Stable methods of statistical models estimation

Scientific monograph to be published:
Radchenko S.G. Stable methods of statistical models estimation: Monograph. – Kiev: PP «Sansparel», 2005. – 504 p.

If you want to purchase this book apply to Radchenko Stanislav Grigoryevich

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Preface

Stability of the mathematical modeling is the main problem of the improperly stipulated systems and as a consequence, the incorrectly formulated tasks. Statistical peculiarities of the scheme of full and fractional factorial experiments are investigated. Conditions of obtaining the «best» statistical models for the fractional factorial experiments are given. Methods for obtaining the stable (robust) successive experiment designs on the basis of multifactor regular designs, quasi-D-optimal designs, designs on the basis of ЛПτ equally distributed sequences and algorithms RASTA1, RASTA2 of the heuristic construction of the quasi-orthogonal and quasi-D-optimal experiment designs are developed.

Formalized choice of the stable structures of the multifactor statistical models, unknown to the researcher is stated, using the developed algorithm RASTA3. The topological method of the stable coefficients estimation of the statistical models in the conditions of the initial multicollinear factors is stated for the first time. Algorithms RASTA4, RASTA5.1 for linear restriction, algorithm RASTA4K for the curvilinear restriction of the factor space and algorithm RASTA10 for the arbitrary convex regions of the factor space are given. Algorithm RASTA14, allowing to establish connections between the pre-image and image having their own coordinate system is designed. Algorithm RASTA13, where the fictitious factors are used, was given for the first time. The use of the complex functions and optimal coordinates of the factor space for stable estimation of the statistical models were presented as well. Applied results, obtained using the developed methods and algorithms as well as the application efficiency of the developed technology for the stable estimation of statistical models were analyzed.

The developed methods and algorithms allow to choose the stable successive multifactor experiment design, to form the stable structure of the regression equation, a priori unknown to the researcher, and to obtain the stable estimation of the coefficients of the statistical models, in conditions of initial multicollinearity of the factors.

Key words:

Regression analysis, statistical models, robust (stable) experiment designs, stable structures of statistical models, stable estimate of the model coefficients.

Contents

Introduction

Chapter 1. The problems of making the formalized decisions in the applied research of the complex systems

1.1. General properties of the complex systems.

1.2. Choice of the description method, adequate to the general defining properties of the described system, process and object.

1.3. Adequacy to the description of the reality by the categories: quotient and general.

1.4. Representation of the system-structural peculiarities of the subject area in mathematical models.

1.5. Software for obtaining formalized solutions.

Conclusions.

Chapter 2. Applied linear regression analysis

2.1. General methods of regression analysis.

2.2. Predetermined system of the regression analysis and its execution in the applied research.

2.3. Orthogonal representation of the main effects and interactions in the multifactor mathematical models.

2.4. Interpretation of the obtained formalized description of the subject area.

2.5. Requirements to the structure of the obtained mathematical models.

2.6. Methods for obtaining the structure of the formalized description of reality.

2.7. Stability (correctness) of the mathematical model structure and values of the coefficients estimation.

2.8. The system of the quality criteria for the multifactor regression equation.

2.9. Testing of the multifactor statistical models by the basic quality criteria.

Conclusions.

Chapter 3. Formation of the best initial conditions to obtain multifactor statistical models

3.1. Principles for obtaining linear polynomial mathematical models relative to parameters (in the material Euclidean space).

3.2. Stability of the mathematical reality modeling.

3.3. The problem of the improperly stipulated systems and as a result of it, the task formulated incorrectly.

3.4. Statistical properties of the schemes of the full and fractional factorial experiment.

3.5. Conditions to obtain «the best» statistical models for the fractional factorial experiment.

3.6. Formation of experiment design on the basis of frequency proportionality of the factor levels.

3.7. Synthesis of the serial multifactorial quasi-D-optimal experiment designs.

3.8. Design of experiment on the basis of ЛПτ equally distributed sequences.

Conclusions.

Chapter 4. Formalized choice of the stable structures of the multifactorial statistical models

4.1. Algorithmic basis of the formalized structure description.

4.2. Criteria for structural element choice of the statistical models and formation of the mathematical model structure.

4.3. Successive search of the mathematical model structure.

Conclusions.

Chapter 5. Topological method of the statistical model stable estimation in the conditions of the initial factor multicollinearity

5.1. Factor space consideration as metrical and topological spaces.

5.2. Topological mapping of the pre-image into the image and mapping conditions.

5.3. Possible modes of topological mapping of the pre-image design of experiment into the image of mathematical modeling.

Conclusions.

Chapter 6. Stable estimation of statistical models with linear restrictions of the factor space

6.1. Algorithms RASTA4 and RASTA5.1 as mapping of the pre-image domain into the image domain under linear restrictions of the factor space.

6.2. Computing experiment of the stable coefficient estimation of the regression models for non-standard regions of the factor space with linear restrictions.

6.3. Usage of the algorithm RASTA4 for the degenerated regions of the factor space with linear restrictions.

Conclusions.

Chapter 7. Stable estimation of the statistical models with curvilinear restrictions of the factor space

7.1. Algorithm RASTA4K as mapping of the pre-image domain into image domain having curvilinear restrictions of the factor space.

7.2. Computing experiment for the stable estimation of the regression model coefficients for non-standard regions of the two-factor space with the curvilinear restrictions.

7.3. Computing experiment for the stable estimation of the regression model coefficients for non-standard regions of three-factor space with the curvilinear restrictions.

7.4. Usage of the algorithm RASTA4K for linear and curvilinear image restrictions.

7.5. General peculiarities of the factor space conversion.

Conclusions.

Chapter 8. Stable estimation of the statistical models in the arbitrary convex regions of the factor space

8.1. Algorithm RASTA10 of the quasi-orthogonal experiment design in the arbitrary convex region of the factor space.

8.2. Realization of the quasi-orthogonal experiment design in the arbitrary convex two-dimensional region of the factor space.

8.3. Informational peculiarities of the multi-factor regression equations, obtained from the initial incorrect conditions.

Conclusions.

Chapter 9. Usage of the fictitious factors, arguments of the complex functions and optimal coordinates of the factor space for the stable estimation of the statistical models

9.1. Algorithm RASTA13 for the stable estimation of coefficient of the statistical models using fictitious factors.

9.2. Usage of arguments of the complex functions for the stable estimation of the statistical models.

9.3. Choice of the optimal coordinates of the factor space for quasi-orthogonal statistical models estimation.

Conclusions.

Chapter 10. Applied solution of problems for the stable estimation of the multifactor statistical models

10.1. Informational correction of the variable systematic errors of the measuring devices and measuring information systems.

10.2. Multifactor mathematical modeling and compromising optimization of the production process by the electroerosive hole piercing.

10.3. Multifactor mathematical modeling of the modular multielement constructions assembly.

10.4. Mathematical modeling of the selective transfer in the measuring devices.

Conclusions.

Conclusions

Appendix A. Catalogue of the multifactorial regular experiment designs.

Appendix B. Catalogue of ЛПτ equally distributed sequences.

Mathematical key terms and concepts.

References.

Index of names.

General index.

Дата публикации:

10 января 2004 года