely rejecting H 0 is T0 ( B) , and of falsely rejecting H a is Ta ( B c ) . If we assume that H 0 is true with prior probability

t ,0dt f, 1 t

then the (upper) Bayes risk of B is 1 1 1 1 T0 ( B)  Ta ( B c ) = T0 ( B)  (1  Va ( B )) . 1 t 1 t 1 t 1 t

(6.31)

Where Va is the conjugate of Ta , i.e. Va 1  Ta ( B c ) . Thus, we are led to minimizing the set function B o tT0 ( B)  Va ( B) , for t fixed. (iii) Dividing a territory Let U be an open and bounded set of Rd, and g i : U o Rd, i=1,2, be continuous functions. Find a closed set A Ž U such that ( ³A g1 ( x)dx)( ³U \ A g 2 ( x)dx)

is maximum over the class F(U) of closed subsets of U. As far as we know, there seems to be no appropriate available variational calculus for the above type of problems. As such, below is our attempt to provide a tool for solving optimization problems of setfunctionals. We are going to define a concept of derivative DF for a setfunction F, defined on some class of subsets of a set U so that when DF(A) = 0, F(A) will be a stationary value for f. Although this can be formulated in a fairly general context, we restrict ourselves to the concrete case of U =Rd, and the domain of F is B(Rd), i.e. F : B o R. In order to extend the classical notion of derivatives of functions to set-functions, we choose to use the symmetric form of derivatives (also called Schwartz derivatives).

84 Chapter 6 Convergence of Random Fuzzy Sets

For F:R o R, defined in some neighborhood of x, and provided that the limit exists, the symmetric (Schwartz) derivative of f at x is by definition f ' ( x)

lim . h o0

f ( x  h)  f ( x  h) 2h

(6.32)

Remark 6.1 This type of symmetric derivatives is useful in numerical estimation since its truncation error is of order of O(h 2 ) . Recall that if exists f ( x  h)  f ( x  h) f ' ( x) lim . , then f ' ( x) leads to the same value. h o0 2h However, the converse is not true: it is possible that f does not have a derivative in the usual sense, but yet it has a derivative f ' ( x) above. Our second idea about extending f ' ( x) to the setting of set-functions is based upon an analogy with partial derivative of a function f ( x1 ,! xn ) of several variables, we need to specify two things: the point ( x1 ,! xn ) at which the derivative is computed, and which variable xi (among x1 ,!, x n ) over which we differentiate. Thus, for F:B o R, the analogy is this. We need to evaluate 1A (t ) for all t  Rd. Viewing the 1A (t ) ’s as “variables”, we need also to specify a particular variable 1A (t ) , or equivalently a t  Rd. Therefore we are led to consider the notion of derivative of F at ( A, t ) for A  B, t  Rd. Sepcifically, Definition 6.1 Let F:B(Rd) o R, and W be the Lebesgue measure on B(Rd). The derivative of F at ( A, t ) , for A  B, t  Rd is defined to be

DF ( A, t )

F ( A * H )  F ( A  H ') H o{t } W (H ) lim

(6.33)

provided the limit exists, where the limit is taken over H such that W ( H ) z 0 , and H o {t} in the Hausdorff metric sense.

6.3 A Variational Calculus of Set Functions

85

Remark 6.2 If || || denotes the usual norm on Rd, then the (extended, pseudo) Hausdorff metric on non-empty subsets of Rd is d ( A, B )

max{sup inf || a  b ||, sup inf || a  b ||} . aA bB

bB aA

Clearly, the definition of DF ( A, t ) is inspired from f ' ( x) in which its numerator f ( x  h)  f ( x) is replaced by F ( A * H )  F ( A \ H ) whereas its denominator 2h (the length of [ h, h] ) is replaced by W (H ) with W (H ) z 0 . ( [ h, h] o {0} , a singleton, corresponds to H o {t} ). Note also that if F is additive, then F ( A * H )  F ( A  H ' ) F ( H ) , and hence the definition of DF ( A, t ) turns into the so-called “general derivative” DF (t )

F (H ) . H o{t },W ( H ) z 0 W ( H ) lim

(6.34)

If F is a measure which is absolutely continuous with respect to W , then dF (t ) is equal almost everywhere to its Radon-Nikodym derivative dt DF (t ) .

Definition 6.2 The set-function F:B o R is said to be differentiable if it has derivative at every point ( A, t )  B u Rd. F is said to be continuously differentiable if it is differentiable and the map t o DF ( A, t ) is continuous for every A  B(Rd)

Example 6.1. (i) In the territorial division problem, the set-functions F1 , F2 , defined on subsets A Ž U , are closely continuously differentiable, where F1 ( A)

³AX1 ( x)dW ( x) , F2 ( A) ³U X 2 ( x)dW ( x)  ³AX 2 ( x)dW ( x)

86 Chapter 6 Convergence of Random Fuzzy Sets

with DF1 ( A, t ) X1 (t ) , DF2 ( A, t ) X 2 (t ) .

And hence F ( A)

F1 ( A) F2 ( A) is also continuously differentiable with

DF ( A, t ) X1 (t ) F2 ( A)  X 2 (t ) F1 ( A)

(6.35)

(ii) The excess mass set-function

H D ( A) ( F  DW )( A) is continuously differentiable if the density f ( x)

dF ( x) is continuous. dt

As such, DH D ( A, t )

f (t )  D .

(6.36)

In the following, the interior, closure and boundary of a set A is denoted as AD , A , G ( A) , respectively. Of course, we assume AD z I . Observe that if F attains its maximum at A, then DF ( A, t ) t 0 or d 0 for all t  AD or all t  ( A' ) D respectively. Indeed assume that A is a set at which F is maximum. For t  AD, the open ball

B (t , r )

{x  R d :|| x  t || r} Ž A for r sufficiently small. Thus, F ( A ‰ B (t , r )) F ( A) and hence DF ( A, t ) t 0 since F ( A) is maximum value. Simi-

larly, for t  ( A' ) D , B(t , r ) Ž A for small J and hence A \ B(t , r ) A , implying that DF ( A, t ) d 0 . The situation for minimum is dual: DF ( A, t ) t 0 or d 0 according to t  AD or t  ( A' ) D . The following result is useful for optimization of set-functions. Theorem 6.1 Let F be continuously differentiable. If F attains maximum (or minimum) at some set A (with AD and ( A' ) D non-empty), then necessarily DF ( A, t )

0 for all t  G ( AD )  G (( A' ) D ) .

6.4 Choquet Weak Convergnece

87

Proof: Suppose F attains its maximum at A. For t  G ( AD ) , t is limit of a sequence t  AD with DF ( A, t n ) t 0 , by continuity. Similarly, for n

D

t  G (( A' ) ) , t is limit of a sequence s n  ( A' ) D with DF ( A, s n ) d 0 for each n, and hence DF ( A, t ) d 0 .

6.4 Choquet Weak Convergnece The material in this section (drawn from Feng and Nguyen [53] and from Ding Feng’s 2004 Ph.D. Thesis) is about convergence of capacity functionals of random sets. The motivation stems from asymptotic aspects in inference and decision-making with coarse data in Biostatistics, set-valued observations, as well as connections between random sets with several emerging uncertainty calculi in intelligent systems such as fuzziness, belief functions and possibility theory. Specifically, we study the counter-part of Billingsley’s Portmanteau Theorem for weak convergence of Choquet integrals. Recall that a random closed set is a random element in the space of all closed sets of the basic setting space. In the classical theory of random sets, the setting space is assumed to be a locally compact, secondcountable Hausdorff (LCSCH) space, see e.g. Matheron [ ]. Throughout this section, U is always considered as a LCSCH space. Especially, the Euclidean space Rd is a nice example of LCSCH spaces. Let F = {all closed sets in U}, K = {all compact sets in U}, G = {all open sets in U}, K0 = K- {I} , and B(U)= {all Borel sets in U}. Let B(F) be the Ӻ-algebra generated by classes FK, K  K and FG , G  g , where Fk

{F  F : F  K z I} , FG

{F  F : F  G z I }

(6.37)

Define a random closed set as a measurable mapping S : (:, 6, P ) o (F, B(F)) from a probability space into a measurable space. Let Q be the

88 Chapter 6 Convergence of Random Fuzzy Sets

probability measure induced on B(F), that is, for each B  B(F), Q( B)

P( S 1 ( B)) .

Note that the distribution of random set S is uniquely determined by its capacity functional T on N defined as: S

P ({Z : S (Z )  K z I}) , K  K

TS ( K )

It is easy to prove that TS defined above satisfies the following properties : (T1) TS is upper semi-continuous (u.s.c) on N , i.e., K n p K in K o TS ( K n ) p TS ( K ) ;

(T2) TS (I )

0 and 0 d TS d 1 ;

(T3) TS is monotone increasing on N and for K1 , K 2 ,!, K n  K, n t 2 , n

TS ( K i ) d i 1

¦ (1)

I z I  (1, 2 ,!,n )

I 1

TS ( * K i ) . iI

Let * denote the class of all the functionals T on K satisfying (T1)-(T3) above. Such fuctionals are called alternating Choquet capacities of infinite order or, for brevity, simply capacity functionals. In the LCSCH space U, the capacity functional T can be extended to Borel sets by T ( B)

sup{T ( K ) : K  K , K  B } for B  B(U)

Note that (T1’) Upper semi-continuity of T on g means that Gn n G in G Ÿ TS (Gn ) n TS (G ) ;

We will use the following lemma in our main results.

6.4 Choquet Weak Convergnece

89

Lemma 6.1 (Choquet’s Theorem). Every probability measure Q on (F, B(F) determines a Choquet capacity functional T on K through the correspondences (C1). T ( K )

Q (FK), K  K, and

(C2). T (G )

Q( FG ) , G  G.

Conversely, ever Choquet capacity functional T on N determines a unique probability measure Q on (F, B(F) that satisfies the above conditions (C1) and (C2). A compact set K  K is called a continuity set for T if and only if T ( K ) T ( K o )) where K o denotes the interior of K. Thus a compact set K

is T-continuous if and only if FK is Q-continuous. Let ST ={K  K:T(K)=T( K o )}.

Results on the weak convergence of random closed sets in U have been discussed in many papers, see, e.g., Salinetti and Wets [76], Norberg [65], and Molchanov [53]. The problem we are going to address is this. Since the analysis of random sets is much simpler in terms of their capacity functionals Tn , T, it is desirable to establish the convergence in distribution in term of the Tn ' s and T, rather than the Qn ' s and Q. The following lemma is given by Molchanov [54].

Lemma 6.2 Let U be a LCSCH space. Then the following statements are equivalent: W Q; (a). Qn o

(b). lim n Tn ( A) T ( A), A  S J .

90 Chapter 6 Convergence of Random Fuzzy Sets

In order to study the counter-part of Billingsley’s Portmanteau Theorem, see Billingsley [31]. We need first an appropriate definition of weak convergence for capacity functionals (which are generalizations of probability measures on (U , B(U). Let Cb (U ) be the space of all bounded realvalued continuous functions on U. For f  Cb (U ) , the Choquet integral of f with respect to T is 0 f ³ fdT : ³0 T ({ f t t})dt  ³f [T ({ f t t})  T (U )]dt .

(6.38)

We say that the sequence of capacity functionals Tn converges, in the Choquet weak sense, to the capacity T in U, if C W ³ fdTn o³ fdT , f  Cb (U ), in symbol, Tn o T , as n o f .

Note that the capacity functional T on the setting space U has the following properties (see, Matheron (1975)): (a) T (G )

sup{T ( K ) : K  K, K  G} , G  G;

(b) T ( K )

inf{T (G ) : G  G, G Š K }, K  K.

Let W f denote the hit-or-miss topology on F generated by two classes, FK, K  K, and FG, G  G. Now observe that F, with the hit-ormiss topology, is compact and metrizable. Thus, the convergence in distribution of a sequence of random closed sets S n to S can be studied in the weak topology on this metric space. A subset B  U is called T-continuous functional closed set if B {x : f ( x) t a} for some a  R, f  Cb (U ) such that T ({x : f ( x) t a}) T ({x : f ( x) ! a}) .

(6.39)

For a given capacity T, a family of capacity functionals A  * is said to be T-tight if for any T-continuous functional closed set B and for each H ! 0 , there exists a compact set K B  B such that

6.4 Choquet Weak Convergnece

sup[ R ( B)  R( B  K B )]  H .

91

(6.40)

RA

Especially, since U is T-continuous functional closed set for any give T, there exists a compact set KU  U such that sup[ R (U )  R ( KU )]  H .

(6.41)

R A

Lemma 6.3 Let T  * and f  Cb (U ) . Then b ³a T ({ f t t})dt

b ³a T ({ f ! t})dt , a, b  ƒ .

Moreover , we have || f || ³|| f || T ({ f t t}) dt  T (U ) || f ||

³ fdT

|| f || ³|| f || T ({ f ! t})dt  T (U ) || f || .

Proof. To see these, simply note that b 1

b ³a T ({ f t t})dt

b lim T ({ f n of a

³

lim ³a  1n T ({ f t t}) dt

n of

n

b

t t  1n }) dt d ³a T ({ f ! t}) dt

b

d ³a T ({ f t t})dt .

Thus

b

³a T ({ f

t t}) dt

b

³a T ({ f

(6.42) ! t})dt .

(6.43)

From this lemma, we know that { f t t} is T-continuous functional closed set almost everywhere on t  ( || f || 1, || f || 1) .

92 Chapter 6 Convergence of Random Fuzzy Sets

Put W={B  B (U ) : B  K}. Say that a class L  W is separating if for any K  K and G  G with K  G, there exists some A  L such that K  A  G . The following proposition is a conclusion of Theorem 2,1 in

Norberg (65).

Proposition 6.1 Suppose U is a LCSCH space. Let S1 S2 ,….be random closed sets in U, T1, T2,…their corresponding capacity functional, and Q1 Q2,….their corresponding probability distributions. If there exists a separating class L  W and a capacity functional R such that R ( A o ) d lim inf Tn ( A) d lim sup Tn ( A) d R ( A) , A  L . n of

n of

then there exists a random set S in U with the probability distributions Q W and hitting functional T satisfying Qn o Q and T = R on K. The main result presented here is that, for the case where U is W LCSCH, Tn C o T is equivalent to the convergence in distribution of corresponding random sets with T-tightness of the sequence {Tn }fn 1 . Let Qn, Q denote probability measures on B(F), and Tn , T denote corresponding capacity functional.

Example 6.2 Let U = R Take Fn:R Æ[0,1] defined by

Fn ( x)

­° x n ® °¯1  x

Then Fn(x) is the distribution of a random varible Xn. Since Fn(x) o 0 for each x R, Xn is not weakly convergent. Let Tn(K) be the capacity functional of random closed set Sn={Xn}. It is easy to see Tn(K) o 0, K  K by 0 d Tn ( K ) d Fn (max(K )) o 0

6.4 Choquet Weak Convergnece

93

Let Qn denote probability measures on B(F) determined by Tn and Q be probability measure on B(F) determined by the capacity 0. Since Tn(K) o W Q by the above Lemma B but 0, K  K , we get Qn o W Tn C o T does not hold.

Indeed, let f(x)=1. Then for 0 0, there exists a q0 such that 0 d T ({x : f ( x) ! a})  T (C q ˆ {x : f ( x) ! a})  H2 , q t q0

By Lemma 6.3 and Lemma 6.4, we have lim sup[Tn ( B )  Tn (Cq ˆ B )] nof

d lim sup[Tn ( B )  Tn (C q ˆ {x : f ( x) ! a})] n of

d lim sup[Tn ( B)  lim inf Tn (C q ˆ {x : f ( x) ! a})] n of

n of

d T ( B )  T (C q ˆ {x : f ( x) ! a}) T ({x : f ( x) ! a})  T (C q ˆ {x : f ( x) ! a})  H2

(6.50)

6.4 Choquet Weak Convergnece

97

Since any family of finitely many of capacity functionals is T-tight, we obtain the sequence {Tn} is T-tight.

W Lemma 6.5 If Qn C o Q and the sequence {Tn }fn 1 is T-tight, then

lim inf ³ fdTn t ³ fdT , f  Cb (U ) . n of

Proof: For any given f  Cb (U ) , Lemma 6.4 implies || f || ³|| f || Tn ({ f ! t})dt  Tn (U ) || f || , n t 1

³ fdTn ³ fdT Note that F

{ f tt }

|| f || ³|| f || Tn ({ f ! t})dt  Tn (U ) || f ||

(6.51)

is open in the F, with the hit-or-miss topology.

W Q, Since Qn o

Billingsley’s Portmanteau Theorem (e.g. see

Billingsley [5] implies lim inf Qn ( F{ f tt } ) t Q( F{ f !t } ) , t  ( || f ||, || f ||) n of

(6.52)

On the other hand, for each İ > 0, by the T-tightness of the sequence {Tn }fn 1 , there exists a compact K  U such that sup[Tn  Tn ( K )]  H n

Simply note that FK is closed in the F , with hit-or-miss topology. For W Qn o Q , by Billingsley’s Portmanteau Theorem, we obtain sup Qn ( Fk ) d Q( Fk ) . By Choquet’s Theorem and Fatou’s Lemma, we obn of

tain lim inf ³ fdTn n of

98 Chapter 6 Convergence of Random Fuzzy Sets || f ||

t lim inf ³|| f ||Tn ({ f ! t}) dt  lim sup Tn (U ) || f || n of

n of

|| f ||

t lim inf ³|| f || Qn ( F{ f !t } )dt  lim sup[Tn ( K  H )] || f || n of

n of

|| f ||

t ³|| f || lim inf Qn ( F{ f !t } )dt  lim sup Qn ( FK ) || f || H || f || n of

n of

|| f ||

t ³|| f || Q( F{ f !t } )dt  Q( FK ) || f || H || f ||

(6.53)

i.e. lim inf ³ fdTn n of

|| f ||

t ³|| f || Q( F{ f !t } )dt  Q( FU ) || f || H || f ||

|| f || ³|| f ||T ( f ! t )dt  T (U ) || f || H || f ||

³ fdt  H || f || Since İ is arbitrary, it implies lim inf ³ fdTn t ³ fdT . n of

(6.54)

W Q and the sequence {Tn }fn 1 is T-tight, then Lemma 6.6 If Qn o

lim sup ³ fdTn d ³ fdT , f  Cb (U ) . n of

Proof: For any fixed f  Cb (U ) , the decreasing function T ({ f ! t}) is Riemann integrable on [ || f ||, || f ||] . By Lemma B, ({ f t t}) is T-

6.4 Choquet Weak Convergnece

continuous functional closed set almost everywhere t  ( || f || 1, || f || 1) . For each İ> 0, there exists a subdivision of  || f || t 0  t1  t 2  "  t j  t j 1  "  t m

99

on

|| f ||

such that (a) { f t t j } is T-continuous functional closed set for j=0,1,2…m-1; (b) the following inequality holds m 1

¦ T { f t t j }{t j 1  t j }  T (U ) || f || ³ fdT  H

(6.55)

j 1

Since the sequence {Tn }fn 1 is T-tight, there exists a compact set K  U such that sup Tn ({ f t t j })  Tn ({ f t t j } ˆ K )  H , j

0,1,2,!, m  1

n

Note that Fu is open and F({ f tt j }ˆ K ) is closed in F, with the hit-or-miss toW Q , by Bilingsley’s Portmanteau Theorem, we obpology. For Qn o

tain lim inf Qn ( FU ) t Q( FU ) , lim inf Qn ( F({ f tt j }ˆK ) ) d Q( F({ f tt j }ˆK ) ) n of

n of

Hence, monotone increasing of each Tn and Choquet’s theorem imply lim sup ³ fdTn n of

|| f ||

lim sup[ ³|| f || Tn ({ f ! t})dt  Tn (U ) || f ||] n of

m 1

t

d lim sup[ ¦ ³t j 1 Tn ({ f t t})dt ]  lim inf Qn ( FU ) || f || n of

j 0

j

n of

100 Chapter 6 Convergence of Random Fuzzy Sets m 1

d lim sup ¦ Tn ({ f t t j })(t j 1  t j )  Q( FU ) || f || j 0 n of m 1

d lim sup[ ¦ Tn ({ f t t j }ˆ K  H )(t j 1  t j )  T (U ) || f || j 0 n of

m 1

d lim sup ¦ [Qn ( F({ f tt j }ˆ K ) ](t j 1  t j )  2H || f || T (U ) || f || j 0

n of m 1

d ¦ Q( F({ f tt j }ˆ K ) )(t j 1  t j )  2H || f || T (U ) || f || j 0

m 1

¦ T ({ f t t j } ˆ K )(t j 1  t j )  2H || f || T (U ) || f ||

(6.56)

j 0

So we obtain m 1

d ¦ T ({ f t t j } ˆ K )(t j 1  t j )  2H || f || T (U ) || f ||

(6.57)

j 0

m 1

d ¦ T ({ f t t j })(t j 1  t j )  T (U ) || f || 2H || f || j 0

 ³ fdT  H  2H || f || .

Since İ is arbitrary, we obtain lim sup ³ fdTn d ³ fdT . n of

Using Lemma E and Lemma F, we obtain

W Q and the sequence {T }f is T-tight , then Theorem 6.3 If Qn o n n 1

C W

Tn o T .

It turns out that the converse of Theorem 6.3 holds, since

6.4 Choquet Weak Convergnece 101 W W o T , then Qn o Q. Lemma 6.7 If Tn C

Proof: We will, at first , prove that the class Sr  W is separating. In fact, let K  K and G  G with K  G . Since U is LCSCH space, by the compactness of K, there exists an open set O  U such that O  K and K  O  G . Note that K and U\O are disjoint closed sets in U .By Urysohn’s lemma, there exists a continuous function f : U o [0,1] such that

f(x) = 0 for x U \ O and f(x)=1 for x  K. Simply note that K  { f t t}  O  G , t  (0,1)

(*)

By lemma B, T { f t t} T { f ! t} is almost everywhere on t  (0,1) Choose B { f t t 0 } for some t0  (0,1) with T{ f t t0 } T{ f ! t0 }

Keeping in mind that { f ! t 0 }  B o  B . We get T ( B ) T ( Int ( B )) T ({ f ! t 0 } . Moreover, B

B  O  K implies B  K and therefore

B  ST . Thus the class ST  W is separating by (*). W o T imOn the other hand, by Lemma C and Lemma D, Tn C

plies, for any lim sup Tn ( K ) d T ( K ) n of

lim inf Tn ( K o ) t T ( K o ) n of

Thus, for K  ST

102 Chapter 6 Convergence of Random Fuzzy Sets

T ( K o )  lim inf Tn ( K o )  lim inf Tn ( K )  lim sup Tn ( K )  T ( K ) n of

n of

n of

By the Proposition 6.1, there exists a random set S’ with the probability distribution P and the corresponding capacity functional R satisfyW ing Qn o P and R = T on K. By Choquet’s Theorem, we have Q = P, W Q i.e. Qn o

Theorem 6.4 Let U be a LCSCH space. Then the following statements are equivalent: W o T ; (a) Tn C W (b) Qn o Q and the sequence {Tn }fn 1 is T-tight;

(c) lim nTn ( K ) d T ( K ) for every compact set K in U, lim nTn (G ) t T (G ) for every open set G in U and the sequence {Tn }fn 1 is T-tight; (d) lim nTn ( A) T ( A) for each A ST , and the sequence {Tn }fn 1 is T-tight.

Corollary 6.1 Let { U , d }be a compact metric space. Then the following statements are equivalent W o T ; (a) Tn C

(b) Qn  o Q ; (c)

lim nTn ( K ) d T ( K ) for

every

compact

set

K

in

U

and

lim nTn (G ) t T (G ) for every open set G in U;

(d) lim n Tn ( A) T ( A) for each A ST . Let C S (U ) denote the space of all nonnegative real-valued, continuous function on U with compact supports.

6.4 Choquet Weak Convergnece 103 W Lemma 6.8 Let U be a LCSCH space. If Qn o Q , then

lim ³ fdTn

n of



³ fdT , f  C S (U ) .

Proof: For any given f  C S (U ) , by definition, we know that {x U : f ( x) z 0}  K , For some K  K. Simply note that Lemma B im-

plies

³ fdTn ³ fdT

|| f || || f || ³0 Tn ({ f t t})dt ³0 Tn ({ f ! t})dt

|| f || || f || ³0 Tn ({ f t t})dt ³0 T ({ f ! t})dt

(6.58)

(6.59)

For each t  (0, || f ||) , the closed set { f t t}  K is compact in the LCSCH space U, so the set F{ f tt }

F  F { f tt } = is closed in the space F

with the hit-or-miss topology. Moreover F { f tt } if open in the space F. On the other hand, we know w ( F{ f tt } )

F{ f tt }  int( F{ f tt } )  F{ f tt }  F{ f !t }

(6.60)

It follows from Choquet’s Theorem that || f ||

|| f ||

0 d ³0 Q[w ( F{ f tt} )]dt d ³0 Q( F{ f tt }  F{ f !t } )dt || f || ³0 {( f t t )}  T ({ f ! t})dt 0

All these mean that F{ f tt } is a G-continuous set almost everywhere on W t  (0, || f ||) .Note the facts Qn d 1 , and the given condition Qn o Q.

By using Lebesgue dominated convergence theorem and Billingsley’s Portmanteau theorem (see, Billingsley [5]), we imply

104 Chapter 6 Convergence of Random Fuzzy Sets

lim[ ³ fdTn  ³ fdT ]

n of

|| f ||

lim ³ [Tn ({ f t t})  T ({ f t t})]dt

n of 0

M

lim ³ [Qn ( F{ f tt } )  Q( F{ f tt } )]dt

n of 0

M [Qn ( F{ f tt } )  Q( F{ f tt } )]dt 0 ³0 nlim of

(6.61)

Let U= ( R m , d ) be the Euclidean space. For any K  k0 j=1,2,…,define