Bayesian bandwidth estimation and semi-metric selection for a functional partial linear model with unknown error density

Abstract

This study examines the optimal selections of bandwidth and semi-metric for a functional partial linear model. Our proposed method begins by estimating the unknown error density using a kernel density estimator of residuals, where the regression function, consisting of parametric and nonparametric components, can be estimated by functional principal component and functional Nadayara-Watson estimators. The estimation accuracy of the regression function and error density crucially depends on the optimal estimations of bandwidth and semi-metric. A Bayesian method is utilized to simultaneously estimate the bandwidths in the regression function and kernel error density by minimizing the Kullback-Leibler divergence. For estimating the regression function and error density, a series of simulation studies demonstrate that the functional partial linear model gives improved estimation and forecast accuracies compared with the functional principal component regression and functional nonparametric regression. Using a spectroscopy dataset, the functional partial linear model yields better forecast accuracy than some commonly used functional regression models. As a by-product of the Bayesian method, a pointwise prediction interval can be obtained, and marginal likelihood can be used to select the optimal semi-metric.

K-Nearest Neighbor Method with Principal Component Analysis for Functional Nonparametric Regression

“…The objective of Tecator data is to permit for the exposure of the proportion of the specific chemical meaning because the examination by chemistry procedure would take more time and be more costly. This instance works out 8,20 when the response variable is scalar and covariates are function. Indeed, the correlation coefficients between 3-contents (Fat, Water, and Protein contents) are given by 𝜌 𝐹𝑎𝑡,𝑊𝑎𝑡𝑒𝑟 = −0.988, 𝜌 𝐹𝑎𝑡,𝑃𝑟𝑜𝑡𝑒𝑖𝑛 = −0.86 and 𝜌 𝑊𝑎𝑡𝑒𝑟,𝑃𝑟𝑜𝑡𝑒𝑖𝑛 = 0.82.…”

“…The soil data samples were obtained from a long-term field experiment at a subarctic fell in Abisko, northern Sweeden. The number of samples is 108, and the wave-length interval of 400-2500 nm (visible and near infrared spectrum) which was scanned at 2 nmintervals with an INR spectrophotometer; for more detail see 20 . Two component varibles, Soil Organic Matter (SOM) was weighted as loss on ignition at 550 0 𝐶, and Eergosterol Concentration (EC) was defined through HPLC.…”

A Literature Review of Semi-functional Partial Linear Regression Models

“… Time Series and Panel Data: The extensions that are related to the time series and forecasting are: the Semi-functional partial linear time series modeling for prediction [31], with autoregressive errors [32,33], with timevarying parameters for latent parameter regimes [34], regularized forecasting via smooth-rough partitioning of the regression coefficients [35].  Bayesian: The Bayesian estimation methods are present in some papers ,but we only mention these two papers in this part: the Bayesian bandwidth estimation and semi-metric selection for a functional partial linear model with unknown error density [36,37].  Spatial: The spatial variability is considered in many research articles such as The partial functional linear spatial regression autoregressive model with spatial dependence responses [38], with two-stage estimator based on quasi-maximum likelihood estimation (QMLE) method and local linear regression method [39], studying the asymptotic normality of the parametric component, and probability convergence with the rate of the nonparametric component [40], B-spline approximation for slope function and residualbased approach for pointwise confidence-intervals [41], the robust spatial autoregressive model with t-distribution error terms with an expectationmaximization algorithm [42].…”