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# Random Integral Equations with Applications to Life Sciences and Engineering

- 1st Edition, Volume 108 - January 1, 1974
- Editors: Chris P. Tsokos, W.J. Padgett
- Language: English
- Hardback ISBN:9 7 8 - 0 - 1 2 - 7 0 2 1 5 0 - 8
- Paperback ISBN:9 7 8 - 0 - 4 4 4 - 5 6 1 7 0 - 1
- eBook ISBN:9 7 8 - 0 - 0 8 - 0 9 5 6 1 7 - 6

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such… Read more

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Request a sales quoteIn this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; andmethods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.As a result, the book represents a blend of new methods in general computational analysis,and specific, but also generic, techniques for study of systems theory ant its particularbranches, such as optimal filtering and information compression.

- Best operator approximation,

- Non-Lagrange interpolation,

- Generic Karhunen-Loeve transform

- Generalised low-rank matrix approximation

- Optimal data compression

- Optimal nonlinear filtering

- Non-Lagrange interpolation,

- Generic Karhunen-Loeve transform

- Generalised low-rank matrix approximation

- Optimal data compression

- Optimal nonlinear filtering

This book is intended for:

Applied mathematicians and Electrical engineers

And:

Statisticians

Applied mathematicians and Electrical engineers

And:

Statisticians

Preface

Contents

1 Overview

I Methods of Operator Approximation in System Modelling

2 Nonlinear Operator Approximation with Preassigned Accuracy

2.1 Introduction

2.2 Generic formulation of the problem

2.3 Operator approximation in space C([0; 1]):

2.4 Operator approximation in Banach spaces by polynomial operators

2.5 Approximation on compact sets in topological vector spaces

2.6 Approximation on noncompact sets in Hilbert spaces

2.7 Special results for maps into Banach spaces

2.8 Concluding remarks

3 Interpolation of Nonlinear Operators 65

3.1 Introduction

3.2 Lagrange interpolation in Banach spaces

3.3 Weak interpolation of nonlinear operators

3.4 Some related results

3.5 Concluding remarks

4 Realistic Operators and their Approximation

4.1 Introduction

4.2 Formalization of concepts related to description of real-world objects

4.3 Approximation of R¡continuous operators

4.4 Concluding remarks

5 Methods of Best Approximation for Nonlinear Operators

5.1 Introduction

5.2 Best Approximation of nonlinear operators in Banach spaces: Deterministic case

5.3 Estimation of mean and covariance matrix for random vectors

5.4 Best Hadamard-quadratic approximation

5.5 Best polynomial approximation

5.6 Best causal approximation

5.7 Best hybrid approximations

5.8 Concluding remarks

II Optimal Estimation of Random Vectors

6 Computational Methods for Optimal Filtering of Stochastic Signals

6.1 Introduction

6.2 Optimal linear Filtering in Finite dimensional vector spaces

6.3 Optimal linear Filtering in Hilbert spaces

6.4 Optimal causal linear Filtering with piecewise constant memory

6.5 Optimal causal polynomial Filtering with arbitrarily variable memory

6.6 Optimal nonlinear Filtering with no memory constraint

6.7 Concluding remarks

7 Computational Methods for Optimal Compression and

Reconstruction of Random Data

7.1 Introduction

7.2 Standard Principal Component Analysis and Karhunen-Loeeve transform (PCA{KLT)

7.3 Rank-constrained matrix approximations

7.4 Generic PCA{KLT

7.5 Optimal hybrid transform based on Hadamard-quadratic approximation

7.6 Optimal transform formed by a combination of nonlinear operators

7.7 Optimal generalized hybrid transform

7.8 Concluding remarks

Bibliography

Index

Contents

1 Overview

I Methods of Operator Approximation in System Modelling

2 Nonlinear Operator Approximation with Preassigned Accuracy

2.1 Introduction

2.2 Generic formulation of the problem

2.3 Operator approximation in space C([0; 1]):

2.4 Operator approximation in Banach spaces by polynomial operators

2.5 Approximation on compact sets in topological vector spaces

2.6 Approximation on noncompact sets in Hilbert spaces

2.7 Special results for maps into Banach spaces

2.8 Concluding remarks

3 Interpolation of Nonlinear Operators 65

3.1 Introduction

3.2 Lagrange interpolation in Banach spaces

3.3 Weak interpolation of nonlinear operators

3.4 Some related results

3.5 Concluding remarks

4 Realistic Operators and their Approximation

4.1 Introduction

4.2 Formalization of concepts related to description of real-world objects

4.3 Approximation of R¡continuous operators

4.4 Concluding remarks

5 Methods of Best Approximation for Nonlinear Operators

5.1 Introduction

5.2 Best Approximation of nonlinear operators in Banach spaces: Deterministic case

5.3 Estimation of mean and covariance matrix for random vectors

5.4 Best Hadamard-quadratic approximation

5.5 Best polynomial approximation

5.6 Best causal approximation

5.7 Best hybrid approximations

5.8 Concluding remarks

II Optimal Estimation of Random Vectors

6 Computational Methods for Optimal Filtering of Stochastic Signals

6.1 Introduction

6.2 Optimal linear Filtering in Finite dimensional vector spaces

6.3 Optimal linear Filtering in Hilbert spaces

6.4 Optimal causal linear Filtering with piecewise constant memory

6.5 Optimal causal polynomial Filtering with arbitrarily variable memory

6.6 Optimal nonlinear Filtering with no memory constraint

6.7 Concluding remarks

7 Computational Methods for Optimal Compression and

Reconstruction of Random Data

7.1 Introduction

7.2 Standard Principal Component Analysis and Karhunen-Loeeve transform (PCA{KLT)

7.3 Rank-constrained matrix approximations

7.4 Generic PCA{KLT

7.5 Optimal hybrid transform based on Hadamard-quadratic approximation

7.6 Optimal transform formed by a combination of nonlinear operators

7.7 Optimal generalized hybrid transform

7.8 Concluding remarks

Bibliography

Index

- No. of pages: 322
- Language: English
- Edition: 1
- Volume: 108
- Published: January 1, 1974
- Imprint: Elsevier Science
- Hardback ISBN: 9780127021508
- Paperback ISBN: 9780444561701
- eBook ISBN: 9780080956176

CT

### Chris P. Tsokos

Chris P. Tsokos is Distinguished University Professor of Mathematics and Statistics at the University of South Florida. Dr. Tsokos’ research has extended into a variety of areas, including stochastic systems, statistical models, reliability analysis, ecological systems, operations research, time series, Bayesian analysis, and mathematical and statistical modelling of global warming, both parametric and nonparametric survival analysis, among others. He is the author of more than 400 research publications in these areas, including Random Integral Equations with Applications to Life Sciences and Engineering, Probability Distribution: An Introduction to Probability Theory with Applications, Mainstreams of Finite Mathematics with Applications, Probability with the Essential Analysis, Applied Probability Bayesian Statistical Methods with Applications to Reliability, and Mathematical Statistics with Applications, among others. Dr. Tsokos is the recipient of many distinguished awards and honors, including Fellow of the American Statistical Association, USF Distinguished Scholar Award, Sigma Xi Outstanding Research Award, USF Outstanding Undergraduate Teaching Award, USF Professional Excellence Award, Pi Mu Epsilon, election to the International Statistical Institute, Sigma Pi Sigma, USF Teaching Incentive Program, and several humanitarian and philanthropic recognitions and awards. He is also serves as Honorary Editor, Chief-Editor, Editor or Associate Editor for more than twelve academic research journals.

Affiliations and expertise

Distinguished University Professor of Mathematics and Statistics at the University of South FloridaWP

### W.J. Padgett

Affiliations and expertise

DEPARTMENT OF MATHEMATICS AND
COMPUTER SCIENCE
UNIVERSITY OF SOUTH CAROLINA
COLUMBIA, SOUTH CAROLINARead

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