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This book describes how model selection and statistical inference can be founded on the shortest code length for the observed data, called the stochastic complexity. This generalization of the algorithmic complexity not only offers an objective view of statistics, where no prejudiced assumptions of 'true' data generating distributions are needed, but it also in one stroke leads to calculable expressions in a range of situations of practical interest and links very closely with mainstream statistical theory. The search for the smallest stochastic complexity extends the classical maximum likelihood technique to a new global one, in which models can be compared regardless of their numbers of parameters. The result is a natural and far reaching extension of the traditional theory of estimation, where the Fisher information is replaced by the stochastic complexity and the Cramer-Rao inequality by an extension of the Shannon-Kullback inequality. Ideas are illustrated with applications from parametric and non-parametric regression, density and spectrum estimation, time series, hypothesis testing, contingency tables, and data compression.
No statistical model is "true" or "false," "right" or "wrong"; the models just have varying performance, which can be assessed. The main theme in this book is to teach modeling based on the principle that the objective is to extract the information from data that can be learned with suggested classes of probability models. The intuitive and fundamental concepts of complexity, learnable information, and noise are formalized, which provides a firm information theoretic foundation for statistical modeling. Although the prerequisites include only basic probability calculus and statistics, a moderate level of mathematical proficiency would be beneficial.
This book gives a reliable review on structure selection of stochastic dynamic systems using information criteria AIC, BIC, o and stochastic complexity. After theoretical investigations many simulations are estimators, which illustrate both the effectiveness and the limitations of these methods. The reader can gain his or her own experience on the"working" of many methods (associated with different parameter estimators) using the demonstration disk which can be run on most IBM-compatible personal computers. The book will be helpful to anybody interested in applying automated methods of model-structure selection inn control engineering, in time series analysis or in signal processing.
Often a statistical analysis involves use of a set of alternative models for the data. A "model-selection criterion" is a formula which provides a figure-of merit for the alternative models. Generally the alternative models will involve different numhers of parameters. Model-selection criteria take into account hoth the goodness-or-fit of a model and the numher of parameters used to achieve that fit. 1.1. SETS OF ALTERNATIVE MODELS Thus the focus in this paper is on data-analytic situations ill which there is consideration of a set of alternative models. Choice of a suhset of explanatory variahles in regression, the degree of a polynomial regression, the number of factors in factor analysis, or the numher of dusters in duster analysis are examples of such situations. 1.2. MODEL SELECTION VERSUS HYPOTHESIS TESTING In exploratory data analysis or in a preliminary phase of inference an approach hased on model-selection criteria can offer advantages over tests of hypotheses. The model-selection approach avoids the prohlem of specifying error rates for the tests. With model selection the focus can he on simultaneous competition between a hroad dass of competing models rather than on consideration of a sequence of simpler and simpler models.