4,518 research outputs found
Decision-theoretic rough sets-based three-way approximations of interval-valued fuzzy sets
In practical situations, interval-valued fuzzy sets are frequently
encountered. In this paper, firstly, we present shadowed sets for interpreting
and understanding interval fuzzy sets. We also provide an analytic solution to
computing the pair of thresholds by searching for a balance of uncertainty in
the framework of shadowed sets. Secondly, we construct errors-based three-way
approximations of interval-valued fuzzy sets. We also provide an alternative
decision-theoretic formulation for calculating the pair of thresholds by
transforming interval-valued loss functions into single-valued loss functions,
in which the required thresholds are computed by minimizing decision costs.
Thirdly, we compute errors-based three-way approximations of interval-valued
fuzzy sets by using interval-valued loss functions. Finally, we employ several
examples to illustrate that how to take an action for an object with
interval-valued membership grade by using interval-valued loss functions
Decision-theoretic rough sets based on time-dependent loss function
A fundamental notion of decision-theoretic rough sets is the concept of loss
functions, which provides a powerful tool of calculating a pair of thresholds
for making a decision with a minimum cost. In this paper, time-dependent loss
functions which are variations of the time are of interest because such
functions are frequently encountered in practical situations, we present the
relationship between the pair of thresholds and loss functions satisfying
time-dependent uniform distributions and normal processes in light of bayesian
decision procedure. Subsequently, with the aid of bayesian decision procedure,
we provide the relationship between the pair of thresholds and loss functions
which are time-dependent interval sets and fuzzy numbers. Finally, we employ
several examples to illustrate that how to calculate the thresholds for making
a decision by using time-dependent loss functions-based decision-theoretic
rough sets
High Granular Operator Spaces, and Less-Contaminated General Rough Mereologies
Granular operator spaces and variants had been introduced and used in
theoretical investigations on the foundations of general rough sets by the
present author over the last few years. In this research, higher order versions
of these are presented uniformly as partial algebraic systems. They are also
adapted for practical applications when the data is representable by data
table-like structures according to a minimalist schema for avoiding
contamination. Issues relating to valuations used in information systems or
tables are also addressed. The concept of contamination introduced and studied
by the present author across a number of her papers, concerns mixing up of
information across semantic domains (or domains of discourse). Rough inclusion
functions (\textsf{RIF}s), variants, and numeric functions often have a direct
or indirect role in contaminating algorithms. Some solutions that seek to
replace or avoid them have been proposed and investigated by the present author
in some of her earlier papers. Because multiple kinds of solution are of
interest to the contamination problem, granular generalizations of RIFs are
proposed, and investigated. Interesting representation results are proved and a
core algebraic strategy for generalizing Skowron-Polkowski style of rough
mereology (though for a very different purpose) is formulated. A number of
examples have been added to illustrate key parts of the proposal in higher
order variants of granular operator spaces. Further algorithms grounded in
mereological nearness, suited for decision-making in human-machine interaction
contexts, are proposed by the present author. Applications of granular
\textsf{RIF}s to partial/soft solutions of the inverse problem are also
invented in this paper.Comment: Research paper: Preprint: Final versio
Dialectics of Counting and the Mathematics of Vagueness
New concepts of rough natural number systems are introduced in this research
paper from both formal and less formal perspectives. These are used to improve
most rough set-theoretical measures in general Rough Set theory (\textsf{RST})
and to represent rough semantics. The foundations of the theory also rely upon
the axiomatic approach to granularity for all types of general \textsf{RST}
recently developed by the present author. The latter theory is expanded upon in
this paper. It is also shown that algebraic semantics of classical \textsf{RST}
can be obtained from the developed dialectical counting procedures. Fuzzy set
theory is also shown to be representable in purely granule-theoretic terms in
the general perspective of solving the contamination problem that pervades this
research paper. All this constitutes a radically different approach to the
mathematics of vague phenomena and suggests new directions for a more realistic
extension of the foundations of mathematics of vagueness from both foundational
and application points of view. Algebras corresponding to a concept of
\emph{rough naturals} are also studied and variants are characterised in the
penultimate section.Comment: This paper includes my axiomatic approach to granules. arXiv admin
note: substantial text overlap with arXiv:1102.255
Related families-based attribute reduction of dynamic covering information systems with variations of object sets
In practice, there are many dynamic covering decision information systems,
and knowledge reduction of dynamic covering decision information systems is a
significant challenge of covering-based rough sets. In this paper, we first
study mechanisms of constructing attribute reducts for consistent covering
decision information systems when adding objects using related families. We
also employ examples to illustrate how to construct attribute reducts of
consistent covering decision information systems when adding objects. Then we
investigate mechanisms of constructing attribute reducts for consistent
covering decision information systems when deleting objects using related
families. We also employ examples to illustrate how to construct attribute
reducts of consistent covering decision information systems when deleting
objects. Finally, the experimental results illustrates that the related
family-based methods are effective to perform attribute reduction of dynamic
covering decision information systems when object sets are varying with time.Comment: arXiv admin note: substantial text overlap with arXiv:1711.0732
An axiomatic approach to the roughness measure of rough sets
In Pawlak's rough set theory, a set is approximated by a pair of lower and
upper approximations. To measure numerically the roughness of an approximation,
Pawlak introduced a quantitative measure of roughness by using the ratio of the
cardinalities of the lower and upper approximations. Although the roughness
measure is effective, it has the drawback of not being strictly monotonic with
respect to the standard ordering on partitions. Recently, some improvements
have been made by taking into account the granularity of partitions. In this
paper, we approach the roughness measure in an axiomatic way. After
axiomatically defining roughness measure and partition measure, we provide a
unified construction of roughness measure, called strong Pawlak roughness
measure, and then explore the properties of this measure. We show that the
improved roughness measures in the literature are special instances of our
strong Pawlak roughness measure and introduce three more strong Pawlak
roughness measures as well. The advantage of our axiomatic approach is that
some properties of a roughness measure follow immediately as soon as the
measure satisfies the relevant axiomatic definition.Comment: to appear in the Fundamenta Informatica
Related family-based attribute reduction of covering information systems when varying attribute sets
In practical situations, there are many dynamic covering information systems
with variations of attributes, but there are few studies on related
family-based attribute reduction of dynamic covering information systems. In
this paper, we first investigate updated mechanisms of constructing attribute
reducts for consistent and inconsistent covering information systems when
varying attribute sets by using related families. Then we employ examples to
illustrate how to compute attribute reducts of dynamic covering information
systems with variations of attribute sets. Finally, the experimental results
illustrates that the related family-based methods are effective to perform
attribute reduction of dynamic covering information systems when attribute sets
are varying with time
Weighting Scheme for a Pairwise Multi-label Classifier Based on the Fuzzy Confusion Matrix
In this work we addressed the issue of applying a stochastic classifier and a
local, fuzzy confusion matrix under the framework of multi-label
classification. We proposed a novel solution to the problem of correcting label
pairwise ensembles. The main step of the correction procedure is to compute
classifier-specific competence and cross-competence measures, which estimates
error pattern of the underlying classifier. At the fusion phase we employed two
weighting approaches based on information theory. The classifier weights
promote base classifiers which are the most susceptible to the correction based
on the fuzzy confusion matrix. During the experimental study, the proposed
approach was compared against two reference methods. The comparison was made in
terms of six different quality criteria. The conducted experiments reveals that
the proposed approach eliminates one of main drawbacks of the original
FCM-based approach i.e. the original approach is vulnerable to the imbalanced
class/label distribution. What is more, the obtained results shows that the
introduced method achieves satisfying classification quality under all
considered quality criteria. Additionally, the impact of fluctuations of data
set characteristics is reduced.Comment: arXiv admin note: substantial text overlap with arXiv:1710.0872
Dialectical Rough Sets, Parthood and Figures of Opposition-1
In one perspective, the main theme of this research revolves around the
inverse problem in the context of general rough sets that concerns the
existence of rough basis for given approximations in a context. Granular
operator spaces and variants were recently introduced by the present author as
an optimal framework for anti-chain based algebraic semantics of general rough
sets and the inverse problem. In the framework, various sub-types of crisp and
non-crisp objects are identifiable that may be missed in more restrictive
formalism. This is also because in the latter cases concepts of complementation
and negation are taken for granted - while in reality they have a complicated
dialectical basis. This motivates a general approach to dialectical rough sets
building on previous work of the present author and figures of opposition. In
this paper dialectical rough logics are invented from a semantic perspective, a
concept of dialectical predicates is formalised, connection with dialetheias
and glutty negation are established, parthood analyzed and studied from the
viewpoint of classical and dialectical figures of opposition by the present
author. Her methods become more geometrical and encompass parthood as a primary
relation (as opposed to roughly equivalent objects) for algebraic semantics.Comment: 41 pages. The second part will appear soo
Feature selection with test cost constraint
Feature selection is an important preprocessing step in machine learning and
data mining. In real-world applications, costs, including money, time and other
resources, are required to acquire the features. In some cases, there is a test
cost constraint due to limited resources. We shall deliberately select an
informative and cheap feature subset for classification. This paper proposes
the feature selection with test cost constraint problem for this issue. The new
problem has a simple form while described as a constraint satisfaction problem
(CSP). Backtracking is a general algorithm for CSP, and it is efficient in
solving the new problem on medium-sized data. As the backtracking algorithm is
not scalable to large datasets, a heuristic algorithm is also developed.
Experimental results show that the heuristic algorithm can find the optimal
solution in most cases. We also redefine some existing feature selection
problems in rough sets, especially in decision-theoretic rough sets, from the
viewpoint of CSP. These new definitions provide insight to some new research
directions.Comment: 23 page
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