Finite Dimensional State-Space Models

2013 ◽  
pp. 17-44
Author(s):  
Brigitte d’Andréa-Novel ◽  
Michel De Lara
Keyword(s):  
State Space ◽  
State Space Models ◽  
Finite Dimensional ◽  
Dimensional State Space

scholarly journals Characteristic phenomena in combustion

1997 ◽  
Vol 1 (2) ◽  
pp. 147-159
Author(s):  
Dirk Meinköhn
Keyword(s):  
State Space ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
The State ◽  
State Variable ◽  
Finite Dimensional ◽  
Dimensional State Space ◽  
Singularity Order ◽  
The Relationship ◽  
Stable Stationary Solutions

For the case of a reaction–diffusion system, the stationary states may be represented by means of a state surface in a finite-dimensional state space. In the simplest example of a single semi-linear model equation given. in terms of a Fredholm operator, and under the assumption of a centre of symmetry, the state space is spanned by a single state variable and a number of independent control parameters, whereby the singularities in the set of stationary solutions are necessarily of the cuspoid type. Certain singularities among them represent critical states in that they form the boundaries of sheets of regular stable stationary solutions. Critical solutions provide ignition and extinction criteria, and thus are of particular physical interest. It is shown how a surface may be derived which is below the state surface at any location in state space. Its contours comprise singularities which correspond to similar singularities in the contours of the state surface, i.e., which are of the same singularity order. The relationship between corresponding singularities is in terms of lower bounds with respect to a certain distinguished control parameter associated with the name of Frank-Kamenetzkii.

Perturbation Theory for Weak Measurements in Quantum Mechanics, Systems with Finite-Dimensional State Space

2018 ◽  
Vol 20 (1) ◽  
pp. 299-335 ◽  
Author(s):  
Miguel Ballesteros ◽  
Nick Crawford ◽  
Martin Fraas ◽  
Jürg Fröhlich ◽  
Baptiste Schubnel
Keyword(s):  
Perturbation Theory ◽  
Quantum Mechanics ◽  
State Space ◽  
Weak Measurements ◽  
Finite Dimensional ◽  
Dimensional State Space

State Dynamics of the Epileptic Brain

2013 ◽  
Author(s):  
Samuel P. Burns ◽  
Sabato Santaniello ◽  
William S. Anderson ◽  
Sridevi V. Sarma
Keyword(s):  
State Space ◽  
State Space Model ◽  
Network Connectivity ◽  
Seizure Onset ◽  
Finite Dimensional ◽  
Dimensional State Space ◽  
Finite Set ◽  
Brain States ◽  
The Brain ◽  
Insight Into

Communication between specialized regions of the brain is a dynamic process allowing for different connections to accomplish different tasks. While the content of interregional communication is complex, the pattern of connectivity (i.e., which regions communicate) may lie in a lower dimensional state-space. In epilepsy, seizures elicit changes in connectivity, whose patterns shed insight into the nature of seizures and the seizure focus. We investigated connectivity in 3 patients by applying network-based analysis on multi-day subdural electrocorticographic recordings (ECoG). We found that (i) the network connectivity defines a finite set of brain states, (ii) seizures are characterized by a consistent progression of states, and (iii) the focus is isolated from surrounding regions at the seizure onset and becomes most connected in the network towards seizure termination. Our results suggest that a finite-dimensional state-space model may characterize the dynamics of the epileptic brain, and may ultimately be used to localize seizure foci.

scholarly journals Spatiotemporal blocking of the bouncy particle sampler for efficient inference in state-space models

2021 ◽  
Vol 31 (5) ◽  
Author(s):  
Jacob Vorstrup Goldman ◽  
Sumeetpal S. Singh
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Simulated Data ◽  
State Space Models ◽  
High Dimensional ◽  
Effective Sample Size ◽  
Stat Assoc ◽  
Dimensional State Space ◽  
State Inference ◽  
New Algorithms

AbstractWe propose a novel blocked version of the continuous-time bouncy particle sampler of Bouchard-Côté et al. (J Am Stat Assoc 113(522):855–867, 2018) which is applicable to any differentiable probability density. This alternative implementation is motivated by blocked Gibbs sampling for state-space models (Singh et al. in Biometrika 104(4):953–969, 2017) and leads to significant improvement in terms of effective sample size per second, and furthermore, allows for significant parallelization of the resulting algorithm. The new algorithms are particularly efficient for latent state inference in high-dimensional state-space models, where blocking in both space and time is necessary to avoid degeneracy of MCMC. The efficiency of our blocked bouncy particle sampler, in comparison with both the standard implementation of the bouncy particle sampler and the particle Gibbs algorithm of Andrieu et al. (J R Stat Soc Ser B Stat Methodol 72(3):269–342, 2010), is illustrated numerically for both simulated data and a challenging real-world financial dataset.

State‐space Models with Finite Dimensional Dependence

2001 ◽  
Vol 22 (6) ◽  
pp. 665-678 ◽  
Author(s):  
Christian Gourieroux ◽  
Joann Jasiak
Keyword(s):  
State Space ◽  
State Space Models ◽  
Finite Dimensional ◽  
Dimensional Dependence

scholarly journals Quantum Aitchison geometry

2021 ◽  
Vol 24 (01) ◽  
pp. 2150001
Author(s):  
Attila Andai ◽  
Attila Lovas
Keyword(s):  
Hilbert Space ◽  
State Space ◽  
Scalar Product ◽  
Likelihood Function ◽  
Real Hilbert Space ◽  
Space Structure ◽  
Finite Dimensional ◽  
Dimensional State Space ◽  
Probability Simplex ◽  
Log Ratio

Multiplying a likelihood function with a positive number makes no difference in Bayesian statistical inference, therefore after normalization the likelihood function in many cases can be considered as probability distribution. This idea led Aitchison to define a vector space structure on the probability simplex in 1986. Pawlowsky-Glahn and Egozcue gave a statistically relevant scalar product on this space in 2001, endowing the probability simplex with a Hilbert space structure. In this paper, we present the noncommutative counterpart of this geometry. We introduce a real Hilbert space structure on the quantum mechanical finite dimensional state space. We show that the scalar product in quantum setting respects the tensor product structure and can be expressed in terms of modular operators and Hamilton operators. Using the quantum analogue of the log-ratio transformation, it turns out that all the newly introduced operations emerge naturally in the language of Gibbs states. We show an orthonormal basis in the state space and study the introduced geometry on the space of qubits in details.

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