"The design of optimal operators takes different forms depending on the random process constituting the scientific model and the operator class of interest. In all cases, operator class and random process must be united in a criterion (cost function) that characterizes the operational objective and, relative to the cost function, an optimal operator found. A common difficulty is uncertainty in the parameters of the scientific model. Then, in addition to optimization relative to the original cost function, optimization must take into account uncertainty relative to an uncertainty class of random processes. If there is a prior distribution (or posterior distribution if data are employed) governing likelihood in the uncertainty class, then one can choose an operator minimizing the expected cost over the uncertainty class. A critical point is that the prior distribution is not on the parameters of the operator model, but on the uncertainty relative to the parameters of the scientific model. The basic principle embodied in the book is to express the optimal operator under the joint probability space formed from the joint internal and external uncertainty in the same form as the optimal operator for a known model by replacing the mathematical structures forming the standard optimal operator with corresponding structures, called effective characteristics, that incorporate model uncertainty. For instance, in Wiener filtering the power spectra might be uncertain and be replaced by effective power spectra in the representation of the Wiener filter"-- Read more... Abstract: In the classical approach to optimal filtering, it is assumed that the stochastic model of the physical process is fully known. With uncertain models, the natural solution is to optimize over both the original objective and the model uncertainty, thereby arriving at optimal robust operators, the topic of this book. Read more... "The design of optimal operators takes different forms depending on the random process constituting the scientific model and the operator class of interest. In all cases, operator class and random process must be united in a criterion (cost function) that characterizes the operational objective and, relative to the cost function, an optimal operator found. A common difficulty is uncertainty in the parameters of the scientific model. Then, in addition to optimization relative to the original cost function, optimization must take into account uncertainty relative to an uncertainty class of random processes. If there is a prior distribution (or posterior distribution if data are employed) governing likelihood in the uncertainty class, then one can choose an operator minimizing the expected cost over the uncertainty class. A critical point is that the prior distribution is not on the parameters of the operator model, but on the uncertainty relative to the parameters of the scientific model. The basic principle embodied in the book is to express the optimal operator under the joint probability space formed from the joint internal and external uncertainty in the same form as the optimal operator for a known model by replacing the mathematical structures forming the standard optimal operator with corresponding structures, called effective characteristics, that incorporate model uncertainty. For instance, in Wiener filtering the power spectra might be uncertain and be replaced by effective power spectra in the representation of the Wiener filter"-- Provided by publisher
In the classical approach to optimal filtering, it is assumed that the stochastic model of the physical process is fully known. For instance, in Wiener filtering it is assumed that the power spectra are known with certainty. The implicit assumption is that the parameters of the model can be accurately estimated. When models are complex or parameter estimation is difficult (or expensive), this assumption is unwarranted. With uncertain models, the natural solution is to optimize over both the original objective and the model uncertainty, thereby arriving atoptimal robust operators, the topic of this book. The book also addresses the correlated problem of optimal experimental design: determining the experiment to perform in order to maximally reduce the uncertainty impacting the operational objective. Model uncertainty impacts a wide spectrum of disciplines: engineering, physics, biology, medicine, and economics. This book aims to provide the reader with a solid theoretical background to the state-of-the art in treating a problem that is only going to grow as our desire to control and make decisions regarding complex systems grows, and to do so by considering a broad set of topics: filtering, control, structural intervention, compression, classification, and clustering.
In the classical approach to optimal filtering, it is assumed that the stochastic model of the physical process is fully known. For instance, in Wiener filtering it is assumed that the power spectra are known with certainty. The implicit assumption is that the parameters of the model can be accurately estimated. When models are complex or parameter estimation is difficult (or expensive), this assumption is unwarranted. With uncertain models, the natural solution is to optimize over both the original objective and the model uncertainty, thereby arriving at optimal robust operators, the topic of this book. The book also addresses the correlated problem of optimal experimental design: determining the experiment to perform in order to maximally reduce the uncertainty impacting the operational objective. Model uncertainty impacts a wide spectrum of disciplines: engineering, physics, biology, medicine, and economics. This book aims to provide the reader with a solid theoretical background to the state-of-the art in treating a problem that is only going to grow as our desire to control and make decisions regarding complex systems grows, and to do so by considering a broad set of topics: filtering, control, structural intervention, compression, classification, and clustering. Copyright......Page 5 Contents......Page 8 Preface......Page 12 Acknowledgments......Page 18 1 Random Functions......Page 20 2 Canonical Expansions......Page 42 3 Optimal Filtering......Page 70 4 Optimal Robust Filtering......Page 112 5 Optimal Experimental Design......Page 168 6 Optimal Classification......Page 216 7 Optimal Clustering......Page 268 References......Page 290 Index......Page 304 About the Author......Page 309
