The Annals of Probability

The local quantization behavior of absolutely continuous probabilities

Siegfried Graf, Harald Luschgy, and Gilles Pagès

Full-text: Open access

Abstract

For a large class of absolutely continuous probabilities $P$ it is shown that, for $r>0$, for $n$-optimal $L^{r}(P)$-codebooks $\alpha_{n}$, and any Voronoi partition $V_{n,a}$ with respect to $\alpha_{n}$ the local probabilities $P(V_{n,a})$ satisfy $P(V_{a,n})\approx n^{-1}$ while the local $L^{r}$-quantization errors satisfy $\int_{V_{n,a}}\|x-a\|^{r}\,dP(x)\approx n^{-(1+r/d)}$ as long as the partition sets $V_{n,a}$ intersect a fixed compact set $K$ in the interior of the support of $P$.

Article information

Source
Ann. Probab., Volume 40, Number 4 (2012), 1795-1828.

Dates
First available in Project Euclid: 4 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1341401149

Digital Object Identifier
doi:10.1214/11-AOP663

Mathematical Reviews number (MathSciNet)
MR2978138

Zentralblatt MATH identifier
1260.60032

Subjects
Primary: 60E99: None of the above, but in this section 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 34A29

Keywords
Vector quantization probability of Voronoi cells inertia of Voronoi cells

Citation

Graf, Siegfried; Luschgy, Harald; Pagès, Gilles. The local quantization behavior of absolutely continuous probabilities. Ann. Probab. 40 (2012), no. 4, 1795--1828. doi:10.1214/11-AOP663. https://projecteuclid.org/euclid.aop/1341401149


Export citation

References

  • [1] Bucklew, J. A. and Wise, G. L. (1982). Multidimensional asymptotic quantization theory with $r$th power distortion measures. IEEE Trans. Inform. Theory 28 239–247.
  • [2] Chernaya, E. V. (1995). An asymptotically sharp estimate for the remainder of weighted cubature formulas that are optimal on certain classes of continuous functions. Ukraïn. Mat. Zh. 47 1405–1415.
  • [3] Chornaya, E. V. (1995). On the optimization of weighted cubature formulae on certain classes of continuous functions. East J. Approx. 1 47–60.
  • [4] Cohn, D. L. (1980). Measure Theory. Birkhäuser, Boston, MA.
  • [5] Delattre, S., Graf, S., Luschgy, H. and Pagès, G. (2004). Quantization of probability distributions under norm-based distortion measures. Statist. Decisions 22 261–282.
  • [6] Fort, J.-C. and Pagès, G. (2002). Asymptotics of optimal quantizers for some scalar distributions. J. Comput. Appl. Math. 146 253–275.
  • [7] Gersho, A. (1979). Asymptotically optimal block quantization. IEEE Trans. Inform. Theory 25 373–380.
  • [8] Graf, S. and Luschgy, H. (2000). Foundations of Quantization for Probability Distributions. Lecture Notes in Math. 1730. Springer, Berlin.
  • [9] Graf, S. and Luschgy, H. (2002). Rates of convergence for the empirical quantization error. Ann. Probab. 30 874–897.
  • [10] Graf, S., Luschgy, H. and Pagès, G. (2008). Distortion mismatch in the quantization of probability measures. ESAIM Probab. Stat. 12 127–153.
  • [11] Gray, R. M. and Neuhoff, D. L. (1998). Quantization. IEEE Trans. Inform. Theory 44 2325–2383.
  • [12] Gruber, P. M. (2004). Optimum quantization and its applications. Adv. Math. 186 456–497.
  • [13] Pagès, G. (1993). Voronoi tessellation, space quantization algorithms and numerical integration. In Proc. ESANN’93 (M. Verleysen, ed.) 221–228. Quorum Editions, Bruxelles.
  • [14] Pagès, G. (1998). A space quantization method for numerical integration. J. Comput. Appl. Math. 89 1–38.
  • [15] Pagès, G. and Printems, J. (2009). Optimal quantization for finance: From random vectors to stochastic processes. In Mathematical Modeling and Numerical Methods in Finance (Special Volume) (A. Bensoussan and Q. Zhang, guest eds.), Coll. Handbook of Numerical Analysis (P. G. Ciarlet, ed.) 595–649. North-Holland, Amsterdam.
  • [16] Pagès, G. and Sagna, A. (2012). Asymptotics of the maximal radius of an $L^{r}$-optimal sequence of quantizers. Bernoulli 18 360–389.
  • [17] Zador, P. L. (1963). Development and evaluation of procedures for quantizing multivariate distributions. Ph.D. thesis, Stanford Univ.
  • [18] Zador, P. L. (1982). Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory 28 139–149.