Local Linear Method Research Articles

A Robust Algorithm for Classification and Diagnosis of Brain Disease Using Local Linear Approximation and Generalized Autoregressive Conditional Heteroscedasticity Model

Regions detection has an influence on the better treatment of brain tumors. Existing algorithms in the early detection of tumors are difficult to diagnose reliably. In this paper, we introduced a new robust algorithm using three methods for the classification of brain disease. The first method is Wavelet-Generalized Autoregressive Conditional Heteroscedasticity-K-Nearest Neighbor (W-GARCH-KNN). The Two-Dimensional Discrete Wavelet (2D-DWT) is utilized as the input images. The sub-banded wavelet coefficients are modeled using the GARCH model. The features of the GARCH model are considered as the main property vector. The second method is the Developed Wavelet-GARCH-KNN (D-WGK), which solves the incompatibility of the WGK method for the use of a low pass sub-band. The third method is the Wavelet Local Linear Approximation (LLA)-KNN, which we used for modeling the wavelet sub-bands. The extracted features were applied separately to determine the normal image or brain tumor based on classification methods. The classification was performed for the diagnosis of tumor types. The empirical results showed that the proposed algorithm obtained a high rate of classification and better practices than recently introduced algorithms while requiring a smaller number of classification features. According to the results, the Low-Low sub-bands are not adopted with the GARCH model; therefore, with the use of homomorphic filtering, this limitation is overcome. The results showed that the presented Local Linear (LL) method was better than the GARCH model for modeling wavelet sub-bands.

Localized linear regression methods for estimating monthly precipitation grids using elevation, rain gauge, and TRMM data

Accurate estimation of the spatial distribution of precipitation is crucial for hydrologic modeling. To achieve the realistic estimation of precipitation, developing a ground-based observatory system is a costly and time-consuming strategy compared with other solutions such as using a combination of satellite- and ground-based observations. In this paper, to improve the estimation accuracy of spatial precipitation variation, various linear regression methods were used that combine digital elevation model (DEM) data, rain gauge observations, and Tropical Rainfall Measuring Mission (TRMM) products. Specifically, fuzzy cluster-based linear regression (FCLR), local multiple linear regression using historical similarity (LMLR-HS), model tree (MT), and moving least squares (MLS) were used in the proposed methodology based on local data behavior. The results were compared with those obtained from multiple linear regression (MLR) methods including simple multiple linear regression (SMLR), robust multiple linear regression (RMLR), and generalized linear model (GLM) for monthly precipitation estimation. The study area was Namak Lake watershed, one of the largest watersheds in Iran. The results, estimated for wet and dry years (years 1999 and 2003, respectively), show superiority of local linear regression methods over the other linear methods. Based on the statistical metrics used for assessing the quality the results, FCLR and MLS outperformed other tested methods.

Adaptive iterative linearization Galerkin methods for nonlinear problems

A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative schemes. In particular, for Lipschitz continuous and strongly monotone operators, we derive a general convergence analysis. Furthermore, in the context of numerical solution schemes for nonlinear partial differential equations, we propose a combination of the iterative linearization approach and the classical Galerkin discretization method, thereby giving rise to the so-called iterative linearization Galerkin (ILG) methodology. Moreover, still on an abstract level, based on two different elliptic reconstruction techniques, we derive a posteriori error estimates which separately take into account the discretization and linearization errors. Furthermore, we propose an adaptive algorithm, which provides an efficient interplay between these two effects. In addition, the ILG approach will be applied to the specific context of finite element discretizations of quasilinear elliptic equations, and some numerical experiments will be performed.

On local linearization method for stochastic differential equations driven by fractional Brownian motion

We propose a local linearization scheme to approximate the solutions of non-autonomous stochastic differential equations driven by fractional Brownian motion with Hurst parameter Toward this end, we approximate the drift and diffusion terms by means of a first-order Taylor expansion. This becomes the original equation into a local fractional linear stochastic differential equation, whose solution can be figured out explicitly. As in the Brownian motion case (i.e., H = 1/2), the rate of convergence, in our case, is twice the one of the Euler scheme. Numerical examples are given to demonstrate the performance of the method.

Numerical spectral examination of EMHD mixed convective flow of second-grade nanofluid towards a vertical Riga plate using an advanced version of the revised Buongiorno’s nanofluid model

Present communication aims to determine the impact of Cattaneo–Christov model and convective boundary on second-grade nanofluid flow alongside a Riga pattern. Zero mass flux is accounted at the solid surface of Riga pattern such that the fraction of nanoparticles maintains itself on strong retardation. The impact of Lorentz forces generated by Riga pate is also an important aspect of the study. The governing nonlinear problem is converted into ordinary problems via suitably adjusted transformations. Spectral local linearization method has been incorporated to find the solutions of the nonlinear problems. Variation in horizontal movement of the nanofluid, thermal distribution and concentration distribution of the nanoparticles has been noted for various fluid parameters. The results are plotted graphically. Outcomes indicate that the horizontal movement gains enhancement for elevated values of modified Hartman factor. Thermal state of the nanofluid and concentration of nanoparticles receive reduction for incremental values of relaxation time parameters. Numerical results for skin friction and heat flux have been reported in tabular form. The CPU run time and residual error are obtained to check the efficiency of the method used for finding the solution.

Wind-tolerant optimal closed loop controller design for a domestic atmospheric research airship

Airships are inherently sensitive to random atmospheric disturbances that could potentially make their data gathering and observation missions a formidable task. In this context robust closed loop feedback controllers are important. The present study is therefore focused on optimal feedback controller design of an indigenous domestically designed airship (DA) for added robustness against atmospheric disturbances. While the general airship six degrees of freedom (6DoF) governing equations of motion are mathematically nonlinear, one often needs to resort to local linearization methods to benefit from proven linear closed loop controller (CLC) design approaches. In this sense an optimal linear quadratic regulator/tracker (LQR/LQT) seems to be a viable alternative for DA control purposes whose implementation relies on local linearization of the 6DoF nonlinear equations around the instantaneous airship trajectory. In order to demonstrate the capabilities of the proposed CLC design against external turbulences, a random wind profile over the DA hull is assumed and the airship behavior is analyzed in regulation as well as the tracking mode. The results show that the proposed CLC design complies with the all mission requirements and adequately reduces the impact of environmental random wind fluctuations. Given the small computational time required for control gain and command determination within the LQR/LQT algorithm against the DA mission flight time, the proposed CLC can be utilized online as a feedback control strategy while the airship is performing any physical atmospheric experiment.

Statistical inferences for single-index models with measurement errors

An important factor in house prices is its location. However, measurement errors arise frequently in the process of observing variables such as the latitude and longitude of the house. The single-index models with measurement errors are used to study the relationship between house location and house price. We obtain the estimators by a SIMEX method based on the local linear method and the estimating equation. To test the significance of the index coefficient and the linearity of the link function, we establish the generalized likelihood ratio (GLR) tests for the models. We demonstrate that the asymptotic null distributions of the established GLR tests follow -distributions which are independent of nuisance parameters or functions. Finally, two simulated examples and a real estate valuation data set are given to illustrate the effect of GLR tests.

Observer-Based Sampled-Data Model-Free Adaptive Control for Continuous-Time Nonlinear Nonaffine Systems With Input Rate Constraints

A sampled-data model-free adaptive control (SMFAC) strategy is proposed for continuous-time nonlinear nonaffine systems with input rate constraints. By using differential and integral mean value theorems as two basic mathematic tools, a sampled-data local dynamic linearization method is proposed at first to transform the continuous-time nonlinear nonaffine model into a sampled-data nonlinear affine I/O model, including a linear parametric term affined to the control input and a nonlinear uncertainty term. On this basis, we consequently propose an observer-based SMFAC (ObSMFAC) scheme, including a sampled-data parameter estimator to estimate the unknown partial derivatives and a sampled-data observer to estimate the residual nonlinear uncertainty, respectively. Note that the sampling period is incorporated explicitly in the proposed ObSMFAC which enhances the control performance by reducing its negative influence on the system stability. The constraint on the input rate is also considered in the control law as the transition condition of the input updating algorithms. The convergence of the proposed ObSMFAC is proved by using the contraction mapping principle. The simulation study demonstrates the theoretical results.

Environmental Modelling through Chaotic Approach for Malaysian West Coast Sea Level.

Sea level forecasting is an essential task for coastal engineering, geodetic application, navigation, and recreational activities. Predicting the behavior of future sea level is important for monitoring and forecasting of changes in fishery and marine ecosystems as well as for protection of coastal. This research is focused on the analysis and prediction of hourly sea level time series data at the benchmark station located in Penang by using chaotic approach. The purpose of this research was to identify the presence of chaotic behavior by the phase space reconstruction and Cao methods, and also a local linear approximation method is applied for prediction purposes. The results notified that the value of correlation coefficient between the observed and predicted time series is 0.8838 which is near to one. This reveals that the local linear approximation method can be used to predict the sea level time series in Malaysia. Certainly, the result of this research is expected to help stakeholder such as Department of Survey and Mapping Malaysia (JUPEM) in having a better sea level management.

Semi-functional partial linear spatial autoregressive model

This paper proposes a semi-functional partial linear spatial autoregressive (SAR) model, in which we allow one of the explanatory variables to be a functional variable, while the dependent variable is scalar. We aim to enable the spatial econometric models to be applied to various areas where data types can be functional data. Based on quasi-maximum likelihood estimation (QMLE) method and local linear regression method, we construct a two-stage estimator to estimate the parameters and nonparametric component. The convergence rate of the estimator of nonparametric component is given. Furthermore, Monte Carlo simulations are performed to investigate our two-stage estimator’s finite sample performance.

Study of Activation Energy on the Movement of Gyrotactic Microorganism in a Magnetized Nanofluids Past a Porous Plate

The present study deals with the swimming of gyrotactic microorganisms in a nanofluid past a stretched surface. The combined effects of magnetohydrodynamics and porosity are taken into account. The mathematical modeling is based on momentum, energy, nanoparticle concentration, and microorganisms’ equation. A new computational technique, namely successive local linearization method (SLLM), is used to solve nonlinear coupled differential equations. The SLLM algorithm is smooth to establish and employ because this method is based on a simple univariate linearization of nonlinear functions. The numerical efficiency of SLLM is much powerful as it develops a series of equations which can be subsequently solved by reutilizing the data from the solution of one equation in the next one. The convergence was improved through relaxation parameters in the study. The accuracy of SLLM was assured through known methods and convergence analysis. A comparison of the proposed method with the existing literature has also been made and found an excellent agreement. It is worth mentioning that the successive local linearization method was found to be very stable and flexible for resolving the issues of nonlinear magnetic materials processing transport phenomena.

Numerical Investigation on the Swimming of Gyrotactic Microorganisms in Nanofluids through Porous Medium over a Stretched Surface

In this article, the effects of swimming gyrotactic microorganisms for magnetohydrodynamics nanofluid using Darcy law are investigated. The numerical results of nonlinear coupled mathematical model are obtained by means of Successive Local Linearization Method. This technique is based on a simple notion of the decoupling systems of equations utilizing the linearization of the unknown functions sequentially according to the order of classifying the system of governing equations. The linearized equations, that developed a sequence of linear differential equations along with variable coefficients, were solved by employing the Chebyshev spectral collocation method. The convergence speed of the SLLM technique can be willingly upgraded by successive applying over relaxation method. The comparison of current study with available published literature has been made for the validation of obtained results. It is found that the reported numerical method is in perfect accord with the said similar methods. The results are displayed through tables and graphs.

Analysis of double-diffusion convection on three-dimensional MHD stagnation point flow of a tangent hyperbolic Casson nanofluid

ABSTRACT In this paper, we present a numerical study of three-dimensional tangent hyperbolic Casson-nanofluid flow past a horizontally stretched surface with a magnetic field and double-diffusion convection. Double-diffusion convection in three-dimensional tangent hyperbolic Casson nanofluid flow together with slip coefficients, to the authors’ observation, has not been studied. Hence, this study discusses the impact of double-diffusion convection on the magnetohydrodynamic stagnation point flow of a tangent hyperbolic Casson nanofluid. The nonlinear flow partial differential equations with the associated boundary conditions are reduced to a coupled system of nonlinear ordinary differential equations using appropriate similarity transformations. The ordinary differential equations are solved using the spectral local linearisation method. We study the significance of the magnetic field, Weissenberg, Prandtl, Soret and Dufour numbers on the fluid properties. The results indicate that increasing the Weissenberg numbers increases the relaxation time and leads to a resistance in the fluid flow in both the tangential and transverse directions. The temperature and concentration profiles are enhanced as a result of increasing the Weissenberg number.

Time-varying income and price elasticities for energy demand: Evidence from a middle-income panel

Abstract We estimate time-varying income and price elasticities for energy demand for a 26-country, middle-income balanced panel that spans 1996–2014. To do so, we employ a recently developed local linear dummy estimation method to estimate the trend and coefficient functions. We find that the price elasticity for energy demand is either insignificant or positive and small. While the income elasticity for energy demand behaves in a non-linear fashion over-time, it is always less than unity and is generally within 0.6–0.8. A GDP elasticity of less than one suggests that these middle-income countries are on the right-hand-side of an inverted-U energy intensity-GDP path that is consistent with the dematerialization process. Also, this finding suggests that energy intensity — but not energy consumption — in these countries will fall with economic growth. Hence, intensity-based targets may be met in a business-as-usual setting, but aggregate or per capita-based carbon emissions targets would likely require policy interventions.

A multivariate cointegration time series model and its applications in analysing stock markets in China

This paper explores nonlinear cointegration between Chinese mainland stock markets and Hong Kong stock market in a multivariate framework for the period January, 1998 to December, 2014 by a nonparametric method. The local linear kernel smoothing method is developed to estimate the unknown function, and the practical problem of implementation is also addressed. Then, a simple nonparametric version of a bootstrap test is adapted for testing misspecification. Furthermore, Some Monte Carlo experiments are presented to examine the finite sample performance of the proposed procedure. Finally, the stock markets data set is discussed in detail by using proposed procedures, showing that Shanghai Stock Index (SHSI) and Shenzhen Component Index (SZCI) can affect Hang Seng Index (HSI), and the influence appears to be a strong nonlinear characteristics.

Testing for covariance matrices in time-varying coefficient panel data models with fixed effects

In this paper, we study the tests for sphericity and identity of covariance matrices in time-varying coefficient high-dimensional panel data models with fixed effects. In order to construct the effective test statistics and avoid the influence of the unknown fixed effects, we apply the difference method to eliminate the dependence of the residual sample, and further construct test statistics using the trace estimators of the covariance matrices. For the estimators of the coefficient functions, we use the local linear dummy variable method. Under some regularity conditions, we study the asymptotic property of the estimators and establish the asymptotic distributions of our proposed test statistics without specifying an explicit relationship between the cross-sectional and the time series dimensions. We further show that the test statistics are asymptotic distribution-free. Subsequently simulation studies are carried out to evaluate our proposed methods. In order to assess the performance of our proposed test method, we compare with the existing test methods in panel data linear models with fixed effects.

Improvement of mapping vegetation cover for arid and semiarid areas using a local nonlinear modelling method and landsat images

Accurately estimating and mapping vegetation cover for monitoring land degradation and desertification of arid and semiarid areas using remotely sensed images is promising but challenging in remote, sparsely vegetated and large areas. In this study, a novel method – geographically weighted logistic regression (GWLR – integrating geographically weighted regression (GWR) and a logistic model) was proposed to improve vegetation cover mapping of Kangbao County, Hebei of China using Landsat 8 image and field data. Additionally, a new method to determine the bandwidth of GWLR is presented. Using cross-validation, GWLR was compared with a globally linear stepwise regression (LSR), a local linear modelling method GWR and a nonparametric method, k-nearest neighbours (kNN) with varying numbers of nearest plots. Results demonstrated (1) the red and near infrared relevant band ratios and vegetation indices significantly improved mapping; (2) the GWLR, GWR and kNN methods led to more accurate predictions than LSR; (3) GWLR reduced overestimations and underestimations compared with LSR, kNN and GWR, and also eliminated negative and very large estimates caused by GWR and LSR; and (4) The maximum distance of spatial autocorrelation could be used to determine the bandwidth for GWLR. Overall, GWLR proved more promising for mapping vegetation cover of arid and semiarid areas.

A Spectral Simulation of Magnetohydrodynamic Convective Nanofluid Slip Flow with Newtonian Heating and Radiative Heat Transfer

In this paper, we present a numerical analysis of natural convection boundary layer flow of a nanofluid over a stretching sheet under the influence of a variable magnetic field in the presence of nonlinear radiative heat transfer, hydrodynamic slip and Newtonian heating effects. The Brownian diffusion and thermophoresis effects are employed to describe the nanofluid flow model consisting of non-linear partial differential equations, which is transformed into a similarity form and then treated with the spectral local-linearization method (SLLM) to present an in-depth analysis of the flow and heat transfer characteristics of the fluid flow. The fluid flow problem finds applications in many engineering devices including geothermal heat source pump and in cooling of electronic devices and stretched wires.

Analytic vs. numerical solutions to a Sturm-Liouville transmission eigenproblem

An elliptic one-dimensional second order boundary value problem involving discontinuous coefficients, with or without transmission conditions, is considered. For the former case by a direct sum spaces method we show that the eigenvalues are real, geometrically simple and the eigenfunctions are orthogonal.
 Then the eigenpairs are computed numerically by a local linear finite element method (FEM) and by some global spectral collocation methods.
 The spectral collocation is based on Chebyshev polynomials (ChC) for problems on bounded intervals respectively on Fourier system (FsC) for periodic problems.
 The numerical stability in computing eigenvalues is investigated by estimating their (relative) drift with respect to the order of approximation. The accuracy in computing the eigenvectors is addressed by estimating their departure from orthogonality as well as by the asymptotic order of convergence. The discontinuity of coefficients in the problems at hand reduces the exponential order of convergence, usual for any well designed spectral algorithm, to an algebraic one.
 As expected, the accuracy of ChC outcomes overpasses by far that of FEM outcomes.

A Spatiotemporal Deformation Modelling Method Based on Geographically and Temporally Weighted Regression

The geographically and temporally weighted regression (GTWR) model is a dynamic model which considers the spatiotemporal correlation and the spatiotemporal nonstationarity. Taking into account these advantages, we proposed a spatiotemporal deformation modelling method based on GTWR. In order to further improve the modelling accuracy and efficiency and considering the application characteristics of deformation modelling, the inverse window transformation method is used to search the optimal fitting window width and furthermore the local linear estimation method is used in the fitting coefficient function. Moreover, a comprehensive model for the statistical tests method is proposed in GTWR. The results of a dam deformation modelling application show that the GTWR model can establish a unified spatiotemporal model which can represent the whole deformation trend of the dam and furthermore can predict the deformation of any point in time and space, with stronger flexibility and applicability. Finally, the GTWR model improves the overall temporal prediction accuracy by 43.6% compared to the single-point time-weighted regression (TWR) model.