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本帖最后由 三T上人 于 2016-7-22 17:23 编辑 <br /><br />Preface
The theories described in the first part of this book summarize the research
work that in past 30-40 years, from different roots and with different aims,
has tried to overcome the boundaries of the classical theory of probability,
both in its objectivist interpretation (relative frequencies of expected
events) and in its subjective, Bayesian or behavioral view. Many compel-
ling and competitive mathematical objects have been proposed in different
areas (robust statistical methods, mathematical logic, artificial intelligence,
generalized information theory). For example, fuzzy sets, bodies of evi-
dence, Choquet capacities, imprecise previsions, possibility distributions,
and sets of desirable gambles.
Many of these new ideas have been tentatively applied in different dis-
ciplines to model the inherent uncertainty in predicting a system’s behavior
or in back analyzing or identifying a system’s behavior in order to obtain
parameters of interest (econometric measures, medical diagnosis, …). In
the early to mid-1990s, the authors turned to random sets as a way to for-
malize uncertainty in civil engineering.
It is far from the intended mission of this book to be an all comprehen-
sive presentation of the subject. For an updated and clear synthesis, the in-
terested reader could for example refer to (Klir 2005). The particular point
of view of the authors is centered on the applications to civil engineering
problems and essentially on the mathematical theories that can be referred
to the general idea of a convex set of probability distributions describing
the input data and/or the final response of systems. In this respect, the the-
ory of random sets has been adopted as the most appropriate and relatively
simple model in many typical problems. However, the authors have tried
to elucidate its connections to the more general theory of imprecise prob-
abilities. If choosing the theory of random sets may lead to some loss of
generality, it will, on the other hand, allow for a self-contained selection of
the arguments and a more unified presentation of the theoretical contents
and algorithms.
Finally, it will be shown that in some (or all) cases the final engineering
decisions should be guided by some subjective judgment in order to obtain
a reasonable compromise between different contrasting objectives (for ex-
ample safety and economy) or to take into account qualitative factors.
Therefore, some formal rules of approximate reasoning or multi-valued
logic will be described and implemented in the applications. These rules
cannot be confined within the boundaries of a probabilistic theory, albeit
extended as indicated above.
Subjects Covered: Within the context of civil engineering, the first chap-
ter provides motivation for the introduction of more general theories of un-
certainty than the classical theory of probability, whose basic definitions
and concepts (à la Kolmogorov) are recalled in the second chapter that also
establishes the nomenclature and notation for the remainder of the book.
Chapter 3 is the main point of departure for this book, and presents the
theory of random sets for one uncertain variable together with its links to
the theory of fuzzy sets, evidence theory, theory of capacities, and impre-
cise probabilities. Chapter 4 expands the treatment to two or more vari-
ables (random relations), whereas the inclusion between random sets (or
relations) is covered in Chapter 5 together with mappings of random sets
and monotonicity of operations on random sets. The book concludes with
Chapter 6, which deals with approximate reasoning techniques. Chapters 3
through 5 should be read sequentially. Chapter 6 may be read after reading
Chapter 3.
Level and Background: The book is written at the beginning graduate
level with the engineering student and practitioner in mind. As a conse-
quence, each definition, concept or algorithm is followed by examples
solved in detail, and cross-references have been introduced to link different
sections of the book. Mathematicians will find excellent presentations in
the books by Molchanov (2005), and Nguyen (2006) where links to the ini-
tial stochastic geometry pathway of Matheron (1975) is recalled and ran-
dom sets are studied as stochastic models.
The authors have equally contributed to the book.
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