×

zbMATH — the first resource for mathematics

A hidden absorbing semi-Markov model for informatively censored temporal data: learning and inference. (English) Zbl 1445.62211
Summary: Modeling continuous-time physiological processes that manifest a patient’s evolving clinical states is a key step in approaching many problems in healthcare. In this paper, we develop the Hidden Absorbing Semi-Markov Model (HASMM): a versatile probabilistic model that is capable of capturing the modern electronic health record (EHR) data. Unlike existing models, the HASMM accommodates irregularly sampled, temporally correlated, and informatively censored physiological data, and can describe non-stationary clinical state transitions. Learning the HASMM parameters from the EHR data is achieved via a novel forward-filtering backward-sampling Monte-Carlo EM algorithm that exploits the knowledge of the end-point clinical outcomes (informative censoring) in the EHR data, and implements the E-step by sequentially sampling the patients’ clinical states in the reverse-time direction while conditioning on the future states. Real-time inferences are drawn via a forward-filtering algorithm that operates on a virtually constructed discrete-time embedded Markov chain that mirrors the patient’s continuous-time state trajectory. We demonstrate the prognostic utility of the HASMM in a critical care prognosis setting using a real-world dataset for patients admitted to the Ronald Reagan UCLA Medical Center. In particular, we show that using HASMMs, a patient’s clinical deterioration can be predicted 8–9 hours prior to intensive care unit admission, with a 22% AUC gain compared to the Rothman index, which is the state-of-the-art critical care risk scoring technology.

MSC:
62M05 Markov processes: estimation; hidden Markov models
68T05 Learning and adaptive systems in artificial intelligence
62N02 Estimation in survival analysis and censored data
62P10 Applications of statistics to biology and medical sciences; meta analysis
Software:
Adam; BNT; GPy; Scikit; TensorFlow
PDF BibTeX XML Cite
Full Text: Link
References:
[1] Mart´ın Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. Tensorflow: A system for large-scale machine learning. In OSDI, volume 16, pages 265–283, 2016.
[2] Ahmed M. Alaa and Mihaela van der Schaar. Balancing suspense and surprise: Timely decision making with endogenous information acquisition. In Advances in Neural Information Processing Systems, pages 2910–2918, 2016.
[3] Ahmed M Alaa, Jinsung Yoon, Scott Hu, and Mihaela van der Schaar. Personalized risk scoring for critical care prognosis using mixtures of gaussian processes. arXiv preprint arXiv:1610.08853, 2016.
[4] Ahmed M. Alaa, Scott Hu, and Mihaela van der Schaar. Learning from clinical judgments: Semi-markov-modulated marked hawkes processes for risk prognosis. Proceedings of the 34th International Conference on Machine Learning (ICML), 2017.
[5] Jeffrey A Bakal, Finlay A McAlister, Wei Liu, and Justin A Ezekowitz.Heart failure re-admission: measuring the ever shortening gap between repeat heart failure hospitalizations. PloS one, 9(9):e106494, 2014.
[6] Jirina Bartkova, Zuzana Hoˇrejˇs´ı, Karen Koed, Alwin Kr¨amer, Frederic Tort, Karsten Zieger, Per Guldberg, Maxwell Sehested, Jahn M Nesland, Claudia Lukas, et al. Dna damage response as a candidate anti-cancer barrier in early human tumorigenesis. Nature, 434 (7035):864–870, 2005.
[7] Kevin Beier, Sabitha Eppanapally, Heidi S Bazick, Domingo Chang, Karthik Mahadevappa, Fiona K Gibbons, and Kenneth B Christopher. Elevation of bun is predictive of longterm mortality in critically ill patients independent of’normal’creatinine. Critical care medicine, 39(2):305, 2011.
[8] Edwin V Bonilla, Kian M Chai, and Christopher Williams. Multi-task gaussian process prediction. In Advances in neural information processing systems, pages 153–160, 2007.
[9] James G Booth and James P Hobert. Maximizing generalized linear mixed model likelihoods with an automated monte carlo em algorithm. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(1):265–285, 1999. · Zbl 0917.62058
[10] Brian S Caffo, Wolfgang Jank, and Galin L Jones. Ascent-based monte carlo expectation– maximization. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(2):235–251, 2005. · Zbl 1075.65011
[11] Lucienne TQ Cardoso, Cintia MC Grion, Tiemi Matsuo, Elza HT Anami, Ivanil AM Kauss, Ludmila Seko, and Ana M Bonametti. Impact of delayed admission to intensive care units on mortality of critically ill patients: a cohort study. Critical Care, 15(1):R28, 2011.
[12] Chris K Carter and Robert Kohn. On gibbs sampling for state space models. Biometrika, 81(3):541–553, 1994. 54 · Zbl 0809.62087
[13] Dustin Charles, Meghan Gabriel, and JaWanna Henry. Electronic capabilities for patient engagement among us non-federal acute care hospitals: 2012-2014. The Office of the National Coordinator for Health Information Technology, 2015.
[14] Zhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, and Yan Liu. Recurrent neural networks for multivariate time series with missing values. arXiv preprint arXiv:1606.01865, 2016.
[15] Baojiang Chen and Xiao-Hua Zhou. Non-homogeneous markov process models with informative observations with an application to alzheimer’s disease. Biometrical Journal, 53 (3):444–463, 2011. · Zbl 1213.62132
[16] Jill M Cholette, Kelly F Henrichs, George M Alfieris, Karen S Powers, Richard Phipps, Sherry L Spinelli, Michael Swartz, Francisco Gensini, L Eugene Daugherty, Emily Nazarian, et al. Washing red blood cells and platelets transfused in cardiac surgery reduces post-operative inflammation and number of transfusions: Results of a prospective, randomized, controlled clinical trial. Pediatric critical care medicine: a journal of the Society of Critical Care Medicine and the World Federation of Pediatric Intensive and Critical Care Societies, 13(3), 2012.
[17] Jesse Davis and Mark Goadrich. The relationship between precision-recall and roc curves. In Proceedings of the 23rd international conference on Machine learning, pages 233–240. ACM, 2006.
[18] Zelalem Getahun Dessie.Multi-state models of hiv/aids by homogeneous semi-markov process. American Journal of Biostatistics, 4(2):21, 2014.
[19] Michael Dewar, Chris Wiggins, and Frank Wood. Inference in hidden markov models with explicit state duration distributions. IEEE Signal Processing Letters, 19(4):235–238, 2012.
[20] Rick Durrett. Probability: theory and examples. Cambridge university press, 2010. · Zbl 1202.60001
[21] Allison A Eddy and Eric G Neilson. Chronic kidney disease progression. Journal of the American Society of Nephrology, 17(11):2964–2966, 2006.
[22] Yohann Foucher, Eve Mathieu, Philippe Saint-Pierre, J Durand, and J Daures. A semimarkov model based on generalized weibull distribution with an illustration for hiv disease. Biometrical journal, 47(6):825, 2005.
[23] Yohann Foucher, Magali Giral, Jean-Paul Soulillou, and Jean-Pierre Daures. A semi-markov model for multistate and interval-censored data with multiple terminal events. application in renal transplantation. Statistics in medicine, 26(30):5381–5393, 2007.
[24] Yohann Foucher, M Giral, JP Soulillou, and JP Daures. A flexible semi-markov model for interval-censored data and goodness-of-fit testing. Statistical methods in medical research, 2008.
[25] Emily Fox, Erik B Sudderth, Michael I Jordan, and Alan S Willsky. Bayesian nonparametric inference of switching dynamic linear models. IEEE Transactions on Signal Processing, 59(4):1569–1585, 2011a. 55
[26] Emily B Fox, Erik B Sudderth, Michael I Jordan, and Alan S Willsky. A sticky hdphmm with application to speaker diarization. The Annals of Applied Statistics, pages 1020–1056, 2011b. · Zbl 1232.62077
[27] Mitchell H Gail and Phuong L Mai. Comparing breast cancer risk assessment models. Journal of the National Cancer Institute, 102(10):665–668, 2010.
[28] Valentine Genon-Catalot, Thierry Jeantheau, Catherine Lar´edo, et al. Stochastic volatility models as hidden markov models and statistical applications. Bernoulli, 6(6):1051–1079, 2000. · Zbl 0966.62048
[29] Konstantinos Georgatzis, Christopher KI Williams, and Christopher Hawthorne. Inputoutput non-linear dynamical systems applied to physiological condition monitoring. Journal of Machine Learning Research, 2016.
[30] Zoubin Ghahramani and Michael I Jordan. Factorial hidden markov models. Machine learning, 29(2-3):245–273, 1997. · Zbl 0892.68080
[31] Marzyeh Ghassemi, Marco AF Pimentel, Tristan Naumann, Thomas Brennan, David A Clifton, Peter Szolovits, and Mengling Feng. A multivariate timeseries modeling approach to severity of illness assessment and forecasting in icu with sparse, heterogeneous clinical data. In Proceedings of the... AAAI Conference on Artificial Intelligence. AAAI Conference on Artificial Intelligence, volume 2015, page 446. NIH Public Access, 2015.
[32] Giacomo Giampieri, Mark Davis, and Martin Crowder. Analysis of default data using hidden markov models. Quantitative Finance, 5(1):27–34, 2005. · Zbl 1118.91321
[33] Florence Gillaizeau, Etienne Dantan, Magali Giral, and Yohann Foucher. A multistate additive relative survival semi-markov model. Statistical methods in medical research, page 0962280215586456, 2015.
[34] Simon J Godsill, Arnaud Doucet, and Mike West. Monte carlo smoothing for nonlinear time series. Journal of the american statistical association, 99(465):156–168, 2004. · Zbl 1089.62517
[35] Peter J Green and David I Hastie. Reversible jump mcmc. Genetics, 155(3):1391–1403, 2009.
[36] Sheffield ML group. Gpy: A gaussian process framework in python. 2012.
[37] Amit Gruber, Yair Weiss, and Michal Rosen-Zvi. Hidden topic markov models. In AISTATS, volume 7, pages 163–170, 2007.
[38] Yann Gu´edon. Exploring the state sequence space for hidden markov and semi-markov chains. Computational Statistics & Data Analysis, 51(5):2379–2409, 2007. · Zbl 1161.62412
[39] Chantal Guihenneuc-Jouyaux, Sylvia Richardson, and Ira M Longini. Modeling markers of disease progression by a hidden markov process: application to characterizing cd4 cell decline. Biometrics, 56(3):733–741, 2000. 56 · Zbl 1060.62619
[40] Tracy D Gunter and Nicolas P Terry. The emergence of national electronic health record architectures in the united states and australia: models, costs, and questions. Journal of medical Internet research, 7(1):e3, 2005.
[41] Alan G Hawkes and David Oakes. A cluster process representation of a self-exciting process. Journal of Applied Probability, pages 493–503, 1974. · Zbl 0305.60021
[42] CharlesO Hershey and Linda Fisher. Why outcome of cardiopulmonary resuscitation in general wards is poor. The Lancet, 319(8262):31–34, 1982.
[43] Asger Hobolth and Jens Ledet Jensen.Summary statistics for endpoint-conditioned continuous-time markov chains. Journal of Applied Probability, pages 911–924, 2011. · Zbl 1231.60071
[44] Helen Hogan, Frances Healey, Graham Neale, Richard Thomson, Charles Vincent, and Nick Black. Preventable deaths due to problems in care in english acute hospitals: a retrospective case record review study. BMJ quality & safety, pages bmjqs–2012, 2012.
[45] William Hoiles and Mihaela van der Schaar. A non-parametric learning method for confidently estimating patient’s clinical state and dynamics. In Advances in Neural Information Processing Systems, pages 2020–2028, 2016.
[46] George Hripcsak, David J Albers, and Adler Perotte. Parameterizing time in electronic health record studies. Journal of the American Medical Informatics Association, 22(4): 794–804, 2015.
[47] Xuelin Huang and Robert A Wolfe. A frailty model for informative censoring. Biometrics, 58(3):510–520, 2002. · Zbl 1210.62129
[48] Aparna V Huzurbazar. Multistate models, flowgraph models, and semi-markov processes. 2004. · Zbl 1066.62103
[49] Christopher H Jackson, Linda D Sharples, Simon G Thompson, Stephen W Duffy, and Elisabeth Couto. Multistate markov models for disease progression with classification error. Journal of the Royal Statistical Society: Series D (The Statistician), 52(2):193– 209, 2003.
[50] Jacques Janssen and R De Dominicis. Finite non-homogeneous semi-markov processes: Theoretical and computational aspects. Insurance: Mathematics and Economics, 3(3): 157–165, 1984. · Zbl 0546.60087
[51] Daniel W Johnson, Ulrich H Schmidt, Edward A Bittner, Benjamin Christensen, Retsef Levi, and Richard M Pino. Delay of transfer from the intensive care unit: a prospective observational study of incidence, causes, and financial impact. Critical Care, 17(4):R128, 2013.
[52] Matthew J Johnson and Alan S Willsky. Bayesian nonparametric hidden semi-markov models. Journal of Machine Learning Research, 14(Feb):673–701, 2013. · Zbl 1320.62050
[53] Pierre Joly and Daniel Commenges. A penalized likelihood approach for a progressive threestate model with censored and truncated data: Application to aids. Biometrics, 55(3): 887–890, 1999. 57 · Zbl 1059.62664
[54] Jared Katzman, Uri Shaham, Jonathan Bates, Alexander Cloninger, Tingting Jiang, and Yuval Kluger. Deep survival: A deep cox proportional hazards network. arXiv preprint arXiv:1606.00931, 2016.
[55] Juliane Kause, Gary Smith, David Prytherch, Michael Parr, Arthas Flabouris, Ken Hillman, et al. A comparison of antecedents to cardiac arrests, deaths and emergency intensive care admissions in australia and new zealand, and the united kingdom—the academia study. Resuscitation, 62(3):275–282, 2004.
[56] Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
[57] Lisa L Kirkland, Michael Malinchoc, Megan O’Byrne, Joanne T Benson, Deanne T Kashiwagi, M Caroline Burton, Prathibha Varkey, and Timothy I Morgenthaler. A clinical deterioration prediction tool for internal medicine patients. American Journal of Medical Quality, 28(2):135–142, 2013.
[58] William A Knaus, Elizabeth A Draper, Douglas P Wagner, and Jack E Zimmerman. Apache ii: a severity of disease classification system. Critical care medicine, 13(10):818–829, 1985.
[59] William A Knaus, Douglas P Wagner, Elizabeth A Draper, Jack E Zimmerman, Marilyn Bergner, Paulo G Bastos, Carl A Sirio, Donald J Murphy, Ted Lotring, and Anne Damiano. The apache iii prognostic system. risk prediction of hospital mortality for critically ill hospitalized adults. Chest Journal, 100(6):1619–1636, 1991.
[60] Vidyadhar G Kulkarni. Modeling and analysis of stochastic systems. CRC Press, 1996. · Zbl 0866.60004
[61] Stephan W Lagakos, Charles J Sommer, and Marvin Zelen. Semi-markov models for partially censored data. Biometrika, 65(2):311–317, 1978. · Zbl 0398.62032
[62] David Lando. On cox processes and credit risky securities. Review of Derivatives research, 2(2-3):99–120, 1998. · Zbl 1274.91459
[63] Laura Landro. Hospitals find new ways to monitor patients 24/7. The Wall Street Journal, 2015.
[64] Jose Leiva-Murillo, AA Rodrguez, and E Baca-Garca.Visualization and prediction of disease interactions with continuous-time hidden markov models. In NIPS 2011 Workshop on Personalized Medicine, 2011.
[65] H Lehman Li-wei, Shamim Nemati, Ryan P Adams, George Moody, Atul Malhotra, and Roger G Mark. Tracking progression of patient state of health in critical care using inferred shared dynamics in physiological time series. In 2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pages 7072–7075. IEEE, 2013.
[66] William A Link. A model for informative censoring. Journal of the American Statistical Association, 84(407):749–752, 1989. 58
[67] Zachary C Lipton, David Kale, and Randall Wetzel. Directly modeling missing data in sequences with rnns: Improved classification of clinical time series. In Machine Learning for Healthcare Conference, pages 253–270, 2016.
[68] Yu-Ying Liu, Shuang Li, Fuxin Li, Le Song, and James M Rehg. Efficient learning of continuous-time hidden markov models for disease progression. In Advances in neural information processing systems, pages 3600–3608, 2015.
[69] Sergio Matos, Surinder S Birring, Ian D Pavord, and H Evans. Detection of cough signals in continuous audio recordings using hidden markov models.IEEE Transactions on Biomedical Engineering, 53(6):1078–1083, 2006.
[70] Raina M Merchant, Lin Yang, Lance B Becker, Robert A Berg, Vinay Nadkarni, Graham Nichol, Brendan G Carr, Nandita Mitra, Steven M Bradley, Benjamin S Abella, et al. Incidence of treated cardiac arrest in hospitalized patients in the united states. Critical care medicine, 39(11):2401, 2011.
[71] Philipp Metzner, Illia Horenko, and Christof Sch¨utte. Generator estimation of markov jump processes based on incomplete observations nonequidistant in time. Physical Review E, 76(6):066702, 2007.
[72] Rui P Moreno, Philipp GH Metnitz, Eduardo Almeida, Barbara Jordan, Peter Bauer, Ricardo Abizanda Campos, Gaetano Iapichino, David Edbrooke, Maurizia Capuzzo, JeanRoger Le Gall, et al. Saps 3-from evaluation of the patient to evaluation of the intensive care unit. part 2: Development of a prognostic model for hospital mortality at icu admission. Intensive care medicine, 31(10):1345–1355, 2005.
[73] RJM Morgan, F Williams, and MM Wright. An early warning scoring system for detecting developing critical illness. Clin Intensive Care, 8(2):100, 1997.
[74] DR Mould. Models for disease progression: new approaches and uses. Clinical Pharmacology & Therapeutics, 92(1):125–131, 2012.
[75] Kevin Murphy et al. The bayes net toolbox for matlab. Computing science and statistics, 33(2):1024–1034, 2001.
[76] Kevin P Murphy. Hidden semi-markov models (hsmms). unpublished notes, 2, 2002.
[77] Uri Nodelman, Christian R Shelton, and Daphne Koller. Expectation maximization and complex duration distributions for continuous time bayesian networks. arXiv preprint arXiv:1207.1402, 2012.
[78] Zdzis law Opial. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society, 73(4):591–597, 1967.
[79] Mari Ostendorf, Vassilios V Digalakis, and Owen A Kimball. 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, 1996. 59
[80] Soren Erik Pedersen, Suzanne S Hurd, Robert F Lemanske, Allan Becker, Heather J Zar, Peter D Sly, Manuel Soto-Quiroz, Gary Wong, and Eric D Bateman. Global strategy for the diagnosis and management of asthma in children 5 years and younger. Pediatric pulmonology, 46(1):1–17, 2011.
[81] Fabian Pedregosa, Ga¨el Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al. Scikit-learn: Machine learning in python. Journal of Machine Learning Research, 12(Oct):2825–2830, 2011. · Zbl 1280.68189
[82] Romain Pirracchio, Maya L Petersen, Marco Carone, Matthieu Resche Rigon, Sylvie Chevret, and Mark J van der Laan. Mortality prediction in intensive care units with the super icu learner algorithm (sicula): a population-based study. The Lancet Respiratory Medicine, 3(1):42–52, 2015.
[83] Andrei D Polyanin and Alexander V Manzhirov. Handbook of integral equations. CRC press, 2008. · Zbl 1154.45001
[84] Ross L Prentice, John D Kalbfleisch, Arthur V Peterson Jr, Nancy Flournoy, Vern T Farewell, and Norman E Breslow. The analysis of failure times in the presence of competing risks. Biometrics, pages 541–554, 1978.
[85] Zhen Qin and Christian R Shelton. Auxiliary gibbs sampling for inference in piecewiseconstant conditional intensity models. In Proceedings of the Thirty-First Conference on Uncertainty in Artificial Intelligence, 2015.
[86] Lawrence R Rabiner. A tutorial on hidden markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2):257–286, 1989.
[87] Rajesh Ranganath, Adler Perotte, No´emie Elhadad, and David Blei. Deep survival analysis. In Machine Learning for Healthcare Conference, pages 101–114, 2016.
[88] Carl Edward Rasmussen. Gaussian processes for machine learning. 2006. · Zbl 1177.68165
[89] Santiago Romero-Brufau, Jeanne M Huddleston, Gabriel J Escobar, and Mark Liebow. Why the c-statistic is not informative to evaluate early warning scores and what metrics to use. Critical Care, 19(1):285, 2015.
[90] Michael J Rothman, Steven I Rothman, and Joseph Beals. Development and validation of a continuous measure of patient condition using the electronic medical record. Journal of biomedical informatics, 46(5):837–848, 2013.
[91] Mohammed Saeed, Christine Lieu, Greg Raber, and Roger G Mark. Mimic ii: a massive temporal icu patient database to support research in intelligent patient monitoring. In Computers in Cardiology, 2002, pages 641–644. IEEE, 2002.
[92] Daniel O Scharfstein and James M Robins. Estimation of the failure time distribution in the presence of informative censoring. Biometrika, 89(3):617–634, 2002. 60 · Zbl 1036.62110
[93] Peter Schulam and Suchi Saria. A framework for individualizing predictions of disease trajectories by exploiting multi-resolution structure. In Advances in Neural Information Processing Systems, pages 748–756, 2015. · Zbl 1404.68117
[94] Padhraic Smyth. Hidden markov models for fault detection in dynamic systems. Pattern recognition, 27(1):149–164, 1994.
[95] Henry T Stelfox, Brenda R Hemmelgarn, Sean M Bagshaw, Song Gao, Christopher J Doig, Cheri Nijssen-Jordan, and Braden Manns. Intensive care unit bed availability and outcomes for hospitalized patients with sudden clinical deterioration. Archives of internal medicine, 172(6):467–474, 2012.
[96] CP Subbe, M Kruger, P Rutherford, and L Gemmel. Validation of a modified early warning score in medical admissions. Qjm, 94(10):521–526, 2001.
[97] MJ Sweeting, VT Farewell, and D De Angelis.Multi-state markov models for disease progression in the presence of informative examination times: An application to hepatitis c. Statistics in medicine, 29(11):1161–1174, 2010.
[98] S Taghipour, D Banjevic, AB Miller, N Montgomery, AKS Jardine, and BJ Harvey. Parameter estimates for invasive breast cancer progression in the canadian national breast screening study. British journal of cancer, 108(3):542–548, 2013.
[99] Hale F Trotter and John W Tukey. Conditional monte carlo for normal samples. In Symposium on Monte Carlo Methods, pages 64–79. Wiley, 1956.
[100] John Varga, Christopher P Denton, and Fredrick M Wigley. Scleroderma: From pathogenesis to comprehensive management. Springer Science & Business Media, 2012.
[101] J-L Vincent, Rui Moreno, Jukka Takala, Sheila Willatts, Arnaldo De Mendon¸ca, Hajo Bruining, CK Reinhart, PeterM Suter, and LG Thijs. The sofa (sepsis-related organ failure assessment) score to describe organ dysfunction/failure. Intensive care medicine, 22(7):707–710, 1996.
[102] Xiang Wang, David Sontag, and Fei Wang. Unsupervised learning of disease progression models. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 85–94. ACM, 2014.
[103] Paul J Werbos. Backpropagation through time: what it does and how to do it. Proceedings of the IEEE, 78(10):1550–1560, 1990.
[104] J Yoon, A Alaa, S Hu, and M van der Schaar. Forecasticu: A prognostic decision support system for timely prediction of intensive care unit admission. pages 1680–1689, 2016.
[105] Lei Yu and Huan Liu. Feature selection for high-dimensional data: A fast correlation-based filter solution. In ICML, volume 3, pages 856–863, 2003.
[106] Shun Yu, Sharon Leung, Moonseong Heo, Graciela J Soto, Ronak T Shah, Sampath Gunda, and Michelle Ng Gong. Comparison of risk prediction scoring systems for ward patients: a retrospective nested case-control study. Critical Care, 18(3):1, 2014. 61
[107] Shun-Zheng Yu. Hidden semi-markov models. Artificial Intelligence, 174(2):215–243, 2010. · Zbl 1344.68181
[108] Yongyue Zhang, Michael Brady, and Stephen Smith. Segmentation of brain mr images through a hidden markov random field model and the expectation-maximization algorithm. IEEE transactions on medical imaging, 20(1):45–57, 2001.
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.