A Key Note on Performance of Smoothing Parameterizations in Kernel Density Estimation

The univariate kernel density estimator requires one smoothing parameter while the bivariate and other higher dimensional kernel density estimators demand more than one smoothing parameter depending on the form of smoothing parameterizations used. The smoothing parameters of the higher dimensional kernels are presented in a matrix called the smoothing matrix. The two forms of parameterizations frequently used in higher dimensional kernel estimators are diagonal or constrained parameterization and full or unconstrained parameterization. While the full parameterization has no restrictions, the diagonal has some form of restrictions. The study investigates the performance of smoothing parameterizations of bivariate kernel estimator using asymptotic mean integrated squared error as error criterion function. The results show that in retention of statistical properties of data and production of smaller values of asymptotic mean integrated squared error as tabulated, the full smoothing parameterization outperforms its diagonal counterpart.Keywords: Smoothing Matrix, Kernel Estimator, Integrated Variance, Integrated Squared Bias, Asymptotic Mean Integration Squared Error (AMISE).

A new family of kernels from the beta polynomial kernels with applications in density estimation

One of the fundamental data analytics tools in statistical estimation is the non-parametric kernel method that involves probability estimates production. The method uses the observations to obtain useful statistical information to aid the practicing statistician in decision making and further statistical investigations. The kernel techniques primarily examine essential characteristics in a data set, and this research aims to introduce new kernel functions that can easily detect inherent properties in any given observations. However, accurate application of kernel estimator as data analytics apparatus requires the kernel function and smoothing parameter that regulates the level of smoothness applied to the estimates. A plethora of kernel functions of different families and smoothing parameter selectors exist in the literature, but no one method is universally acceptable in all situations. Hence, more kernel functions with smoothing parameter selectors have been propounded customarily in density estimation. This article proposes a distinct kernel family from the beta polynomial kernel family using the exponential progression in its derivation. The newly proposed kernel family was evaluated with simulated and life data. The outcomes clearly indicated that this kernel family could compete favorably well with other kernel families in density estimation. A further comparison of numerical results of the new family and the existing beta family revealed that the new family outperformed the classical beta kernel family with simulation and real data examples with the aid of asymptotic mean integrated squared error (AMISE) as criterion function. The information obtained from the data analysis of this research could be used for decision making in an organization, especially when human and material resources are to be considered. In addition, Kernel functions are vital tools for data analysis and data visualization; hence the newly proposed functions are vital exploratory tools

Takagi-Sugeno Integrated Fuzzy System in Subsurface Identification

This study investigates the possibility of using the rule-based fuzzy (FZ) inference method to analyse petrophysical data (DT). Some well logs (WL) DT provided by Shell Producing Development Company (SPDC), Nigeria, were utilised for this study. The exploration WL DT were clustered using an unsupervised neural network. The rule-based lithology (LTG) procedures were established from the training DT sets, and the procedure strength is weighted. The Takagi-Sugeno inference arrangement and the centroid of extent defuzzification technique were employed for the FZ inference. It was observed that FZ inference systems provide fast and comprehensive details of the LTG and fluid content of the subsurface structure of the petrophysical DT that was interpreted

On Hybridizations of Fourth Order Kernel of the Beta Polynomial Family.

The usual second order nonparametric kernel estimators are of wide uses in data analysis and visualization but constrained with slow convergence rate. Higher order kernels provide a faster convergence rates and are known to be bias reducing kernels. In this paper, we propose a hybrid of the fourth order kernel which is a merger of two successive fourth order kernels and the statistical properties of these hybrid kernels were study. The results of our simulation reveals that the proposed higher order hybrid kernels outperformed their corresponding parent’s kernel functions using the asymptotic mean integrated squared error

A Comparative Study of Higher Order Kernel Estimation and Kernel Density Derivative Estimation of the Gaussian Kernel Estimator with Data Application

Higher-order kernel estimation and kernel density derivative estimation are techniques for reducing the asymptotic mean integrated squared error in nonparametric kernel density estimation. A reduction in the error criterion is an indication of better performance. The estimation of kernel function relies greatly on bandwidth and the identified reduction methods in the literature are bandwidths reliant for their implementation. This study examines the performance of higher order kernel estimation and kernel density derivatives estimation techniques with reference to the Gaussian kernel estimator owing to its wide applicability in real-life-settings. The explicit expressions for the bandwidth selectors of the two techniques in relation to the Gaussian kernel and the bandwidths were accurately obtained. Empirical results using two data sets obviously revealed that kernel density derivative estimation outperformed the higher order kernel estimation excellently well with the asymptotic mean integrated squared error as the criterion function

Comparison of kernel density function in detecting effects of daily emission of Sulphur (IV) oxide from an industrial plant

Air pollution is a major concern of environmentalists because of the importance of air to man and other living organisms. This paper is about the investigation on the effects of daily emission of Sulphur (IV) oxide from an industrial pollutant using a nonparametric estimator which is the kernel estimator. Nonparametric estimators are free from distributional assumptions owing to the fact that most real-life data are not from a particular family of distribution. The functionality of this estimator is contingent on the smoothing parameter also called the bandwidth that determines the degree of the smoothness applied when analyzing the data. The bandwidth is extrapolated by minimizing the asymptotic mean integrated squared error which is the objective function of the kernel estimator. In this investigation, we selected some kernel functions of the beta family with the Gaussian kernel and obtained their bandwidths or smoothing parameters with respect to their distribution. The result of the analysis showed that an increase in number of tons of Sulphur (IV) oxide was associated with higher concentration level of the gas which suggests a potential danger of the gas to humans, animals and plants in the environment