scholarly journals Matrix Algebraic Properties of the Fisher Information Matrix of Stationary Processes

Entropy ◽  
2014 ◽  
Vol 16 (4) ◽  
pp. 2023-2055 ◽  
Author(s):  
André Klein
Keyword(s):  
Fisher Information ◽  
Fisher Information Matrix ◽  
Information Matrix ◽  
Stationary Processes ◽  
Algebraic Properties

scholarly journals Optimal design for population pk/pd models

2012 ◽  
Vol 51 (1) ◽  
pp. 115-130
Author(s):  
Sergei Leonov ◽  
Alexander Aliev
Keyword(s):  
Optimal Design ◽  
Fisher Information ◽  
Fisher Information Matrix ◽  
Information Matrix ◽  
Optimal Sampling ◽  
Design Algorithm ◽  
Sampling Schemes ◽  
Population Pk ◽  
Different Types

ABSTRACT We provide some details of the implementation of optimal design algorithm in the PkStaMp library which is intended for constructing optimal sampling schemes for pharmacokinetic (PK) and pharmacodynamic (PD) studies. We discuss different types of approximation of individual Fisher information matrix and describe a user-defined option of the library.

Singularities Affect Dynamics of Learning in Neuromanifolds

2006 ◽  
Vol 18 (5) ◽  
pp. 1007-1065 ◽  
Author(s):  
Shun-ichi Amari ◽  
Hyeyoung Park ◽  
Tomoko Ozeki
Keyword(s):  
Model Selection ◽  
Parameter Space ◽  
Fisher Information ◽  
Fisher Information Matrix ◽  
Information Matrix ◽  
Gaussian Mixture ◽  
Multilayer Perceptrons ◽  
Hierarchical Systems ◽  
Space Forms ◽  
Training Error

The parameter spaces of hierarchical systems such as multilayer perceptrons include singularities due to the symmetry and degeneration of hidden units. A parameter space forms a geometrical manifold, called the neuromanifold in the case of neural networks. Such a model is identified with a statistical model, and a Riemannian metric is given by the Fisher information matrix. However, the matrix degenerates at singularities. Such a singular structure is ubiquitous not only in multilayer perceptrons but also in the gaussian mixture probability densities, ARMA time-series model, and many other cases. The standard statistical paradigm of the Cramér-Rao theorem does not hold, and the singularity gives rise to strange behaviors in parameter estimation, hypothesis testing, Bayesian inference, model selection, and in particular, the dynamics of learning from examples. Prevailing theories so far have not paid much attention to the problem caused by singularity, relying only on ordinary statistical theories developed for regular (nonsingular) models. Only recently have researchers remarked on the effects of singularity, and theories are now being developed. This article gives an overview of the phenomena caused by the singularities of statistical manifolds related to multilayer perceptrons and gaussian mixtures. We demonstrate our recent results on these problems. Simple toy models are also used to show explicit solutions. We explain that the maximum likelihood estimator is no longer subject to the gaussian distribution even asymptotically, because the Fisher information matrix degenerates, that the model selection criteria such as AIC, BIC, and MDL fail to hold in these models, that a smooth Bayesian prior becomes singular in such models, and that the trajectories of dynamics of learning are strongly affected by the singularity, causing plateaus or slow manifolds in the parameter space. The natural gradient method is shown to perform well because it takes the singular geometrical structure into account. The generalization error and the training error are studied in some examples.

A multiobjective sensor placement optimization for SHM systems considering Fisher information matrix and mode shape interpolation

2018 ◽  
Vol 35 (2) ◽  
pp. 519-535 ◽  
Author(s):  
Guilherme Ferreira Gomes ◽  
Fabricio Alves de Almeida ◽  
Patricia da Silva Lopes Alexandrino ◽  
Sebastiao Simões da Cunha ◽  
Bruno Silva de Sousa ◽  
...  
Keyword(s):  
Fisher Information ◽  
Mode Shape ◽  
Fisher Information Matrix ◽  
Information Matrix ◽  
Sensor Placement ◽  
Placement Optimization ◽  
Sensor Placement Optimization ◽  
Shape Interpolation
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