Computational Statistics-Anuj

Computational Statistics-Anuj - COMPUTATIONAL METHODS IN...

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COMPUTATIONAL METHODS IN STATISTICS Anuj Srivastava Department of Statistics Florida State University Tallahassee, FL 32306 August 25, 2008
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2 PREFACE Statistics plays an important role in scientific studies, in a wide variety of applications. From the core statistical elements (observe, analyze and infer) to complicated design of experiments, statistics is finding newer roles in new areas. We believe that the most important reason be- hind advancement of statistics, and its applications in important and challenging problems, is the growth in computing power. The phenomenal advances in architectures supporting faster processing, and the simultaneous decline in cost of desktop computing have enabled diverse practitioners to use sophisticated statistical procedures. Applications with massive datasets (of the order of terrabytes) or real-time applications (requiring inferences in fractions of millisec- onds) or applications with complicated probability models, can all employ reasonable scientific frameworks and still provide satisfactory results. In this course, we focus on the basic issues and standard practices in application of computational techniques in standard statistical analysis. The following outlines the basic philosophy behind this course: 1. The main focus is on ability to convert ideas into algorithms and practice their computer implementations. In this sense, this course complements the knowledge of statistical theory acquired in other mathematics/statistics courses. 2. Theoretical results, although important, are not emphasized here and instead the emphasis is on taking computational solutions and implementing them into algorithms and programs. 3. Through implementations and examples, we will learn, verify and appreciate various as- sumptions that are needed in deriving some theoretical results. One example is to establish the conditions under which Markov chains generate samples from given probability distri- butions. 4. This course is designed to be self-contained. Each chapter will start with the notations, definitions and the theory needed to set up computational goals. The rest of the chapter will deal with deriving algorithms and practice programs. 5. The main programming environment used here is matlab. We will start with the representation of real number numbers of a digital storage device of a computer, called the floating point representation. This leads to the study of errors introduced in the computations due to floating point arithmetic. Understanding of these errors will guide us in selecting proper algorithms for computations of sample statistics (mean, variance) given the availability of memory, processing and storage resources. The next topic is solving for (linear) regression between the observed vector and a set of predictor vectors. In Chapter 2, we will start by re-visiting some basic definitions and rules from linear algebra. Then, we will pose the problem of multiple linear regression and present
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