Probability density function

Abstract: The probability of a configuration is given in classical theory by the Boltzmann formula exp [— V/hT] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.

Abstract: This note discusses some aspects of the estimation of the density function of a univariate probability distribution. All estimates of the density function satisfying relatively mild conditions are shown to be biased. The asymptotic mean square error of a particular class of estimates is evaluated.

Abstract: This is a new edition of a book on random data analysis which has been on the market since 1966 and which was extensively revised in 1971. The book has been a bestseller since. It has been fully updated to cover new procedures developed in the last 15 years and extends the discussion to a broad range of applied fields, such as aerospace, automotive industries or biomedical research. The primary purpose of this book is to provide a practical reference and tool for working engineers and scientists investigating dynamic data or using statistical methods to solve engineering problems. It is comprehensive and self-contained and expands the coverage of the theory, including derivations of the key relationships in probability and random-process theory not usually found to such extent in a book of this kind. It could well be used as a teaching textbook for advanced courses on the analysis of random processes. The first four chapters present the background material on descriptions of data, properties of linear systems and statistical principles. They also include probability distribution formulas for one-, two- and higher-order changes of variables. Chapter five gives a comprehensive discussion of stationary random-process theory, including material on wave-number spectra, level crossings and peak values of normally distributed random data. Chapters six and seven develop mathematical relationships for the detailed analysis of single input/output and multiple input/output linear systems including algorithms. In chapters eight and nine important practical formulas to determine statistical errors in estimates of random data parameters and linear system properties from measured data are derived. Chapter ten deals with data aquisition and processing, including data qualification. Chapter eleven describes methods of data analysis such as data preparation, Fourier transforms, probability density functions, auto- and cross-correlation, spectral functions, joint record functions and multiple input/output functions. Chapter twelve shows how to handle nonstationary data analysis, classification of nonstationary data, probability structure of nonstationary data, calculation of nonstationary mean values or mean square values, correlation structures of nonstationary data and spectral structures of nonstationary data. The last chapter deals with the Hilbert transform including applications for both nondispersive and dispersive propagation problems. All chapters include many illustrations and references as well as examples and problem sets. This allows the reader to use the book for private study purposes. Altogether the book can be recommended for practical working engineers and scientists to support their daily work, as well as for university readers as a teaching textbook in advanced courses. M Krystek

Abstract: This chapter contains sections titled: The probability density, Fourier transforms and characteristic functions, Joint statistics and statistical independence, Correlation functions and spectra, The central limit theorem

Abstract: Nonparametric density gradient estimation using a generalized kernel approach is investigated. Conditions on the kernel functions are derived to guarantee asymptotic unbiasedness, consistency, and uniform consistency of the estimates. The results are generalized to obtain a simple mcan-shift estimate that can be extended in a k -nearest-neighbor approach. Applications of gradient estimation to pattern recognition are presented using clustering and intrinsic dimensionality problems, with the ultimate goal of providing further understanding of these problems in terms of density gradients.