SCIENCE CHINA Information Sciences, Volume 59, Issue 5: 052201(2016) https://doi.org/10.1007/s11432-015-5304-z

Iterative parameter estimate with batched binary-valued observations

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  • ReceivedNov 24, 2015
  • AcceptedFeb 2, 2016
  • PublishedApr 12, 2016


in this paper, we consider linear system identification with batched binary-valued observations. We constructed an iterative parameter estimate algorithm to achieve the maximum likelihood (ML) estimate. The first interesting result was that there exists at most one finite ML solution for this specific maximum likelihood problem, which was induced by the fact that the Hessian matrix of the log-likelihood function was negative definite under binary data and gaussian system noises. The global concave property and local strongly concave property of the log-likelihood function were obtained. Under mild conditions on the system input, we proved that the ML function has a unique maximum point. The second main result was that the ML estimate was consistent under persistent excitation inputs, which infers the effectiveness of ML estimate. Finally, the proposed iterative estimate algorithm converged to a fixed vector with an exponential rate that was proved by constructing a Lyapunov function. A more interesting result was that the limit of the iterative algorithm achieved the maximization of the ML function. Numerical simulations are illustrated to support the theoretical results obtained in this paper well.

Funded by

National Basic Research Program of China(973 Program)

"source" : null , "contract" : "2014CB845301"}]

National Natural Science Foundation of China(61174042)

National Natural Science Foundation of China(11171333)



This work was supported by National Natural Science Foundation of China (Grant Nos. 61174042, 11171333) and National Basic Research Program of China (973 Program) (Grant No. 2014CB845301).


[1] Wang L Y, Zhang J F, Yin G, et al. System Identification With Quantized Observations. Boston: BirkhAauser, 2010. Google Scholar

[2] Wang L Y, Zhang J F, Yin G G. IEEE Trans Autom Control, 2003, 48: 1892-1907 Google Scholar

[3] Ribeiro A, Giannakis G B, Roumeliotis S I. IEEE Trans Signal Process, 2006, 54: 4782-4795 Google Scholar

[4] Marelli D, You K Y, Fu M Y. Automatica, 2013, 49: 360-369 Google Scholar

[5] Chen T S, Zhao Y L, Ljung L. Impulse response estimation with binary measurements: a regularized fir model approach. In: Proceedings of the 16th IFAC Symposium on System Identification, Brussels, 2012. 113-118. Google Scholar

[6] Godoya B I, Goodwin G C, Agueroa J C, et al. Automatica, 2011, 47: 1905-1915 Google Scholar

[7] Severini T A. Likelihood Methods in Statistics. Oxford: Oxford University Press, 2000. Google Scholar

[8] Mclachlan G J, Krishnan T. The EM Algorithm and Extensions. 2nd ed. Hoboken: John Wiley & Sons Inc, 2008. Google Scholar

[9] Newcomb S. American J Math, 1886, 8: 343-366 Google Scholar

[10] Beale E M L, Little R J A. J Royal Stat Soc Ser B (Methodol), 1975, 37: 129-145 Google Scholar

[11] Buck S F. J Royal Stat Soc B (Methodol), 1960, 22: 302-306 Google Scholar

[12] Hartley H O. Biometrics, 1958, 14: 174-194 Google Scholar

[13] Healy M J R, Westmacott M. Appl Stat, 1966, 5: 203-206 Google Scholar

[14] Mckendrick A G. Applications of mathematics to medical problems. In: Proceedings of the Edinburgh Mathematical Society. Cambridge: Cambridge University Press, 1926, 44: 98-130. Google Scholar

[15] Orchard T, Woodbury M A. A missing information principle: theory and applications. In: Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press, 1972, 1: 697-715. Google Scholar

[16] Dempster A P, Laird N M, Rubin D B. J Royal Stat Soc Ser B (Methodol), 1977, 39: 1-38 Google Scholar

[17] Render R A, Walker H F. Soc Indust Appl Math, 1984, 26: 195-239 Google Scholar

[18] Wolynetz M S. J Royal Stat Soc Ser C (Appl Stat), 1979, 28: 185-195 Google Scholar

[19] Byrne W. IEEE Trans Neural Netw, 1992, 3: 612-620 Google Scholar

[20] Boyles R A. J Royal Stat Soc Ser B (Methodol), 1983, 45: 47-50 Google Scholar

[21] Wu C F. Annals Stat, 1983, 11: 95-103 Google Scholar

[22] Ljung L. System Identification. Boston: BirkhAauser, 1998. Google Scholar

[23] Bi W J, Zhao Y L, Liu C X, et al. Set-valued analysis for genome-wide association studies of complex diseases. In: Proceedings of the 32st Chinese Control Conference, Xi'an, 2013. 8262-8267. Google Scholar

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