scholarly journals The Maximal Strichartz Family of Gaussian Distributions: Fisher Information, Index of Dispersion, and Stochastic Ordering

2016 ◽  
Vol 2016 ◽  
pp. 1-17
Author(s):  
Alessandro Selvitella
Keyword(s):  
Fisher Information ◽  
Fisher Information Matrix ◽  
Information Matrix ◽  
Stochastic Order ◽  
Stochastic Ordering ◽  
Gaussian Distributions ◽  
Information Index ◽  
The Family ◽  
Statistical Dispersion ◽  
Fisher Information Index

We define and study several properties of what we callMaximal Strichartz Family of Gaussian Distributions. This is a subfamily of the family of Gaussian Distributions that arises naturally in the context of theLinear Schrödinger Equationand Harmonic Analysis, as the set of maximizers of certain norms introduced by Strichartz. From a statistical perspective, this family carries with itself some extrastructure with respect to the general family of Gaussian Distributions. In this paper, we analyse this extrastructure in several ways. We first compute theFisher Information Matrixof the family, then introduce some measures ofstatistical dispersion, and, finally, introduce aPartial Stochastic Orderon the family. Moreover, we indicate how these tools can be used to distinguish between distributions which belong to the family and distributions which do not. We show also that all our results are in accordance with the dispersive PDE nature of the family.

Planning of refracted starlight observation based on observability analysis

2021 ◽  
pp. 095441002110301
Author(s):  
Chengzhen Wu ◽  
Xueying An ◽  
Dingjie Wang ◽  
Hongbo Zhang
Keyword(s):  
Fisher Information ◽  
Information Matrix ◽  
Quantitative Relationship ◽  
Navigation Problem ◽  
Information Index ◽  
Observability Analysis ◽  
Observation Sequence ◽  
Fisher Information Index ◽  
Selection Of ◽  
Observation Scheme

In traditional observation schemes of stellar refraction navigation, the accuracy was limited due to unreasonable observation directions. In order to ameliorate this situation, a method of refracted starlight observation based on observability analysis is proposed. The function of this method is optimally generating an observation attitude sequence according to standard trajectories of spacecraft so that the selection of a refracted starlight observation sequence can be realized. Specifically, the improvement of Fisher information matrix calculation enables this method to be qualified for the navigation problem with unsteady measurement quantities as well as the non-fully observability which is defined as the capability of estimating the system state through measurements in finite time. Here, we construct a quantitative relationship between refracted starlight measurements and system observability by means of Fisher information index ( FII). Next, the observation scheme is retrieved by searching the maximum value of the optimized variable, which includes the ( FII). Finally, we resort to the extended Kalman filter to accomplish typical trajectory navigation simulations of the observation scheme. The results indicate that our method brings more accuracy than traditional ones in estimation of position and velocity of the optimal observation scheme.

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.

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