Browsing by Subject "Markov models"

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  • Dudel, Christian; Myrskylä, Mikko (BioMed Central, 2020)
    Abstract Background Markov models are a key tool for calculating expected time spent in a state, such as active life expectancy and disabled life expectancy. In reality, individuals often enter and exit states recurrently, but standard analytical approaches are not able to describe this dynamic. We develop an analytical matrix approach to calculating the expected number and length of episodes spent in a state. Methods The approach we propose is based on Markov chains with rewards. It allows us to identify the number of entries into a state and to calculate the average length of episodes as total time in a state divided by the number of entries. For sampling variance estimation, we employ the block bootstrap. Two case studies that are based on published literature illustrate how our methods can provide new insights into disability dynamics. Results The first application uses a classic textbook example on prednisone treatment and liver functioning among liver cirrhosis patients. We replicate well-known results of no association between treatment and survival or recovery. Our analysis of the episodes of normal liver functioning delivers the new insight that the treatment reduced the likelihood of relapse and extended episodes of normal liver functioning. The second application assesses frailty and disability among elderly people. We replicate the prior finding that frail individuals have longer life expectancy in disability. As a novel finding, we document that frail individuals experience three times as many episodes of disability that were on average twice as long as the episodes of nonfrail individuals. Conclusions We provide a simple analytical approach for calculating the number and length of episodes in Markov chain models. The results allow a description of the transition dynamics that goes beyond the results that can be obtained using standard tools for Markov chains. Empirical applications using published data illustrate how the new method is helpful in unraveling the dynamics of the modeled process.
  • Dudel, Christian; Myrskylä, Mikko (2020)
    Background Markov models are a key tool for calculating expected time spent in a state, such as active life expectancy and disabled life expectancy. In reality, individuals often enter and exit states recurrently, but standard analytical approaches are not able to describe this dynamic. We develop an analytical matrix approach to calculating the expected number and length of episodes spent in a state. Methods The approach we propose is based on Markov chains with rewards. It allows us to identify the number of entries into a state and to calculate the average length of episodes as total time in a state divided by the number of entries. For sampling variance estimation, we employ the block bootstrap. Two case studies that are based on published literature illustrate how our methods can provide new insights into disability dynamics. Results The first application uses a classic textbook example on prednisone treatment and liver functioning among liver cirrhosis patients. We replicate well-known results of no association between treatment and survival or recovery. Our analysis of the episodes of normal liver functioning delivers the new insight that the treatment reduced the likelihood of relapse and extended episodes of normal liver functioning. The second application assesses frailty and disability among elderly people. We replicate the prior finding that frail individuals have longer life expectancy in disability. As a novel finding, we document that frail individuals experience three times as many episodes of disability that were on average twice as long as the episodes of nonfrail individuals. Conclusions We provide a simple analytical approach for calculating the number and length of episodes in Markov chain models. The results allow a description of the transition dynamics that goes beyond the results that can be obtained using standard tools for Markov chains. Empirical applications using published data illustrate how the new method is helpful in unraveling the dynamics of the modeled process.
  • Jitta, Aditya; Klami, Arto (2017)
    Markov models are natural tools for modeling trajectories, following the principle that recent location history is predictive of near-future directions. In this work we study Markov models for describing and predicting human movement in indoor spaces, with the goal of modeling the movement on a coarse scale to protect the privacy of the individuals. Modern positioning devices, however, provide location information on a much more finer scale. To utilize this additional information we develop a novel family of partially hidden Markov models that couple each observed state with an auxiliary side information vector characterizing the movement within the coarse grid cell. We implement the model as a non-parametric Bayesian model and demonstrate it on real-world trajectory data collected in a hypermarket.
  • Silva, Jorge M.; Pinho, Eduardo; Matos, Sergio; Pratas, Diogo (2020)
    Sources that generate symbolic sequences with algorithmic nature may differ in statistical complexity because they create structures that follow algorithmic schemes, rather than generating symbols from a probabilistic function assuming independence. In the case of Turing machines, this means that machines with the same algorithmic complexity can create tapes with different statistical complexity. In this paper, we use a compression-based approach to measure global and local statistical complexity of specific Turing machine tapes with the same number of states and alphabet. Both measures are estimated using the best-order Markov model. For the global measure, we use the Normalized Compression (NC), while, for the local measures, we define and use normal and dynamic complexity profiles to quantify and localize lower and higher regions of statistical complexity. We assessed the validity of our methodology on synthetic and real genomic data showing that it is tolerant to increasing rates of editions and block permutations. Regarding the analysis of the tapes, we localize patterns of higher statistical complexity in two regions, for a different number of machine states. We show that these patterns are generated by a decrease of the tape's amplitude, given the setting of small rule cycles. Additionally, we performed a comparison with a measure that uses both algorithmic and statistical approaches (BDM) for analysis of the tapes. Naturally, BDM is efficient given the algorithmic nature of the tapes. However, for a higher number of states, BDM is progressively approximated by our methodology. Finally, we provide a simple algorithm to increase the statistical complexity of a Turing machine tape while retaining the same algorithmic complexity. We supply a publicly available implementation of the algorithm in C++ language under the GPLv3 license. All results can be reproduced in full with scripts provided at the repository.