Local Polynomial Regression Method Research Articles

Coverage error optimal confidence intervals for local polynomial regression

This paper studies higher-order inference properties of nonparametric local polynomial regression methods under random sampling. We prove Edgeworth expansions for t statistics and coverage error expansions for interval estimators that (i) hold uniformly in the data generating process, (ii) allow for the uniform kernel, and (iii) cover estimation of derivatives of the regression function. The terms of the higher-order expansions, and their associated rates as a function of the sample size and bandwidth sequence, depend on the smoothness of the population regression function, the smoothness exploited by the inference procedure, and on whether the evaluation point is in the interior or on the boundary of the support. We prove that robust bias corrected confidence intervals have the fastest coverage error decay rates in all cases, and we use our results to deliver novel, inference-optimal bandwidth selectors. The main methodological results are implemented in companion R and Stata software packages.

Local polynomial regression for pooled response data

We propose local polynomial estimators for the conditional mean of a continuous response when only pooled response data are collected under different pooling designs. Asymptotic properties of these estimators are investigated and compared. Extensive simulation studies are carried out to compare finite sample performance of the proposed estimators under various model settings and pooling strategies. We apply the proposed local polynomial regression methods to two real-life applications to illustrate practical implementation and performance of the estimators for the mean function.

Analyzing Intracellular Gradients in Pollen Tubes.

Conspicuous intracellular gradients manifest and/or drive intracellular polarity in pollen tubes. However, quantifying these gradients raises multiple technical challenges. Here we present a sensible computational protocol to analyze gradients in growing pollen tubes and to filter nonrepresentative time points. As an example, we use imaging data from pollen tubes expressing a genetically encoded ratiometric Ca2+ probe, Yellow CaMeleon 3.6, from which a kymograph is extracted. The tip of the pollen tube is detected with CHUKNORRIS, our previously published methodology, allowing the reconstruction of the intracellular gradient through time. Statistically confounding time points, such as growth arrest where gradients are highly oscillatory, are filtered out and a mean spatial profile is estimated with a local polynomial regression method. Finally, we estimate the gradient slope by the linear portion of the decay in mean fluorescence, offering a quantitative method to detect phenotypes of gradient steepness, location, intensity, and variability. The data manipulation protocol proposed can be achieved in a simple and efficient manner using the statistical programming language R, opening paths to perform high-throughput spatiotemporal phenotyping of intracellular gradients in apically growing cells.

Nprobust: Nonparametric Kernel-Based Estimation and Robust Bias-Corrected Inference

Nonparametric kernel density and local polynomial regression estimators are very popular in statistics, economics, and many other disciplines. They are routinely employed in applied work, either as part of the main empirical analysis or as a preliminary ingredient entering some other estimation or inference procedure. This article describes the main methodological and numerical features of the software package nprobust, which offers an array of estimation and inference procedures for nonparametric kernel-based density and local polynomial regression methods, implemented in both the R and Stata statistical platforms. The package includes not only classical bandwidth selection, estimation, and inference methods (Wand and Jones 1995; Fan and Gijbels 1996), but also other recent developments in the statistics and econometrics literatures such as robust bias-corrected inference and coverage error optimal bandwidth selection (Calonico, Cattaneo, and Farrell 2018, 2019a). Furthermore, this article also proposes a simple way of estimating optimal bandwidths in practice that always delivers the optimal mean square error convergence rate regardless of the specific evaluation point, that is, no matter whether it is implemented at a boundary or interior point. Numerical performance is illustrated using an empirical application and simulated data, where a detailed numerical comparison with other R packages is given.

Business Cycle Estimation with High-Pass and Band-Pass Local Polynomial Regression

Filters constructed on the basis of standard local polynomial regression (LPR) methods have been used in the literature to estimate the business cycle. We provide a frequency domain interpretation of the contrast filter obtained by the difference of a series and its long-run LPR component and show that it operates as a kind of high-pass filter, so that it provides a noisy estimate of the cycle. We alternatively propose band-pass local polynomial regression methods aimed at isolating the cyclical component. Results are compared to standard high-pass and band-pass filters. Procedures are illustrated using the US GDP series.

Business Cycle Estimation with High-Pass and Band-Pass Local Polynomial Regression

Filters constructed on the basis of standard local polynomial regression (LPR) methods have been used in the literature to estimate the business cycle. We provide a frequency domain interpretation of the contrast filter obtained by the difference between a series and its long-run LPR component and show that it operates as a kind of high-pass filter, meaning it provides a noisy estimate of the cycle. We alternatively propose band-pass local polynomial regression methods aimed at isolating the cyclical component. Results are compared to standard high-pass and band-pass filters. Procedures are illustrated using the US GDP series.

Identifying transient patterns of in vivo muscle behaviors during isometric contraction by local polynomial regression

Abstract Polynomial regression is the most common method to estimate the relationship between muscle signals and torque during muscle contraction, but it is not capable of characterizing important transient patterns in the signal–torque relationship that only exist during short bursts of torque but may convey detailed information of muscle behavior. In this study, we proposed an integrated data analysis approach based on local polynomial regression (LPR) to identify transient patterns in the signal–torque relationship. For each subject, the LPR method can represent electromyography (EMG), mechanomyography (MMG) and ultrasonography (US) features as nonlinear functions of torque and can further estimate the derivatives of these signal–torque nonlinear functions. Further, a number of break points can be detected from the derivatives of the signal–torque relationships at the group level, and they can segment the signal–torque relationships into several stages, where multimodal features change with torque in different dynamic manners. Eight subjects performed isometric ramp contraction of knee up to 90% of the maximal voluntary contraction (MVC). EMG, MMG and US were simultaneously recorded from the rectus femoris muscle. Results showed that, for each feature, the whole torque range were clearly segmented into several distinct stages by the proposed method and the feature–torque relationship could be approximately described by a piecewise linear function with different slopes at different stages. A critical break-point of 20% MVC was detected during the isometric contraction for all muscle signals. As compared with the conventional regression methods, the proposed LPR-based data analysis approach can effectively identify stage-dependent transient patterns in the feature–torque relationships, providing deeper insights into the motor unit activation strategy.

Non-consistent cell-average multiresolution operators with application to image processing

In recent years different techniques to process signal and image have been designed and developed. In particular, multiresolution representations of data have been studied and used successfully for several applications such as compression, denoising or inpainting. A general framework about multiresolution representation has been presented by Harten (1996) [20]. Harten’s schemes are based on two operators: decimation, D, and prediction, P, that satisfy the consistency property DP=I, where I is the identity operator. Recently, some new classes of multiresolution operators have been designed using learning statistical tools and weighted local polynomial regression methods obtaining filters that do not satisfy this condition. We show some proposals to solve the consistency problem and analyze its properties. Moreover, some numerical experiments comparing our methods with the classical methods are presented.

Modelling impacts of regulation on flows to the Lowbidgee floodplain of the Murrumbidgee River, Australia

Summary Major wetland systems have been significantly affected by alteration of flows by dams and subsequent abstraction upstream around the world. Estimating the level of this impact is particularly difficult where there is high flow variability, such as dryland rivers in Australia. Such information remains critical for assessing ecological impacts to ecosystems (e.g. long-lived flood-dependent trees). To determine effects of flow reduction to a large floodplain wetland, we built statistical flow models, integrated flow and flood modelling (IFFM), for the extensive and ecologically important Lowbidgee wetland, supplied by the Murrumbidgee River, based on local annual rainfall and upstream flow data (1880–2010). Large volumes of water are diverted upstream primarily for irrigation and to Australia’s capital city, Canberra, achieved with 26 large dams: Burrinjuck Dam, Snowy Mountains Scheme dams and other upper catchment dams. We identified two periods using structural change analysis, low (before 1957) and high (after 1958) regulated periods; the year marked significant alteration in monthly flow at Redbank gauge, within the Lowbidgee wetland, after which most major dams were built. To determine differences in flow between these periods, we developed two models for three flow gauges on the lower Murrumbidgee River: Hay, Maude and Redbank. The latter two were within the floodplain wetland. The models were based on annual rainfall from stations in the upper catchment and flow, using LOESS and leave-one-out samples without overfitting. This flexible local polynomial regression method was a useful approach to modelling complex processes without theoretical models. The proposed low and high regulated models performed well using the goodness of fit (Nash–Sutcliffe coefficient of efficiency ⩾91.5%) and Kolmogorov–Smirnov test ( p > 0.9999), i.e., there was no significant difference between the observed and fitted distributions of flow data at the Hay, Maude and Redbank flow gauges for both before 1957 and after 1958 periods. And these models were used to extend and predict annual low and high regulated flows from 1880 to 2010, where the Kolmogorov–Smirnov test ( p

Solution of an Integral-Differential Equation Arising in Oscillating Magnetic Fields Using Local Polynomial Regression

An integrodifferential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields is considered. The local polynomial regression method (LPR) is used to solve this equation. The reliability of this method and the reduction in the size of computational domain give this method a wider applicability. Several representative examples are given to reconfirm the efficiency of these algorithms. The results of applying this theory to the integro-differential equation with time-periodic coefficients reveal that LPR method possesses very high accuracy, adaptability, and efficiency.

Two-Stage Local Polynomial Regression Method for Image Heteroscedastic Noise Removal

In this paper, we introduce the extension of local polynomial fitting to the linear heteroscedastic regression model and its applications in digital image heteroscedastic noise removal. For better image noise removal with heteroscedastic energy, firstly, the local polynomial regression is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. Due to non-parametric technique of local polynomial estimation, we do not need to know the heteroscedastic noise function. Therefore, we improve the estimation precision, when the heteroscedastic noise function is unknown. Numerical simulations results show that the proposed method can improve the image quality of heteroscedastic noise energy.

Local Polynomial Regression Solution for Partial Differential Equations with Initial and Boundary Values

Local polynomial regression (LPR) is applied to solve the partial differential equations (PDEs). Usually, the solutions of the problems are separation of variables and eigenfunction expansion methods, so we are rarely able to find analytical solutions. Consequently, we must try to find numerical solutions. In this paper, two test problems are considered for the numerical illustration of the method. Comparisons are made between the exact solutions and the results of the LPR. The results of applying this theory to the PDEs reveal that LPR method possesses very high accuracy, adaptability, and efficiency; more importantly, numerical illustrations indicate that the new method is much more efficient than B-splines and AGE methods derived for the same purpose.

On Kernel Selection of Multivariate Local Polynomial Modelling and its Application to Image Smoothing and Reconstruction

This paper studies the problem of adaptive kernel selection for multivariate local polynomial regression (LPR) and its application to smoothing and reconstruction of noisy images. In multivariate LPR, the multidimensional signals are modeled locally by a polynomial using least-squares (LS) criterion with a kernel controlled by a certain bandwidth matrix. Based on the traditional intersection confidence intervals (ICI) method, a new refined ICI (RICI) adaptive scale selector for symmetric kernel is developed to achieve a better bias-variance tradeoff. The method is further extended to steering kernel with local orientation to adapt better to local characteristics of multidimensional signals. The resulting multivariate LPR method called the steering-kernel-based LPR with refined ICI method (SK-LPR-RICI) is applied to the smoothing and reconstruction problems in noisy images. Simulation results show that the proposed SK-LPR-RICI method has a better PSNR and visual performance than conventional LPR-based methods in image processing.

Modelling flow to and inundation of the Macquarie Marshes in arid Australia

AbstractBuilding of dams and subsequent abstraction upstream of rivers significantly affects major wetland systems. Knowledge of anthropogenic impacts of river regulation, relative to stochastic variation, is essential if managers are able to adequately manage flows to wetlands. We built annual flow and inundated area models for the Macquarie Marshes supplied by the Macquarie River, based on available annual rainfall data (1879–2006). We used LOESS, the flexible local polynomial regression method, ideal for modelling complex processes where no theoretical models exist. To compare effects of river regulation and abstraction, we developed two models, before river regulation (‘natural’ or ‘unregulated’) and after river regulation (‘regulated’). The division was when Burrendong dam, the most significant regulatory structure affecting flows, was built in 1963 and completed in 1967. After this point, water was diverted upstream, predominantly for irrigation. We developed ‘natural’ and ‘regulated’ flow models for three flow gauges Dubbo, Warren and Oxley based on annual rainfall at rainfall stations in the upper catchment and flow, and then inundated area models based on flow and local rainfall to fit annual inundated area data (1979–2006) in the Macquarie Marshes, using LOESS and leave‐one‐out samples without overfitting. We evaluated the performance of the proposed models and used these models to predict annual flows and inundated areas from 1879 to 2006. Comparison was done in terms of ‘natural’ and ‘regulated’ data using Kolmogorov–Smirnov test, identifying significant reductions in flow and wetland area. Copyright © 2009 John Wiley & Sons, Ltd.

Robust non-parametric smoothing of non-stationary time series

Motivated by the need of extracting local trends and low frequency components in non-stationary time series, this paper discusses methods of robust non-parametric smoothing. Basic approach is the combination of the parametric M-estimation with kernel and local polynomial regression methods. The result is an iterative estimator that retains a linear structure, but has kernel weights also in the direction of the prediction errors. The design of smoothing coefficients is carried out with robust cross-validation criteria and rules of thumb. The method works well both to remove the influence of patches of outliers and to detect the local breaks and persistent structural change in time series.

The optimized interpolation of fish positions and speeds in an array of fixed acoustic receivers

Abstract Hedger, R. D., Martin, F., Dodson, J, J., Hatin, D., Caron, F., and Whoriskey, F. G. 2008. The optimized interpolation of fish positions and speeds in an array of fixed acoustic receivers. – ICES Journal of Marine Science, 65: 1248–1259. The principal method for interpolating the positions and speeds of tagged fish within an array of fixed acoustic receivers is the weighted-mean method, which uses a box-kernel estimator, one of the simplest smoothing options available. This study aimed to determine the relative error of alternative, non-parametric regression methods for estimating these parameters. It was achieved by predicting the positions and speeds of three paths made through a dense array of fixed acoustic receivers within a coastal embayment (Gaspé Bay, Québec, Canada) by a boat with a GPS trailing an ultrasonic transmitter. Transmitter positions and speeds were estimated from the receiver data using kernel estimators, with box and normal kernels and the kernel size determined arbitrarily, and by several non-parametric methods, i.e. a kernel estimator, a smoothing spline, and local polynomial regression, with the kernel size or smoothing span determined by cross-validation. Prediction error of the kernel estimator was highly dependent upon kernel size, and a normal kernel produced less error than the box kernel. Of the methods using cross-validation, local polynomial regression produced least error, suggesting it as the optimal method for interpolation. Prediction error was also strongly dependent on array density. The local polynomial regression method was used to determine the movement patterns of a sample of tagged Atlantic salmon (Salmo salar) smolt and kelt, and American eel (Anguilla rostrata). Analysis of the estimates from local polynomial regression suggested that this was a suitable method for monitoring patterns of fish movement.

Intraday stock prices, volume, and duration: a nonparametric conditional density analysis

We investigate the distribution of high-frequency price changes, conditional on trading volume and duration between trades, on four stocks traded on the New York Stock Exchange. The conditional probabilities are estimated nonparametrically using local polynomial regression methods. We find substantial skewness in the distribution of price changes, with the direction of skewness dependent on the sign of trade. We also find that the probability of larger price changes increases with volume, but only for trades that occur with longer durations. The distribution of price changes vary with duration primarily when volume is high.

A Markov model approach shows a large variation in the length of S phase in MCF‐7 breast cancer cells

The potential doubling time of a tumor has been suggested to be a measurement of tumor aggressiveness; therefore, it is of interest to find reliable methods to estimate this time. Because of variability in length of the various cell cycle phases, stochastic modeling of the cell cycle might be a suitable approach. The relative movement curve and the DNA synthesis time were estimated by using local polynomial regression methods. Further, the rate of nucleotide incorporation was estimated by using a Markov pure birth process with one absorbing state to model the progression of the DNA distribution through S phase. An estimate of the DNA synthesis time, with confidence intervals, was obtained from the relative movement curve. The Markov approach provided an estimate of the distribution of the time to complete S phase given the initial distribution. Using the Markov approach we also made an estimate of the mean number of active replicons during S phase. A Markov pure birth process has shown to be useful to model the progression of cells through S phase and to increase knowledge about the variability in the length of S phase and a large variation is shown.

Electrical Conductivities and Critical Micelle Concentrations (Determined by the Local Polynomial Regression Method) of Imipramine and Clomipramine Hydrochlorides from (283 to 313) K

Electrical conductivity measurements for aqueous solutions of imipramine and clomipramine hydrochlorides have been carried out from (283 to 313) K. From these data, critical micelle concentrations (CMCs) have been obtained by application of the local polynomial regression method, which is based on a nonparametric estimation of the regression function. The method is extremely flexible because it does not impose any parametric model on the subjacent structure of the data but rather allows the data to speak for themselves. The variation of the CMC with temperature was a second-order polynomial, with different trends for imipramine (convex) and clomipramine (concave). This distinct behavior was analyzed by the effects of dehydration of the headgroups and thermal solubility of molecules.

A nonparametric approach to calculate critical micelle concentrations: the local polynomial regression method.

The application of a statistical method, the local polynomial regression method, (LPRM), based on a nonparametric estimation of the regression function to determine the critical micelle concentration (cmc) is presented. The method is extremely flexible because it does not impose any parametric model on the subjacent structure of the data but rather allows the data to speak for themselves. Good concordance of cmc values with those obtained by other methods was found for systems in which the variation of a measured physical property with concentration showed an abrupt change. When this variation was slow, discrepancies between the values obtained by LPRM and others methods were found.