Distances and Riemannian metrics for multivariate spectral densities

IEEE Transactions on Automatic Control

Volumn 57, Issue 7, 2012, Pages 1723-1735

  Jiang, Xianhua ^a   Ning, Lipeng ^a   Georgiou, Tryphon T ^a  

^a University of Minnesota   (United States)

Author keywords

Geodesics; metrics; multivariable power spectra; spectral analysis

Indexed keywords

CLASSICAL PREDICTIONS; DIVERGENCE MEASURES; EXPLICIT FORMULA; FISHER-RAO METRIC; GEODESIC DISTANCES; GEODESICS; METRICS; MULTI VARIABLES; POWER-SPECTRA; PRACTICAL PROBLEMS; RIEMANNIAN METRICS;

POWER SPECTRAL DENSITY; POWER SPECTRUM; SPECTRAL DENSITY; SPECTRUM ANALYSIS;

GEOMETRY;

EID: 84862992932     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2012.2183171     Document Type: Article

References (24)

1. M. Basseville, "Distance measures for signal processing and pattern recognition," Signal Process., vol. 18, no. 4, pp. 349-369, 1989. (Pubitemid 20636472)
2. R. Gray,A. Buzo, A. Gray, Jr., and Y. Matsuyama, "Distortion measures for speech processing," IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-28, no. 4, pp. 367-376, Aug. 1980.
3. C. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," Bull. Calcutta Math. Soc., vol. 37, pp. 81-91, 1945.
4. S.-I. Amari and H. Nagaoka, "Methods of information geometry," Amer. Math. Soc., 2000.
5. N. Cencov, "Statistical decision rules and optimal inference," Amer. Math. Soc., no. 53, 1982.
6. R. Kass and P. Vos, Geometrical Foundations of Asymptotic Inference. New York: Wiley, 1997.
7. T. Georgiou, "Distance and Riemannian metrics for spectral density functions," IEEE Trans. Signal Process., vol. 55, no. 8, pp. 3995-4003, Aug. 2007. (Pubitemid 47261522)
8. A. Ferrante, C. Masiero, and M. Pavon, "Time and spectral domain relative entropy: A new approach to multivariate spectral estimation," Arxiv preprint arXiv:1103.5602, 2011.
9. A. Ferrante, M. Pavon, and F. Ramponi, "Hellinger versus Kullback-Leibler multivariable spectrum approximation," IEEE Trans. Autom. Control, vol. 53, no. 4, pp. 954-967, Apr. 2008. (Pubitemid 351795855)
10. B. Musicus and R. Johnson, "Multichannel relative-entropy spectrum analysis," IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-34, no. 3, pp. 554-564, Jun. 1986.
11. P. Enqvist and J. Karlsson, "Minimal Itakura-Saito distance and covariance interpolation," in Proc. 47th IEEE Conf. Decision Control, 2008, pp. 137-142.
12. T. Georgiou, "Relative entropy and the multivariable multidimensional moment problem," IEEE Trans. Inf. Theory, vol. 52, no. 3, pp. 1052-1066, Mar. 2006. (Pubitemid 46444891)
13. R. Bhatia, Positive Definite Matrices. Princeton, NJ: Princeton Univ Press, 2007.
14. M. Pinsker, "Information and information stability of random variables and processes," in Izv. Akad. Nauk. SSSR, Moscow, Russia, 1960.
15. S.Yu and P. Mehta, "TheKullback-Leibler rate pseudo-metric for comparing dynamical systems," IEEE Trans. Autom. Control, vol. 55, no. 7, pp. 1585-1598, Jul. 2010.
16. N. Wiener and P. Masani, "The prediction theory of multivariate stochastic processes, Part I," Acta Math., vol. 98, pp. 111-150, 1957.
17. P. Masani, "Recent trends in multivariable prediction theory," in Multivariate Analysis, P. R. Krishnaiah, Ed. New York: Academic, 1966, pp. 351-382.
18. T. Georgiou, "The Carathéodory-Fejér-Pisarenko decomposition and its multivariable counterpart," IEEE Trans. Autom. Control, vol. 52, no. 2, pp. 212-228, Feb. 2007. (Pubitemid 46444277)
19. P. Petersen, Riemannian Geometry. Berlin, Germany: Springer-Verlag, 2006.
20. J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects. New York: Birkhäuser, 1997.
21. M. Moakher, "A differential geometric approach to the geometric mean of symmetric positive-definite matrices," SIAM J. Matrix Anal. Applicat., vol. 26, no. 3, pp. 735-747, 2005. (Pubitemid 41049842)
22. A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis, Volume 2. Menlo Park, CA: Graylock, 1961.
23. R. Horn and C. Johnson, Topics in Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1994.
24. R. Horn and C. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 2005.