Hidden semi-Markov models.

*(English)*Zbl 1344.68181Summary: As an extension to the popular hidden Markov model (HMM), a hidden semi-Markov model (HSMM) allows the underlying stochastic process to be a semi-Markov chain. Each state has variable duration and a number of observations being produced while in the state. This makes it suitable for use in a wider range of applications. Its forward-backward algorithms can be used to estimate/update the model parameters, determine the predicted, filtered and smoothed probabilities, evaluate goodness of an observation sequence fitting to the model, and find the best state sequence of the underlying stochastic process. Since the HSMM was initially introduced in 1980 for machine recognition of speech, it has been applied in thirty scientific and engineering areas, such as speech recognition/synthesis, human activity recognition/prediction, handwriting recognition, functional MRI brain mapping, and network anomaly detection. There are about three hundred papers published in the literature. An overview of HSMMs is presented in this paper, including modelling, inference, estimation, implementation and applications. It first provides a unified description of various HSMMs and discusses the general issues behind them. The boundary conditions of HSMM are extended. Then the conventional models, including the explicit duration, variable transition, and residential time of HSMM, are discussed. Various duration distributions and observation models are presented. Finally, the paper draws an outline of the applications.

##### MSC:

68Txx | Artificial intelligence |

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

68Q87 | Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) |

68-02 | Research exposition (monographs, survey articles) pertaining to computer science |

##### Keywords:

hidden Markov model (HMM); hidden semi-Markov model (HSMM); explicit duration HMM; variable duration HMM; forward-backward algorithm; Viterbi algorithm##### Software:

hsmm
Full Text:
DOI

##### References:

[1] | K. Achan, S. Roweis, A. Hertzmann, B. Frey, A segment-based probabilistic generative model of speech, in: Proc. of the 2005 IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 5, Philadelphia, PA, 2005, pp. 221-224 |

[2] | B. Ait-el-Fquih, F. Desbouvries, Kalman filtering for triplet Markov chains: Applications and extensions, in: Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, ICASSP 05, vol. 4, Philadelphia, USA, 2005, pp. 685-688 |

[3] | C. Alasseur, L. Husson, F. Perez-Fontan, Simulation of rain events time series with Markov model, in: Proc. of 15th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC 2004, vol. 4, 2004, pp. 2801-2805 |

[4] | Amara, N.B.; Belaid, A., Printed PAW recognition based on planar hidden Markov models, (), 220-224 |

[5] | Ariki, Y.; Jack, M.A., Enhanced time duration constraints in hidden Markov modelling for phoneme recognition, Electronics letters, 25, 13, 824-825, (22 June 1989) |

[6] | Askar, M.; Derin, H., A recursive algorithm for the Bayes solution of the smoothing problem, IEEE trans. automat. contr., AC-26, 558-561, (Apr. 1981) |

[7] | S.C. Austin, F. Fallside, Frame compression in hidden Markov models, in: Proc. of 1988 International Conference on Acoustics, Speech, and Signal Processing, ICASSP-88, 11-14 April 1988, pp. 477-480 |

[8] | Aydin, Z.; Altunbasak, Y.; Borodovsky, M., Protein secondary structure prediction for a single-sequence using hidden semi-Markov models, BMC bioinformatics, 7, 178, (2006), Available: |

[9] | M. Azimi, P. Nasiopoulos, R.K. Ward, Online identification of hidden semi-Markov models, in: Proceedings of the 3rd International Symposium on Image and Signal Processing and Analysis, ISPA 2003, vol. 2, 18-20 Sept. 2003, pp. 991-996 · Zbl 1370.93295 |

[10] | M. Azimi, P. Nasiopoulos, R.K. Ward, A new signal model and identification algorithm for hidden semi-Markov signals, in: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 2004, (ICASSP ’04), vol. 2, 17-21 May 2004, pp. ii-521-ii-524 |

[11] | Azimi, M.; Nasiopoulos, P.; Ward, R.K., Offline and online identification of hidden semi-Markov models, IEEE transactions on signal processing, 53, 8, 2658-2663, (Aug. 2005), Part 1 |

[12] | Bae, K.; Mallick, B.K.; Elsik, C.G., Prediction of protein interdomain linker regions by a nonstationary hidden Markov model, Journal of the American statistical association, 103, 483, 1085-1099, (Sep. 2008) |

[13] | Baum, L.E.; Petrie, T., Statistical inference for probabilistic functions of finite state Markov chains, Ann. math. stat., 37, 1554-1563, (1966) · Zbl 0144.40902 |

[14] | Bechhoefer, E.; Bernhard, A.; He, D.; Banerjee, P., Use of hidden semi-Markov models in the prognostics of shaft failure, in: Proceedings of the American Helicopter Society 62th Annual Forum, Phoenix, AZ, 2006. Available: |

[15] | Benouareth, A.; Ennaji, A.; Sellami, M., Arabic handwritten word recognition using HMMs with explicit state duration, EURASIP journal on advances signal processing, 2008, 1-13, (2008) · Zbl 1143.68571 |

[16] | R. Bippus, V. Margner, Script recognition using inhomogeneous P2DHMM and hierarchical search space reduction, in: Proceedings of the Fifth International Conference on Document Analysis and Recognition, 1999 (ICDAR ’99), 20-22 Sept. 1999, pp. 773-776 |

[17] | A. Bonafonte, X. Ros, J.B. Marino, An efficient algorithm to find the best state sequence in HSMM, in: Proceedings of Eurospeech’93, Berlin, 1993, pp. 1547-1550 |

[18] | A. Bonafonte, J. Vidal, A. Nogueiras, Duration modeling with expanded HMM applied to speech recognition, in: Fourth International Conference on Spoken Language, 1996 (ICSLP ’96), vol. 2, 3-6 Oct. 1996, pp. 1097-1100 |

[19] | Borodovsky, M.; Lukashin, A.V., Genemark.hmm: new solutions for gene finding, Nucleic acids res., 26, 1097-1100, (1998) |

[20] | Boutillon, E.; Gross, W.J.; Gulak, P.G., VLSI architectures for the MAP algorithm, IEEE transactions on communications, 51, 2, 175-185, (Feb. 2003) |

[21] | Bouyahia, Z.; Benyoussef, L.; Derrode, S., Change detection in synthetic aperture radar images with a sliding hidden Markov chain model, Journal of applied remote sensing, 2, 023526, (2008) |

[22] | Bulla, J.; Bulla, I., Stylized facts of financial time series and hidden semi-Markov models, Computational statistics and data analysis, 51, 4, 2192-2209, (December 2006) |

[23] | Bulla, J.; Bulla, I.; Nenadic, O., HSMMâ€”an R package for analyzing hidden semi-Markov models, Computational statistics and data analysis, (2009) · Zbl 05689616 |

[24] | Burge, C.; Karlin, S., Prediction of complete gene structures in human genomic DNA, J. mol. biol., 268, 78-94, (1997) |

[25] | D. Burshtein, Robust parametric modeling of durations in hidden Markov models, in: Proc. of 1995 International Conference on Acoustics, Speech, and Signal Processing, (ICASSP-95), vol. 1, 9-12 May 1995, p. 548 |

[26] | Burshtein, D., Robust parametric modeling of durations in hidden Markov models, IEEE transactions on speech and audio processing, 4, 3, 240-242, (May 1996) |

[27] | J. Cai, Z.-Q. Liu, Integration of structural and statistical information for unconstrained handwritten numeral recognition, in: Proceedings of Fourteenth International Conference on Pattern Recognition, 1998, vol. 1, 16-20 Aug. 1998, pp. 378-380 |

[28] | Cai, J.; Liu, Z.-Q., Integration of structural and statistical information for unconstrained handwritten numeral recognition, IEEE transactions on pattern analysis and machine intelligence, 21, 3, 263-270, (March 1999) |

[29] | Cappe, O.; Moulines, E.; Ryden, T., Inference in hidden Markov models, (2005), Springer New York · Zbl 1080.62065 |

[30] | Chaubert-Pereira, F.; Guedon, Y.; Lavergne, C.; Trottier, C., Markov and semi-Markov switching linear mixed models for identifying forest tree growth components, Research Report. Available: · Zbl 1203.62211 |

[31] | Chen, K.; Hasegawa-Johnson, M.; Cohen, A.; Borys, S.; Kim, S.-S.; Cole, J.; Choi, J.-Y., Prosody dependent speech recognition on radio news corpus of American English, IEEE transactions on audio, speech, and language processing, 14, 1, 232-245, (Jan. 2006), see also IEEE Transactions on Speech and Audio Processing |

[32] | M.-Y. Chen, A. Kundu, S.N. Srihari, Handwritten word recognition using continuous density variable duration hidden Markov model, in: Proc. of 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP-93), vol. 5, 27-30 April 1993, pp. 105-108 |

[33] | M.-Y. Chen, A. Kundu, S.N. Srihari, Variable duration hidden Markov model and morphological segmentation for handwritten word recognition, in: Proc. of 1993 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR ’93), 15-17 June 1993, pp. 600-601 |

[34] | M.-Y. Chen, A. Kundu, A complement to variable duration hidden Markov model in handwritten word recognition, in: Proceedings of IEEE International Conference on Image Processing, 1994 (ICIP-94), vol. 1, 13-16 Nov. 1994, pp. 174-178 |

[35] | Chen, M.Y.; Kundu, A.; Srihari, S.N., Variable duration hidden Markov model and morphological segmentation for handwritten word recognition, IEEE trans. image processing, 4, 1675-1688, (Dec. 1995) |

[36] | Chien, J.-T.; Huang, C.-H., Bayesian learning of speech duration models, IEEE transactions on speech and audio processing, 11, 6, 558-567, (Nov. 2003) |

[37] | J.-T. Chien, C.-H. Huang, Bayesian duration modeling and learning for speech recognition, in: Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing, 2004 (ICASSP ’04), vol. 1, 17-21 May 2004, pp. I-1005-I-1008 |

[38] | Chung, P.C.; Liu, C.D., A daily behavior enabled hidden Markov model for human behavior understanding, Pattern recognition, 41, 1572-1580, (2008) · Zbl 1140.68458 |

[39] | M. Codogno, L. Fissore, Duration modelling in finite state automata for speech recognition and fast speaker adaptation, in: Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’87), vol. 12, Apr. 1987, pp. 1269-1272 |

[40] | Cohen, Y.; Erell, A.; Bistritz, Y., Enhancement of connected words in an extremely noisy environment, IEEE transactions on speech and audio processing, 5, 2, 141-148, (March 1997) |

[41] | A. Culotta, A. McCallum, Confidence estimation for information extraction, in: Human Language Technology Conference (HLT), 2004 |

[42] | Dempster, A.P.; Laird, N.M.; Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm, Journal of the royal statistical society, series B (methodological), 39, 1-38, (1977) · Zbl 0364.62022 |

[43] | Deng, L.; Aksmanovic, M.; Sun, X.; Wu, J., Speech recognition using hidden Markov models with polynomial regression functions as nonstationary states, IEEE transactions on speech and audio processing, 2, 4, 507-520, (1994) |

[44] | Deng, L.; Aksmanovic, M., Speaker-independent phonetic classification using hidden Markov models with mixtures of trend functions, IEEE transactions on speech and audio processing, 5, 4, 319-324, (July 1997) |

[45] | Devijver, P.A., Baum’s forward-backward algorithm revisited, Pattern recognition letters, 3, 369-373, (1985) · Zbl 0593.62083 |

[46] | P.M. Djuric, J.-H. Chun, Estimation of nonstationary hidden Markov models by MCMC sampling, in: 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’99), vol. 3, 15-19 March 1999, pp. 1737-1740 |

[47] | Djuric, P.M.; Chun, J.-H., An MCMC sampling approach to estimation of nonstationary hidden Markov models, IEEE transactions on signal processing, 50, 5, 1113-1123, (May 2002) |

[48] | Dong, M.; He, D.; Banerjee, P.; Keller, J., Equipment health diagnosis and prognosis using hidden semi-Markov models, The international journal of advanced manufacturing technology, 30, 7-8, 738-749, (October 2006), (12) |

[49] | Dong, M.; He, D., A segmental hidden semi-Markov model (HSMM)-based diagnostics and prognostics framework and methodology, Mechanical systems and signal processing, 21, 5, 2248-2266, (July 2007) |

[50] | Dong, M.; He, D., Hidden semi-Markov model-based methodology for multi-sensor equipment health diagnosis and prognosis, European journal of operational research, 178, 3, 858-878, (May 2007) |

[51] | Dong, M., A novel approach to equipment health management based on auto-regressive hidden semi-Markov model (AR-HSMM), Science in China series F: information sciences, 51, 9, 1291-1304, (Sep. 2008) |

[52] | Dong, M.; Yang, D.; Kuang, Y.; He, D.; Erdal, S.; Kenski, D., PM2. 5 concentration prediction using hidden semi-Markov model-based times series data mining, Expert systems with applications, 36, 9046-9055, (2009) |

[53] | J. Dumont, A.I. Hernandez, J. Fleureau, G. Carrault, Modelling temporal evolution of cardiac electrophysiological features using hidden semi-Markov models, in: Proceedings: Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Aug. 2008, pp. 165-168 |

[54] | T.V. Duong, H.H. Bui, D.Q. Phung, S. Venkatesh, Activity recognition and abnormality detection with the switching hidden semi-Markov model, in: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005 (CVPR 2005), vol. 1, 20-25 June 2005, pp. 838-845 |

[55] | T.V. Duong, D.Q. Phung, H.H. Bui, S. Venkatesh, Efficient Coxian duration modelling for activity recognition in smart environments with the hidden semi-Markov model, in: Proceedings of the 2005 International Conference on Intelligent Sensors, Sensor Networks and Information Processing Conference, 2005, 5-8 Dec. 2005, pp. 277-282 |

[56] | T.V. Duong, D.Q. Phung, H.H. Bui, S. Venkatesh, Human behavior recognition with generic exponential family duration modeling in the hidden semi-Markov model, in: Proc. of 18th International Conference on Pattern Recognition, 2006 (ICPR 2006), vol. 3, 20-24 Aug. 2006, pp. 202-207 |

[57] | Ephraim, Y.; Merhav, N., Hidden Markov processes, IEEE trans. information theory, 48, 6, 1518-1569, (June 2002) |

[58] | S. Faisan, L. Thoraval, J.-P. Armspach, F. Heitz, Hidden semi-Markov event sequence models: Application to brain functional MRI sequence analysis, in: Proceedings of 2002 International Conference on Image Processing, vol. 1, 22-25 Sept. 2002, pp. I-880-I-883 |

[59] | Faisan, S.; Thoraval, L.; Armspach, J.-P.; Metz-Lutz, M.-N.; Heitz, F., Unsupervised learning and mapping of active brain functional MRI signals based on hidden semi-Markov event sequence models, IEEE transactions on medical imaging, 24, 2, 263-276, (Feb. 2005) |

[60] | Ferguson, J.D., Variable duration models for speech, (), 143-179 |

[61] | L. Finesso, Consistent estimation of the order for Markov and hidden Markov chains, Ph.D. dissertation, Univ. Maryland, College Park, 1990 |

[62] | J. Ford, V. Krishnamurthy, J.B. Moore, Adaptive estimation of hidden semi-Markov chains with parameterised transition probabilities and exponential decaying states, in: Proc. of Conf. on Intell. Signal Processing and Communication Systems (ISPACS), Sendai, Japan, Oct. 1993, pp. 88-92 |

[63] | M. Gales, S. Young, The theory of segmental hidden Markov models, Technical Report CUED/F-INFENG/TR 133, Cambridge University, Engineering Department, 1993 |

[64] | X. Ge, P. Smyth, Deformable Markov model templates for time-series pattern matching, in: Proc. of the 6th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Boston, MA, 2000, pp. 81-90 |

[65] | Ge, X.; Smyth, P., Segmental semi-Markov models for change-point detection with applications to semiconductor manufacturing, (March 2000), Technical Report UCI-ICS 00-08 |

[66] | Ghahramani, Z., An introduction to hidden Markov models and Bayesian networks, International journal of pattern recognition and artificial intelligence, 15, 1, 9-42, (2001) |

[67] | Gu, H.-Y.; Tseng, C.-Y.; Lee, L.-S., Isolated-utterance speech recognition using hidden Markov models with bounded state durations, IEEE transactions on signal processing, 39, 8, 1743-1752, (Aug. 1991), see also IEEE Transactions on Acoustics, Speech, and Signal Processing |

[68] | Y. Guedon, C. Cocozza-Thivent, Use of the Derin’s algorithm in hidden semi-Markov models for automatic speech recognition, in: Proc. of 1989 International Conference on Acoustics, Speech, and Signal Processing (ICASSP-89), 23-26 May 1989, pp. 282-285 |

[69] | Guedon, Y.; Barthelemy, D.; Caraglio, Y.; Costes, E., Pattern analysis in branching and axillary flowering sequences, Journal of theoretical biology, 212, 4, 481-520, (Oct. 2001) |

[70] | Guedon, Y., Estimating hidden semi-Markov chains from discrete sequences, Journal of computational and graphical statistics, 12, 3, 604-639, (2003) |

[71] | Guedon, Y., Hidden hybrid Markov/semi-Markov chains, Computational statistics and data analysis, 49, 3, 663-688, (June 2005) |

[72] | Guedon, Y., Exploring the state sequence space for hidden Markov and semi-Markov chains, Computational statistics and data analysis, 51, 5, 2379-2409, (Feb. 2007) |

[73] | T. Hanazawa, K. Kita, S. Nakamura, T. Kawabata, K. Shikano, ATR HMM-LR continuous speech recognition system, in: Proc. of 1990 International Conference on Acoustics, Speech, and Signal Processing, 1990, ICASSP-90, 3-6 April 1990, pp. 53-56 |

[74] | J. He, H. Leich, A unified way in incorporating segmental feature and segmental model into HMM, in: Proc. of 1995 International Conference on Acoustics, Speech, and Signal Processing, 1995, ICASSP-95, vol. 1, 9-12 May 1995, pp. 532-535 |

[75] | H. He, S. Wu, P. Banerjee, E. Bechhoefer, Probabilistic model based algorithms for prognostics, in: Proc. of 2006 IEEE Aerospace Conference, 4-11 March 2006 |

[76] | J.L. Hieronymus, D. McKelvie, F. McInnes, Use of acoustic sentence level and lexical stress in HSMM speech recognition, in: Proc. of 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, 1992, ICASSP-92, vol. 1, 23-26 March 1992, pp. 225-227 |

[77] | W.J. Holmes, M.J. Russell, Experimental evaluation of segmental HMMs, in: Proc. of International Conference on Acoustics, Speech, and Signal Processing, 1995, ICASSP-95, vol. 1, 9-12 May 1995, pp. 536-539 |

[78] | Holmes, W.J.; Russell, M.J., Probabilistic-trajectory segmental hmms, Computer speech and language, 13, 1, 3-37, (1999) |

[79] | S. Hongeng, R. Nevatia, Large-scale event detection using semi-hidden Markov models, in: Proceedings of Ninth IEEE International Conference on Computer Vision, 13-16 Oct. 2003, pp. 1455-1462 |

[80] | Hongeng, S.; Nevatia, R.; Bremond, F., Video-based event recognition: activity representation and probabilistic methods, Comp. vis. and image understanding, 96, 129-162, (2004) |

[81] | J. Hu, R. Kashi, G. Wilfong, Document classification using layout analysis, in: Proc. of First Intl. Workshop on Document Analysis and Understanding for Document Databases, Florence, Italy, September 1999 |

[82] | Hu, J.; Kashi, R.; Wilfong, G., Comparison and classification of documents based on layout similarity, Information retrieval, 2, 2, 227-243, (May 2000) |

[83] | Huang, X.D., Phoneme classification using semicontinuous hidden Markov models, IEEE transactions on signal processing, 40, 5, 1062-1067, (May 1992), see also IEEE Transactions on Acoustics, Speech, and Signal Processing |

[84] | N.P. Hughes, S.J. Roberts, L. Tarassenko, Semi-supervised learning of probabilistic models for ECG segmentation, in: Proc. of 26th Annual International Conference of the Engineering in Medicine and Biology Society, 2004, EMBC 2004, vol. 1, 2004, pp. 434-437 |

[85] | Hughes, N.P.; Tarassenko, L.; Roberts, S.J., Markov models for automated ECG interval analysis, Advances in neural information processing systems, (2003), Available: |

[86] | Johnson, M.T., Capacity and complexity of HMM duration modeling techniques, IEEE signal processing letters, 12, 5, 407-410, (May 2005) |

[87] | Katagiri, S.; Lee, C.-H., A new hybrid algorithm for speech recognition based on HMM segmentation and learning vector quantization, IEEE transactions on speech and audio processing, 1, 4, 421-430, (Oct. 1993) |

[88] | C.C. Ke, J. Llinas, Literature survey on ground target tracking problems, Research Project Report, Center for Multisource Information Fusion, State University of New York at Buffalo, 1999 |

[89] | Kim, S.; Smyth, P., Segmental hidden Markov models with random effects for waveform modeling, Journal of machine learning research, 7, 945-969, (2006) · Zbl 1222.68351 |

[90] | W.-G. Kim, J.-Y. Yoon, D.H. Youn, HMM with global path constraint in Viterbi decoding for isolated word recognition, in: Proc. ICASSP 1994, 1994, pp. 605-608 |

[91] | H. Kobayashi, S.-Z. Yu, Hidden semi-Markov models and efficient forward-backward algorithms, in: 2007 Hawaii and SITA Joint Conference on Information Theory, Honolulu, Hawaii, 29-31 May 2007, pp. 41-46 |

[92] | V. Krishnamurthy, J.B. Moore, Signal processing of semi-Markov models with exponentially decaying states, in: Proceedings of the 30th Conference on Decision and Control, Brighton, England, Dec. 1991, pp. 2744-2749 |

[93] | Krishnamurthy, V.; Moore, J.B.; Chung, S.H., Hidden fractal model signal processing, Signal processing, 24, 2, 177-192, (Aug. 1991) |

[94] | D. Kulp, D. Haussler, M.G. Reese, F.H. Eeckman, A generalized hidden Markov model for the recognition of human genes in DNA, in: Proc. 4th Int. Conf. Intell. Syst. Molecular Bio., 1996, pp. 134-142 |

[95] | A. Kundu, Y. He, M.-Y. Chen, Efficient utilization of variable duration information in HMM based HWR systems, in: Proceedings of International Conference on Image Processing, 1997, vol. 3, 26-29 Oct. 1997, pp. 304-307 |

[96] | Kundu, A.; He, Y.; Chen, M.-Y., Alternatives to variable duration HMM in handwriting recognition, IEEE transactions on pattern analysis and machine intelligence, 20, 11, 1275-1280, (Nov. 1998) |

[97] | Kwon, O.W.; Un, C.K., Context-dependent word duration modelling for Korean connected digit recognition, Electronics letters, 31, 19, 1630-1631, (Sept. 1995) |

[98] | P. Lanchantin, W. Pieczynski, Unsupervised nonstationary image segmentation using triplet Markov chains, in: Proc. of Advanced Concepts for Intelligent Vision Systems (ACVIS 04), Brussels, Belgium, Aug. 31-Sept. 3, 2004 · Zbl 1373.62431 |

[99] | Lanchantin, P.; Pieczynski, W., Unsupervised restoration of hidden nonstationary Markov chain using evidential priors, IEEE transactions on signal processing, 53, 8, 3091-3098, (2005) · Zbl 1373.62431 |

[100] | Lanchantin, P.; Lapuyade-Lahorguea, J.; Pieczynski, W., Unsupervised segmentation of triplet Markov chains hidden with long-memory noise, Signal processing, 88, 5, 1134-1151, (May 2008) |

[101] | J. Lapuyade-Lahorgue, W. Pieczynski, Unsupervised segmentation of hidden semi-Markov non-stationary chains, in: Twenty Sixth International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, MaxEnt 2006, Paris, France, 8-13 July 2006 |

[102] | K. Laurila, Noise robust speech recognition with state duration constraints, in: Proc. of 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing, 1997, ICASSP-97, vol. 2, 21-24 April 1997, pp. 871-874 |

[103] | C.-H. Lee, F.K. Soong, B.-H. Juang, A segment model based approach to speech recognition, in: Proc. Int’l. Conf. on Acoust., Speech and Signal Processing, 1988, pp. 501-504 |

[104] | Lee, C.-H.; Rabiner, L.R., A frame-synchronous network search algorithm for connected word recognition, IEEE transactions on acoustics, speech, and signal processing, 37, 11, 1649-1658, (Nov. 1989), see also IEEE Transactions on Signal Processing |

[105] | Leland, W.; Taqqu, M.; Willinger, W.; Wilson, D., On the self-similar nature of Ethernet traffic (extended version), IEEE/ACM transactions on networking, 2, 1, 1-15, (February 1994) |

[106] | Levinson, S.E.; Rabiner, L.R.; Sondhi, M.M., An introduction to the application of the theory of probabilistic functions of a Markov process in automatic speech recognition, B.s.t.j., 62, 1035-1074, (1983) · Zbl 0507.68058 |

[107] | Levinson, S.E., Continuously variable duration hidden Markov models for automatic speech recognition, Computer speech and language, 1, 1, 29-45, (1986) |

[108] | S.E. Levinson, Continuously variable duration hidden Markov models for speech analysis, in: Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP ’86., vol. 11, Apr. 1986, pp. 1241-1244 |

[109] | S.E. Levinson, A. Ljolje, L.G. Miller, Large vocabulary speech recognition using a hidden Markov model for acoustic/phonetic classification, in: Proc. of 1988 International Conference on Acoustics, Speech, and Signal Processing, ICASSP-88, 11-14 April 1988, pp. 505-508 |

[110] | S.E. Levinson, M.Y. Liberman, A. Ljolje, L.G. Miller, Speaker independent phonetic transcription of fluent speech for large vocabulary speech recognition, in: Proc. of 1989 International Conference on Acoustics, Speech, and Signal Processing, ICASSP-89, 23-26 May 1989, pp. 441-444 |

[111] | M. Li, S.-Z. Yu, A network-wide traffic anomaly detection method based on HSMM, in: Proc. of 2006 International Conference on Communications, Circuits and Systems Proceedings, vol. 3, June 2006, pp. 1636-1640 |

[112] | H.-P. Lin, M.-J. Tseng, F.-S. Tsai, A non-stationary hidden Markov model for satellite propagation channel modeling, in: Proceedings of 2002 IEEE 56th Vehicular Technology Conference, VTC 2002-Fall, vol. 4, 24-28 Sept. 2002, pp. 2485-2488 |

[113] | X.B. Liu, D.S. Yang, X.O. Chen, New approach to classification of Chinese folk music based on extension of HMM, in: International Conference on Audio, Language and Image Processing, ICALIP 2008, 7-9 July 2008, pp. 1172-1179 |

[114] | Z. Liu, J.X. Yu, L. Chen, D. Wu, Detection of shape anomalies: A probabilistic approach using hidden Markov models, in: IEEE 24th International Conference on Data Engineering, ICDE 2008, 7-12 April 2008, pp. 1325-1327 |

[115] | Ljolje, A.; Levinson, S.E., Development of an acoustic-phonetic hidden Markov model for continuous speech recognition, IEEE transactions on signal processing, 39, 1, 29-39, (Jan. 1991), see also IEEE Transactions on Acoustics, Speech, and Signal Processing |

[116] | W.-Z. Lu, S.-Z. Yu, An HTTP flooding detection method based on browser behavior, in: Proc. of 2006 International Conference on Computational Intelligence and Security, vol. 2, Nov. 2006, pp. 1151-1154 |

[117] | W.-Z. Lu, S.-Z. Yu, Clustering web traffic of request bursts, in: Proc. of 2006 IEEE Region 10 Conference on Communications, TENCON 2006, Nov. 2006, pp. 1-4 |

[118] | E. Marcheret, M. Savic, Random walk theory applied to language identification, in: Proc. of 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-97, vol. 2, 21-24 April 1997, pp. 1119-1122 |

[119] | E. Marhasev, M. Hadad, G.A. Kaminka, Non-stationary hidden semi-Markov models in activity recognition, in: Proceedings of the AAAI Workshop on Modeling Others from Observations (MOO-06), 2006 |

[120] | B.L. Mark, Z.R. Zaidi, Robust mobility tracking for cellular networks, in: Proc. of IEEE International Conference on Communications, 2002, ICC 2002, vol. 1, April 28-May 2, 2002, pp. 445-449 |

[121] | McLachlan, G.J.; Krishnan, T., The EM algorithm and extensions, (2008), Wiley New York · Zbl 1165.62019 |

[122] | C.D. Mitchell, R.A. Helzerman, L.H. Jamieson, M.P. Harper, A parallel implementation of a hidden Markov model with duration modeling for speech recognition, in: Proceedings of the Fifth IEEE Symposium on Parallel and Distributed Processing, 1993, 1-4 Dec. 1993, pp. 298-306 |

[123] | C. Mitchell and L. Jamieson, Modeling duration in a hidden Markov model with the exponential family, in: Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-93, 1993, pp. 331-334 |

[124] | Mitchell, C.; Harper, M.; Jamieson, L., On the complexity of explicit duration hmms, IEEE transactions on speech and audio processing, 3, 2, 213-217, (May 1995) |

[125] | Moore, M.D.; Savic, M.I., Speech reconstruction using a generalized HSMM (GHSMM), Digital signal processing, 14, 1, 37-53, (2004) |

[126] | Murphy, K.P., Hidden semi-Markov models (HSMMs), (Nov. 2002) |

[127] | S. Nakagawa, Y. Hashimoto, A method for continuous speech segmentation using HMM, in: Proc. of 9th International Conference on Pattern Recognition, vol. 2, 14-17 Nov. 1988, pp. 960-962 |

[128] | P. Natarajan, R. Nevatia, Hierarchical multi-channel hidden semi-Markov models, in: The Twentieth International Joint Conference on Artificial Intelligence, Hyderabad, India, Jan. 2007, pp. 2562-2567 |

[129] | P. Natarajan, R. Nevatia, Coupled hidden semi-Markov models for activity recognition, in: IEEE Workshop on Motion and Video Computing, 2007, WMVC ’07, Feb. 2007 |

[130] | P. Natarajan, R. Nevatia, Online, real-time tracking and recognition of human actions, in: IEEE Workshop on Motion and Video Computing, WMVC 2008, 8-9 Jan. 2008, pp. 1-8 |

[131] | Niwase, N.; Yamagishi, J.; Kobayashi, T., Human walking motion synthesis with desired pace and stride length based on HSMM, IEICE transactions on information and systems, (2005) |

[132] | T. Nose, J. Yamagishi, T. Kobayashi, A style control technique for speech synthesis using multiple regression HSMM, in: Proc. INTERSPEECH 2006-ICSLP, Sept. 2006, pp. 1324-1327 |

[133] | Nose, T.; Yamagishi, J.; Masuko, T.; Kobayashi, T., A style control technique for HMM-based expressive speech synthesis, IEICE transactions on information and systems, E90-D, 9, 1406-1413, (Sept. 2007) |

[134] | T. Nose, Y. Kato, T. Kobayashi, A speaker adaptation technique for MRHSMM-based style control of synthetic speech, in: Proc. ICASSP 2007, vol. IV, Apr. 2007, pp. 833-836 |

[135] | Ostendorf, M.; Roukos, S., A stochastic segment model for phoneme-based continuous speech recognition, IEEE transactions on acoustics, speech, and signal processing, 37, 12, 1857-1869, (Dec. 1989), see also IEEE Transactions on Signal Processing |

[136] | Ostendorf, M.; Digalakis, V.V.; Kimball, O.A., From HMM’s to segment models: A unified view of stochastic modeling for speech recognition, IEEE transactions on speech and audio processing, 4, 5, 360-378, (Sep. 1996) |

[137] | K. Oura, H. Zen, Y. Nankaku, A. Lee, K. Tokuda, Hidden semi-Markov model based speech recognition system using weighted finite-state transducer, in: IEEE International Conference on Acoustics, Speech, and Signal Processing, 2006, ICASSP 2006, vol. 1, 14-19 May 2006, pp. I-33-I-36 |

[138] | Park, Y.K.; Un, C.K.; Kwon, O.W., Modeling acoustic transitions in speech by modified hidden Markov models with state duration and state duration-dependent observation probabilities, IEEE transactions on speech and audio processing, 4, 5, 389-392, (Sept. 1996) |

[139] | K. Park, G.T. Kim, M.E. Crovella, On the effect of traffic self-similarity on network performance, in: Proceedings of SPIE International Conference on Performance and Control of Network Systems, November 1997, pp. 296-310 |

[140] | M. Pavel, T.L. Hayes, A. Adami, H.B. Jimison, J. Kaye, Unobtrusive assessment of mobility, in: 28th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, New York, NY, 30 August-3 September 2006 |

[141] | G. Peng, B. Zhang, W. S.-Y. Wang, Performance of mandarin connected digit recognizer with word duration modeling, in: ASR2000 - Automatic Speech Recognition: Challenges for the New Millenium, Paris, France, 18-20 Sep. 2000, pp. 140-144 |

[142] | D. Phung, T. Duong, H. Bui, S. Venkatesh, Activity recognition and abnormality detection with the switching hidden semi-Markov model, in: Int. Conf. on Comp. Vis. & Pat. Recog, 2005 |

[143] | D.Q. Phung, T.V. Duong, S. Venkatesh, H.H. Bui, Topic transition detection using hierarchical hidden Markov and semi-Markov models, in: Proceedings of the 13th Annual ACM International Conference, 2005, pp. 11-20 |

[144] | W. Pieczynski, C. Hulard, T. Veit, Triplet Markov chains in hidden signal restoration, in: SPIE’s International Symposium on Remote Sensing, Crete, Greece, 22-27 September 2002 |

[145] | W. Pieczynski, Modeling nonstationary hidden semi-Markov chains with triplet Markov chains and theory of evidence, in: 2005 IEEE/SP 13th Workshop on Statistical Signal Processing, 17-20 July 2005, pp. 727-732 |

[146] | W. Pieczynski, F. Desbouvries, On triplet Markov chains, in: International Symposium on Applied Stochastic Models and Data Analysis (ASMDA 2005), Brest, France, May 2005 · Zbl 1031.62080 |

[147] | Pieczynski, W., Multisensor triplet Markov chains and theory of evidence, International journal of approximate reasoning, 45, 1, 1-16, (May 2007) |

[148] | Pikrakis, A.; Theodoridis, S.; Kamarotos, D., Classification of musical patterns using variable duration hidden Markov models, IEEE transactions on audio, speech, and language processing, 14, 5, 1795-1807, (Sept. 2006), see also IEEE Transactions on Speech and Audio Processing |

[149] | J.A. du Preez, Modelling durations in hidden Markov models with application to word spotting, in: Proceedings of South African Symposium on Communications and Signal Processing, 1991, COMSIG 1991, 30 Aug. 1991, pp. 1-5 |

[150] | Rabiner, L.R., A tutorial on hidden Markov models and selected application in speech recognition, Proceedings of the IEEE, 77, 2, 257-286, (Feb. 1989) |

[151] | P. Ramesh, J.G. Wilpon, Modeling state durations in hidden Markov models for automatic speech recognition, in: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-92, vol. 1, 23-26 March 1992, pp. 381-384 |

[152] | N. Ratnayake, M. Savic, J. Sorensen, Use of semi-Markov models for speaker-independent phoneme recognition, in: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-92, vol. 1, 23-26 March 1992, pp. 565-568 |

[153] | Riska, A.; Squillante, M.; Yu, S.-Z.; Liu, Z.; Zhang, L., Matrix-analytic analysis of a MAP/PH/1 queue fitted to web server data, (), 333-356 · Zbl 1066.90516 |

[154] | Russell, M.J., Reducing computational load in segmental hidden Markov model decoding for speech recognition, Electronics letters, 41, 25, 1408-1409, (Dec. 2005) |

[155] | M.J. Russell, R.K. Moore, Explicit modelling of state occupancy in hidden Markov models for automatic speech recognition, in: Proc. IEEE Int. Conf. Acoust. Speech Signal Processing, vol. 10, Apr. 1985, pp. 5-8 |

[156] | M.J. Russell, A. Cook, Experimental evaluation of duration modelling techniques for automatic speech recognition, in: Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing, 1987, pp. 2376-2379 |

[157] | M.J. Russell, A segmental HMM for speech pattern modelling, in: 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-93, vol. 2, 27-30 April 1993, pp. 499-502 |

[158] | Salzenstein, F.; Collet, C.; Lecam, S.; Hatt, M., Non-stationary fuzzy Markov chain, Pattern recognition letters, 28, 16, 2201-2208, (Dec. 2007) |

[159] | Sansom, J.; Thomson, P., Fitting hidden semi-Markov models to breakpoint rainfall data, J. appl. probab. A, 38, 2001, 142-157, (2001) · Zbl 1008.62110 |

[160] | Sansom, J.; Thompson, C.S., Spatial and temporal variation of rainfall over New Zealand, J. geophys. res. D, 113, 6, (Apr. 2008) |

[161] | S. Sarawagi, W.W. Cohen, Semi-Markov conditional random fields for information extraction, in: Advances in Neural Information Processing Systems, vol. 17, NIPS, 2004 |

[162] | Schmidler, S.C.; Liu, J.S.; Brutlag, D.L., Bayesian segmentation of protein secondary structure, J. comp. biol., 7, 233-248, (2000) |

[163] | A. Senior, J. Subrahmonia, K. Nathan, Duration modeling results for an on-line handwriting recognizer, in: Proceedings of 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-96, vol. 6, 7-10 May 1996, pp. 3482-3485 |

[164] | Sin, B.; Kim, J.H., Nonstationary hidden Markov model, Signal processing, 46, 31-46, (1995) · Zbl 0880.60075 |

[165] | R.N.V. Sitaram, T.V. Sreenivas, Phoneme recognition in continuous speech using large inhomogeneous hidden Markov models, in: 1994 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-94, vol. i, 19-22 April 1994, pp. I/41-I/44 |

[166] | R. Sitaram, T. Sreenivas, Connected phoneme HMMs with implicit duration modelling for better speech recognition, in: Proceedings of 1997 International Conference on Information, Communications and Signal Processing, ICICS 1997, 9-12 Sept. 1997, pp. 1024-1028 |

[167] | Squire, K., HMM-based semantic learning for a mobile robot, Ph.D. dissertation, University of Illinois at Urbana-Champaign, 2004. Available: |

[168] | K. Squire, S.E. Levinson, Recursive maximum likelihood estimation for hidden semi-Markov models, in: 2005 IEEE Workshop on Machine Learning for Signal Processing, 28-30 Sept. 2005, pp. 329-334 |

[169] | M. Tachibana, J. Yamagishi, T. Masuko, T. Kobayashi, Performance evaluation of style adaptation for hidden semi-Markov model based speech synthesis, in: INTERSPEECH-2005, 2005, pp. 2805-2808 |

[170] | Tachibana, M.; Yamagishi, J.; Masuko, T.; Kobayashi, T., A style adaptation technique for speech synthesis using HSMM and suprasegmental features, IEICE transactions on information and systems, E89-D, 3, 1092-1099, (2006) |

[171] | M. Tachibana, S. Izawa, T. Nose, T. Kobayashi, Speaker and style adaptation using average voice model for style control in hmm-based speech synthesis, in: Proc. ICASSP 2008, pp. 4633-4636 |

[172] | Tan, X.R.; Xi, H.S., Hidden semi-Markov model for anomaly detection, Applied mathematics and computation, 205, 562-567, (2008) · Zbl 1173.68424 |

[173] | V. Ter-Hovhannisyan, Unsupervised and semi-supervised training methods for eukaryotic gene prediction, Ph.D. dissertation, Georgia Institute of Technology, 2008 |

[174] | Thoraval, L.; Carrault, G.; Mora, F., Continuously variable duration hidden Markov models for ECG segmentation, (), 529-530 |

[175] | Thoraval, L.; Carrault, G.; Bellanger, J.J., Heart signal recognition by hidden Markov models: the ECG case, Meth. inform. med., 33, 10-14, (1994) |

[176] | Thoraval, L., Technical Report: Hidden semi-Markov event sequence models, 2002. Available: |

[177] | Tuan, T.; Park, K., Multiple time scale congestion control for self-similar network traffic, Performance evaluation, 36, 359-386, (1999) · Zbl 1051.68534 |

[178] | D. Tweed, R. Fisher, J. Bins, T. List, Efficient hidden semi-Markov model inference for structured video sequences, in: 2nd Joint IEEE International Workshop on Visual Surveillance and Performance Evaluation of Tracking and Surveillance, 2005, 15-16 Oct. 2005 pp. 247-254 |

[179] | Vaseghi, S.V., Hidden Markov models with duration-dependent state transition probabilities, Electronics letters, 27, 8, 625-626, (April 1991) |

[180] | S.V. Vaseghi, P. Conner, On increasing structural complexity of finite state speech models, in: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-92, vol. 1, 23-26 March 1992, pp. 537-540 |

[181] | Vaseghi, S.V., State duration modeling in hidden Markov models, Signal processing, 41, 1, 31-41, (1995) · Zbl 0875.68752 |

[182] | Wang, X., Durationally constrained training of hmm without explicit state durational pdf, (), 111-130 |

[183] | X. Wang, L.F.M. ten Bosch, L.C.W. Pols, Integration of context-dependent durational knowledge into HMM-based speech recognition, in: Proceedings of Fourth International Conference on Spoken Language, 1996, ICSLP 96, vol. 2, 3-6 Oct. 1996, pp. 1073-1076 |

[184] | Wang, J.B.; Athitsos, V.; Sclaroff, S.; Betke, M., Detecting objects of variable shape structure with hidden state shape models, IEEE transactions on pattern analysis and machine intelligence, 30, 3, 477-492, (March 2008) |

[185] | C. Wellington, A. Courville, A. Stentz, Interacting Markov random fields for simultaneous terrain modeling and obstacle detection, in: Proceedings of Robotics: Science and Systems, 2005 |

[186] | Wu, C.-H.; Hsia, C.-C.; Liu, T.-H.; Wang, J.-F., Voice conversion using duration-embedded bi-HMMs for expressive speech synthesis, IEEE transactions on audio, speech, and language processing, 14, 4, 1109-1116, (July 2006), see also IEEE Transactions on Speech and Audio Processing |

[187] | Y. Xie, S.-Z. Yu, A dynamic anomaly detection model for web user behavior based on HsMM, in: 10th International Conference on Computer Supported Cooperative Work in Design, May 2006, pp. 1-6 |

[188] | Y. Xie, S.-Z. Yu, A novel model for detecting application layer DDoS attacks, in: First International Multi-Symposiums on Computer and Computational Sciences, IMSCCS ’06, vol. 2, 20-24 April 2006, pp. 56-63 |

[189] | J. Yamagishi, T. Kobayashi, Adaptive training for hidden semi-Markov model, in: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005 (ICASSP ’05), vol. 1, 18-23 March 2005, pp. 365-368 |

[190] | J. Yamagishi, K. Ogata, Y. Nakano, J. Isogai, T. Kobayashi, HSMM-based model adaptation algorithms for average-voice-based speech synthesis, in: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 2006, ICASSP 2006, vol. 1, 14-19 May 2006, pp. I-77-I-80 |

[191] | Yamagishi, J.; Kobayashi, T., Average-voice-based speech synthesis using HSMM-based speaker adaptation and adaptive training, IEICE transactions on information and systems, E90-D, 2, 533-543, (2007) |

[192] | T. Yamazaki, N. Niwase, J. Yamagishi, T. Kobayashi, Human walking motion synthesis based on multiple regression hidden semi-Markov model, in: International Conference on Cyberworlds, 23-25 Nov. 2005 |

[193] | P. Yang, G. Dumont, J.M. Ansermino, An adaptive Cusum test based on a hidden semi-Markov model for change detection in non-invasive mean blood pressure trend, in: Proceedings of the 28th IEEE EMBS Annual International Conference, New York City, USA, Aug. 30-Sept. 3 2006, pp. 3395-3398 |

[194] | N.B. Yoma, F.R. McInnes, M.A. Jack, Weighted Viterbi algorithm and state duration modelling for speech recognition in noise, in: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 1998, ICASSP ’98, vol. 2, 12-15 May 1998, pp. 709-712 |

[195] | Yoma, N.B.; McInnes, F.R.; Jack, M.A.; Stump, S.D.; Ling, L.L., On including temporal constraints in viterbi alignment for speech recognition in noise, IEEE transactions on speech and audio processing, 9, 2, 179-182, (Feb. 2001) |

[196] | Yoma, N.B.; Sanchez, J.S., MAP speaker adaptation of state duration distributions for speech recognition, IEEE transactions on speech and audio processing, 10, 7, 443-450, (Oct. 2002) |

[197] | S.-Z. Yu, B. L. Mark, H. Kobayashi, Mobility tracking and traffic characterization for efficient wireless internet access, in: IEEE MMT’2000, Multiaccess, Mobility and Teletraffic in Wireless Communications, vol. 5, Duck Key, Florida, 3-6 Dec. 2000, pp. 279-290 |

[198] | S.-Z. Yu, Z. Liu, M. Squillante, C. Xia, L. Zhang, A hidden semi-Markov model for web workload self-similarity, in: 21st IEEE International Performance, Computing, and Communications Conference, IPCCC 2002, Phoenix, Arizona, 3-5 April 2002, pp. 65-72 |

[199] | Yu, S.-Z.; Kobayashi, H., An efficient forward-backward algorithm for an explicit duration hidden Markov model, IEEE signal processing letters, 10, 1, 11-14, (Jan. 2003) |

[200] | Yu, S.-Z.; Kobayashi, H., A hidden semi-Markov model with missing data and multiple observation sequences for mobility tracking, Signal processing, 83, 2, 235-250, (Feb. 2003) |

[201] | S.-Z. Yu, Multiple tracking based anomaly detection of mobile nodes, in: 2nd International Conference on Mobile Technology, Applications and Systems, 2005, 15-17 Nov. 2005, pp. 5-9 |

[202] | Yu, S.-Z.; Kobayashi, H., Practical implementation of an efficient forward-backward algorithm for an explicit-duration hidden Markov model, IEEE transactions on signal processing, 54, 5, 1947-1951, (January 2006) |

[203] | Yun, Y.-S.; Oh, Y.-H., A segmental-feature HMM for speech pattern modeling, IEEE signal processing letters, 7, 6, 135-137, (2000) |

[204] | H. Zen, K. Tokuda, T. Masuko, T. Kobayashi, T. Kitamura, Hidden semi-Markov model based speech synthesis, in: Proc. of 8th International Conference on Spoken Language Processing, ICSLP, Jeju Island, Korea, 4-8 Oct. 2004, pp. 1393-1396 |

[205] | W. Zhang, F. Chen, W. Xu, E. Zhang, Real-time video intelligent surveillance system, in: 2006 IEEE International Conference on Multimedia and Expo, July 2006, pp. 1021-1024 |

[206] | W. Zhang, F. Chen, W. Xu, Y. Du, Learning human activity containing sparse irrelevant events in long sequence, in: 2008 Congress on Image and Signal Processing, CISP’08, 2008, pp. 211-215 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.