Local Polynomial Modeling Research Articles

A new 3D phase unwrapping method by region partitioning and local polynomial modeling in abdominal quantitative susceptibility mapping.

Accurate phase unwrapping is a critical prerequisite for successful applications in phase-related MRI, including quantitative susceptibility mapping (QSM) and susceptibility weighted imaging. However, many existing 3D phase unwrapping algorithms face challenges in the presence of severe noise, rapidly changing phase, and open-end cutline. In this study, we introduce a novel 3D phase unwrapping approach utilizing region partitioning and a local polynomial model. Initially, the method leverages phase partitioning to create initial regions. Noisy voxels connecting areas within these regions are excluded and grouped into residual voxels. The connected regions within the region of interest are then reidentified and categorized into blocks and residual voxels based on voxel count thresholds. Subsequently, the method sequentially performs inter-block and residual voxel phase unwrapping using the local polynomial model. The proposed method was evaluated on simulation and in vivo abdominal QSM data, and was compared with the classical Region-growing, Laplacian_based, Graph-cut, and PRELUDE methods. Simulation experiments, conducted under different signal-to-noise ratios and phase change levels, consistently demonstrate that the proposed method achieves accurate unwrapping results, with mean error ratios not exceeding 0.01%. In contrast, the error ratios of Region-growing (N/A, 84.47%), Laplacian_based (20.65%, N/A), Graph-cut (2.26%, 20.71%), and PRELUDE (4.28%, 10.33%) methods are all substantially higher than those of the proposed method. In vivo abdominal QSM experiments further confirm the effectiveness of the proposed method in unwrapping phase data and successfully reconstructing susceptibility maps, even in scenarios with significant noise, rapidly changing phase, and open-end cutline in a large field of view. The proposed method demonstrates robust and accurate phase unwrapping capabilities, positioning it as a promising option for abdominal QSM applications.

New 3D phase-unwrapping method based on voxel clustering and local polynomial modeling: application to quantitative susceptibility mapping.

To develop an accurate and robust 3-dimensional (3D) phase-unwrapping method that works effectively in the presence of severe noise, disconnected regions, rapid phase changes, and open-ended lines for quantitative susceptibility mapping (QSM). We developed a 3D phase-unwrapping method based on voxel clustering and local polynomial modeling named CLOSE3D, which firstly explores the 26-neighborhood to calculate local variation of the phasor and the phase, and then according to the local variation of the phasor, clusters the phase data into easy-to-unwrap blocks and difficult-to-unwrap residual voxels. Next, CLOSE3D sequentially performs intrablock, interblock, and residual-voxel unwrapping by using the region-growing local polynomial modeling method. CLOSED3D was evaluated in simulation and using in vivo brain QSM data, and was compared with classical region-growing and region-expanding labeling for unwrapping estimates methods. The simulation experiments showed that CLOSE3D achieved accurate phase-unwrapping results with a mean error ratio <0.39%, even in the presence of serious noise, disconnected regions, and rapid phase changes. The error ratios of region-growing (P=0.000 and P=0.000) and region-expanding labeling for unwrapping estimates (P=0.007, P=0.014) methods were both significantly higher than that of CLOSE3D, when the noise level was ≥60%. The results of the in vivo brain QSM experiments showed that CLOSE3D unwrapped the phase data and accurately reconstructed quantitative susceptibility data, even with serious noise, rapid-varying phase, or an open-ended cutline. CLOSE3D achieves phase unwrapping with high accuracy and robustness, which will help phase-related 3D magnetic resonance imaging (MRI) applications such as QSM and susceptibility weighted imaging.

Local polynomial smoothing based on the Kaplan–Meier estimate

The local polynomial modeling of the Kaplan–Meier estimate for random designs under the right censored data setting is investigated in detail. Two classes of boundary aware estimates are developed: estimates of the distribution function and its derivatives of any arbitrary order and estimates of integrated distribution function derivative products. Their statistical properties are quantified analytically and their implementation is facilitated by the development of corresponding data driven plug-in bandwidth selectors. The asymptotic rate of convergence of the plug-in rule for the estimates of the distribution function and its derivatives is quantified analytically. Numerical evidence is also provided on its finite sample performance. A real life data analysis illustrates how the methodological advances proposed herein help to generate additional insights in comparison to existing methods.

Two-dimensional recursive least squares identification based on local polynomial modeling for batch processes

In this paper, a two-dimensional recursive least squares identification method based on local polynomial modeling for batch processes is proposed. Using local polynomial modeling method to parameterize the time-varying characteristics of batch processes, a two-dimensional cost function along both time and batch directions is minimized to design the recursive least squares identification algorithm. Because this proposed method employs local polynomial modeling and utilizes two-dimensional data information to estimate model parameters, it can effectively improve the estimation accuracy and accelerate the convergence rate. Furthermore, the convergence property of the proposed method is analyzed. Finally, the simulation results show the superiority of the proposed method.

A Variable Regularized Recursive Subspace Model Identification Algorithm With Extended Instrumental Variable and Variable Forgetting Factor

This paper proposes a new smoothly clipped absolute deviation (SCAD) regularized recursive subspace model identification algorithm with square root(SR) extended instrumental variable (EIV) and locally optimal variable forgetting factor (LOFF). The proposed algorithm is based on a new local polynomial modeling (LPM)-based variable FF EIV projection approximation subspace tracking (PAST) and multivariate recursive least squares algorithms for estimating respectively the subspace of the extended observability matrix, and the input matrix and feedthrough matrix. The SREIV-PAST algorithm offers improved resilience to additive noise over the PAST algorithm and it is implemented using the more stable square-root form. The asymptotic mean square error of the corresponding LPM-based model is derived and minimized at each time instant to obtain the proposed LOFF for improving the convergence speed and steady state error. A recursive bi-iteration singular value decomposition (SVD) algorithm is also proposed for recursive computation of the pseudo-inverse of the state transition matrix and its eigenvalues. This facilitates online estimation of its model order using classical model selection criteria. Moreover, a new criterion based on the percentage of explained variance is also proposed with improved performance. The SCAD regularization is proposed for automatic model selection of the input and feedthrough matrices, as it is asymptotically unbiased. Efficient techniques for incorporating SCAD and estimating other required quantities are also developed. The proposed algorithms are evaluated using computer simulations under stationary and nonstationary environments and a real dataset on wing flutter data. Results show that the proposed algorithms offer improved convergence speed and steady state performance over the conventional algorithms and it also provides an online estimate of the system model order.

A New Variable Forgetting Factor and Variable Regularized Square Root Extended Instrumental Variable PAST Algorithm With Applications

This article proposes a square-root (SR) extended instrumental variable (EIV) projection approximation subspace tracking (PAST) algorithm with variable forgetting factor (VFF) and variable regularization (VR). A new local polynomial modeling (LPM) based VFF is proposed by minimizing the mean squares deviation of the EIV linear model and the IV-PAST algorithm. A new variable $\ell 2$ regularization term is also derived to reduce the variance of the estimator resulting from possibly ill conditioned covariance matrix at low input signal level. An SR version of the algorithm is developed to improve the numerical stability of the algorithm and avoid the problem of loss of positive definiteness of the inverse covariance matrix. The proposed LOFF-VR-SREIV-PAST algorithm can be implemented by both the conventional EIV-PAST algorithm and numerically more stable hyperbolic rotations. Furthermore, the convergence of the proposed VFF-EIV-PAST algorithm using the ordinary differential equation method is analyzed. Its application to the estimation and tracking of direction of arrival under spatial color sensor noise in both stationary and nonstationary scenarios are presented. Simulations demonstrate that the proposed algorithm yields improved performance over the conventional PAST and EIV-PAST algorithms, especially at medium to low signal-to-noise ratio, which is more frequently encountered in practical situations.

A New Local Polynomial Modeling Based Variable Forgetting Factor and Variable Regularized PAST Algorithm for Subspace Tracking

This paper proposes a new local polynomial modeling based variable forgetting factor (VFF) and variable regularized (VR) projection approximation subspace tracking (PAST) algorithm, which is based on a novel VR-VFF recursive least squares (RLS) algorithm with multiple outputs. The subspace to be estimated is modeled as a local polynomial model so that a new locally optimal forgetting factor (LOFF) can be obtained by minimizing the resulting mean square deviation of the RLS algorithm after using the projection approximation. An $l_2$ -regularization term is also incorporated to the LOFF-PAST algorithm to reduce the estimation variance of the subspace during signal fading. The proposed LOFF-VR-PAST algorithm can be implemented by the conventional RLS algorithm as well as the numerically more stable QR decomposition. Applications of the proposed algorithms to subspace-based direction-of-arrival estimation under stationary and nonstationary environments are presented to validate their effectiveness. Simulation results show that the proposed algorithms offer improved performance over the conventional PAST algorithm and a comparable performance to the Kalman filter with variable measurement subspace tracking algorithm, which requires a considerably higher arithmetic complexity. The new LOFF-VR-RLS algorithm may also be applicable to other RLS problems involving multiple outputs.

Mix local polynomial and spline truncated: the development of nonparametric regression model

Regression analysis is a statistical method used to model response variables with predictor variables. The approach to regression analysis can be done by parametric, semiparametric and nonparametric. The nonparametric approach is more complex than parametric. Some nonparametric approaches include local and spline polynomial. Local polynomial is the development of the taylor series which has an arbited fixed point, while the splines are pieces of the spline function. In both spline and local polynomial modeling requires a smoothing parameterb determined by GCV (General Cross Validation) method. In this paper, we developed a method of combining local and spline polynomials. The parameter estimation uses the OLS (Ordinary Least Square) method, yielding an estimator (XTKhX)-1XTKhy, where X is a predictor variable with combination of polynomial and spline function, whereas Kh is a kernel function used for the combination of local polynomial and spline function.

A variable forgetting factor diffusion recursive least squares algorithm for distributed estimation

Distributed recursive least squares (RLS) algorithms have superior convergence properties compared to the least mean squares (LMS) counterpart. However, with a fixed forgetting factor (FF), they are not suitable for tracking time-varying (TV) parameters. This paper proposes a novel diffusion variable FF RLS (Diff-VFF-RLS) algorithm based on a local polynomial modeling (LPM) of the unknown TV system. The diffusion RLS solution is derived analytically such that the estimation deviation from the true value is investigated. Based on the analysis and the LPM of the TV system, a new optimal VFF formula that tries to minimize the estimation deviation is obtained. Simulations are conducted to verify the theoretical analysis in terms of the steady-state mean square deviation (MSD) and the VFF formula. Results also show that the convergence and tracking performance of the proposed algorithm compares favorably with conventional ones.

Bias Compensation for Rational Polynomial Coefficients of High-Resolution Satellite Imagery by Local Polynomial Modeling

The Rational Function Model (RFM) is a widely used generic sensor model for georeferencing satellite images. Owing to inaccurate measurement of satellite orbit and attitude, the Rational Polynomial Coefficients (RPCs) provided by image vendors are commonly biased and cannot be directly used for high-precision remote-sensing applications. In this paper, we propose a new method for the bias compensation of RPCs using local polynomial models (including the local affine model and the local quadratic model), which provides the ability to correct non-rigid RPC deformations. Performance of the proposed approach was evaluated using a stereo triplet of ZY-3 satellite images and compared with conventional global-polynomial-based models (including the global affine model and the global quadratic model). The experimental results show that, when the same polynomial form was used, the correction residuals of the local model could be notably smaller than those of the global model, which indicates that the new method has great ability to remove complex errors existed in vendor-provided RPCs. In the experiments of this study, the accuracy of the local affine model was nearly 15% better than that of the global affine model. Performance of the local quadratic model was not as good as the local affine model when the number of Ground Control Points (GCPs) was less than 10, but it improved rapidly with an increase in the number of redundant observations. In the test scenario with 15 GCPs, the accuracy of the local quadratic model was about 9% and 27% better than those of the local affine model and the global quadratic model, respectively.

PEMODELAN REGRESI NONPARAMETRIK MENGGUNAKAN PENDEKATAN POLINOMIAL LOKAL PADA BEBAN LISTRIK DI KOTA SEMARANG

Semarang is the provincial capital of Central Java, with infrastructure and economic’s growth was high. The phenomenon of power outages that occurred in Semarang, certainly disrupted economic development in Semarang. Large electrical energy consumed by industrial-scale consumers and households in the San Francisco area, monitored or recorded automatically and presented into a historical data load power consumption. Therefore, this study modeling the load power consumption at a time when not influenced by the use of electrical load (t-1)-th. Modeling using nonparametric regression approach with Local polynomial. In this study, the kernel used is a Gaussian kernel. In local polynomial modeling, determined optimum bandwidth. One of the optimum bandwidth determination using the Generalized Cross Validation (GCV). GCV values obtained amounted to 1425.726 with a minimum bandwidth of 394. Modelling generate local polynomial of order 2 with MSE value of 1408.672. Keywords: electrical load, local polinomial, gaussian kernel, GCV.

A New Local Polynomial Modeling-Based Variable Forgetting Factor RLS Algorithm and Its Acoustic Applications

This paper proposes a new class of local polynomial modeling (LPM)-based variable forgetting factor (VFF) recursive least squares (RLS) algorithms called the LPM-based VFF RLS (LVFF-RLS) algorithms. It models the time-varying channel coefficients as local polynomials so as to obtain the expressions of the bias and variance terms in the mean square error (MSE) of the RLS algorithm. A new locally optimal VFF (LOVFF) is then derived by minimizing the resulting MSE and the theoretical analysis is found to be in good agreement with experimental results. Methods for estimating the parameters involved in this LOVFF are also developed, resulting in an improved RLS algorithm with VFF. The algorithm is further extended to include variable regularization and a QR decomposition (QRD) version which is numerically more stable and amenable to multiplier-less implementation using coordinate rotation digital computer (CORDIC) algorithm. Applications of these algorithms to frequency estimation and adaptive beamforming in time-varying speech and audio signals are also presented to illustrate the effectiveness of the proposed algorithms. Simulations show that the convergence and tracking performance of the proposed algorithms compare favorably with conventional algorithms.

A generalization of inverse distance weighting method via kernel regression and its application to surface modeling

The inverse-distance weighting (IDW) method is considered as one of the most popular deterministic methods and is widely applied to a variety of areas because of its low computational cost and easy implementation. In this paper, we show that the classical IDW is essentially a zeroth-order local kernel regression method with an inverse distance weight function. Thus, it suffers from various shortcomings, such as the boundary bias. Considering the advantages of the local polynomial modeling technique in statistics, the classical IDW was generalized into a higher-order regression by the Taylor expansion and then computed by means of a weighted least-squares method. Surface modeling of rainfall fields in China indicated that the generalized IDWs with the first- and second-orders are more accurate than the classical IDW in terms of root mean square error (RMSE). The example of digital elevation model construction with a group of sample points showed that the two generalized IDWs have better RMSE and mean error than the classical IDW. Furthermore, the second-order IDW has a better performance than the ordinary kriging in terms of RMSE. A theoretical analysis demonstrated that the gradient-plus-inverse distance squared method presented by Nalder and Wein (Agric For Meteorol 92(4): 211–225, 1998) is a first-order form of the generalized IDW expanded on spatial coordinates and elevation. In a word, the generalized IDW can incorporate multiple covariates, which can better explain the interpolation procedure and might improve its accuracy.

Nonparametric estimation of nonlinear dynamics by metric-based local linear approximation

This paper discusses nonparametric estimation of nonlinear dynamical system models by a method of metric-based local linear approximation. We assume no functional form of a given model but estimate it from experimental data by approximating the curve implied by the function by the tangent plane around the neighborhood of a tangent point. To specify an appropriate neighborhood, we prepare a metric defined over the Euclidean space in which the curve exists and then evaluate the closeness to the tangent point according to the distances. The proposed method differs from the first order polynomial modeling in discerning the metric and the weighting function, but the first order polynomial modeling with Gaussian kernels is shown to be a special version of the proposed method. Simulation studies and application to ECG signals show the proposed method is easy to manipulate and has performance comparable to or better than the first order local polynomial modeling.

Designing of Child Growth Chart Based on Multi-Response Local Polynomial Modeling

Problem statements: Anthropometry measures used to measure physical children growth are not onlyweight but also height and head circumference. In this study we develop the estimation of multi-response localpolynomial regression and apply it to design growth chart for children up to five years old based on three responsevariables i.e., weight, height and head circumference. Approach: Based on local polynomial estimator, wedescribe the estimation of multi-response nonparametric regression model by using weighted least squared. Themodel is applied to design health card of children up to five years old by using children data in Surabaya, Indonesia. Generalized Cross Validation (GCV) method is used to determine the order of local polynomial fit and the bandwidthfor each response variable. Results: We formulate the multi-response local polynomial modeling and give a design of health card of children up to five years old in Surabayacity, Indonesia. Conclusion: The child growth chart based on multi-response local polynomial modeling showsincreasing of children nutrition in Surabaya 2010.Because of the strong correlations among all three response variables,the simultaneosly approach for model estimationis better than partly single response approach. The result of simultaneosly model estimation based on multi-response local polynomial modeling satisfies goodness of fit criterion i.e., mean squared error value tend to zero and determination coefficient value tend to one.

Local Polynomial Modeling and Variable Bandwidth Selection for Time-Varying Linear Systems

This paper proposes a local polynomial modeling (LPM) approach and variable bandwidth selection (VBS) algorithm for identifying time-varying linear systems (TVLSs). The proposed method models the time-varying coefficients of a TVLS locally by polynomials, which can be estimated by least squares estimation with a kernel having a certain bandwidth. The asymptotic behavior of the proposed LPM estimator is studied, and the existence of an optimal local bandwidth which minimizes the local mean-square error is established. A new data-driven VBS algorithm is then proposed to estimate this optimal variable bandwidth adaptively and locally. An individual bandwidth is assigned for each coefficient instead of the whole coefficient vector so as to improve the accuracy in fast-varying systems encountered in fault detection and other applications. Important practical issues such as online implementation are also discussed. Simulation results show that the LPM-VBS method outperforms conventional TVLS identification methods, such as the recursive least squares algorithm and generalized random walk Kalman filter/smoother, in a wide variety of testing conditions, in particular, at moderate to high signal-to-noise ratio. Using local linearization, the LPM method is further extended to identify time-varying systems with mild nonlinearities. Simulation results show that the proposed LPM-VBS method can achieve a satisfactory performance for mildly nonlinear systems based on appropriate linearization. Finally, the proposed method is applied to a practical problem of voltage-flicker-tracking problem in power systems. The usefulness of the proposed approach is demonstrated by its improved performance over other conventional methods.

Local Polynomial Modeling of Time-Varying Autoregressive Models With Application to Time–Frequency Analysis of Event-Related EEG

This paper proposes a new local polynomial modeling (LPM) method for identification of time-varying autoregressive (TVAR) models and applies it to time-frequency analysis (TFA) of event-related electroencephalogram (ER-EEG). The LPM method models the TVAR coefficients locally by polynomials and estimates the polynomial coefficients using weighted least-squares with a window having a certain bandwidth. A data-driven variable bandwidth selection method is developed to determine the optimal bandwidth that minimizes the mean squared error. The resultant time-varying power spectral density estimation of the signal is capable of achieving both high time resolution and high frequency resolution in the time-frequency domain, making it a powerful TFA technique for nonstationary biomedical signals like ER-EEG. Experimental results on synthesized signals and real EEG data show that the LPM method can achieve a more accurate and complete time-frequency representation of the signal.

Semiparametric indirect utility and consumer demand

A semiparametric model of consumer demand is considered. In the model, the indirect utility function is specified as a partially linear, where utility is nonparametric in expenditure and parametric (with fixed- or varying-coefficients) in prices. Because the starting point is a model of indirect utility, rationality restrictions like homogeneity and Slutsky symmetry are easily imposed. The resulting model for expenditure shares (as functions of expenditures and prices) is locally given by a fraction whose numerator is partially linear, but whose denominator is nonconstant and given by the derivative of the numerator. The basic insight is that given a local polynomial model for the numerator, the denominator is given by a lower order local polynomial. The model can thus be estimated using modified versions of local polynomial modeling techniques. For inference, a new asymmetric version of the wild bootstrap is introduced. Monte Carlo evidence that the proposed technique’s work is provided as well as an implementation of the model on Canadian consumer expenditure and price micro-data.

Estimation in covariate-adjusted regression

The method of covariate-adjusted regression was recently proposed for situations where both predictors and response in a regression model are not directly observed, but are observed after being contaminated by unknown functions of a common observable confounder in a multiplicative fashion. One example is data collected for a study on diabetes, where the variables of interest, systolic and diastolic blood pressures and glycosolated hemoglobin levels are known to be influenced by an observable confounder, body mass index. An estimation procedure based on equidistant binning (EB), currently available, gives consistent estimators for the regression coefficients adjusted for the confounder. In this paper, we propose two new estimation procedures based on nearest-neighbor binning (NB) and local polynomial modeling (LP). Even though, the three methods perform similarly in terms of their bias, it is shown through simulation studies that NB has smaller variance compared to EB, and LP yields substantially lower variance relative to the two binning methods for small to moderate sample sizes. The consistency and convergence rates of the proposed estimators of LP, with the smallest MSE, are also established. We illustrate the proposed method of LP with the above-mentioned diabetes data, where the goal is to uncover the regression relation between the response, glycosolated hemoglobin levels, and the predictors, systolic and diastolic blood pressures, adjusted for body mass index.

Parametric Statistical Modeling by Minimum Integrated Square Error

The likelihood function plays a central role in parametric and Bayesian estimation, as well as in nonparametric function estimation via local polynomial modeling. However, integrated square error has enjoyed a long tradition as the goodness-of-fit criterion of choice in nonparametric density estimation. In this article, I investigate the use of integrated square error, or L2 distance, as a theoretical and practical estimation tool for a variety of parametric statistical models. I show that the asymptotic inefficiency of the parameters estimated by minimizing the integrated square error or L2estimation (L2E) criterion versus the maximum likelihood estimator is roughly that of the median versus the mean. I demonstrate by example the well-known result that minimum distance estimators, including L2E, are inherently robust; however, L2E does not require specification of any tuning factors found in robust likelihood algorithms. L2E is particularly appropriate for analyzing massive datasets in which data cleaning is impractical and statistical efficiency is a secondary concern. Setting up the L2E criterion is relatively simple even with some very complex model specifications. Specific problems studied in this article include univariate density estimation, mixture density estimation, multivariate regression estimation, and robust estimation of the mean and covariance. John Tukey had a pivotal role in both nonparametric and robust estimation. This article is dedicated to his memory.