3,088 research outputs found
Multivariate integration of functions depending explicitly on the minimum and the maximum of the variables
By using some basic calculus of multiple integration, we provide an
alternative expression of the integral in which the minimum and the maximum are replaced
with two single variables. We demonstrate the usefulness of that expression in
the computation of orness and andness average values of certain aggregation
functions. By generalizing our result to Riemann-Stieltjes integrals, we also
provide a method for the calculation of certain expected values and
distribution functions.Comment: 15 page
Invariant functionals on completely distributive lattices
In this paper we are interested in functionals defined on completely
distributive lattices and which are invariant under mappings preserving
{arbitrary} joins and meets. We prove that the class of nondecreasing invariant
functionals coincides with the class of Sugeno integrals associated with
-valued capacities, the so-called term functionals, thus extending
previous results both to the infinitary case as well as to the realm of
completely distributive lattices. Furthermore, we show that, in the case of
functionals over complete chains, the nondecreasing condition is redundant.
Characterizations of the class of Sugeno integrals, as well as its superclass
comprising all polynomial functionals, are provided by showing that the
axiomatizations (given in terms of homogeneity) of their restriction to
finitary functionals still hold over completely distributive lattices. We also
present canonical normal form representations of polynomial functionals on
completely distributive lattices, which appear as the natural extensions to
their finitary counterparts, and as a by-product we obtain an axiomatization of
complete distributivity in the case of bounded lattices
Weighted lattice polynomials
We define the concept of weighted lattice polynomial functions as lattice
polynomial functions constructed from both variables and parameters. We provide
equivalent forms of these functions in an arbitrary bounded distributive
lattice. We also show that these functions include the class of discrete Sugeno
integrals and that they are characterized by a median based decomposition
formula.Comment: Revised version (minor changes
Weighted lattice polynomials of independent random variables
We give the cumulative distribution functions, the expected values, and the
moments of weighted lattice polynomials when regarded as real functions of
independent random variables. Since weighted lattice polynomial functions
include ordinary lattice polynomial functions and, particularly, order
statistics, our results encompass the corresponding formulas for these
particular functions. We also provide an application to the reliability
analysis of coherent systems.Comment: 14 page
On indefinite sums weighted by periodic sequences
For any integer we provide a formula to express indefinite sums of
a sequence weighted by -periodic sequences in terms of
indefinite sums of sequences , where
. When explicit expressions for the latter sums are
available, this formula immediately provides explicit expressions for the
former sums. We also illustrate this formula through some examples
Computing subsignatures of systems with exchangeable component lifetimes
The subsignatures of a system with continuous and exchangeable component
lifetimes form a class of indexes ranging from the Samaniego signature to the
Barlow-Proschan importance index. These indexes can be computed through
explicit linear expressions involving the values of the structure function of
the system. We show how the subsignatures can be computed more efficiently from
the reliability function of the system via identifications of variables,
differentiations, and integrations
Algorithms and formulas for conversion between system signatures and reliability functions
The concept of signature is a useful tool in the analysis of semicoherent
systems with continuous and i.i.d. component lifetimes, especially for the
comparison of different system designs and the computation of the system
reliability. For such systems, we provide conversion formulas between the
signature and the reliability function through the corresponding vector of
dominations and we derive efficient algorithms for the computation of any of
these concepts from the other. We also show how the signature can be easily
computed from the reliability function via basic manipulations such as
differentiation, coefficient extraction, and integration
Structure functions and minimal path sets
In this short note we give and discuss a general multilinear expression of
the structure function of an arbitrary semicoherent system in terms of its
minimal path and cut sets. We also examine the link between the number of
minimal path and cut sets consisting of 1 or 2 components and the concept of
structure signature of the system
On modular decompositions of system signatures
Considering a semicoherent system made up of components having i.i.d.
continuous lifetimes, Samaniego defined its structural signature as the
-tuple whose -th coordinate is the probability that the -th component
failure causes the system to fail. This -tuple, which depends only on the
structure of the system and not on the distribution of the component lifetimes,
is a very useful tool in the theoretical analysis of coherent systems.
It was shown in two independent recent papers how the structural signature of
a system partitioned into two disjoint modules can be computed from the
signatures of these modules. In this work we consider the general case of a
system partitioned into an arbitrary number of disjoint modules organized in an
arbitrary way and we provide a general formula for the signature of the system
in terms of the signatures of the modules.
The concept of signature was recently extended to the general case of
semicoherent systems whose components may have dependent lifetimes. The same
definition for the -tuple gives rise to the probability signature, which may
depend on both the structure of the system and the probability distribution of
the component lifetimes. In this general setting, we show how under a natural
condition on the distribution of the lifetimes, the probability signature of
the system can be expressed in terms of the probability signatures of the
modules. We finally discuss a few situations where this condition holds in the
non-i.i.d. and nonexchangeable cases and provide some applications of the main
results
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