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 |  موضوع: رسالة دكتوراة بعنوان Design and Analysis of Experiments for Model Discrimination in Uniresponse and Multiresponse Systems الخميس 02 ديسمبر 2021, 9:51 pm | |
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أخواني في الله
أحضرت لكم
رسالة دكتوراة بعنوان
Design and Analysis of Experiments for Model Discrimination in Uniresponse and Multiresponse Systems
A Thesis Submitted
In Partial Fulfilment of the Requirements
For the Degree of
Doctor of Philosophy
By
Santokh Singh
To the
Departmbnt of Mathemetigs
Indian Institute of Technology Kanpur
August, 1986
و المحتوى كما يلي :
table of Contents
' List of Tables
List of Figures
Some Notations and Symbols
Synopsis
Chapter Page
1 INTRODUCTION AND REVIEW OF THE PREVIOUS WORK 1
1.1 Systems and Models 1
1.2 The Modelling Process
1.3 Mathematization of a Process
1.4 Review of the Previous Work in Model
Discrimination
1.4.1 Review of the Previous Works
Uniresponse Case
1.4.2 Review of the Previous Works
Multiresponse Case
METHODOLOGY FOR MODEL DISCRIMINATION S
DESIGN AND ANALYSIS
An Approach to the Problem of Model
Discrimination
Search for an Appropriate Function
2.2.1 Metric Properties of the
Function, K
2.2.2 Utilizing the Distance
Function, K, in Model
Discrimination
Model Discriminations Nonsequential
Model Discriminations Sequential
2.4.1 Discrimination Criterion *
An Index
2.4.2 Design Criterion s A Weighted
Function
Development of the Discrimination
Criterion
2.5.1 Some Assumptions
2.5.2 Evaluating the Akinness Between
Two Normal Populations
Chapter Page
2.5.3 An Important Observation
2.5.4 Distance for Assessment of the
Appropriateness of a Model
2.6 Termination Criterion
2.7 Testing the Significance of the
Distance Between Two Models
2.8 A Model Adequacy Criterion
DESIGN OF EXPERIMENTS FOR MODEL DISCRIMINATION
IN UNIRESPONSE SYSTEMS
3.1 Basic Assumptions 66
3.2 Case 1 s Known, Homogeneous Variances
of Error s
3.3 Case 2 s Unknown, Homogeneous
Variances of Errors
3.4 Case 3 ! Unknown, Heterogeneous
Variances of Errors
4 DESIGN OF EXPERIMENTS FOR MODEL DISCRIMINATION
IN MULTIRESPONSE SYSTEMS
4.1 Basic Assumptions
4.2 Case 1 s Known, Equal Covariance
Matrices of Errors
4.3 Case 2 * Unknown, Equal Covariance
Matrices of Errors
4.4 Case 3 * Unknown, Unequal Covariance
Matrices of Errors
SOME THEORY OF ESTIMATION
5.1 Estimation of Model Parameters 125
5.2 Estimation of Variance(s)/Covariances
of Error(s)
5.3 Estimation of Eu and Under
Different Types of Assumptions
APPLICATION EXAMPLES
6.1 The Scheme for Implementation of the
Discrimination Procedure
Chapter Page
6.1.1 Simulation of Data 139
6.1.2 Estimation. of Model Parameters
6.1.3 Discriminating Among Model
and Design of Additional
Experiments
6.1.4 Optimization of Criterion
Functions
6.2 Discrimination Among Univariate Models
6.2.1 Linear Models * Known,Equal
Error Variances
6.2.2 Nonlinear Models i Unknown,
Homogeneous Error Variances
6.2.3 Nonlinear Models s Unknown,
Homogeneous Error Variances
6.3 Discrimination Among Multivariate Models
6.3.1 Nonlinear Models s Known, Equal
Error Covariance Matrices
6.3.2 Nonlinear Models s Unknown,
Unequal Covariance Matrices
of Errors 182
APPENDIX i
A COMBINING TWO QUADRATIC FORMS
SOME RESULTS OF MATRIX ALGEBRA
JOINT CUMULANT GENERATING FUNCTION
C.l Univariate Case
C.2 Multivariate Case
SOME USEFUL RESULTS AND FORMULAE
185 i
B 188
C 194
194
198
D 203
REFERENCES
207LIST OF TABLES
TABLE Page
Values of the coefficients,
aip *
Values of the coefficients,
aiks»
Estimates of model parameters at
different stages
Sequential discrimination among
polynomial models * the present (-3)
and the Box-Hill procedures(True model*M ) 148
Weights for the criterion function 0
at different stagesj discrimination
among polynomial models
Sequential discrimination among
nonlinear models : the present,
the Box-Hill, and the Buzzi et al.
procedures(True model*M(5))
Weights for the criterion function 0
at different stagesj discrimination
among nonlinear univariate models
Sequential discrimination among
nonlinear univariate models s
the present and the Box-Hill
procedures(True modelsM(2))
Sequential discrimination among
nonlinear bivariate models t
the present procedure(True model* M
Weights used in the criterion
function 0 at different stagesj
discrimination among nonlinear
bivariate models
Sequential discrimination among .
nonlinear bivariate models s
the procedure of
(True model
Discrimination index and posterior
probability in discriminating
among nonlinear bivariate models
Buzzi et al' .
182LIST OF FIGURES
FUGURE Page
6.1,1 Scheme for illustration of sequential
discrimination 140
6.2.1 Status of models as determined by
discrimination index; linear
univariate models. True model!
Model 3
Change of weights w.
to second stage; linear univariate
models
150
6,2.2 from first
u,v;k
153
6,2.3 Progress in discrimination through
the proposed procedure; nonlinear
univariate models. Time models Model 5 163
6.2.4 Change in weights wu vjk
to second stage; nonlinear univariate
models
from first
166
6.2.5 Performance of discrimination index
in identifying the true model;
Model 2 170
6.2.6 Change in the posterior probabilities
at different stages. True models Models
Performance of the proposed procedure
in discriminating among nonlinear
bivariate models. True model s
Model 1
6.3.2 Change in weights w from first
to second stage; nonlinear bivariate
models 178SOME NOTATIONS AND SYMBOLS
Notations
ABS absolute value
E expectation operator
exponential
the expectation tinder model u
exp
E (u)
sup supremum
tr(A) trace of the matrix A
r-variate t-distribution
criterion proposed in literature
the criterion proposed in this work
A 0
Symbols
(u) (as superscript); under model u
(on top) maximum likelihood, estimator. (m.l*.es)
(on top) some estimator
(on top ) some estimator also used for some constants
(on top ) some estimator
(at bottom) a vector
(at bottom ) a two dimension array
a statistical population
product
a process
determinant A
proportional
the summation
log likelihood
estimation criterion function
transpose of A
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