79 research outputs found
Weakly fuzzy topological entropy
summary:In 2005, İ. Tok fuzzified the notion of the topological entropy R. A. Adler et al. (1965) using the notion of fuzzy compactness of C. L. Chang (1968). In the present paper, we have proposed a new definition of the fuzzy topological entropy of fuzzy continuous mapping, namely weakly fuzzy topological entropy based on the notion of weak fuzzy compactness due to R. Lowen (1976) along with its several properties. We have shown that the topological entropy R. A. Adler et al. (1965) of continuous mapping , where is compact, is equal to the weakly fuzzy topological entropy of . We have also established an example that shows that the fuzzy topological entropy of İ. Tok (2005) cannot give such a bridge result to the topological entropy of Adler et al. (1965). Moreover, our definition of the weakly fuzzy topological entropy can be applied to find the topological entropy (namely weakly fuzzy topological entropy ) of the mapping (where is either compact or weakly fuzzy compact), whereas the topological entropy of Adler does not exist for the mapping (where is non-compact weakly fuzzy compact). Finally, a product theorem for the weakly fuzzy topological entropy has been established
Transitivity in Fuzzy Hyperspaces
Given a metric space (X, d), we deal with a classical problem in the theory of hyperspaces:
how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity)
between a discrete dynamical system f : (X, d) → (X, d) and its natural extension to the hyperspace
are related. In this context, we consider the Zadeh’s extension fbof f to F(X), the family of all normal
fuzzy sets on X, i.e., the hyperspace F(X) of all upper semicontinuous fuzzy sets on X with compact
supports and non-empty levels and we endow F(X) with different metrics: the supremum metric
d∞, the Skorokhod metric d0, the sendograph metric dS and the endograph metric dE. Among other
things, the following results are presented: (1) If (X, d) is a metric space, then the following conditions
are equivalent: (a) (X, f) is weakly mixing, (b) ((F(X), d∞), fb) is transitive, (c) ((F(X), d0), fb) is
transitive and (d) ((F(X), dS)), fb) is transitive, (2) if f : (X, d) → (X, d) is a continuous function,
then the following hold: (a) if ((F(X), dS), fb) is transitive, then ((F(X), dE), fb) is transitive, (b) if
((F(X), dS), fb) is transitive, then (X, f) is transitive; and (3) if (X, d) be a complete metric space, then
the following conditions are equivalent: (a) (X × X, f × f) is point-transitive and (b) ((F(X), d0) is
point-transitive
One-class classifiers based on entropic spanning graphs
One-class classifiers offer valuable tools to assess the presence of outliers
in data. In this paper, we propose a design methodology for one-class
classifiers based on entropic spanning graphs. Our approach takes into account
the possibility to process also non-numeric data by means of an embedding
procedure. The spanning graph is learned on the embedded input data and the
outcoming partition of vertices defines the classifier. The final partition is
derived by exploiting a criterion based on mutual information minimization.
Here, we compute the mutual information by using a convenient formulation
provided in terms of the -Jensen difference. Once training is
completed, in order to associate a confidence level with the classifier
decision, a graph-based fuzzy model is constructed. The fuzzification process
is based only on topological information of the vertices of the entropic
spanning graph. As such, the proposed one-class classifier is suitable also for
data characterized by complex geometric structures. We provide experiments on
well-known benchmarks containing both feature vectors and labeled graphs. In
addition, we apply the method to the protein solubility recognition problem by
considering several representations for the input samples. Experimental results
demonstrate the effectiveness and versatility of the proposed method with
respect to other state-of-the-art approaches.Comment: Extended and revised version of the paper "One-Class Classification
Through Mutual Information Minimization" presented at the 2016 IEEE IJCNN,
Vancouver, Canad
Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems
[EN] Let X be a compact metric space and a continuous map f:X-->X which defines a discrete dynamical system (X,f). The map f induces two natural maps, namely \bar{f}:K(X)-->K(X) on the hyperspace K(X) of non-empty compact subspaces of X and the Zadeh¿s extension \hat{f}:F(X)-->F(X) on the space F(X) of normal fuzzy set. In this work, we analyze the interaction of some orbit tracing dynamical properties, namely the specification and shadowing properties of the discrete dynamical system (X,f) and its induced discrete dynamical systems (K(X),\bar{f}) and (F(X),\hat{f}). Adding an algebraic structure yields stronger conclusions, and we obtain a full characterization of the specification property in the hyperspace, in the fuzzy space, and in the phase space X if we assume that the later is a convex compact subset of a (metrizable and complete) locally convex space and f is a linear operator.This work was supported by MCIN/AEI/10.13039/501100011033, Project PID2019-105011GB-I00, and by Generalitat Valenciana, Project PROMETEU/2021/070.Bartoll Arnau, S.; Martínez Jiménez, F.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2022). Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems. Axioms. 11(12):1-11. https://doi.org/10.3390/axioms11120733111111
Toward a multilevel representation of protein molecules: comparative approaches to the aggregation/folding propensity problem
This paper builds upon the fundamental work of Niwa et al. [34], which
provides the unique possibility to analyze the relative aggregation/folding
propensity of the elements of the entire Escherichia coli (E. coli) proteome in
a cell-free standardized microenvironment. The hardness of the problem comes
from the superposition between the driving forces of intra- and inter-molecule
interactions and it is mirrored by the evidences of shift from folding to
aggregation phenotypes by single-point mutations [10]. Here we apply several
state-of-the-art classification methods coming from the field of structural
pattern recognition, with the aim to compare different representations of the
same proteins gathered from the Niwa et al. data base; such representations
include sequences and labeled (contact) graphs enriched with chemico-physical
attributes. By this comparison, we are able to identify also some interesting
general properties of proteins. Notably, (i) we suggest a threshold around 250
residues discriminating "easily foldable" from "hardly foldable" molecules
consistent with other independent experiments, and (ii) we highlight the
relevance of contact graph spectra for folding behavior discrimination and
characterization of the E. coli solubility data. The soundness of the
experimental results presented in this paper is proved by the statistically
relevant relationships discovered among the chemico-physical description of
proteins and the developed cost matrix of substitution used in the various
discrimination systems.Comment: 17 pages, 3 figures, 46 reference
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Fuzzy fractals, chaos, and noise
To distinguish between chaotic and noisy processes, the authors analyze one- and two-dimensional chaotic mappings, supplemented by the additive noise terms. The predictive power of a fuzzy rule-based system allows one to distinguish ergodic and chaotic time series: in an ergodic series the likelihood of finding large numbers is small compared to the likelihood of finding them in a chaotic series. In the case of two dimensions, they consider the fractal fuzzy sets whose {alpha}-cuts are fractals, arising in the context of a quadratic mapping in the extended complex plane. In an example provided by the Julia set, the concept of Hausdorff dimension enables one to decide in favor of chaotic or noisy evolution
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