Optimal quantization for a probability measure on a nonuniform stretched Sierpiński triangle

Megha Pandey; Mrinal Kanti Roychowdhury

doi:10.3934/dcdss.2025087

Article Contents

2025, Volume 18, Issue 12: 3964-3981. Doi: 10.3934/dcdss.2025087

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Optimal quantization for a probability measure on a nonuniform stretched Sierpiński triangle

Megha Pandey^1, , and
Mrinal Kanti Roychowdhury^2,

1.
School of Mathematics Northwest University Xi'an, Shaanxi Province, 710069, China
2.
School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA

^*Corresponding author: Megha Pandey
^*Corresponding author: Megha Pandey

Received: March 2024

Revised: May 2025

Early access: June 3, 2025

Published online: June 3, 2025

Abstract / Introduction Full Text(HTML) Figure(5) / Table(1) Related Papers Cited by

Abstract

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. In this paper, we have considered a Borel probability measure $ P $ on $ \mathbb R^2 $, which has support a nonuniform stretched Sierpiński triangle generated by a set of three contractive similarity mappings on $ \mathbb R^2 $. For this probability measure, we investigate the optimal sets of $ n $-means and the $ n $th quantization errors for all positive integers $ n $.

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Mathematics Subject Classification: 60E05, 28A80, 94A34.

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Figure 1. Some basic triangles with their vertices that construct the stretched Sierpiński triangle

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Figure 2. Optimal configuration of $ n $ points for $ 1\leq n\leq 6 $

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Figure 3. Optimal configuration of $ n $ points for $ n = 7 $

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Figure 4. Optimal configuration of $ n $ points for $ n = 8 $

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Figure 5. Tree diagram of the optimal sets from $ \alpha_8 $ to $ \alpha_{21} $

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Table 1. Number of $ \alpha_n $ in the range $ 5\leq n\leq 82 $

$ n $	$ \text{ card }(\mathcal{C}_n) $	$ n $	$ \text{ card }(\mathcal{C}_n) $	$ n $	$ \text{ card }(\mathcal{C}_n) $	$ n $	$ \text{ card }(\mathcal{C}_n) $	$ n $	$ \text{ card }(\mathcal{C}_n) $	$ n $	$ \text{ card }(\mathcal{C}_n) $
5	1	18	4	31	6	44	1	57	495	70	56
6	1	19	6	32	4	45	8	58	792	71	28
7	2	20	4	33	1	46	28	59	924	72	8
8	1	21	1	34	6	47	56	60	792	73	1
9	1	22	1	35	15	48	70	61	495	74	1
10	2	23	6	36	20	49	56	62	220	75	12
11	1	24	15	37	15	50	28	63	66	76	66
12	1	25	20	38	6	51	8	64	12	77	220
13	4	26	15	39	1	52	1	65	1	78	495
14	6	27	6	40	1	53	1	66	8	79	792
15	4	28	1	41	4	54	12	67	28	80	924
16	1	29	1	42	6	55	66	68	56	81	792
17	1	30	4	43	4	56	220	69	70	82	495

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