Random Distortion Testing and Optimality of Thresholding Tests

This paper addresses the problem of testing whether the Mahalanobis distance between a random signal Θ and a known deterministic model θ0 exceeds some given non-negative real number or not, when Θ has unknown probability distribution and is observed in additive independent Gaussian noise with positive definite covariance matrix. When Θ is deterministic unknown, we prove the existence of thresholding tests on the Mahalanobis distance to θ0 that have specified level and maximal constant power (MCP). The MCP property is a new optimality criterion involving Wald's notion of tests with uniformly best constant power ( UBCP) on ellipsoids for testing the mean of a normal distribution. When the signal is random with unknown distribution, constant power maximality extends to maximal constant conditional power (MCCP) and the thresholding tests on the Mahalanobis distance to θ0 still verify this novel optimality property. Our results apply to the detection of signals in independent and additive Gaussian noise. In particular, for a large class of possible model mismatches, MCCP tests can guarantee a specified false alarm probability, in contrast to standard Neyman-Pearson tests that may not respect this constraint.

[1]  Marco Lops,et al.  Asymptotically optimum radar detection in compound-Gaussian clutter , 1995 .

[2]  Steven M. Kay,et al.  Optimal invariant detection of a sinusoid with unknown parameters , 2002, IEEE Trans. Signal Process..

[3]  J.R. Gabriel,et al.  On the relationship between the GLRT and UMPI tests for the detection of signals with unknown parameters , 2005, IEEE Transactions on Signal Processing.

[4]  Abdeldjalil Aïssa-El-Bey,et al.  Contribution of statistical tests to sparseness-based blind source separation , 2012, EURASIP J. Adv. Signal Process..

[5]  Louis L. Scharf,et al.  The adaptive coherence estimator: a uniformly most-powerful-invariant adaptive detection statistic , 2005, IEEE Transactions on Signal Processing.

[6]  Stephen P. Boyd,et al.  Optimal Estimation of Deterioration From Diagnostic Image Sequence , 2009, IEEE Transactions on Signal Processing.

[7]  A. Wald Tests of statistical hypotheses concerning several parameters when the number of observations is large , 1943 .

[8]  H. Vincent Poor,et al.  Detection of Stochastic Processes , 1998, IEEE Trans. Inf. Theory.

[9]  Stephen E. Fienberg,et al.  Testing Statistical Hypotheses , 2005 .

[10]  Mitra Fouladirad,et al.  Optimal statistical fault detection with nuisance parameters , 2005, Autom..

[11]  Vikram Krishnamurthy Optimal Threshold Policies for Multivariate Stopping-Time POMDPs , 2009, ECSQARU.

[12]  Belkacem Ould Bouamama,et al.  Robust Monitoring of an Electric Vehicle With Structured and Unstructured Uncertainties , 2009, IEEE Transactions on Vehicular Technology.

[13]  Van Trees,et al.  Detection, Estimation, and Modulation Theory. Part 1 - Detection, Estimation, and Linear Modulation Theory. , 1968 .

[14]  Igor V. Nikiforov,et al.  Non-Bayesian Detection and Detectability of Anomalies From a Few Noisy Tomographic Projections , 2007, IEEE Transactions on Signal Processing.

[15]  S. Kay,et al.  An Invariance property of the generalized likelihood ratio test , 2003, IEEE Signal Processing Letters.

[16]  Dominique Pastor,et al.  A sharp upper bound for the probability of error of the likelihood ratio test for detecting signals in white Gaussian noise , 2002, IEEE Trans. Inf. Theory.

[17]  E. L’her,et al.  Automatic detection of AutoPEEP during controlled mechanical ventilation , 2012, Biomedical engineering online.

[18]  Mark E. Campbell,et al.  Efficient Unbiased Tracking of Multiple Dynamic Obstacles Under Large Viewpoint Changes , 2011, IEEE Transactions on Robotics.

[19]  E. Conte,et al.  Adaptive matched filter detection in spherically invariant noise , 1996, IEEE Signal Processing Letters.

[20]  S. Bose,et al.  A maximal invariant framework for adaptive detection with structured and unstructured covariance matrices , 1995, IEEE Trans. Signal Process..

[21]  Antonio De Maio,et al.  Adaptive Detection in Gaussian Interference With Unknown Covariance After Reduction by Invariance , 2010, IEEE Transactions on Signal Processing.

[22]  Michèle Basseville,et al.  Detection of abrupt changes: theory and application , 1993 .

[23]  L. Scharf,et al.  A review of matched and adaptive subspace detectors , 2000, Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No.00EX373).

[24]  Antonio De Maio,et al.  On the Invariance, Coincidence, and Statistical Equivalence of the GLRT, Rao Test, and Wald Test , 2010, IEEE Transactions on Signal Processing.

[25]  Giuseppe Ricci,et al.  Recursive estimation of the covariance matrix of a compound-Gaussian process and its application to adaptive CFAR detection , 2002, IEEE Trans. Signal Process..

[26]  Florent Retraint,et al.  $\varepsilon$ -Optimal Non-Bayesian Anomaly Detection for Parametric Tomography , 2008, IEEE Transactions on Image Processing.

[27]  Calyampudi R. Rao Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[28]  Hoon Sohn,et al.  A review of structural health monitoring literature 1996-2001 , 2002 .

[29]  R. Muirhead Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[30]  Jonathan Lawry,et al.  Symbolic and Quantitative Approaches to Reasoning with Uncertainty , 2009 .

[31]  J. Norris Appendix: probability and measure , 1997 .

[32]  H. Vincent Poor,et al.  An Introduction to Signal Detection and Estimation , 1994, Springer Texts in Electrical Engineering.

[33]  H. Vincent Poor,et al.  An introduction to signal detection and estimation (2nd ed.) , 1994 .

[34]  M. L. Eaton Multivariate statistics : a vector space approach , 1985 .

[35]  Louis L. Scharf,et al.  Adaptive subspace detectors , 2001, IEEE Trans. Signal Process..

[36]  P. Mahalanobis On the generalized distance in statistics , 1936 .

[37]  Louis L. Scharf,et al.  Signal detection in Gaussian noise of unknown level: An invariance application , 1971, IEEE Trans. Inf. Theory.

[38]  Sarangapani Jagannathan,et al.  A Model-Based Fault-Detection and Prediction Scheme for Nonlinear Multivariable Discrete-Time Systems With Asymptotic Stability Guarantees , 2010, IEEE Transactions on Neural Networks.

[39]  Louis L. Scharf,et al.  The CFAR adaptive subspace detector is a scale-invariant GLRT , 1999, IEEE Trans. Signal Process..

[40]  Louis L. Scharf,et al.  Matched subspace detectors , 1994, IEEE Trans. Signal Process..

[41]  E. J. Kelly An Adaptive Detection Algorithm , 1986, IEEE Transactions on Aerospace and Electronic Systems.

[42]  Antonio De Maio,et al.  CFAR detection of multidimensional signals: an invariant approach , 2003, IEEE Trans. Signal Process..

[43]  R. Engle Wald, likelihood ratio, and Lagrange multiplier tests in econometrics , 1984 .