Local Linear Method Research Articles

Local linear estimation of the trimmed regression for censored data

We introduce a non-parametric estimation of the trimmed regression by using the local linear method of a censored scalar response variable, given a functional covariate. The main result of this work is the establishment of almost complete convergence for the constructed estimator. A simulation study is carried out to compare the finite sample performance based on the mean square error between the classic local linear regression estimator and the trimmed local linear regression estimator. Moreover, a real data study is used to illustrate our methodology.

Semiparametric spatio‐temporal models with unknown and banded autoregressive coefficient matrices

We consider a new class of semiparametric spatio‐temporal models with unknown and banded autoregressive coefficient matrices. The setting represents a type of sparse structure in order to include as many panels as possible. We apply the local linear method and least squares method for Yule–Walker equation to estimate trend function and spatio‐temporal autoregressive coefficient matrices respectively. We also balance the over‐determined and under‐determined phenomena in part by adjusting the order of extracting sample information. Both the asymptotic normality and convergence rates of the proposed estimators are established. We demonstrate, using both simulation and case studies, that the proposed estimators are stable among different sample sizes, and more efficient than the traditional method with known spatial weight matrices.

Rheological Analysis of Suspended Single-Walled Carbon Nanotubes in a Walters' B fluid

Objective: In this paper, we present a rheological analysis of suspended single-walled Carbon nanotubes in a Walters' B fluid. Methods: We assume that the viscosity varies exponentially as a function of the temperature use the Reynolds model of viscosity for the study. A variable thermal conductivity is also assumed along with buoyancy, magnetic field, viscous dissipation and a convective boundary condition. The system of nonlinear coupled equations is solved using the spectral local linearization method. Results: Our solutions are validated using through comparison with previously published results for a given set of conditions. Conclusion: The study finds that good agreement is achieved.

Spectral methods to solve nonlinear problems: A review

The study is intended to give a brief overview of recent papers on spectral methods in different types of fluid flow especially nanofluid flow. The survey does not presume to be exhaustive; hence some selection is necessary. The papers reviewed herein relate to the recent use of some spectral based algorithms, notably the spectral relaxation method, the spectral local linearization method, the spectral quasi-linearization method, the successive linearization method, the piecewise successive linearization method and the spectral homotopy analysis method, etc. We review the important papers and problems where these methods have been used. The paper is divided into sections related to the use of each method. In addition to reviewing the journal articles in the main body of the paper, we also give details of important books published on spectral methods in fluid dynamics.

Semiparametric Spatial Autoregressive Panel Data Model with Fixed Effects and Time-Varying Coefficients

This article considers a semiparametric spatial autoregressive (SAR) panel data model with fixed effects and time-varying coefficients. The time-varying coefficients are allowed to follow unknown functions of time, while the other parameters are assumed to be unknown constants. We propose a local linear quasi-maximum likelihood estimation method to obtain consistent estimators for the SAR coefficient, the variance of the error term, and the nonparametric time-varying coefficients. The asymptotic properties of the proposed estimators are also established. Monte Carlo simulations are conducted to evaluate the finite sample performance of our proposed method. We apply the proposed model to study labor compensation in Chinese cities. The results show significant spatial dependence among cities and the impacts of capital, investment, and the economy’s structure on labor compensation change over time.

Dynamics of radiative-reactive Walters-b fluid due to mixed convection conveying gyrotactic microorganisms, tiny particles experience haphazard motion, thermo-migration, and Lorentz force

The significance of the dynamics of mixture of polymethyl methacrylate and pyridine at 25 C containing 30.5 g of polymer per litre with density 0.98 g per litre has led to countable published facts with the intention to come up with insightful suggestions and recommendations for the expert in biofuel, bioengineering, biomedical, and mechanical engineering applications. However, nothing is known on the increasing bioconvection Rayleigh number, velocity ratio, Lorentz force, Prandtl number, bioconvection Peclet number, bioconvection Lewis number, and microorganisms concentration Biot number. The Spectral Local Linearization Method (SLLM) and MATLAB in-built package were adopted to solve the dimensionless form of the corresponding governing equation for the dynamics of radiative-reactive Walters-B fluid due to mixed convection when tiny particles experience haphazard motion, thermo-migration, and Lorentz force are significant and analyzed. It can be concluded that the Nusselt number proportional to the heat transfer rate can be declined by rising Lorentz force. The microorganism’s flux at the wall is a decreasing property of Prandtl number and bioconvection Peclet number. Increasement in the temperature distribution of the fluid flow is a property that could be enhanced through enhancement in the Lorentz force.

Overlapping grid spectral collocation approach for electrical MHD bioconvection Darcy–Forchheimer flow of a Carreau–Yasuda nanoliquid over a periodically accelerating surface

AbstractThe focus of this study is on computational grid‐manipulation to enhance the accuracy, convergence, and computational efficiency of the multidomain bivariate spectral local linearization method (MD‐BSLLM). The improved method is used in the scrutiny of Darcy–Forchheimer bioconvection flow of Carreau–Yasuda nanofluid induced by an oscillatory moving surface with cross‐diffusion, activation energy with binary chemical reactive species, combined electrical and magnetohydrodynamic field effects. The proposed method is utilized in solving the nondimensional form of the flow equations. Sensitivity and error analyses are provided to aid an understanding of the efficiency, stability, convergence rate, and accuracy of the iterative scheme. The impact of different parameter values on fluid properties and transport phenomena is discussed. Numerical simulation has indicated that the overlapping grid MD‐BSLLM is computational efficacious, and produces stable and sufficiently accurate results using a few collocation nodes in each respective subinterval and the entire computational domain. Other findings include the fact that fluid properties are enhanced with injection while flow characteristics are improved with suction. Using the Darcy–Forchheimer model in the flow analysis improves the temperature of the nanofluid. The imposition of electric field augments nanofluid velocity, the amplitude of skin friction coefficient, rates of mass, and motile microorganisms transfer while reducing the rate of heat transfer. The considered flow analysis can contribute towards engineering solicitations in paper production, polymer solution, metal extrusion, and glass blowing.

Second order expansions of estimators in nonparametric moment conditions models with weakly dependent data

This paper considers estimation of nonparametric moment conditions models with weakly dependent data. The estimator is based on a local linear version of the generalized empirical likelihood approach, and is an alternative to the popular local linear generalized method of moment estimator. The paper derives uniform convergence rates and pointwise asymptotic normality of the resulting local linear generalized empirical likelihood estimator. The paper also develops second order stochastic expansions (under a standard undersmoothing condition) that explain the better finite sample performance of the local linear generalized empirical likelihood estimator compared to that of the efficient local linear generalized method of moments estimator, and can be used to obtain (second order) bias corrected estimators. Monte Carlo simulations and an empirical application illustrate the competitive finite sample properties and the usefulness of the proposed estimators and second order bias corrections.

Simultaneous variable selection and structural identification for time‐varying coefficient models

Time‐varying coefficient models are important tools in time series analysis due to their flexibility to fit non‐stationary data. To improve the accuracy of these models, it is important to identify covariates with null, constant and time‐varying effects and to estimate their coefficients. This article proposes a combination of the local linear smoothing method and the adaptive group lasso penalty approach to achieve covariate identification and coefficient estimation. The penalty term consists of two parts. The first term penalizes the norm of the coefficient function, which is used to select relevant variables. The second term penalizes the norm of the derivative function, which assesses the constancy of the coefficient functions. The asymptotic properties of the proposed methodology are established. Performance of the proposed method is demonstrated using simulated data along with an application to the analysis of the air quality and health data in Hong Kong.

Numerical study of a nonlinear density relation and binary reaction on nanofluid flow over a heated vertical surface with sinusoidal wall temperature

AbstractThis study discusses the impact of the nonlinear relationship between nonlinear density‐temperature and density‐concentration on a nanofluid flow with Arrhenius activation energy and sinusoidal wall temperature variations. The highly nonlinear partial differential equations that model the nanofluid flow are solved using the bivariate spectral local linearization method. In literature, it is known that certain thermal systems perform better at high temperatures. This performance may be influenced by the nonlinear temperature‐concentration‐dependent density relation, which accounts for optimal thermal and solutal transport in such systems. This study explores the impact of some physical parameters on the fluid transport properties, and the results show, among other findings, that the nonlinear density‐temperature leads to increased fluid velocity. In contrast, the nonlinear density‐concentration reduces the fluid speed. This study provides new insights into the impact of nonlinear density‐temperature and density‐concentration on fluid properties. Additionally, there is a development of destructive chemical reactions in the nanoparticle volume fraction profiles due to the Arrhenius activation energy.

Asymptotic properties of a nonparametric conditional density estimator in the local linear estimation for functional data via a functional single-index model

This paper deals with the conditional density estimator of a real response variable given a functional random variable (i.e., takes values in an infinite-dimensional space). Specifically, we focus on the functional index model, this approach represents a good compromise between nonparametric and parametric models. Then we give under general conditions and when the variables are independent, the quadratic error and asymptotic normality of estimator by local linear method, based on the single-index structure. Finally, we complete these theoretical advances by some simulation studies showing both the practical result of the local linear method and the good behaviour for finite sample sizes of the estimator and of the Monte Carlo methods to create functional pseudo-confidence area.

Traffic Flow Prediction in Urban Area Using Inverse Approach of Chaos Theory

Traffic congestions problem could affect everyday life especially in urban area. In order to solve the issue, an excellent traffic flow prediction needs to be developed for a better traffic management. Hence, this study was conducted in order to predict traffic flow by using the data of total volume of vehicles per hour at two main roads located in urban areas namely Selangor and Kuala Lumpur, Malaysia by using application of chaos theory. Phase space reconstruction was used to determine the chaotic behaviour of the total volume of vehicles per hour data. The reconstruction of phase space involves a single variable of the total volume of vehicles per hour data to m-dimensional phase space. Meanwhile, the inverse approach as well as local linear approximation method was used to develop prediction model of the traffic flow time series data. This study found that (i) the time series data were chaotic behaviour based on the phase space plot and (ii) inverse approach can provide prediction on the traffic flow time series data besides give excellent prediction with the value of correlation coefficient more than 0.7500. Hence, inverse approach of chaos theory can develop to prediction model towards the traffic flow in urban area; thus may help the local authorities to provide good traffic management.

Stagnation point MHD slip-flow of viscoelastic nanomaterial over a stretched inclined cylindrical surface in a porous medium with dual stratification

This communication discusses the magnetohydrodynamic (MHD) Stagnation point flow of a viscoelastic nanofluid past an inclined cylinder stretching linearly. A modified Darcy's law with the effect of Arrhenius activation energy is considered. The cylinder wall is both thermally and concentration-wise stratified, the fluid is electrically conducting and under a uniform magnetic field. Reduction of the PDEs boundary value problem to ODEs and solutions of the resulting equations are obtained by employing similarity transformations as well as Spectral Local Linearization Method (SLLM). The effects of apposite parameters on the fluid velocity, temperature and concentration are discussed through graphs. Similarly, their effects on the skin-friction coefficient, Nusselt and Sherwood numbers are elucidated via tables. There is the emergence of the modified Darcy parameter and it reduced the nanofluid velocity and depleted the thickness of the boundary layer. Both thermal and solutal stratified parameters were seen to dilute the species concentration within the flow system. Furthermore, the activation energy enhanced the species concentration levels of the nanofluid.

Mixed geographically and temporally weighted regression for spatio-temporal deformation modelling

When the regression coefficient of independent variable has both global stationarity and spatio-temporal non-stationarity properties, the deformation model based on the geographically and temporally weighted regression (GTWR) will no longer be applicable. In order to resolve this problem, we propose an improved method to establish the spatio-temporal deformation model using mixed geographically and temporally weighted regression (MGTWR). In this method, both the global regression coefficient and the variable regression coefficient are selected for regression coefficient hypothesis test, and the local linear two-step estimation method is used to fit the MGTWR model. A dam deformation modelling example shows that the MGTWR model improves the average prediction accuracy by 57.6% compared to the GTWR model when the regression coefficients have both global stationarity and spatio-temporal non-stationarity properties.

Evaluation and Numerical Simulation of Music Education Informationization Based on the Local Linear Regression Method

The study of this theoretical problem enables sparse or dense functional data, including educational information evaluation data. The choice of different weights is subjected to principal component analysis. The evaluation of music education informatization level mainly evaluates the status quo of music education informatization development, provides a basis for formulating and adjusting music education informatization development policies, and provides support for educational decision-making, to promote the sustainable and balanced development of music education informatization. The evaluation of music education informatization has become the key promotion work of music education informatization at this stage. This paper studies the convergence rate of functional principal components based on the local linear method under general weighting conditions. First, we introduce the related research on the estimation of mean and covariance function under general weighting. Secondly, for principal functional components under general weighting, namely, eigenvalues and eigen functions, the text gives the corresponding estimated values and derives its strong uniform convergence rate. Finally, the convergence rate was verified by simulation research. The estimation methods and conclusions of this article enrich the research of functional linear regression models and will help analyze the complex and changeable problems encountered in the application of music education information.

Double smoothing local linear estimation in nonlinear time series

We generalize the double smoothing local linear regression method to nonparametric regression of time series. Under a strong mixing condition for the dependence of the time series, we show that after another round of smoothing based on the local linear regression estimates, the double smoothing local linear estimate will have reduced asymptotic bias, while keeping the variance at the same asymptotic order. The asymptotic bias reduces from the order of h 2 for the local linear estimates to h 4 for the double smoothing local linear estimates, where h is the bandwidth. Hence the double smoothing local linear method produces more optimal estimates in terms of mean squared error. Simulation studies and real time series data analysis confirm the advantages of the double smoothing method compared to the local linear method.

Computational aspects of the kNN local linear smoothing for some conditional models in high dimensional statistics

We consider the problem of nonparametric estimation of certain conditional models when the explanatory variable is functional. The models studied are the Conditional Expectation (CE), the Conditional Distribution Function (CDF) and the Conditional Probability Density (CPD). The estimators are constructed by combining the local linear method and the k-Nearest Neighbors (kNN) smoothing approach. For each of the three models, we define an optimal estimator associated with the best number of neighbors chosen using two bandwidth selection procedures which are the cross-validation criterion and the bootstrap smoothing rule. The asymptotic properties of these optimal kNN Local Linear Estimators ( NN- LLE) are established under some standard conditions. The performances of the finite samples of these estimators are then examined and compared through a Monte-carlo study. In addition, an application on the riboflavin content in the yogurt using the Near-infrared curves is carried out to demonstrate the usefulness of the models proposed in the point-wise prediction as well as for the predictive interval.

To trust or not to trust an explanation: using LEAF to evaluate local linear XAI methods.

The main objective of eXplainable Artificial Intelligence (XAI) is to provide effective explanations for black-box classifiers. The existing literature lists many desirable properties for explanations to be useful, but there is a scarce consensus on how to quantitatively evaluate explanations in practice. Moreover, explanations are typically used only to inspect black-box models, and the proactive use of explanations as a decision support is generally overlooked. Among the many approaches to XAI, a widely adopted paradigm is Local Linear Explanations—with LIME and SHAP emerging as state-of-the-art methods. We show that these methods are plagued by many defects including unstable explanations, divergence of actual implementations from the promised theoretical properties, and explanations for the wrong label. This highlights the need to have standard and unbiased evaluation procedures for Local Linear Explanations in the XAI field. In this paper we address the problem of identifying a clear and unambiguous set of metrics for the evaluation of Local Linear Explanations. This set includes both existing and novel metrics defined specifically for this class of explanations. All metrics have been included in an open Python framework, named LEAF. The purpose of LEAF is to provide a reference for end users to evaluate explanations in a standardised and unbiased way, and to guide researchers towards developing improved explainable techniques.

Recognition of interface and category of roadway roof strata based on drilling parameters

Measurement while drilling is an important part of the intelligent development of coal mines and the development trend of roof formation detection. The main purpose of this paper is to study the best identification method of rock interface and category based on drilling parameters. Firstly, the borehole drilling device was independently developed; Secondly, 4 concrete blocks were poured to simulate the different rock formations; Thirdly, displacement, rotational speed, torque, and sound pressure level data were collected during drilling the concrete blocks, and the original data was processed by the local linear kernel estimation method; Finally, K-means clustering and change points detection method, was used to identify the rock formation interface respectively, and BP neural network and support vector machine method was used to identify the rock category respectively. Experimental results show that the local linear kernel estimation method based on exponentially weighted loss function can better reflect the response characteristics of drilling parameters. The change point detection method based on the penetration rate is better for the rock interface prediction (the sum of the two interface errors is 10.25 mm). Besides, with the penetration rate and sound pressure level as input parameters, BP neural network and support vector machine can achieve the operational classification of rock formations. This research is of great significance for understanding the prediction of the interface and category of rock formation in coal mines, which can not only promote safe mining but also reflect efficient coal production.

Numerical study and error estimation in power-law nanofluid flow over vertical frustum of a cone

The aim of this article is to analyze the mixed convective Ostwald–de Waele power-law nanofluid flow over vertical frustum of a cone in a non-Darcy porous medium using an efficient numerical technique. The involved power-law nanofluid model utilizes water as the base fluid, and Ti-alloy (Ti6Al4V) and multi-wall carbon nanotubes (MWCNTs) as the nanoparticles. The solution of resultant non-similarity equations subjected to boundary conditions is described using the local non-similarity technique along with an efficient spectral local linearization method. The error estimation is provided to show the efficiency of above-mentioned solution procedure. A detailed explanation about the impact of nanoparticle volume fraction on the dimensionless velocity and temperature profiles along with heat transfer rate and skin friction coefficient is also provided for both the opposing and aiding flow cases. On comparison of the present results in particular cases with the relevant published data, it is assured that this method gives highly accurate outcomes for this kind of very complex fluid flow problems. The domination of dilatant nanofluid over pseudo-plastic nanofluid in both the aiding and opposing flow cases is noticed for velocity profiles, and the velocity is decreased with an increment in the nanoparticle volume fraction. Also, the variation in profiles with a streamwise coordinate $$\xi$$ shows non-similar nature of the problem. The use of Ti-alloy and MWCNTs in this work makes it very profitable in various important sectors like aerospace and medical sector.