Estimating the square root of a density via compactly supported wavelets

Pinheiro, Aluísio; Vidaković, Brani

doi:10.1016/s0167-9473(97)00013-3

Cited by 56 publications

(44 citation statements)

References 10 publications

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“…The definition domain considered was M", n= 1,2,... and the minimax rates were in terms of functional norms of /, obtained by measuring the size of the wavelet coefficients of / in appropriate weighted sequence spaces. In [1,2], for n = 1, were considered different wavelet approximations of the density of / that preserve its non-negativity and equality of the integral to 1. Moreover, in [1] it was also proved that the statistical estimator based on the new positivity-and integral-preserving wavelet approximation, also achieves certain assymptotically minimax rates.…”

Section: Resultsmentioning

confidence: 99%

“…Another example is non-negative non-parametric regression-function estimation in positron-emission tomography (PET) imaging. In [1,2] shape-preserving statistical density estimators were proposed by considering A/7 G L2, where f &L\ is an unknown density. For this approach, optimal estimation rates of the risk were obtained under the assumptions that ^/f belongs, more specifically, to certain function spaces with additional smoothness, continuously embedded in L2.…”

Section: Introductionmentioning

confidence: 99%

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Relations between functional norms of a non-negative function and its square root on the positive cone of Besov and Triebel-Lizorkin spaces

Dechevsky¹,

Grip²,

Venkov³

et al. 2009

AIP Conference Proceedings

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Abstract. In this communication we study in detail the relations between the smoothness of/ and -s/f in the case when the smoothness of the univariate non-negative functions / is measured via Besov and Triebel-Lizorkin space scales. The results obtained can be considered also as embedding theorems for usual Besov and Triebel-Lizorkin spaces and their analogues in Hellinger metric. These results can be used in constrained approximation using wavelets, with applications to probability density estimation in speech recognition, non-negative non-parametric regression-function estimation in positron-emission tomography (PET) imaging, shape/order-preserving and/or one-sided approximation and many others.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Relations between functional norms of a non-negative function and its square root on the positive cone of Besov and Triebel-Lizorkin spaces

Dechevsky¹,

Grip²,

Venkov³

et al. 2009

AIP Conference Proceedings

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show abstract

“…For each value of j 0 and j 1 estimate the wavelet density coefficients of the expansion [16,13]. This will give you the likelihood term needed for (14) .…”

Section: And the Geometry Of Square-root Wavelet Densitiesmentioning

confidence: 99%

An information geometry approach to shape density Minimum Description Length model selection

Peter

Rangarajan

2011

2011 IEEE International Conference on Computer Vision Workshops (ICCV Workshops)

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For advantages such as a richer representation power and inherent robustness to noise, probability density functions are becoming a staple for complex problems in shape analysis. We consider a principled and geometric approach to selecting the model order for a class of shape density models where the square-root of the distribution is expanded in an orthogonal series. The free parameters associated with these estimators can then be rigorously selected using the Minimum Description Length (MDL) criterion for model selection. Under these models, it is shown that the MDL has a closed-form representation, atypical for most applications of MDL in density estimation. We provide a straightforward application of our derivations by using this closed-from MDL criterion to select the optimal multiresolution level(s) for a class of square-root, wavelet density estimators. Experimental evaluation of our technique is conducted on one and two dimensional density estimation problems in shape analysis, with comparative analysis against other popular model selection criteria such as Bayesian and Akaike information criteria.

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“…To ensure non-negativity, we approximate the density indirectly via its square root utilizing the Hellinger metric. This concept was also used for density approximation with wavelets [10]. Unfortunately employing wavelet representations does result in a closed form Bayesian estimator.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Multidimensional Bayesian Estimation with Fourier Densities

Brunn¹,

Sawo²,

Hanebeck³

2006

Proceedings of the 45th IEEE Conference on Decision and Control

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Abstract-Efficiently implementing nonlinear Bayesian estimators is still an unsolved problem, especially for the multidimensional case. A trade-off between estimation quality and demand on computational resources has to be found. Using multidimensional Fourier series as representation for probability density functions, so called Fourier densities, is proposed. To ensure non-negativity, the approximation is performed indirectly via Ψ-densities, of which the absolute square represent the Fourier density. It is shown that Ψ-densities can be determined using the efficient fast Fourier transform algorithm and their coefficients have an ordering with respect to the Hellinger metric. Furthermore, the multidimensional Bayesian estimator based on Fourier densities is derived in closed form. That allows an efficient realization of the Bayesian estimator where the demands on computational resources are adjustable.

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Estimating the square root of a density via compactly supported wavelets

Cited by 56 publications

References 10 publications

Relations between functional norms of a non-negative function and its square root on the positive cone of Besov and Triebel-Lizorkin spaces

Relations between functional norms of a non-negative function and its square root on the positive cone of Besov and Triebel-Lizorkin spaces

An information geometry approach to shape density Minimum Description Length model selection

Nonlinear Multidimensional Bayesian Estimation with Fourier Densities

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