On Cramér-Rao Lower Bounds with Random Equality Constraints

Numerous works have shown the versatility of deterministic constrained Cramér-Rao bound for estimation performance analysis and design of a system of measurements. Indeed, most of factors impacting the asymptotic estimation performance of the parameters of interest can be taken into account via equality constraints. In this communication, we introduce a new constrained Cramér-Rao- like bound for observations where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on random variables as well. In this setting, it is now possible to consider random equality constraints, i.e., equality constraints on the unknown deterministic parameters depending on the random parameters, which can not be addressed with the usual constrained Cramér-Rao bound. The usefulness of the proposed bound is illustrated by way of a coupled canonical polyadic model with linear constraints applied to the hyperspectral super-resolution problem.

[1]  H. V. Trees,et al.  Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking , 2007 .

[2]  Souleymen Sahnoun,et al.  Joint Source Estimation and Localization , 2015, IEEE Transactions on Signal Processing.

[3]  David S. Slepian,et al.  Estimation of signal parameters in the presence of noise , 1954, Trans. IRE Prof. Group Inf. Theory.

[4]  Pierre Comon,et al.  Performance bounds for coupled models , 2016, 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM).

[5]  R. W. Miller,et al.  A modified Cramér-Rao bound and its applications (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[6]  M. Fréchet Sur l'extension de certaines evaluations statistiques au cas de petits echantillons , 1943 .

[7]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[8]  Pierre Comon,et al.  Tensor Decompositions, State of the Art and Applications , 2002 .

[9]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[10]  Christian Jutten,et al.  Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects , 2015, Proceedings of the IEEE.

[11]  Thomas L. Marzetta,et al.  Detection, Estimation, and Modulation Theory , 1976 .

[12]  Thomas L. Marzetta,et al.  Parameter estimation problems with singular information matrices , 2001, IEEE Trans. Signal Process..

[13]  F. Glave,et al.  A new look at the Barankin lower bound , 1972, IEEE Trans. Inf. Theory.

[14]  L. Wald,et al.  Fusion of satellite images of different spatial resolutions: Assessing the quality of resulting images , 1997 .

[15]  Harry L. Van Trees,et al.  Optimum Array Processing , 2002 .

[16]  Charles R. Johnson,et al.  Matrix Analysis, 2nd Ed , 2012 .

[17]  Rasmus Bro,et al.  MULTI-WAY ANALYSIS IN THE FOOD INDUSTRY Models, Algorithms & Applications , 1998 .

[18]  Gary A. Shaw,et al.  Hyperspectral Image Processing for Automatic Target Detection Applications , 2003 .

[19]  Umberto Mengali,et al.  The modified Cramer-Rao bound and its application to synchronization problems , 1994, IEEE Trans. Commun..

[20]  Henry Arguello,et al.  An analysis of spectral variability in hyperspectral imagery: a case study of stressed oil palm detection in Colombia , 2019, International Journal of Remote Sensing.

[21]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[22]  B. C. Ng,et al.  On the Cramer-Rao bound under parametric constraints , 1998, IEEE Signal Processing Letters.

[23]  Wing-Kin Ma,et al.  Hyperspectral Super-Resolution: A Coupled Tensor Factorization Approach , 2018, IEEE Transactions on Signal Processing.

[24]  Lawrence P. Seidman,et al.  A useful form of the Barankin lower bound and its application to PPM threshold analysis , 1969, IEEE Trans. Inf. Theory.

[25]  Eric Chaumette,et al.  Versatility of constrained CRB for system analysis and design , 2014, IEEE Transactions on Aerospace and Electronic Systems.

[26]  Eric Chaumette,et al.  Lower bounds for non standard deterministic estimation , 2016, 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM).

[27]  M. Haardt,et al.  Performance Bounds for Coupled CP Model in the Framework of Hyperspectral Super-Resolution , 2019, 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[28]  David Brie,et al.  Coupled Tensor Low-rank Multilinear Approximation for Hyperspectral Super-resolution , 2019, ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[29]  Alfred O. Hero,et al.  Lower bounds for parametric estimation with constraints , 1990, IEEE Trans. Inf. Theory.

[30]  Joseph Tabrikian,et al.  General Classes of Performance Lower Bounds for Parameter Estimation—Part I: Non-Bayesian Bounds for Unbiased Estimators , 2010, IEEE Transactions on Information Theory.

[31]  Fulvio Gini,et al.  On the use of Cramer-Rao-like bounds in the presence of random nuisance parameters , 2000, IEEE Trans. Commun..

[32]  Claudia Biermann,et al.  Mathematical Methods Of Statistics , 2016 .

[33]  Pierre Comon,et al.  Tensors : A brief introduction , 2014, IEEE Signal Processing Magazine.

[34]  Thomas L. Marzetta,et al.  A simple derivation of the constrained multiple parameter Cramer-Rao bound , 1993, IEEE Trans. Signal Process..

[35]  Brian M. Sadler,et al.  Maximum-Likelihood Estimation, the CramÉr–Rao Bound, and the Method of Scoring With Parameter Constraints , 2008, IEEE Transactions on Signal Processing.

[36]  Hagit Messer,et al.  Notes on the Tightness of the Hybrid CramÉr–Rao Lower Bound , 2009, IEEE Transactions on Signal Processing.

[37]  Peter M. Schultheiss,et al.  Array shape calibration using sources in unknown locations-Part I: Far-field sources , 1987, IEEE Trans. Acoust. Speech Signal Process..

[38]  Eric Chaumette,et al.  New Results on Deterministic Cramér–Rao Bounds for Real and Complex Parameters , 2012, IEEE Transactions on Signal Processing.

[39]  Pierre Comon,et al.  Performance estimation for tensor CP decomposition with structured factors , 2015, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[40]  Eric Chaumette,et al.  On Lower Bounds for Nonstandard Deterministic Estimation , 2017, IEEE Transactions on Signal Processing.