Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later

Moler, Cleve B.; Loan, Charles Van

doi:10.1137/s00361445024180

Cited by 1,979 publications

(1,513 citation statements)

References 85 publications

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“…This can be computed by methods described by Moler and Van Loan (2003); Kalbfleisch and Lawless (1985) and Jackson (2011) discuss algorithms for the computation and maximization of (3). The msm package in R (Jackson 2011) is one of several that will do this, and it also handles piecewise-constant models.…”

Section: Maximum Likelihood Estimationmentioning

confidence: 99%

Estimation and assessment of markov multistate models with intermittent observations on individuals

Lawless

Rad

2014

Lifetime Data Anal

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SummaryMultistate models provide important methods of analysis for many life history processes, and this is an area where John Klein made numerous contributions. When individuals in a study group are observed continuously so that all transitions between states, and their times, are known, estimation and model checking is fairly straightforward. However, individuals in many studies are observed intermittently, and only the states occupied at the observation times are known. We review methods of estimation and assessment for Markov models in this situation. Numerical studies that show the effects of inter-observation times are provided, and new methods for assessing fit are given. An illustration involving viral load dynamics for HIV-positive persons is presented.

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Section: Maximum Likelihood Estimationmentioning

confidence: 99%

Estimation and assessment of markov multistate models with intermittent observations on individuals

Lawless

Rad

2014

Lifetime Data Anal

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“…The computation of matrix exponentials is not an easy task (Moler and Loan, 2003). Fortunately, a software package designed for Markov chain models, expokit (Sidje, 1998), has been implemented in MatLab.…”

Section: Matrix Exponential Methodsmentioning

confidence: 99%

Stochastic modelling of prey depletion processes

Clerc

Davison

Bersier

2009

Journal of Theoretical Biology

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The modelling of prey-predator interactions is of major importance for the understanding of population dynamics. Classically, these interactions are modelled using ordinary differential equations, but this approach has the drawbacks of assuming continuous population variables and of being deterministic. We propose a general approach to stochastic modelling based on the concept of functional response for a prey depletion process with a constant number of predators. Our model could involve any kind of functional response, and permits a likelihood-based approach to statistical modelling and stable computation using matrix exponentials. To illustrate the method we use the Holling-Juliano functional response and compare the outcomes of our model with a deterministic counterpart considered by Schenk and Bacher [2002. Functional response of a generalist insect predator to one of its prey species in the field. Journal of Animal Ecology 71 (3), 524-531], who observed the depletion of Cassida rubiginosa due to its exclusive predator, Polistes dominulus. The predation was found to be Holling type III, reflecting the ability of the predator to regulate its prey. Our approach corroborates this result, but suggests that the prey depletion census should have been performed more often, and that predation features were significantly different between the two years for which data are available.

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“…Specifically, W i = v i y T i , where v j is the right eigenvector corresponding to the jth eigenvalue and y T j is the jth row of V −1 where V is the matrix whose columns are the right eigenvectors ofB (Moler and Van Loan, 2003). It then follows from the orthogonality of the eigenvectors that W i W j = 0 for i j.…”

Section: Early Time Approximationmentioning

confidence: 99%

The effect of clumped population structure on the variability of spreading dynamics

Black

House

Keeling

et al. 2014

Journal of Theoretical Biology

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Processes that spread through local contact, including outbreaks of infectious diseases, are inherently noisy, and are frequently observed to be far noisier than predicted by standard stochastic models that assume homogeneous mixing. One way to reproduce the observed levels of noise is to introduce significant individual-level heterogeneity with respect to infection processes, such that some individuals are expected to generate more secondary cases than others. Here we consider a population where individuals can be naturally aggregated into clumps (subpopulations) with stronger interaction within clumps than between them. This clumped structure induces significant increases in the noisiness of a spreading process, such as the transmission of infection, despite complete homogeneity at the individual level. Given the ubiquity of such clumped aggregations (such as homes, schools and workplaces for humans or farms for livestock) we suggest this as a plausible explanation for noisiness of many epidemic time series.

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Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later

Cited by 1,979 publications

References 85 publications

Estimation and assessment of markov multistate models with intermittent observations on individuals

Estimation and assessment of markov multistate models with intermittent observations on individuals

Stochastic modelling of prey depletion processes

The effect of clumped population structure on the variability of spreading dynamics

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