61 research outputs found
An overview of the quantization for mixed distributions
The basic goal of quantization for probability distribution is to reduce the
number of values, which is typically uncountable, describing a probability
distribution to some finite set and thus approximation of a continuous
probability distribution by a discrete distribution. Mixed distributions are an
exciting new area for optimal quantization. In this paper, we have determined
the optimal sets of -means, the th quantization error, and the
quantization dimensions of different mixed distributions. Besides, we have
discussed whether the quantization coefficients for the mixed distributions
exist. The results in this paper will give a motivation and insight into more
general problems in quantization of mixed distributions.Comment: arXiv admin note: text overlap with arXiv:1701.0416
Nonhomogeneous distributions and optimal quantizers for Sierpi\'nski carpets
The purpose of quantization of a probability distribution is to estimate the
probability by a discrete probability with finite support. In this paper, a
nonhomogeneous probability measure on which has support the
Sierpi\'nski carpet generated by a set of four contractive similarity mappings
with equal similarity ratios has been considered . For this probability
measure, the optimal sets of -means and the th quantization errors are
investigated for all .Comment: arXiv admin note: text overlap with arXiv:1603.0073
Optimal quantization for the Cantor distribution generated by infinite similutudes
Let be a Borel probability measure on generated by an
infinite system of similarity mappings such that
, where for each and , .
Then, the support of is the dyadic Cantor set generated by the
similarity mappings such that
and for all .
In this paper, using the infinite system of similarity mappings associated with the probability vector , for all , we determine the optimal sets of -means
and the th quantization errors for the infinite self-similar measure .
The technique obtained in this paper can be utilized to determine the optimal
sets of -means and the th quantization errors for more general infinite
self-similar measures
The quantization of the standard triadic Cantor distribution
The quantization scheme in probability theory deals with finding a best
approximation of a given probability distribution by a probability distribution
that is supported on finitely many points. For a given , let be a set of contractive similarity mappings such that
for all , and
let . Then, is a unique Borel
probability measure on such that has support the Cantor set
generated by the similarity mappings for . In this paper,
for the probability measure , when , we investigate the optimal sets of
-means and the th quantization errors for all . We further show
that the quantization coefficient does not exist though the quantization
dimension exists
Optimal quantization for infinite nonhomogeneous distributions on the real line
Quantization for probability distributions concerns the best approximation of
a -dimensional probability distribution by a discrete probability with a
given number of supporting points. In this paper, an infinitely generated
nonhomogeneous Borel probability measure is considered on . For
such a probability measure , an induction formula to determine the optimal
sets of -means and the th quantization error for every natural number
is given. In addition, using the induction formula we give some results and
observations about the optimal sets of -means for all .Comment: arXiv admin note: text overlap with arXiv:1512.0037
Quantization dimension for Gibbs-like measures on cookie-cutter sets
In this paper using Banach limit we have determined a Gibbs-like measure
supported by a cookie-cutter set which is generated by a single
cookie-cutter mapping . For such a measure and
we have shown that there exists a unique such that
is the quantization dimension function of the probability measure
, and established its functional relationship with the temperature
function of the thermodynamic formalism. The temperature function is commonly
used to perform the multifractal analysis, in our context of the measure
. In addition, we have proved that the -dimensional lower
quantization coefficient of order of the probability measure is positive.Comment: arXiv admin note: text overlap with arXiv:1203.272
Canonical sequences of optimal quantization for condensation measures
Let ,
where , for all , and be a Borel probability measure on with compact
support. Such a measure is called a condensation measure, or an an
inhomogeneous self-similar measure, associated with the condensation system
. Let denote the
quantization dimension of a measure if it exists. Let be the
unique number such that . In
this paper, we have considered four different self-similar measures
satisfying ,
, and . For each measure
we show that there exist two sequences and , which we call
as canonical sequences. With the help of the canonical sequences, we obtain a
closed formula to determine the optimal sets of -means and th
quantization errors for the condensation measure for each . Then, we
show that for each measure the quantization dimension of the
condensation measure exists, and satisfies: .
Moreover, we show that for , the -dimensional lower and
upper quantization coefficients are finite, positive and unequal; on the other
hand, for , the -dimensional lower quantization
coefficient is infinity. This shows that for , the
-dimensional lower and upper quantization coefficients can be either
finite, positive and unequal, or it can be infinity
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