Kernel Estimators in Industrial Applications

The specification, based on experimental data, of functions which characterize an object under investigation, constitutes one of the main tasks in modern science and technological problems. A typical example here is the estimation of density function of random variable distribution from any given sample. The classical procedures rely here on arbitrary assumption of the form of this function, and then in specification of its parameters. These are called parametric methods. A valuable advantage is their theoretical and calculational simplicity, as well as their being commonly known and present in subject literature. Nowadays – along with the dynamic development of computer systems – nonparametric methods, whose main feature constitutes a lack of arbitrary assumptions of the form of a density function, are used more and more often. In a probabilistic approach, kernel estimators are becoming the principal method in this subject. Although their concept is relatively simple and interpretation transparent, the applications are impossible without a high class of computer which, even until recently, significantly hindered theoretical, and especially practical research. In this chapter, first – in Section 2 – the basics of kernel estimators methodology are presented in a form suitable for researchers without thorough knowledge in the area of advanced statistical methods. So, the fundamental definitions of a kernel estimator are described, as are one-dimensional, and also radial and product kernels for the multidimensional case, suboptimal – in a mean-square sense – methods for calculation of functions and parameters occurring there, as well as procedures of smoothing parameter modification and linear transformation. Thanks to today’s availability and the possibilities of contemporary computer systems as well as the automation of metrological and data gathering processes, the universal character of kernel estimators allows for their broad application in various problems of modern science and technology, particularly those of an industrial nature. In Section 3 of this chapter, uses of kernel estimators are described for the following subjects: