Recursions for the MMPP Score Vector and Observed Information Matrix

REFERENCES

• Abramowitz , M. ; Stegun , I. Handbook of Mathematical Functions; , 10th Ed. ; Dover Publications : New York , 1972 .
   Google Scholar
• Brewer , J. The derivative of the exponential matrix with respect to a matrix . IEEE Transactions on Automatic Control 1977 , 22 ( 4 ), 656 – 657 .
   Web of Science ®Google Scholar
• Burzykowski , T. ; Szubiakowski , J. ; Rydèn , T. Analysis of photon count data from single-molecule fluorescence experiments . Chemical Physics 2003 , 288 , 291 – 307 .
   Web of Science ®Google Scholar
• Chen , B. ; Zadrozny , P.A. Analytic derivatives of the matrix exponential for estimation of linear continuous-time models . Journal of Economic Dynamics and Control 2001 , 25 ( 12 ), 1867 – 1879 .
   Web of Science ®Google Scholar
• Davison , A.C. ; Ramesh , N.I. Some models for discretized series of events . Journal of the American Statistical Association 1996 , 91 , 601 – 609 .
   Web of Science ®Google Scholar
• Ephraim , Y. ; Merhav , N. Hidden Markov processes . IEEE Trans. Inform. Theory 2002 , 48 , 1518 – 1569 .
   Web of Science ®Google Scholar
• Ephraim , Y. ; Roberts , W.J.J. An EM Algorithm for Markov modulated Markov processes . IEEE Trans. Sig. Proc. 2009 , 57 , 463 – 470 .
   Web of Science ®Google Scholar
• Fischer , W. ; Meier-Hellstern , K. The Markov-modulated Poisson process (MMPP) cookbook . Performance Evaluation 1992 , 18 , 149 – 171 .
   Web of Science ®Google Scholar
• Gerald , C.F. ; Wheatley , P.O. Applied Numerical Analysis; , 4th Ed. ; Addison-Wesley : New York , 1989 .
   Google Scholar
• Holst , U. ; Lindgren , G. Recursive estimation in mixture models with Markov regime . IEEE Transactions on Information Theory 1991 , 37 ( 6 ), 1683 – 1690 .
   Web of Science ®Google Scholar
• Klimeš , L. Accuracy of elastic finite differences in smooth media . Pure and Applied Geophysics 1996 , 148 ( 1/2 ), 39 – 76 .
   Web of Science ®Google Scholar
• Kou , S.C. ; Xie , X.S. ; Liu , J.S. Bayesian analysis of single-molecule experimental data . Journal of the Royal Statistical Society: Series C (Applied Statistics) 2005 , 54 , 469 – 506 .
   Web of Science ®Google Scholar
• Krishnamurthy , V. ; Dey , S. Reduced spatio-temporal complexity MMPP and image-based tracking filters for maneuvering targets . IEEE Transactions on Aerospace and Electronic Systems 2003 , 39 , 1277 – 1291 .
   Web of Science ®Google Scholar
• Lancaster , P. ; Tismenetsky , M. The Theory of Matrices, Second Edition: with Applications ; Academic Press : London , 1985 .
   Google Scholar
• LeGland , F. ; Mèvel , L. Recursive identification of HMMs with observations in a finite set . Proceedings of the 34th IEEE Conference on Decision and Control 1995 , 1 , 216 – 221 .
   Google Scholar
• Linton , O. ; McCrorie , J.R. Differentiation of an exponential matrix function . Econometric Theory 1995 , 11 ( 5 ), 1182 – 1185 .
   Web of Science ®Google Scholar
• Lystig , T.C. ; Hughes , J.P. Exact computation of the observed information matrix for hidden Markov models . Journal of Computational & Graphical Statistics 2002 , 11 , 678 – 689 .
   Web of Science ®Google Scholar
• Magnus , J.R. ; Neudecker , H. The commutation matrix: Some properties and applications. The Annals of Statistics 1979, 7 (2), 381–394.
   Web of Science ®Google Scholar
• Magnus , J.R. ; Neudecker , H. Matrix Differential Calculus with Applications in Statistics and Econometrics ; Wiley : New York , 2001 .
   Google Scholar
• Miao , D.W.-C. Markov Modulated Poisson Processing in Credit Risk Modeling ; University of Oxford , Transfer Report : Oxford, U.K , 2005 .
   Google Scholar
• Moler , C. ; Van Loan , C.F. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later . SIAM Review 2003 , 45 ( 1 ), 3 – 49 .
   Web of Science ®Google Scholar
• Muscariello , L. ; Mellia , M. ; Meo , M. ; Marsan , M.A. ; Cigno , R.L. Markov models of internet traffic and a new hierarchical MMPP model . Computer Communications 2005 , 28 , 1835 – 1851 .
   Web of Science ®Google Scholar
• Najfeld , I. ; Havel , T.F. Derivatives of the matrix exponential and their computation . Advances in Applied Mathematics 1995 , 16 , 321 – 375 .
   Web of Science ®Google Scholar
• Ramesh , N.I. Temporal modelling of short-term rainfall using Cox processes . Environmetrics 1998 , 9 , 629 – 643 .
   Web of Science ®Google Scholar
• Roberts , W.J.J. ; Ephraim , Y. ; Dieguez , E. On Rydèn's EM algorithm for estimating MMPPs . IEEE Signal Processing Letters 2006 , 13 ( 6 ), 373 – 376 .
   Web of Science ®Google Scholar
• Roberts , W.J.J. ; Ephraim , Y. An EM algorithm for ion-channel current estimation . IEEE Transactions on Signal Processing 2008 , 56 ( 1 ), 26 – 33 .
   Web of Science ®Google Scholar
• Rydèn , T. Parameter estimation for Markov modulated Poisson processes . Stochastic Models 1994 , 10 ( 4 ), 795 – 829 .
   Google Scholar
• Rydèn , T. An EM algorithm for estimation in Markov-modulated Poisson processes . Computational Statistics & Data Analysis 1996 , 21 , 431 – 447 .
   Web of Science ®Google Scholar
• Rydèn , T. EM versus Markov chain Monte Carlo for estimation of hidden Markov models: A computational perspective . Bayesian Analysis 2008 , 3 ( 4 ), 659 – 688 .
   Web of Science ®Google Scholar
• Scott , S.L. Bayesian analysis of a two-state Markov modulated Poisson process . Journal of Computational and Graphical Statistics 1999 , 8 , 662 – 670 .
   Web of Science ®Google Scholar
• Sherlock , C. Methodology for inference on the Markov modulated Poisson process and theory for optimal scaling of the random walk metropolis ; Lancaster University : Lancaster , U.K. , Ph.D. Thesis , 2006 .
   Google Scholar
• Skaug , H. Markov modulated Poisson processes for clustered line transect data . Environmental and Ecological Statistics 2006 , 13 , 199 – 211 .
   Web of Science ®Google Scholar
• Van Loan , C.F. Computing integrals involving the matrix exponential . IEEE Transactions on Automatic Control 1978 , 23 ( 3 ), 395 – 404 .
   Web of Science ®Google Scholar
• Vetter , W. Derivative operations on matrices . IEEE Transactions on Automatic Control 1970 , 15 ( 2 ), 241 – 244 .
   Web of Science ®Google Scholar
• Willy , C.J. Recursive Parameter Estimation of the Markov-Modulated Poisson Process ; George Washington University : Washington , D.C., D.Sc. Diss. , 2008 .
   Google Scholar
• Willy , C.J. ; Roberts , W.J.J. ; Mazzuchi , T.A. ; Sarkani , S. On recursive MMPP parameter estimation . IEEE Signal Processing Letters 2008 , 15 , 883 – 886 .
   Web of Science ®Google Scholar