Topic

# Mathematical model

About: Mathematical model is a(n) research topic. Over the lifetime, 11695 publication(s) have been published within this topic receiving 247233 citation(s). The topic is also known as: theoretical model & theoretical models.

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Abstract: Models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit. Multivariate models with general covariance structure are often difficult to apply to highly unbalanced data, whereas two-stage random-effects models can be used easily. In two-stage models, the probability distributions for the response vectors of different individuals belong to a single family, but some random-effects parameters vary across individuals, with a distribution specified at the second stage. A general family of models is discussed, which includes both growth models and repeated-measures models as special cases. A unified approach to fitting these models, based on a combination of empirical Bayes and maximum likelihood estimation of model parameters and using the EM algorithm, is discussed. Two examples are taken from a current epidemiological study of the health effects of air pollution.

7,960 citations

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TL;DR: A Bayesian calibration technique which improves on this traditional approach in two respects and attempts to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best‐fitting parameter values is presented.

Abstract: We consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models. Such models, implemented as computer codes, are often generic in the sense that by a suitable choice of some of the model's input parameters the code can be used to predict the behaviour of the system in a variety of specific applications. However, in any specific application the values of necessary parameters may be unknown. In this case, physical observations of the system in the specific context are used to learn about the unknown parameters. The process of fitting the model to the observed data by adjusting the parameters is known as calibration. Calibration is typically effected by ad hoc fitting, and after calibration the model is used, with the fitted input values, to predict the future behaviour of the system. We present a Bayesian calibration technique which improves on this traditional approach in two respects. First, the predictions allow for all sources of uncertainty, including the remaining uncertainty over the fitted parameters. Second, they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best-fitting parameter values. The method is illustrated by using data from a nuclear radiation release at Tomsk, and from a more complex simulated nuclear accident exercise.

3,190 citations

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29 Mar 1991

Abstract: Mathematical Models and Engineering Decisions. Generalized Solutions Based on the Principle of Virtual Work. Finite Element Discretizations in One Dimension. Extensions and Their Convergence Rates in One Dimension. Two-Dimensional Linear Elastostatic Problems. Element-Level Basis Functions in Two Dimensions. Computation of Stiffness Matrices and Load Vectors for Two Dimensional Elastostatic Problems. Potential Flow Problems. Assembly, Constraint Enforcement, and Solution. Extensions and Their Convergence Rates in Two Dimensions. Computation of Displacements, Stresses and Stress Resultants. Computation of the Coefficients of Asymptotic Expansions. Three-Dimensional Linear Elastostatic Problems. Models for Plates and Shells. Miscellaneous Topics. Estimation and Control of Errors of Discretization. Mathematical Models. Appendices. Index.

2,747 citations

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01 Dec 1984

TL;DR: A survey of the technology of modal testing, a new method for describing the vibration properties of a structure by constructing mathematical models based on test data rather than using conventional theoretical analysis.

Abstract: A survey of the technology of modal testing, a new method for describing the vibration properties of a structure by constructing mathematical models based on test data rather than using conventional theoretical analysis. Shows how to build a detailed mathematical model of a test structure and analyze and modify the structure to improve its dynamics. Covers techniques for measuring the mode, shapes, and frequencies of practical structures from turbine blades to suspension bridges.

2,505 citations