Variable bandwidth kernel regression estimation

In this paper we propose a variable bandwidth kernel regression estimator for $i.i.d.$ observations in $\mathbb{R}^2$ to improve the classical Nadaraya-Watson estimator. The bias is improved to the order of $O(h_n^4)$ under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kernel method over the classical kernel method.

Random Noise Suppression of Magnetic Resonance Sounding Data with Intensive Sampling Sparse Reconstruction and Kernel Regression Estimation

The magnetic resonance sounding (MRS) method is a non-invasive, efficient and advanced geophysical method for groundwater detection. However, the MRS signal received by the coil sensor is extremely susceptible to electromagnetic noise interference. In MRS data processing, random noise suppression of noisy MRS data is an important research aspect. We propose an approach for intensive sampling sparse reconstruction (ISSR) and kernel regression estimation (KRE) to suppress random noise. The approach is based on variable frequency sampling, numerical integration and statistical signal processing combined with kernel regression estimation. In order to realize the approach, we proposed three specific sparse reconstructions, namely rectangular sparse reconstruction, trapezoidal sparse reconstruction and Simpson sparse reconstruction. To solve the distortion of peaks and valleys after sparse reconstruction, we introduced the KRE to deal with the processed data by the ISSR. Further, the simulation and field experiments demonstrate that the ISSR-KRE approach is a feasible and effective way to suppress random noise. Besides, we find that rectangular sparse reconstruction and trapezoidal sparse reconstruction are superior to Simpson sparse reconstruction in terms of noise suppression effect, and sampling frequency is positively correlated with signal-to-noise improvement ratio (SNIR). In one case of field experiment, the standard deviation of noisy MRS data was reduced from 1200.80 nV to 570.01 nV by the ISSR-KRE approach. The proposed approach provides theoretical support for random noise suppression and contributes to the development of MRS instrument with low power consumption and high efficiency. In the future, we will integrate the approach into MRS instrument and attempt to utilize them to eliminate harmonic noise from power line.

Kernel Regression Estimation for Functional Data

This article provides an overview of recent nonparametric and semiparametric advances in kernel regression estimation for functional data. In particular, it considers the various statistical techniques based on kernel smoothing ideas that have recently been developed for functional regression estimation problems. The article first examines nonparametric functional regression modelling before discussing three popular functional regression estimates constructed by means of kernel ideas, namely: the Nadaraya-Watson convolution kernel estimate, the kNN functional estimate, and the local linear functional estimate. Uniform asymptotic results are then presented. The article proceeds by reviewing kernel methods in semiparametric functional regression such as single functional index regression and partial linear functional regression. It also looks at the use of kernels for additive functional regression and concludes by assessing the impact of kernel methods on practical real-data analysis involving functional (curves) datasets.