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 $n$-means, the $n$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 $P$ on $\mathbb R^2$ 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 $n$-means and the $n$th quantization errors are investigated for all $n\geq 2$.Comment: arXiv admin note: text overlap with arXiv:1603.0073

Optimal quantization for the Cantor distribution generated by infinite similutudes

Let $P$ be a Borel probability measure on $\mathbb R$ generated by an infinite system of similarity mappings $\{S_j : j\in \mathbb N\}$ such that $P=\sum_{j=1}^\infty \frac 1{2^j} P\circ S_j^{-1}$, where for each $j\in \mathbb N$ and $x\in \mathbb R$, $S_j(x)=\frac 1{3^{j}}x+1-\frac 1 {3^{j-1}}$. Then, the support of $P$ is the dyadic Cantor set $C$ generated by the similarity mappings $f_1, f_2 : \mathbb R \to \mathbb R$ such that $f_1(x)=\frac 13 x$ and $f_2(x)=\frac 13 x+\frac 23$ for all $x\in \mathbb R$. In this paper, using the infinite system of similarity mappings $\{S_j : j\in \mathbb N\}$ associated with the probability vector $(\frac 12, \frac 1{2^2}, \cdots)$, for all $n\in \mathbb N$, we determine the optimal sets of $n$-means and the $n$th quantization errors for the infinite self-similar measure $P$. The technique obtained in this paper can be utilized to determine the optimal sets of $n$-means and the $n$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 $k\geq 2$, let $\{S_j : 1\leq j\leq k\}$ be a set of $k$ contractive similarity mappings such that $S_j(x)=\frac 1 {2k-1} x +\frac{2 (j-1)} {2k-1}$ for all $x\in \mathbb R$, and let $P= \frac 1 k \sum_{j=1}^kP\circ S_j^{-1}$. Then, $P$ is a unique Borel probability measure on $\mathbb R$ such that $P$ has support the Cantor set generated by the similarity mappings $S_j$ for $1\leq j\leq k$. In this paper, for the probability measure $P$, when $k=3$, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. 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 $d$-dimensional probability distribution $P$ by a discrete probability with a given number $n$ of supporting points. In this paper, an infinitely generated nonhomogeneous Borel probability measure $P$ is considered on $\mathbb R$. For such a probability measure $P$, an induction formula to determine the optimal sets of $n$-means and the $n$th quantization error for every natural number $n$ is given. In addition, using the induction formula we give some results and observations about the optimal sets of $n$-means for all $n\geq 2$.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 $\mu_h$ supported by a cookie-cutter set $E$ which is generated by a single cookie-cutter mapping $f$. For such a measure $\mu_h$ and $r\in (0, +\infty)$ we have shown that there exists a unique $\kappa_r \in (0, +\infty)$ such that $\kappa_r$ is the quantization dimension function of the probability measure $\mu_h$, 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 $\mu_h$. In addition, we have proved that the $\kappa_r$-dimensional lower quantization coefficient of order $r$ 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 $P:=\frac 1 3 P\circ S_1^{-1}+\frac 13 P\circ S_2^{-1}+\frac 13\nu$, where $S_1(x)=\frac 15 x$, $S_2(x)=\frac 1 5 x+\frac 45$ for all $x\in \mathbb R$, and $\nu$ be a Borel probability measure on $\mathbb R$ with compact support. Such a measure $P$ is called a condensation measure, or an an inhomogeneous self-similar measure, associated with the condensation system $(\{S_1, S_2\}, (\frac 13, \frac 13, \frac 13), \nu)$. Let $D(\mu)$ denote the quantization dimension of a measure $\mu$ if it exists. Let $\kappa$ be the unique number such that $(\frac 13 (\frac 15)^2)^{\frac {\kappa}{2+\kappa}}+(\frac 13 (\frac 15)^2)^{\frac {\kappa}{2+\kappa}}=1$. In this paper, we have considered four different self-similar measures $\nu:=\nu_1, \nu_2, \nu_3, \nu_4$ satisfying $D(\nu_1)>\kappa$, $D(\nu_2)\kappa$, and $D(\nu_4)=\kappa$. For each measure $\nu$ we show that there exist two sequences $a(n)$ and $F(n)$, which we call as canonical sequences. With the help of the canonical sequences, we obtain a closed formula to determine the optimal sets of $F(n)$-means and $F(n)$th quantization errors for the condensation measure $P$ for each $\nu$. Then, we show that for each measure $\nu$ the quantization dimension $D(P)$ of the condensation measure $P$ exists, and satisfies: $D(P)=\max\{\kappa, D(\nu)\}$. Moreover, we show that for $D(\nu_1)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients are finite, positive and unequal; on the other hand, for $\nu=\nu_2, \nu_3, \nu_4$, the $D(P)$-dimensional lower quantization coefficient is infinity. This shows that for $D(\nu)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients can be either finite, positive and unequal, or it can be infinity