Multiscale Poisson Intensity and Density Estimation

The nonparametric Poisson intensity and density estimation methods studied in this paper offer near minimax convergence rates for broad classes of densities and intensities with arbitrary levels of smoothness. The methods and theory presented here share many of the desirable features associated with...

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Bibliographic Details
Published in:IEEE transactions on information theory 2007-09, Vol.53 (9), p.3171-3187
Main Authors: Willett, R.M., Nowak, R.D.
Format: Article
Language:English
Subjects:
Applied sciences
Classification and Regression Tree (CART) algorithm
complexity regularization
Convergence
Data smoothing
Density
Errors
Estimating techniques
Estimators
Exact sciences and technology
Heart
Image converters
Information theory
Information, signal and communications theory
Mathematical analysis
Minimax techniques
Miscellaneous
nonparametric estimation
piecewise-polynomial approximation
platelets
Poisson distribution
Polynomials
Recursive estimation
Regression tree analysis
Signal processing
Signal resolution
Smoothness
Spatial resolution
Studies
Telecommunications and information theory
Two dimensional
Upper bound
wavelets
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Description
Summary:The nonparametric Poisson intensity and density estimation methods studied in this paper offer near minimax convergence rates for broad classes of densities and intensities with arbitrary levels of smoothness. The methods and theory presented here share many of the desirable features associated with wavelet-based estimators: computational speed, spatial adaptivity, and the capability of detecting discontinuities and singularities with high resolution. Unlike traditional wavelet-based approaches, which impose an upper bound on the degree of smoothness to which they can adapt, the estimators studied here guarantee nonnegativity and do not require any a priori knowledge of the underlying signal's smoothness to guarantee near-optimal performance. At the heart of these methods lie multiscale decompositions based on free-knot, free-degree piecewise-polynomial functions and penalized likelihood estimation. The degrees as well as the locations of the polynomial pieces can be adapted to the observed data, resulting in near-minimax optimal convergence rates. For piecewise-analytic signals, in particular, the error of this estimator converges at nearly the parametric rate. These methods can be further refined in two dimensions, and it is demonstrated that platelet-based estimators in two dimensions exhibit similar near-optimal error convergence rates for images consisting of smooth surfaces separated by smooth boundaries.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2007.903139