Optimal quantization for nonuniform Cantor distributions

Abstract

Let P be a Borel probability measure on R such that P = 1 4 P • S −1 1 + 3 4 P • S −1 2 , where S 1 and S 2 are two similarity mappings on R such that S 1 (x) = 1 4 x and S 2 (x) = 1 2 x + 1 2 for all x ∈ R. Such a probability measure P has support the Cantor set generated by S 1 and S 2 . For this probability measure, in this paper, we give an induction formula to determine the optimal sets of n-means and the nth quantization errors for all n ≥ 2. We have shown that the same induction formula also works for the Cantor distribution P := ψ 2 P • S −1 1 + ψ 4 P • S −1 2 supported by the Cantor set generated by S 1 (x) = 1 3 x and S 2 (x) = 1 3 x + 2 3 for all x ∈ R, where ψ is the square root of the Golden ratio 1 2 ( √ 5 − 1). In addition, we give a counter example to show that the induction formula does not work for all Cantor distributions. Using the induction formula we obtain some results and observations which are also given in this paper.2010 Mathematics Subject Classification. 60Exx, 28A80, 94A34.

Quantization for Infinite Affine Transformations

“…S. Graf and H. Luschgy determined the optimal sets of n-means and the nth quantization errors for the Cantor distribution, for all n ≥ 1, completing its quantization program [10]. This result has been extended to the setting of a nonuniform Cantor distribution by L. Roychowdhury [11]. Analogously, the Cantor dust is generated by the contractive mappings {S i } 4 i=1 on R 2 , where S 1 (x 1 , x 2 ) = 1 3 (x 1 , x 2 ), S 2 (x 1 , x 2 ) = 1 3 (x 1 , x 2 ) + ( 2 3 , 0), S 3 (x 1 , x 2 ) = 1 3 (x 1 , x 2 ) + (0, 2 3 ), and S 4 (x 1 , x 2 ) = 1 3 (x 1 , x 2 ) + ( 2 3 , 2 3 ).…”

Quantization for the mixtures of uniform distributions on connected and disconnected line segments

“…x 2 dP (x) < ∞ such a set α always exists (see [AW,GKL,GL1,GL2]). For some recent work in this direction one can see [CR,DR1,DR2,GL3,L1,R1,R2,R3,R4,R5,R6,RR1].…”

Quantization for the mixtures of overlap probability distributions

“…The elements of an optimal set of n-means are called optimal quantizers. For some work in this direction, one can see [CR,DR1,DR2,GL3,L1,R1,R2,R3,R4,R5,R6,RR1].…”