Higher-Order Markov Chains

Ching, Wai‐Ki; Huang, Ximin; Ng, Michael K.; Siu, Tak Kuen

doi:10.1007/978-1-4614-6312-2_6

Cited by 27 publications

(22 citation statements)

References 47 publications

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“…Before starting the Monte Carlo procedures, we set X to the averaged value of actual signals for a given word. 6) From the simulated signals n X , ACVFs are calculated using Equation (1) and compared with those obtained in step As Equation 9shows, signals t X are considered to be consequences of the time-varying conditional probabilities for word occurrence given by Equation (9) in the framework of additive binary Markov chain theory. In this sense, the unobserved time-varying probabilities given by Equation 9 We further calculated the time-varying probabilities of n X being 1 for typical Type-I words selected from five academic books.…”

Section: ( )mentioning

confidence: 99%

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Origin of Dynamic Correlations of Words in Written Texts

Ogura¹,

Amano²,

Kondo³

2019

JDAIP

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In a previous study, we introduced dynamical aspects of written texts by regarding serial sentence number from the first to last sentence of a given text as discretized time. Using this definition of a textual timeline, we defined an autocorrelation function (ACF) for word occurrences and demonstrated its utility both for representing dynamic word correlations and for measuring word importance within the text. In this study, we seek a stochastic process governing occurrences of a given word having strong dynamic correlations. This is valuable because words exhibiting strong dynamic correlations play a central role in developing or organizing textual contexts. While seeking this stochastic process, we find that additive binary Markov chain theory is useful for describing strong dynamic word correlations, in the sense that it can reproduce characteristics of autocovariance functions (an unnormalized version of ACFs) observed in actual written texts. Using this theory, we propose a model for time-varying probability that describes the probability of word occurrence in each sentence in a text. The proposed model considers hierarchical document structures such as chapters, sections, subsections, paragraphs, and sentences. Because such a hierarchical structure is common to most documents, our model for occurrence probability of words has a wide range of universality for interpreting dynamic word correlations in actual written texts. The main contributions of this study are, therefore, finding usability of the additive binary Markov chain theory to analyze dynamic correlations in written texts and offering a new model of word occurrence probability in which common hierarchical structure of documents is taken into account.

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Section: ( )mentioning

confidence: 99%

“…As that figure shows, the two features described above are common among all cases, indicating that the two features are substantial for all Type-I words having Figure 7. Conditional probabilities of X n being 1 calculated using Equation (9). The procedures for obtaining these conditional probabilities are the same as those for Figure 6.…”

Section: ( )mentioning

confidence: 99%

Origin of Dynamic Correlations of Words in Written Texts

Ogura¹,

Amano²,

Kondo³

2019

JDAIP

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“…beyond the only two training dates allowed by the current software packages. In this sense, the integration of higherorder Markov chains (Ching et al 2013) into LUCC modelling tools could be a potential path to consider, because the successive adjustments of the Markovian matrices proposed here cannot constitute a robust methodology applicable to all cases and all types of thematic data.…”

Section: Discussionmentioning

confidence: 99%

Improving Business-as-Usual Scenarios in Land Change Modelling by Extending the Calibration Period and Integrating Demographic Data

Mejean

Paegelow

Saqalli

et al. 2019

Lecture Notes in Geoinformation and Cartography

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Land use and land cover change (LUCC) models are increasingly being used to anticipate the future of territories, particularly through the prospective scenario method. In the case of so-called trend or Business-as-Usual (BAU) scenarios, the aim is to observe the current dynamics and to extend them into the future. However, as they are implemented as baseline simulation in most current software packages, BAU scenarios are calibrated from a training period built from only two dates. We argue that this limits the quantitative estimation of future change intensity, and we illustrate it from a simple model of deforestation in Northern Ecuadorian Amazon using the Land Change Modeler (LCM) software package. This paper proposes a contribution to improve BAU scenarios calibration by mainly two enhancements: taking into account a longer calibration period for estimating change quantities and the integration of thematic data in change probabilities matrices. We thus demonstrate the need to exceed the linear construction of BAU scenarios as well as the need to integrate thematic and particularly socio-demographic data into the estimation of future quantities of change. The spatial aspects of our quantitative adjustments are discussed and tend to show that improvements in the quantitative aspects should not be dissociated from an improvement in the spatial allocation of changes, which may lead to a decrease in the predictive accuracy of the simulations.

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“…A typical practice is to independently train several Markov chains with the corresponding subsets of data. However, this approach usually requires a large training dataset to prevent over-fitting of the model [18]. The problem of over-fitting may become severe when a Markov chain is in a higher order, where the number of parameters increases exponentially with the order of Markov chain.…”

Section: Likelihood Distribution: Gaussian Process Markov Chainsmentioning

confidence: 99%

“…A common choice of available solutions is to assume Markov property [2][3][4][5][6], which asserts that the next state only depends on nearby states, and is conditionally independent of the previous states. For nowadays applications of Markov models, firstorder Markov property [18], which mentions that the probability distribution of future state is dependent only upon the present state, is the most frequently used, but is not likely satisfied in our case. Therefore, we relaxed the assumption to allow the future state depending on the past states.…”

Section: Markov Chains For Drawing Processesmentioning

confidence: 99%