2,603 research outputs found
Improved Chebyshev inequality: new probability bounds with known supremum of PDF
In this paper, we derive new probability bounds for Chebyshev's inequality if
the supremum of the probability density function is known. This result holds
for one-dimensional or multivariate continuous probability distributions with
finite mean and variance (covariance matrix). We also show that the similar
result holds for specific discrete probability distributions.Comment: 7 pages, 2 figure
A mean value theorem for systems of integrals
More than a century ago, G. Kowalewski stated that for each n continuous
functions on a compact interval [a,b], there exists an n-point quadrature rule
(with respect to Lebesgue measure on [a,b]), which is exact for given
functions. Here we generalize this result to continuous functions with an
arbitrary positive and finite measure on an arbitrary interval. The proof
relies on a version of Caratheodory's convex hull theorem for a continuous
curve, that we also prove in the paper. As applications, we give a
representation of the covariance for two continuous functions of a random
variable, and a most general version of Gruess' inequality.Comment: 7 page
Transductive-Inductive Cluster Approximation Via Multivariate Chebyshev Inequality
Approximating adequate number of clusters in multidimensional data is an open
area of research, given a level of compromise made on the quality of acceptable
results. The manuscript addresses the issue by formulating a transductive
inductive learning algorithm which uses multivariate Chebyshev inequality.
Considering clustering problem in imaging, theoretical proofs for a particular
level of compromise are derived to show the convergence of the reconstruction
error to a finite value with increasing (a) number of unseen examples and (b)
the number of clusters, respectively. Upper bounds for these error rates are
also proved. Non-parametric estimates of these error from a random sample of
sequences empirically point to a stable number of clusters. Lastly, the
generalization of algorithm can be applied to multidimensional data sets from
different fields.Comment: 16 pages, 5 figure
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