Problems in the Theory of Optimal Systems

Pugachev, V.S.

doi:10.1016/b978-0-08-010421-8.50022-9

Cited by 6 publications

(6 citation statements)

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“…Whereas in this paper we have considered using SDE prior knowledge to construct prior distributions governing uncertainty classes of feature-label distributions, it seems feasible to use SDE knowledge to construct prior distributions governing uncertainty classes of random-process characteristics in the case of optimal filtering. Of course, one must confront the increased abstraction presented by canonical representation of random processes [ 24 , 25 ]; nevertheless, so long as one remains in the framework of second-order canonical expansions, it should be doable.…”

Section: Discussionmentioning

confidence: 99%

Incorporating prior knowledge induced from stochastic differential equations in the classification of stochastic observations

Zollanvari

Dougherty

2016

J Bioinform Sys Biology

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In classification, prior knowledge is incorporated in a Bayesian framework by assuming that the feature-label distribution belongs to an uncertainty class of feature-label distributions governed by a prior distribution. A posterior distribution is then derived from the prior and the sample data. An optimal Bayesian classifier (OBC) minimizes the expected misclassification error relative to the posterior distribution. From an application perspective, prior construction is critical. The prior distribution is formed by mapping a set of mathematical relations among the features and labels, the prior knowledge, into a distribution governing the probability mass across the uncertainty class. In this paper, we consider prior knowledge in the form of stochastic differential equations (SDEs). We consider a vector SDE in integral form involving a drift vector and dispersion matrix. Having constructed the prior, we develop the optimal Bayesian classifier between two models and examine, via synthetic experiments, the effects of uncertainty in the drift vector and dispersion matrix. We apply the theory to a set of SDEs for the purpose of differentiating the evolutionary history between two species.Electronic supplementary materialThe online version of this article (doi:10.1186/s13637-016-0036-y) contains supplementary material, which is available to authorized users.

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Section: Discussionmentioning

confidence: 99%

Incorporating prior knowledge induced from stochastic differential equations in the classification of stochastic observations

Zollanvari

Dougherty

2016

J Bioinform Sys Biology

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“…In order to describe the randomness of the distortion to the yarn path and cross-section caused by yarn squeeze uncertainty, according to the representation method of canonical expansion of any random function [ 29 ], the projection of random path vector in x – y and x – z planes, cross-section size ( a , b ) or cross-section torsion angle

can be expressed in the form of a pure deterministic component and a pure random component.

where

represents the projection of the path vector in the x – y and x – z planes,

represents the cross-section size,

is the cross-section torsion angle,

is the specified deterministic function,

is the deterministic coordinate basis function,

is the set of orthogonal zero-mean random values and

is the running parameter.…”

Section: Characteristic Parameters and Properties Of Distortion Yarnmentioning

confidence: 99%

Nonlinear Mechanical Property of 3D Braided Composites with Multi-Types Micro-Distortion: A Quantitative Evaluation

et al. 2023

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A new alternative calculation procedure is developed to quantify the effect of yarn distortion characteristics on the mechanical properties of three-dimensional (3D) braided carbon/resin composites. Firstly, the multi-type yarn distortion characteristics factors including path, cross-section shape and cross-section torsion effects are described based on the stochastic theory. Then, the multiphase finite element method is employed to overcome the complex discretization in traditional numerical analysis, and the parametric studies including multi-type yarn distortion and different braided geometrical parameters on the resulting mechanical properties are performed. It is shown that the proposed procedure can simultaneously capture the yarn path and cross-section distortion characteristics caused by the mutual squeeze of component materials, which is difficult to characterize by experimental methods. In addition, it is found that even small distortions of yarn may significantly affect the mechanical properties for 3D braided composites, and the 3D braided composites with different braiding geometric parameters will show different sensitivity to the distortion characteristics factors of yarn. The procedure, which could be implemented into commercial finite element codes, is an efficient tool for the design and structural optimization analysis of a heterogeneous material with anisotropic properties or complex geometries.

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“…Integral canonical expansions are formed via a kernel a ( t, ξ ) by defining

Z (ξ) = \int_{T} X (t) \bar{a (t, ξ)} d t .

Three conditions are necessary and sufficient for a canonical expansion to result [34], [35]. In [19], these conditions are extended to IBR filtering by taking the expectations of the involved covariances with respect to the prior distribution and take the form

x (t, ξ) = \frac{1}{I (ξ)} \int_{T} a (v, ξ) E_{π} [R_{X_{θ}} (t, v)] d v,

\int_{T} \bar{a (t, ξ)} x (t, ξ^{'}) d t = δ (ξ - ξ^{'}),

\int_{Φ} x (t, ξ) \bar{a (t^{'}, ξ)} d ξ = δ (t - t^{'}),

where the intensity of the white noise is given by

I (ξ) = \int_{Φ} \int_{T} \int_{T} E_{π} [R_{X_{θ}} (t, t^{'})] \bar{a (t, ξ)} a (t^{'}, ξ^{'}) d t d t^{'} d ξ^{'} .

…”

Section: Intrinsically Bayesian Robust Filtersmentioning

confidence: 99%

Bayesian Regression With Network Prior: Optimal Bayesian Filtering Perspective

Qian

Dougherty

2016

IEEE Trans. Signal Process.

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The recently introduced intrinsically Bayesian robust filter (IBRF) provides fully optimal filtering relative to a prior distribution over an uncertainty class ofjoint random process models, whereas formerly the theory was limited to model-constrained Bayesian robust filters, for which optimization was limited to the filters that are optimal for models in the uncertainty class. This paper extends the IBRF theory to the situation where there are both a prior on the uncertainty class and sample data. The result is optimal Bayesian filtering (OBF), where optimality is relative to the posterior distribution derived from the prior and the data. The IBRF theories for effective characteristics and canonical expansions extend to the OBF setting. A salient focus of the present work is to demonstrate the advantages of Bayesian regression within the OBF setting over the classical Bayesian approach in the context otlinear Gaussian models.

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Problems in the Theory of Optimal Systems

Cited by 6 publications

References 0 publications

Incorporating prior knowledge induced from stochastic differential equations in the classification of stochastic observations

Incorporating prior knowledge induced from stochastic differential equations in the classification of stochastic observations

Nonlinear Mechanical Property of 3D Braided Composites with Multi-Types Micro-Distortion: A Quantitative Evaluation

Bayesian Regression With Network Prior: Optimal Bayesian Filtering Perspective

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