Local Polynomial Research Articles

Spatial analysis of seasonal precipitation using various interpolation methods in the Euphrates basin, Turkey

It is vital to accurately map the spatial distribution of precipitation, which is widely used in many fields such as hydrology, climatology, meteorology, ecology, and agriculture. This study aimed to reveal the spatial distribution of seasonal, long-term average precipitation in the Euphrates Basin with various interpolation methods. For this reason, Simple Kriging, Ordinary Kriging, Universal Kriging, Ordinary CoKriging, Empirical Bayesian Kriging, Radial Basis Functions (Completely Regularized Spline, Thin Plate Spline, Multiquadratic, Inverse Multiquadratic, Spline with Tensor), Local Polynomial Interpolation, Global Polynomial Interpolation, and Inverse Distance Weighting methods have been applied in the Geographical Information Systems environment. Long-term seasonal precipitation averages between 1966 and 2017 are presented as input for predicting precipitation maps. The accuracy of the precipitation prediction maps was based on linear regression analysis, root mean square error (RMSE), mean absolute error (MAE), correlation coefficient (R), and determination coefficient (R2) values obtained from the cross-validation tests. The most suitable method was chosen for the interpolation method that gives the lowest RMSE, MAE, and the largest R and R2. As a result of the study, Ordinary CoKriging in spring and winter precipitation, Local Polynomial Interpolation in summer precipitation, and Ordinary Kriging in autumn precipitation were the most appropriate estimation methods.

Partial-wave approach to the Stark resonance problem of the water molecule

A partial-wave method is developed to deal with small molecules dominated by a central atom as an extension of earlier single-center methods. In particular, a model potential for the water molecule is expanded over a basis of spherical harmonics. A finite-element method is employed to generate local polynomial functions in subintervals to represent the radial part of the wave function. The angular parts of the wave function are represented by spherical harmonics. The problem of Stark resonances is treated with the exterior complex scaling method which incorporates a wave-function discontinuity at the scaling radius. The resultant non-Hermitian matrix eigenvalue problem yields resonance positions and widths (decay rates). We present these dc Stark shifts and exponential decay rates for the valence orbitals $1{b}_{1}, 3{a}_{1}$, and the bonding orbital $1{b}_{2}$. Furthermore, comparison is made with total molecular decay rates and dc shifts obtained recently within the Hartree-Fock and coupled-cluster approaches.

Semiparametric Time-Series Model Using Local Polynomial: An Application on the Effects of Financial Risk Factors on Crop Yield

This paper proposes a semiparametric local polynomial estimator for modelling agricultural time-series. We consider the modelling of the crop yield variable according to determined financial risk factors in Turkey. The derivation of a semiparametric local polynomial estimator is provided with its fundamental statistical properties to estimate the semiparametric time-series model. This paper attaches importance to precision agriculture (PA) and therefore a local polynomial technique is considered due to some advantages it has over alternative methods. The introduced estimator provides less estimation risk, involving both parametric and nonparametric components that allow the estimator to represent the data structure better. From that, it can be said that the proposed estimator and model is beneficial to agricultural researchers for financial decision-making processes.

Renormalization of determinant lines in quantum field theory

On a compact manifold $M$, we consider the affine space $A$ of non self-adjoint perturbations of some invertible elliptic operator acting on sections of some Hermitian bundle, by some differential operator of lower order. We construct and classify all complex analytic functions on the Fr\'echet space $A$ vanishing exactly over non invertible elements, having minimal order and which are obtained by local renormalizations, a concept coming from quantum field theory, called renormalized determinants. The additive group of local polynomial functionals of finite degrees acts freely and transitively on the space of renormalized determinants. We provide different representations of the renormalized determinants in terms of spectral zeta determinants, Gaussian Free Fields, infinite product and renormalized Feynman amplitudes in perturbation theory in position space \`a la Epstein-Glaser. Specializing to the case of Dirac operators coupled to vector potentials and reformulating our results in terms of determinant line bundles, we prove our renormalized determinants define some complex analytic trivializations of some holomorphic line bundle over $A$ relating our results to a conjectural picture from some unpublished notes by Quillen [52] from April 1989.

Feature Set for Classification of Man-Made Underwater Objects in Optical and SAS Data

This paper proposes a feature characterization method that can discriminate between man-made and natural objects on the seabed using a pair of SAS and optical images. Feature matching methods, like scale-invariant feature transform (SIFT), usually utilize gradient information to detect feature points, whereas contour-based features often use geometrical descriptors such as roughness, circularity, and solidity. However, due to the different characteristics of sonar and optical imagery, current descriptors cannot be directly applied over sonar-optical image pairs. The clarity of the water (muddy water, concentration of particles that scatter light) for optical images, and the stability of the platform, and reflections from seabed for SAS images are important parameters that distort the two images and make it hard to match between the two image sources. This paper proposes a feature characterization method that can identify man-made objects in underwater optic-SAS image pairs. Two new contour-based feature descriptors are introduced. The first is the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">entropy angle feature</i> , which aims to reflect the entropy of the angles that originate from points on the contour. The second feature is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">local curve fitting</i> , which characterizes the radial-polar space of the local curve polynomial fitting. The descriptors are designed to characterize man-made objects whose contour is expected to be smoother than natural objects such as rocks. Experimental results on a real, large dataset comprised of 1,519 optic-SAS image pairs, are shown to verify that the proposed method has a high level of classification accuracy and good discrimination between man-made and natural objects.

Computational analysis of the behavior of stochastic volatility models with financial applications

Volatility estimation is an important issue in certain aspects of the financial community, such as risk management and asset pricing. It is known that stock returns often exhibit volatility clustering and the tails of the distributions of these series are fatter than the normal distribution. As a response to the need of these issues, the high unconditional volatility of assets encourages the users to predict their price in an ever changing market environment. Our main focus in this paper is to study the behavior of returns and volatility dynamics of some general stochastic economic models. First, we apply the local polynomial kernel smoothing method based on nonparametric regression to estimate the mean and the variance of the returns. We then implement and develop an empirical likelihood procedure in terms of conditional variance on daily log returns for inference on the nonparametric stochastic volatility as well as to construct a confidence interval for the volatility function. It appears that the proposed algorithm is applicable to some popular financial models and represents a good fit for the behavior observed in the stock and cryptocurrency markets. Some numerical results in connection to real data on the S&P 500 index and highly volatile Bitcoin dataset are also illustrated.

Weighted local polynomial estimations of a non-parametric function with censoring indicators missing at random and their applications

Weighted local polynomial estimations of a non-parametric function with censoring indicators missing at random and their applications

Nonparametric statistical learning based on modal regression

In this article, we propose a novel nonparametric statistical learning tool based on modal regression, which can complement the standard mean and quantile regression and has broad applicability in various fields. We first propose a local polynomial modal regression which focuses on the most likely conditional value (conditional mode) of a dependent variable Y given covariates x, and has several superiorities over conditional mean or quantiles, such as resistance to outliers and some forms of measurement error and having shorter prediction intervals when data are skewed. We employ the idea of local polynomial regression to estimate the modal regression nonparametrically. To broaden the applicability of the new technique to multivariate data or functional/longitudinal data, we further develop a varying coefficient modal regression. A Monte Carlo simulation study and an analysis of health care expenditure data demonstrate that the new proposed nonparametric modal regression is a promising complementary nonparametric data analysis tool to conventional nonparametric mean or quantile regression.

MODELING OF LOCAL POLYNOMIAL KERNEL NONPARAMETRIC REGRESSION FOR COVID DAILY CASES IN SEMARANG CITY, INDONESIA

Coronavirus disease 2019 (COVID-19) is an infectious disease caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) which was recently discovered. Coronavirus disease is now a pandemic that occurs in many countries in the world, one of which is Indonesia. One of the cities in Indonesia that has found many COVID cases is Semarang city, located in Central Java. Data on cases of COVID patients in Semarang City which are measured daily do not form a certain distribution pattern. We can build a model with a flexible statistical approach without any assumptions that must be used, namely the nonparametric regression. The nonparametric regression in this research using Local Polynomial Kernel approach. Determination of the polynomial order and optimal bandwidth in Local Polynomial Kernel Regression modeling use the GCV (Generalized Cross Validation) method. The data used this research are data on the number of COVID patients daily cases in Semarang, Indonesia. Based on the results of the application of the COVID patient daily cases in Semarang City, the optimal bandwidth value is 0.86 and the polynomial order is 4 with the minimum GCV is 3179.568 so that the model estimation results the MSE is 2922.22 and the determination coefficient is 97%. The estimation results show the highest number of Corona in the Semarang City at the beginning of July 2020. After the corona case increased in July, while the corona case in August decreased.

A novel approach for temporal simulations with very high-order finite volume schemes on polyhedral unstructured grids

In the finite volume method (FVM), the temporal term of the equation must be integrated in space with the same accuracy as the other spatial schemes from the governing equation. The classic way to solve this is to consider that the computational value inside the cell presents its mean value. In this work, a new operator is proposed, which converts point values into mean ones enabling simpler high-order spatial schemes from the pointwise framework for unsteady problems. One advantage of these schemes is the low number of contributions to the global matrix when compared to the mean counterpart.In the 2D space, this mean operator uses the weighted least-squares method to compute local polynomials and the Gauss quadrature to integrate cell inertial momentums. The method was verified for fourth, sixth and eighth orders in both Cartesian and unstructured meshes when using the Crank-Nicolson time discretization. Dirichlet and Neumann boundary conditions were also successfully verified. Additionally, the numerical spatial error evolution with the solver runtime and the required memory are studied as efficiency metrics of the implemented high-order schemes. High-order schemes provide faster and more accurate results than the second-order counterpart. To reduce the number of required timesteps, high-order backward differentiation formula (BDF) schemes were verified and they lead to considerable time savings when using high-order spatial schemes.

Estimation of Finite Population Totals in High Dimensional Spaces

In this paper, the problem of Nonparametric Estimation of Finite Population Totals in high dimensional datasets is considered. A robust estimator of the Finite Population Total based on Feedforward Backpropagation Neural Network is derived with the aid of a Super-Population Model. This current study is motivated by the fact that Local Polynomials and Kernel methods have in preceding related studies, been shown to provide good estimators for Finite Population Totals but in low dimensions. Even in these situations however, bias at boundary points presents a big challenge when using these estimators in estimating Finite Population parameters. The challenge worsens as the dimension of regressors increase. This is because as the dimension of the Regressor Vectors grows, the Sparseness of the Regressors’ values in the design space becomes unfeasible, resulting in a decrease in the fastest achievable rates of convergence of the Regression Function Estimators towards the target curve, rendering Kernel Methods and Local Polynomials ineffective to address these challenges. This study considers the technique of Artificial Neural Networks which yields robust estimators in high dimensions and reduces the estimation bias with marginal increase in variance. This is due to its Multi-Layer Structure, which can approximate a wide range of functions to any required level of precision. The estimator’s properties are developed, and a comparison with existing estimators was conducted to evaluate the estimator’s performance using real data sets acquired from the United Nations Development Programme 2020. The estimation approach performs well in an example using data from a United Nations Development Programme 2020 on the study of Human Development Index against other factors. The theoretical and practical results imply that the Neural Network estimator is highly recommended for survey sampling estimation of the finite population total.

Functional estimation of anisotropic covariance and autocovariance operators on the sphere

We propose nonparametric estimators for the second-order central moments of possibly anisotropic spherical random fields, within a functional data analysis context. We consider a measurement framework where each random field among an identically distributed collection of spherical random fields is sampled at a few random directions, possibly subject to measurement error. The collection of random fields could be i.i.d. or serially dependent. Though similar setups have already been explored for random functions defined on the unit interval, the nonparametric estimators proposed in the literature often rely on local polynomials, which do not readily extend to the (product) spherical setting. We therefore formulate our estimation procedure as a variational problem involving a generalized Tikhonov regularization term. The latter favours smooth covariance/autocovariance functions, where the smoothness is specified by means of suitable Sobolev-like pseudo-differential operators. Using the machinery of reproducing kernel Hilbert spaces, we establish representer theorems that fully characterize the form of our estimators. We determine their uniform rates of convergence as the number of random fields diverges, both for the dense (increasing number of spatial samples) and sparse (bounded number of spatial samples) regimes. We moreover demonstrate the computational feasibility and practical merits of our estimation procedure in a simulation setting, assuming a fixed number of samples per random field. Our numerical estimation procedure leverages the sparsity and second-order Kronecker structure of our setup to reduce the computational and memory requirements by approximately three orders of magnitude compared to a naive implementation would require.

Lpdensity: Local Polynomial Density Estimation and Inference

Density estimation and inference methods are widely used in empirical work. When the underlying distribution has compact support, conventional kernel-based density estimators are no longer consistent near or at the boundary because of their well-known boundary bias. Alternative smoothing methods are available to handle boundary points in density estimation, but they all require additional tuning parameter choices or other typically ad hoc modifications depending on the evaluation point and/or approach considered. This article discusses the R and Stata package lpdensity implementing a novel local polynomial density estimator proposed and studied in Cattaneo, Jansson, and Ma (2020, 2022), which is boundary adaptive and involves only one tuning parameter. The methods implemented also cover local polynomial estimation of the cumulative distribution function and density derivatives. In addition to point estimation and graphical procedures, the package offers consistent variance estimators, mean squared error optimal bandwidth selection, robust bias-corrected inference, and confidence bands construction, among other features. A comparison with other density estimation packages available in R using a Monte Carlo experiment is provided.

Local polynomial bi-responses multi-predictors nonparametric regression for predicting the maturity of mango (Gadung Klonal 21): a theoretical discussion and simulation

Local polynomial bi-responses multi-predictors nonparametric regression for predicting the maturity of mango (Gadung Klonal 21): a theoretical discussion and simulation

Estimation of Critical Collapse Solutions to Black Holes with Nonlinear Statistical Models

The self-similar gravitational collapse solutions to the Einstein-axion-dilaton system have already been found out. Those solutions become invariants after combining the spacetime dilation with the transformations of internal SL(2, R). We apply nonlinear statistical models to estimate the functions that appear in the physics of Black Holes of the axion-dilaton system in four dimensions. These statistical models include polynomial regression, nonparametric kernel regression and local polynomial regression models. Through various numerical studies, we reached accurate numerical and closed-form continuously differentiable estimates for the functions appearing in the metric and equations of motion.

Learning the smoothness of noisy curves with application to online curve estimation

Combining information both within and across trajectories, we propose a simple estimator for the local regularity of the trajectories of a stochastic process. Independent trajectories are measured with errors at randomly sampled time points. The proposed approach is model-free and applies to a large class of stochastic processes. Non-asymptotic bounds for the concentration of the estimator are derived. Given the estimate of the local regularity, we build a nearly optimal local polynomial smoother from the curves from a new, possibly very large sample of noisy trajectories. We derive non-asymptotic pointwise risk bounds uniformly over the new set of curves. Our estimates perform well in simulations, in both cases of differentiable or non-differentiable trajectories. Real data sets illustrate the effectiveness of the new approaches.

Multiscale regression on unknown manifolds

&lt;abstract&gt;&lt;p&gt;We consider the regression problem of estimating functions on $ \mathbb{R}^D $ but supported on a $ d $-dimensional manifold $ \mathcal{M} ~~\subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $ \mathcal{M} $ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $ \mathbb{R}^D $. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.&lt;/p&gt;&lt;/abstract&gt;

Propagation Map Reconstruction via Interpolation Assisted Matrix Completion

Constructing a propagation map from a set of scattered measurements finds important applications in many areas, such as localization, spectrum monitoring and management. Classical interpolation-type methods have poor performance in regions with very sparse measurements. Recent advance in matrix completion has the potential to reconstruct a propagation map from sparse measurements, but the spatial resolution is limited. This paper proposes to integrate interpolation with matrix completion to exploit both the spatial correlation and the potential low rank structure of the propagation map. The proposed method first enriches matrix observations using interpolation, and develops the statistics of the interpolation error based on a local polynomial regression model. Then, two uncertainty-aware matrix completion algorithms are developed to exploit the interpolation error statistics. It is numerically demonstrated that the proposed method outperforms Kriging and other state-of-the-art schemes, and reduces the mean squared error (MSE) of propagation map reconstruction by 10%-50% for a medium to large number of measurements.

PEMODELAN HARGA EMAS DUNIA MENGGUNAKAN METODE NONPARAMETRIK POLINOMIAL LOKAL DILENGKAPI GUI R

Investing in gold is a flexible choice because it can be sold at any time and used as an emergency fund. Investors should have the knowledge to predict data from time to time to achieve investment goals. One of the statistical methods for time series data modeling is ARIMA. The ARIMA model is strict with the assumptions that the data must be stationary, the residuals must be normally distributed, independent, and with constant variance, so an alternative model is proposed, namely nonparametric regression model, which has no modeling assumptions requirement. In this study, the daily world gold price data will be modeled using a local polynomial nonparametric model as an alternative because the assumptions in the ARIMA are not fulfilled. The data is divided into 2 parts, namely in sample data from January 2, 2020 to November 30, 2020 to form a model and out sample data from December 1, 2020 to December 31, 2020 used for evauation of model performance based on MAPE values. The chosen best model is the local polynomial model with Gaussian kernel function of degree 5, bandwidth of 373, and local point of 1744 with an MSE value of 482.6420. The local polynomial model out sample data MAPE value is 0.61%, indicating that the model has excellent forecasting capability. In this study, Graphical User Interface (GUI) using R software with the help of shiny package is also built, making data analyzing easier and generating more interactive display output.

WENO3-NN: A maximum-order three-point data-driven weighted essentially non-oscillatory scheme

Neural networks have become more and more relevant for computational fluid dynamics. In recent works, neural network based weighted essentially non-oscillatory schemes have been developed. Challenges faced with such schemes are to ensure maximum-order convergence on narrow stencils and the ENO property. In this work, we use a neural network as a weighting function in the WENO scheme and address these shortcomings. Based on the input stencil, the neural network calculates a convex combination of local interpolation polynomials. We use a Galilean invariant embedding in the input layer and introduce an additional loss on the reconstruction weights, such that the WENO scheme inherently recognizes a smooth input function and achieves maximum-order convergence. The performance of the WENO3-NN scheme is demonstrated for one- and two-dimensional test cases, including strong shocks and shock-density wave interactions. The WENO3-NN scheme shows very good generalizability across all benchmark cases and different resolutions, and exhibits a performance similar to or better than the classical WENO5-JS scheme. By analyzing the approximate dispersion relation of the WENO3-NN scheme, we find that the neural network scheme learns a highly non-trivial dispersion-dissipation relation. Especially, data-driven schemes may introduce vanishing dissipation near the cutoff wavenumber which is counterintuitive to classical discretization-design principles.