Polynomial Smoothing Research Articles

Improved convergence bounds for two-level methods with an aggressive coarsening and massive polynomial smoothing

An improved convergence bound for the polynomially accelerated two-level method of Brousek et al. [Electron. Trans. Numer. Anal., 44 (2015), pp. 401–442, Section 5] is proven. This method is a reinterpretation of the smoothed aggregation method with an aggressive coarsening and massive polynomial smoothing of Vaněk, Brezina, and Tezaur [SIAM J. Sci. Comput., 21 (1999), pp. 900–923], and its convergence rate estimate is improved here quantitatively. Next, since the symmetrization of the method requires two solutions of the coarse problem, a modification of the method is proposed that does not have this disadvantage, and a qualitatively better convergence result for the modification is established. In particular, it is shown that a bound of the convergence rate of the method with a multiply (k-times) smoothed prolongator is asymptotically inversely proportional to d^{2k}, where d is the degree of the smoothing polynomial. In earlier works, this acceleration effect is only quadratic. Finally, for another modified multiply smoothed method, it is proved that this convergence improvement is not limited only to an asymptotic regime but holds true everywhere.

Identification and estimation of nonseparable single-index models in panel data with correlated random effects

The identification in a nonseparable single-index models with correlated random effects is considered in panel data with a fixed number of time periods. The identification assumption is based on the correlated random effects structure. Under this assumption, the parameters of interest are identified up to a multiplicative constant and could be estimated by an average difference of derivatives estimator based on the local polynomial smoothing. We suggest to linearly combine the estimators obtained for different orders of differences and derive the variance-minimizing weighting scheme. The asymptotic distribution of the proposed estimators is derived both for stationary and non-stationary explanatory variables along with a test of the stationarity. Finally, Monte Carlo experiments reveal finite sample properties of the proposed estimator.

Intervehicle Range Estimation From Periodic Broadcasts

Dedicated short-range communications enables vehicular communication using periodic broadcast messages. We propose to use these periodic broadcasts to perform intervehicle ranging. Motivated by this scenario, we study the general problem of range estimation between pairs of moving vehicles using periodic broadcasts. Each vehicle has its own independent and unsynchronized clock, which can exhibit significant drift between consecutive periodic broadcast transmissions. Consequently, both the clock offsets and drifts need to be taken into account in addition to the vehicle motion to accurately estimate the vehicle ranges. We develop a range estimation algorithm using local polynomial smoothing of the vehicle motion. The proposed algorithm can be applied to networks with arbitrary number of vehicles and requires no additional message exchanges apart from the periodic broadcasts. We experimentally validate our algorithm and show that its performance is close to that of unicast round-trip time ranging. In particular, we are able to achieve submeter ranging accuracies in vehicular scenarios. Our scheme requires additional timestamp information to be transmitted as part of the broadcast messages, and we develop a novel timestamp compression algorithm to minimize the resulting overhead.

The role of media in preventing student suicides: A Hong Kong experience

IntroductionIn view of the surge of student suicides in March 2016, a large-scale media involvement was engaged to minimize copycat effects and to extensively spread more preventive information. MethodsTrend differences between student suicides in overall as well as youth suicides between 2003 and 2016 were examined. Impacts of media involvement were examined as to the changes of their intensities in reporting student suicides in different manners. Local polynomial smoothing method was used to estimate the intensities throughout the study period. ResultsThe intensity of students’ suicides has been slowly increasing since 2006, which was in contrast to the decreasing trend of overall suicide intensity. After our engagement with the media, acute descriptive-reporting in student suicide news has sharply dropped and remained low since then, whereas preventive-reporting has increased sharply in March and April 2016, but dropped back to normal afterwards. The higher intensity of preventive-reporting seemed to have a protective effect on student suicide. LimitationsData of student suicides in 2016 were substantially extracted from the local news, which may not contain complete information. The analyses suggested, but could not confirm, a causal relationship between the changes of student suicide news-reporting and the incidents of suicides themselves. ConclusionsThe recent surge of student suicides in Hong Kong was statistically abnormal. Media engagement has changed local suicide news-reporting to become more preventive, which could have contributed to the drop of student suicides. It is essential to sustain the media's coverage of preventive reporting to maintain its enduring effect.

Robust estimators for additive models using backfitting

ABSTRACTAdditive models provide an attractive setup to estimate regression functions in a nonparametric context. They provide a flexible and interpretable model, where each regression function depends only on a single explanatory variable and can be estimated at an optimal univariate rate. Most estimation procedures for these models are highly sensitive to the presence of even a small proportion of outliers in the data. In this paper, we show that a relatively simple robust version of the backfitting algorithm (consisting of using robust local polynomial smoothers) corresponds to the solution of a well-defined optimisation problem. This formulation allows us to find mild conditions to show Fisher consistency and to study the convergence of the algorithm. Our numerical experiments show that the resulting estimators have good robustness and efficiency properties. We illustrate the use of these estimators on a real data set where the robust fit reveals the presence of influential outliers.

Multigrid methods for convection–diffusion problems discretized by a monotone scheme

We study multigrid (MG) methods for the solution of systems of linear algebraic equations obtained from a stable discretization of convection–diffusion problems by an exponential fitting scheme. The latter ensures the stability of the simplest possible coarse grid operators obtained from Galerkin projections based on graph matching. Linear and nonlinear MG preconditioners are defined in the framework of algebraic multilevel iteration. The option of using polynomial smoothers is investigated in context of nonsymmetric problems and a systematic performance comparison is presented for various algorithms on a representative set of two- and three-dimensional test problems.

Nearest neighbor estimates of regression

New nearest neighbor estimators of the nonparametric regression function and its derivatives are developed. Asymptotic normality is obtained for the proposed estimators over the interior points and the boundary region. Connections with other estimators such as local polynomial smoothers are established. The proposed estimators are boundary adaptive and extensions of the Stute estimators. Asymptotic minimax risk properties are also established for the proposed estimators. Simulations are conducted to compare the performance of the proposed estimators with others.

Studies on the importance of chain length of alkanols on the thermodynamic and transport properties of liquid mixtures at various temperatures

Densities (ρ), speeds of sound (u), and viscosities (η) are reported for binary mixtures of N,N-diethyl aniline with higher 1-alkanols (1-octanol, 1-nonanol and 1-decanol) over the entire composition range of mole fraction at T=(303.15–318.15) K and at atmospheric pressure 0.1MPa. The excess properties such as excess molar volume, excess isentropic compressibility and deviation in viscosity are calculated from the experimental density, speed of sound and viscosity. Excess properties are correlated using the Redlich–Kister polynomial smoothing equation. The excess partial molar volumes and excess partial molar isentropic compressibilities are calculated for all the binary systems throughout the composition range and also at infinite dilutions. The results are analysed in terms of rupture of hydrogen bonded chain of the dipolar interaction between solute and alkan-1-ol exceeds the intermolecular interaction through dipole-dipole and hydrogen bonding between N,N-diethyl aniline and 1-alkanol molecules. The VE results are analysed in the light of Prigogine-Flory-Patterson theory. Analysis of each of the three contributions viz. interactional, free volume and P∗ to VE has shown that interactional contribution are positive for all systems, the free volume effect are negative for all the mixtures and P∗ contribution are positive for all the mixtures except 1-octanol which has a negative internal pressure. The variations of these parameters with composition and temperature discussed in terms of intermolecular interactions prevailing in these mixtures.

Background Subtraction of Raman Spectra Based on Iterative Polynomial Smoothing

In this paper, a novel background subtraction algorithm is presented that can automatically recover Raman signal. This algorithm is based on an iterative polynomial smoothing method that highly reduces the need for experience and a priori knowledge. First, a polynomial filter is applied to smooth the input spectrum (the input spectrum is just an original spectrum at the first iteration). The output curve of the filter divides the original spectrum into two parts, top and bottom. Second, a proportion is calculated between the lowest point of the signal in the bottom part and the highest point of the signal in the top part. The proportion is a key index that decides whether to go into a new iteration. If a new iteration is needed, the minimum value between the output curve and the original spectrum forms a new curve that goes into the same filter in the first step and continues as another iteration until no more iteration is needed to finally get the background of the original spectrum. Results from the simulation experiments not only show that the iterative polynomial smoothing algorithm achieves good performance, processing time, cost, and accuracy of recovery, but also prove that the algorithm adapts to different background types and a large signal-to-noise ratio range. Furthermore, real measured Raman spectra of organic mixtures and non-organic samples are used to demonstrate the application of the algorithm.

Multiscale local polynomial decompositions using bandwidths as scales

The multiscale local polynomial transform, developped in this paper, combines the benefits from local polynomial smoothing with sparse multiscale decompositions. The contribution of the paper is twofold. First, it focusses on the bandwidths used throughout the transform. These bandwidths operate as user controlled scales in a multiscale analysis, which is explained to be of particular interest in the case of nonequispaced data. The paper presents both a likelihood based optimal bandwidth selection and a fast, heuristic approach. The second contribution of the paper is the combination of local polynomial smoothing with orthogonal prefilters, similar to Daubechies' wavelet filters, but defined on irregularly spaced covariate values.

Partially linear single-index proportional hazards model with current status data

A partially linear single-index proportional hazards model with current status data is introduced, where the cumulative hazard function is assumed to be nonparametric and a nonlinear link function is assumed to take a parametric spline function. Efficient estimation and effective algorithm are established. Polynomial spline smoothing is invoked for the estimation of the cumulative baseline hazard function with monotonicity constraint on the functional, while a simultaneous sieve maximum likelihood (SML) estimation is proposed to estimate regression parameters. The proposed SML estimator for the parameter vector is shown to be asymptotically normal and semiparametric efficient. The spline estimator of the functional of the cumulative hazard function is shown to achieve the optimal nonparametric rate of convergence. A simulation study is conducted to examine the finite sample performance of the proposed estimators and algorithm, and an analysis of renal function recovery data is presented.

The Error Theory of Polynomial Smoothing Functions for Support Vector Machines

The Error Theory of Polynomial Smoothing Functions for Support Vector Machines

A new algorithm of image segmentation using curve fitting based higher order polynomial smoothing

Image segmentation plays an efficient role in image analysis which discriminates the objects from its background in pixel level. In accordance with the application, image segmentation is widely spread over various fields. The motivation of this paper is to focus on the application of statistical analysis in image segmentation. In this paper, we have incorporated curve fitting technique on an image to acquire the segmented image thereby extracting information from the images. By using higher order polynomial smoothing curve, appropriate result is obtained from detection of the object. Furthermore, we have calculated the image quality metrics which is a method of statistical analysis to get the quality measures and performance analysis of images. Extensive experiments show that the proposed approach outperforms the existing approaches namely histogram based segmentation, edge detection based segmentation, Ostu’s segmentation and Watershed segmentation. The outcome is derived by applying the proposed algorithm and results obtained are appreciable.

A pointing facilitation system for motor-impaired users combining polynomial smoothing and time-weighted gradient target prediction models

ABSTRACTAs computers become more and more essential for everyday life, people who cannot use them are missing out on an important tool. The predominant method of interaction with a screen is a mouse, and difficulty in using a mouse can be a huge obstacle for people who would otherwise gain great value from using a computer. If mouse pointing were to be made easier, then a large number of users may be able to begin using a computer efficiently where they may previously have been unable to. The present article aimed to improve pointing speeds for people with arm or hand impairments. The authors investigated different smoothing and prediction models on a stored data set involving 25 people, and the best of these algorithms were chosen. A web-based prototype was developed combining a polynomial smoothing algorithm with a time-weighted gradient target prediction model. The adapted interface gave an average improvement of 13.5% in target selection times in a 10-person study of representative users of the system. A demonstration video of the system is available at https://youtu.be/sAzbrKHivEY

A three-step estimation procedure using local polynomial smoothing for inconsistently sampled longitudinal data.

Parametric mixed-effects models are useful in longitudinal data analysis when the sampling frequencies of a response variable and the associated covariates are the same. We propose a three-step estimation procedure using local polynomial smoothing and demonstrate with data where the variables to be assessed are repeatedly sampled with different frequencies within the same time frame. We first insert pseudo data for the less frequently sampled variable based on the observed measurements to create a new dataset. Then standard simple linear regressions are fitted at each time point to obtain raw estimates of the association between dependent and independent variables. Last, local polynomial smoothing is applied to smooth the raw estimates. Rather than use a kernel function to assign weights, only analytical weights that reflect the importance of each raw estimate are used. The standard errors of the raw estimates and the distance between the pseudo data and the observed data are considered as the measure of the importance of the raw estimates. We applied the proposed method to a weight loss clinical trial, and it efficiently estimated the correlation between the inconsistently sampled longitudinal data. Our approach was also evaluated via simulations. The results showed that the proposed method works better when the residual variances of the standard linear regressions are small and the within-subjects correlations are high. Also, using analytic weights instead of kernel function during local polynomial smoothing is important when raw estimates have extreme values, or the association between the dependent and independent variable is nonlinear. Copyright © 2016 John Wiley & Sons, Ltd.

Robust preprocessing for stimulus-based functional MRI of the moving fetus.

Fetal motion manifests as signal degradation and image artifact in the acquired time series of blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) studies. We present a robust preprocessing pipeline to specifically address fetal and placental motion-induced artifacts in stimulus-based fMRI with slowly cycled block design in the living fetus. In the proposed pipeline, motion correction is optimized to the experimental paradigm, and it is performed separately in each phase as well as in each region of interest (ROI), recognizing that each phase and organ experiences different types of motion. To obtain the averaged BOLD signals for each ROI, both misaligned volumes and noisy voxels are automatically detected and excluded, and the missing data are then imputed by statistical estimation based on local polynomial smoothing. Our experimental results demonstrate that the proposed pipeline was effective in mitigating the motion-induced artifacts in stimulus-based fMRI data of the fetal brain and placenta.

A Descriptive Study of the Vital Health Indicators of Four Sub‐Saharan African Countries, 1960–2012

BackgroundThe importance of country‐level vital health statistics is recognized globally because of their use in initiating publichealth and development policies. Over the past few decades, economic growth has occurred in sub‐Saharan African countries, resulting in benefits to the population, such as increased access to quality health care and optimum nutrition. However, attaining the best possible levels of infant nutrition and maternal health remains a challenge.MethodsThe main objective of this study is to present aset of detailed time series analyses from 1960 to 2012 for selected health and wellness indicators, using country‐level aggregated data collected by the World Development Index (WDI) database of the World Bank, with specific focus on the countries of Ghana, Togo, Burkina Faso, and the Ivory Coast. The vital health indicators collected for this region can be used to analyze the health status of the vulnerable populations, epidemiological patterns, and the impact of economic growth on health outcomes; thus, we use these indicators to calculate descriptive statistics and identify trends that may inform hypotheses aimed at improving infant nutrition and maternal health. Gaps in the longitudinal record were imputed via polynomial smoothing. Linear trends over time and differences among countries were assessed using simple ANOVA models, and associations within countries were calculated using Spearman's rank correlation.ResultsSome general trends are common across the countries under consideration, but there are divergences. For example, although there has been a steep decline in infant mortality rates in all four countries since 1960, those of Burkina Faso and the Ivory Coast generally tended to be higher over the period of study (p<0.001). Likewise, life expectancy at birth among women, while generally increasing over the period of study, reversed course between the mid‐1980s and 2000 for all four countries. However, this decrease in life expectancy was significantly more pronounced in the Ivory Coast (p<0.001). Interestingly, the vital indicators in Ghana are more strongly associated(either directly on inversely) with each other over time than in Burkina Faso, Togo or the Ivory Coast; this is due to more consistent improvements in these measures over time and more frequent reporting to the WDI. Indeed, all four of these countries had several measures with less than four records across the period of interest.ConclusionsSub‐Saharan Africa has seen a general improvement in selected health indicators, but it has been mixed and inconsistent across countries. Research efforts in this area are hindered by alack of data consistently reported over time.Support or Funding InformationNone

A Bioequivalence Test by the Direct Comparison of Concentration-versus-Time Curves Using Local Polynomial Smoothers.

In order to test if two chemically or pharmaceutically equivalent products have the same efficacy and/or toxicity, a bioequivalence (BE) study is conducted. The 80%/125% rule is the most commonly used criteria for BE and states that BE cannot be claimed unless the 90% CIs for the ratio of selected pharmacokinetics (PK) parameters of the tested to the reference drug are within 0.8 to 1.25. Considering that estimates of these PK parameters are derived from the concentration-versus-time curves, a direct comparison between these curves motivates an alternative and more flexible approach to test BE. Here, we propose to frame the BE test in terms of an equivalence of concentration-versus-time curves which are constructed using local polynomial smoother (LPS). A metric is presented to quantify the distance between the curves and its 90% CIs are calculated via bootstrapping. Then, we applied the proposed procedures to data from an animal study and found that BE between a generic drug and its brand name cannot be concluded, which was consistent with the results by applying the 80%/125% rule. However, the proposed procedure has the advantage of testing only on a single metric, instead of all PK parameters.

JOINT STRUCTURE SELECTION AND ESTIMATION IN THE TIME-VARYING COEFFICIENT COX MODEL.

Time-varying coefficient Cox model has been widely studied and popularly used in survival data analysis due to its flexibility for modeling covariate effects. It is of great practical interest to accurately identify the structure of covariate effects in a time-varying coefficient Cox model, i.e. covariates with null effect, constant effect and truly time-varying effect, and estimate the corresponding regression coefficients. Combining the ideas of local polynomial smoothing and group nonnegative garrote, we develop a new penalization approach to achieve such goals. Our method is able to identify the underlying true model structure with probability tending to one and simultaneously estimate the time-varying coefficients consistently. The asymptotic normalities of the resulting estimators are also established. We demonstrate the performance of our method using simulations and an application to the primary biliary cirrhosis data.

Estimation of semi-varying coefficient models with nonstationary regressors

ABSTRACTWe study a semivarying coefficient model where the regressors are generated by the multivariate unit root I(1) processes. The influence of the explanatory vectors on the response variable satisfies the semiparametric partially linear structure with the nonlinear component being functional coefficients. A semiparametric estimation methodology with the first-stage local polynomial smoothing is applied to estimate both the constant coefficients in the linear component and the functional coefficients in the nonlinear component. The asymptotic distribution theory for the proposed semiparametric estimators is established under some mild conditions, from which both the parametric and nonparametric estimators are shown to enjoy the well-known super-consistency property. Furthermore, a simulation study is conducted to investigate the finite sample performance of the developed methodology and results.