Foundations of quantization for probability distributions / Siegfried Graf, Harald Luschgy.Material type: TextSeries: Lecture notes in mathematics (Springer-Verlag) ; 1730.Publication details: Berlin ; New York : Springer, ©2000. Description: 1 online resource (x, 230 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9783540455776; 3540455779Subject(s): Distribution (Probability theory) | Cluster analysis | Fractals | Coding theory | Cluster analysis | Coding theory | Distribution (Probability theory) | Fractals | WaarschijnlijkheidsverdelingenGenre/Form: Electronic books. Additional physical formats: Print version:: Foundations of quantization for probability distributions.DDC classification: 510 s | 519.2/4 LOC classification: QA3.L28 no. 1730 | QA273.6Online resources: Click here to access online
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Includes bibliographical references (pages 215-224) and index.
I. General properties of the quantization for probability distributions: Voronoi partitions. Centers and moments of probability distributions. The quantization problem. Basic properties of optimal quantizers. Uniqueness and optimality in one dimension -- II. Asymptotic quantization for nonsingular probability distributions: Asymptotics for the quantization error. Asymptotically optimal quantizers. Regular quantizers and quantization coefficients. Random quantizers and quantization coefficients. Asymptotics for the covering radius -- III. Asymptotic quantization for singular probability distributions: The quantization dimension. Regular sets and measures of dimension D. Rectifiable curves. Self-similar sets and measures.
Due to the rapidly increasing need for methods of data compression, quantization has become a flourishing field in signal and image processing and information theory. The same techniques are also used in statistics (cluster analysis), pattern recognition, and operations research (optimal location of service centers). The book gives the first mathematically rigorous account of the fundamental theory underlying these applications. The emphasis is on the asymptotics of quantization errors for absolutely continuous and special classes of singular probabilities (surface measures, self-similar measures) presenting some new results for the first time. Written for researchers and graduate students in probability theory the monograph is of potential interest to all people working in the disciplines mentioned above.