The Kronecker product and stochastic automata networks

Langville, Amy N.

doi:10.1016/s0377-0427(03)00931-2

Cited by 26 publications

(30 citation statements)

References 11 publications

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“…The Kronecker product is also known as the tensor product or direct product, among others [15]. The Kronecker product is a useful device to generate realistic network models and analyse network properties, see [16].…”

Section: The Kronecker Productmentioning

confidence: 99%

“…The Kronecker product is a useful device to generate realistic network models and analyse network properties, see [16]. The Kronecker product is an important mechanism in the study of certain applied problems involving large scale Markov chains, see [15]. Further reading on the Kronecker product and applications of its properties can be found in [15], [16] and [19].…”

Section: The Kronecker Productmentioning

confidence: 99%

“…The graph K(6, 2), which is also a srg (15,6,1,3). The vertices are labelled as the 2-element subsets of the set {1, 2, 3, 4, 5, 6}.…”

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confidence: 99%

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Edge-minimal graphs of exponent 2

O'Mahony

Quinlan

2018

Linear Algebra and its Applications

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Section: The Kronecker Productmentioning

confidence: 99%

Section: The Kronecker Productmentioning

confidence: 99%

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Edge-minimal graphs of exponent 2

O'Mahony

Quinlan

2018

Linear Algebra and its Applications

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“…Convergence of (2) is assured under the condition that Ψ ⊗ I is a contraction, that is, ∥Ψ ⊗ I∥ ≤ 1. We exploit the following properties of the Kronecker product ( [26], [27]):…”

Section: Problem Formulationmentioning

confidence: 99%

Distributed Fault Detection Using Consensus of Markov Chains

Jovanović

Pollett

2014

Intelligent Systems, Control and Automation: Science and Engineering

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We propose a fault detection procedure appropriate for use in a variety of industrial engineering contexts, which employs consensus among a group of agents about the state of a system.Markov chains are used to model subsystem behaviour, and consensus is reached by way of an iterative method based on estimates of a mixture of the transition matrices of these chains. To deal with the case where system states cannot be observed directly, we extended the procedure to accommodate Hidden Markov Models. Index TermsFault detection, consensus algorithm, mixtures of Markov chains, the EM algorithm, Hidden Markov Model (HMM), multi-agent systems.

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“…A way of resolving these two problems has been found in the parametrization of the spatiotemporal covariance matrix through a Kronecker product (KP) (Langville and Stewart, 2004;Van Loan, 2000) of a spatial and a temporal covariance matrix, reducing its dimensionality considerably (de Munck et al, 1992Huizenga et al, 2002). The KP parametrization assumes that an arbitrary spatiotemporal correlation can be modeled as a product of a spatial and a temporal factor.…”

Section: Introductionmentioning

confidence: 99%

The spatiotemporal MEG covariance matrix modeled as a sum of Kronecker products

2005

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The single Kronecker product (KP) model for the spatiotemporal covariance of MEG residuals is extended to a sum of Kronecker products. This sum of KP is estimated such that it approximates the spatiotemporal sample covariance best in matrix norm. Contrary to the single KP, this extension allows for describing multiple, independent phenomena in the ongoing background activity. Whereas the single KP model can be interpreted by assuming that background activity is generated by randomly distributed dipoles with certain spatial and temporal characteristics, the sum model can be physiologically interpreted by assuming a composite of such processes. Taking enough terms into account, the spatiotemporal sample covariance matrix can be described exactly by this extended model.In the estimation of the sum of KP model, it appears that the sum of the first 2 KP describes between 67% and 93%. Moreover, these first two terms describe two physiological processes in the background activity: focal, frequency-specific alpha activity, and more widespread non-frequency-specific activity. Furthermore, temporal nonstationarities due to trial-to-trial variations are not clearly visible in the first two terms, and, hence, play only a minor role in the sample covariance matrix in terms of matrix power. Considering the dipole localization, the single KP model appears to describe around 80% of the noise and seems therefore adequate. The emphasis of further improvement of localization accuracy should be on improving the source model rather than the covariance model. D

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The Kronecker product and stochastic automata networks

Cited by 26 publications

References 11 publications

Edge-minimal graphs of exponent 2

Edge-minimal graphs of exponent 2

Distributed Fault Detection Using Consensus of Markov Chains

The spatiotemporal MEG covariance matrix modeled as a sum of Kronecker products

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