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Model Order Selection for Short Data: An Exponential Fitting Test (EFT)

Abstract

High-resolution methods for estimating signal processing parameters such as bearing angles in array processing or frequencies in spectral analysis may be hampered by the model order if poorly selected. As classical model order selection methods fail when the number of snapshots available is small, this paper proposes a method for noncoherent sources, which continues to work under such conditions, while maintaining low computational complexity. For white Gaussian noise and short data we show that the profile of the ordered noise eigenvalues is seen to approximately fit an exponential law. This fact is used to provide a recursive algorithm which detects a mismatch between the observed eigenvalue profile and the theoretical noise-only eigenvalue profile, as such a mismatch indicates the presence of a source. Moreover this proposed method allows the probability of false alarm to be controlled and predefined, which is a crucial point for systems such as RADARs. Results of simulations are provided in order to show the capabilities of the algorithm.

References

  1. 1.

    Yin YQ, Krishnaiah PR: On some nonparametric methods for detection of the number of signals. IEEE Transactions on Acoustics, Speech, and Signal Processing 1987,35(11):1533–1538. 10.1109/TASSP.1987.1165063

    MathSciNet  Article  Google Scholar 

  2. 2.

    Scharf LL, Tufts DW: Rank reduction for modeling stationary signals. IEEE Transactions on Acoustics, Speech, and Signal Processing 1987,35(3):350–355. 10.1109/TASSP.1987.1165136

    Article  Google Scholar 

  3. 3.

    Anderson TW: Asymptotic theory for principal component analysis. Annals of Mathematical Statistics 1963, 34: 122–148. 10.1214/aoms/1177704248

    MathSciNet  Article  Google Scholar 

  4. 4.

    James AT: Test of equality of latent roots of the covariance matrix. Journal of Multivariate Analysis 1969, 205–218.

    Google Scholar 

  5. 5.

    Chen W, Wong KM, Reilly J: Detection of the number of signals: a predicted eigen-threshold approach. IEEE Transactions on Signal Processing 1991,39(5):1088–1098. 10.1109/78.80959

    Article  Google Scholar 

  6. 6.

    Stoica P, Selén Y: Model-order selection: a review of information criterion rules. IEEE Signal Processing Magazine 2004,21(4):36–47. 10.1109/MSP.2004.1311138

    Article  Google Scholar 

  7. 7.

    Akaike H: A new look at the statistical model identification. IEEE Transactions on Automatic Control 1974,19(6):716–723. 10.1109/TAC.1974.1100705

    MathSciNet  Article  Google Scholar 

  8. 8.

    Rissanen J: Modeling by shortest data description length. Automatica 1978,14(5):465–471. 10.1016/0005-1098(78)90005-5

    Article  Google Scholar 

  9. 9.

    Wax M, Kailath T: Detection of signals by information theoretic criteria. IEEE Transactions on Acoustics, Speech, and Signal Processing 1985,33(2):387–392. 10.1109/TASSP.1985.1164557

    MathSciNet  Article  Google Scholar 

  10. 10.

    Wax M, Ziskind I: Detection of the number of coherent signals by the MDL principle. IEEE Transactions on Acoustics, Speech, and Signal Processing 1989,37(8):1190–1196. 10.1109/29.31267

    Article  Google Scholar 

  11. 11.

    Kaveh M, Wang H, Hung H: On the theoretical performance of a class of estimators of the number of narrow-band sources. IEEE Transactions on Acoustics, Speech, and Signal Processing 1987,35(9):1350–1352. 10.1109/TASSP.1987.1165277

    Article  Google Scholar 

  12. 12.

    Wong KM, Zhang Q-T, Reilly J, Yip P: On information theoretic criteria for determining the number of signals in high resolution array processing. IEEE Transactions on Acoustics, Speech, and Signal Processing 1990,38(11):1959–1971. 10.1109/29.103097

    Article  Google Scholar 

  13. 13.

    Wu Q, Fuhrmann D: A parametric method for determining the number of signals in narrow-band direction finding. IEEE Transactions on Signal Processing 1991,39(8):1848–1857. 10.1109/78.91155

    Article  Google Scholar 

  14. 14.

    Djurić PM: Model selection based on asymptotic Bayes theory. Proceedings of the 7th IEEE SP Workshop on Statistical Signal and Array Processing, June 1994, Quebec City, Quebec, Canada 7–10.

    Google Scholar 

  15. 15.

    Bishop WB, Djurić PM: Model order selection of damped sinusoids in noise by predictive densities. IEEE Transactions on Signal Processing 1996,44(3):611–619. 10.1109/78.489034

    Article  Google Scholar 

  16. 16.

    Van Trees HL: Optimum Array Processing, Detection, Estimation and Modulation Theory. Volume 4. John Wiley & Sons, New York, NY, USA; 2002.

    Google Scholar 

  17. 17.

    Di A: Multiple source location - a matrix decomposition approach. IEEE Transactions on Acoustics, Speech, and Signal Processing 1985,33(5):1086–1091. 10.1109/TASSP.1985.1164700

    Article  Google Scholar 

  18. 18.

    Wu H-T, Yang J-F, Chen F-K: Source number estimators using transformed Gerschgorin radii. IEEE Transactions on Signal Processing 1995,43(6):1325–1333. 10.1109/78.388844

    Article  Google Scholar 

  19. 19.

    Liavas AP, Regalia PA: On the behavior of information theoretic criteria for model order selection. IEEE Transactions on Signal Processing 2001,49(8):1689–1695. 10.1109/78.934138

    Article  Google Scholar 

  20. 20.

    Quinlan A, Barbot J-P, Larzabal P: Automatic determination of the number of targets present when using the time reversal operator. The Journal of the Acoustical Society of America 2006,119(4):2220–2225. 10.1121/1.2180207

    Article  Google Scholar 

  21. 21.

    Tanter M, Thomas J-L, Fink M: Time reversal and the inverse filter. The Journal of the Acoustical Society of America 2000,108(1):223–234. 10.1121/1.429459

    Article  Google Scholar 

  22. 22.

    Grouffaud J, Larzabal P, Clergeot H: Some properties of ordered eigenvalues of a Wishart matrix: application in detection test and model order selection. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '96), May 1996, Atlanta, Ga, USA 5: 2463–2466.

    Google Scholar 

  23. 23.

    Valaee S, Kabal P: An information theoretic approach to source enumeration in array signal processing. IEEE Transactions on Signal Processing 2004,52(5):1171–1178. 10.1109/TSP.2004.826168

    MathSciNet  Article  Google Scholar 

  24. 24.

    Champagne B: Adaptive eigendecomposition of data covariance matrices based on first-order perturbations. IEEE Transactions on Signal Processing 1994,42(10):2758–2770. 10.1109/78.324741

    Article  Google Scholar 

  25. 25.

    Ziskind I, Wax M: Maximum likelihood localization of multiple sources by alternating projection. IEEE Transactions on Acoustics, Speech, and Signal Processing 1988,36(10):1553–1560. 10.1109/29.7543

    Article  Google Scholar 

  26. 26.

    Johnson NL, Kotz S: Distributions in Statistics: Continuous Multivariate Distributions. John Wiley & Sons, New York, NY, USA; 1972. chapter 38–39

    Google Scholar 

  27. 27.

    Krishnaiah PR, Schurmann FJ: On the evaluation of some distribution that arise in simultaneous tests of the equality of the latents roots of the covariance matrix. Journal of Multivariate Analysis 1974, 4: 265–282. 10.1016/0047-259X(74)90033-5

    MathSciNet  Article  Google Scholar 

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Correspondence to Angela Quinlan.

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Quinlan, A., Barbot, J., Larzabal, P. et al. Model Order Selection for Short Data: An Exponential Fitting Test (EFT). EURASIP J. Adv. Signal Process. 2007, 071953 (2006). https://doi.org/10.1155/2007/71953

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Keywords

  • Computational Complexity
  • False Alarm
  • Selection Method
  • Gaussian Noise
  • Quantum Information