41,843 research outputs found
Cuntz-Krieger algebras and wavelets on fractals
We consider representations of Cuntz--Krieger algebras on the Hilbert space
of square integrable functions on the limit set, identified with a Cantor set
in the unit interval. We use these representations and the associated
Perron-Frobenius and Ruelle operators to construct families of wavelets on
these Cantor sets.Comment: 37 pages, LaTe
CuntzâKrieger Algebras and Wavelets on Fractals
We consider representations of CuntzâKrieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated PerronâFrobenius and Ruelle operators to construct families of wavelets on these Cantor sets
On the incorporation of interval-valued fuzzy sets into the Bousi-Prolog system: declarative semantics, implementation and applications
In this paper we analyse the benefits of incorporating interval-valued fuzzy
sets into the Bousi-Prolog system. A syntax, declarative semantics and im-
plementation for this extension is presented and formalised. We show, by using
potential applications, that fuzzy logic programming frameworks enhanced with
them can correctly work together with lexical resources and ontologies in order
to improve their capabilities for knowledge representation and reasoning
A Distance-Based Test of Association Between Paired Heterogeneous Genomic Data
Due to rapid technological advances, a wide range of different measurements
can be obtained from a given biological sample including single nucleotide
polymorphisms, copy number variation, gene expression levels, DNA methylation
and proteomic profiles. Each of these distinct measurements provides the means
to characterize a certain aspect of biological diversity, and a fundamental
problem of broad interest concerns the discovery of shared patterns of
variation across different data types. Such data types are heterogeneous in the
sense that they represent measurements taken at very different scales or
described by very different data structures. We propose a distance-based
statistical test, the generalized RV (GRV) test, to assess whether there is a
common and non-random pattern of variability between paired biological
measurements obtained from the same random sample. The measurements enter the
test through distance measures which can be chosen to capture particular
aspects of the data. An approximate null distribution is proposed to compute
p-values in closed-form and without the need to perform costly Monte Carlo
permutation procedures. Compared to the classical Mantel test for association
between distance matrices, the GRV test has been found to be more powerful in a
number of simulation settings. We also report on an application of the GRV test
to detect biological pathways in which genetic variability is associated to
variation in gene expression levels in ovarian cancer samples, and present
results obtained from two independent cohorts
Data granulation by the principles of uncertainty
Researches in granular modeling produced a variety of mathematical models,
such as intervals, (higher-order) fuzzy sets, rough sets, and shadowed sets,
which are all suitable to characterize the so-called information granules.
Modeling of the input data uncertainty is recognized as a crucial aspect in
information granulation. Moreover, the uncertainty is a well-studied concept in
many mathematical settings, such as those of probability theory, fuzzy set
theory, and possibility theory. This fact suggests that an appropriate
quantification of the uncertainty expressed by the information granule model
could be used to define an invariant property, to be exploited in practical
situations of information granulation. In this perspective, a procedure of
information granulation is effective if the uncertainty conveyed by the
synthesized information granule is in a monotonically increasing relation with
the uncertainty of the input data. In this paper, we present a data granulation
framework that elaborates over the principles of uncertainty introduced by
Klir. Being the uncertainty a mesoscopic descriptor of systems and data, it is
possible to apply such principles regardless of the input data type and the
specific mathematical setting adopted for the information granules. The
proposed framework is conceived (i) to offer a guideline for the synthesis of
information granules and (ii) to build a groundwork to compare and
quantitatively judge over different data granulation procedures. To provide a
suitable case study, we introduce a new data granulation technique based on the
minimum sum of distances, which is designed to generate type-2 fuzzy sets. We
analyze the procedure by performing different experiments on two distinct data
types: feature vectors and labeled graphs. Results show that the uncertainty of
the input data is suitably conveyed by the generated type-2 fuzzy set models.Comment: 16 pages, 9 figures, 52 reference
A new fuzzy set merging technique using inclusion-based fuzzy clustering
This paper proposes a new method of merging parameterized fuzzy sets based on clustering in the parameters space, taking into account the degree of inclusion of each fuzzy set in the cluster prototypes. The merger method is applied to fuzzy rule base simplification by automatically replacing the fuzzy sets corresponding to a given cluster with that pertaining to cluster prototype. The feasibility and the performance of the proposed method are studied using an application in mobile robot navigation. The results indicate that the proposed merging and rule base simplification approach leads to good navigation performance in the application considered and to fuzzy models that are interpretable by experts. In this paper, we concentrate mainly on fuzzy systems with Gaussian membership functions, but the general approach can also be applied to other parameterized fuzzy sets
A Functional Wavelet-Kernel Approach for Continuous-time Prediction
We consider the prediction problem of a continuous-time stochastic process on
an entire time-interval in terms of its recent past. The approach we adopt is
based on functional kernel nonparametric regression estimation techniques where
observations are segments of the observed process considered as curves. These
curves are assumed to lie within a space of possibly inhomogeneous functions,
and the discretized times series dataset consists of a relatively small,
compared to the number of segments, number of measurements made at regular
times. We thus consider only the case where an asymptotically non-increasing
number of measurements is available for each portion of the times series. We
estimate conditional expectations using appropriate wavelet decompositions of
the segmented sample paths. A notion of similarity, based on wavelet
decompositions, is used in order to calibrate the prediction. Asymptotic
properties when the number of segments grows to infinity are investigated under
mild conditions, and a nonparametric resampling procedure is used to generate,
in a flexible way, valid asymptotic pointwise confidence intervals for the
predicted trajectories. We illustrate the usefulness of the proposed functional
wavelet-kernel methodology in finite sample situations by means of three
real-life datasets that were collected from different arenas
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