Kernel Smoothing Research Articles

NONPARAMETRIC REGRESSION ANALYSIS BASED ON RATIONAL (Padé) APPROXIMATION FOR CENSORED-DATA

This paper considers the estimation of a nonparametric regression model with randomly right-censored data. To estimate the model, rational (Padé) approximation based on truncated total least squares (P-TTLS) is used as a smoothing method. Because of censored, data points cannot be used directly in modeling process, a data transformation is needed for overcoming this problem. As known, synthetic data transformation assigns censored points as zero and gives additional magnitudes to uncensored ones associated with Kaplan-Meier distribution of the censored dataset. Thus, the differences between censored and uncensored observations grow which causes a kind of spatial variation in the shape of data. In this paper, to bring a solution to this problematic situation, P-TTLS is used that works well on spatial variation. Also, to see the performance of the P-TTLS on censored data modeling, a simulation study is carried out and it is compared with the benchmarked kernel smoothing (B-KS) method to observe how P-TTLS behaves.

Nonparametric Mass Imputation for Data Integration.

Data integration combining a probability sample with another nonprobability sample is an emerging area of research in survey sampling. We consider the case when the study variable of interest is measured only in the nonprobability sample, but comparable auxiliary information is available for both data sources. We consider mass imputation for the probability sample using the nonprobability data as the training set for imputation. The parametric mass imputation is sensitive to parametric model assumptions. To develop improved and robust methods, we consider nonparametric mass imputation for data integration. In particular, we consider kernel smoothing for a low-dimensional covariate and generalized additive models for a relatively high-dimensional covariate for imputation. Asymptotic theories and variance estimation are developed. Simulation studies and real applications show the benefits of our proposed methods over parametric counterparts.

A new family of kernels from the beta polynomial kernels with applications in density estimation

One of the fundamental data analytics tools in statistical estimation is the non-parametric kernel method that involves probability estimates production. The method uses the observations to obtain useful statistical information to aid the practicing statistician in decision making and further statistical investigations. The kernel techniques primarily examine essential characteristics in a data set, and this research aims to introduce new kernel functions that can easily detect inherent properties in any given observations. However, accurate application of kernel estimator as data analytics apparatus requires the kernel function and smoothing parameter that regulates the level of smoothness applied to the estimates. A plethora of kernel functions of different families and smoothing parameter selectors exist in the literature, but no one method is universally acceptable in all situations. Hence, more kernel functions with smoothing parameter selectors have been propounded customarily in density estimation. This article proposes a distinct kernel family from the beta polynomial kernel family using the exponential progression in its derivation. The newly proposed kernel family was evaluated with simulated and life data. The outcomes clearly indicated that this kernel family could compete favorably well with other kernel families in density estimation. A further comparison of numerical results of the new family and the existing beta family revealed that the new family outperformed the classical beta kernel family with simulation and real data examples with the aid of asymptotic mean integrated squared error (AMISE) as criterion function. The information obtained from the data analysis of this research could be used for decision making in an organization, especially when human and material resources are to be considered. In addition, Kernel functions are vital tools for data analysis and data visualization; hence the newly proposed functions are vital exploratory tools.

Nonparametric location–scale model for the joint forecasting of $$\hbox {SO}_{{2}}$$ and $$\hbox {NO}_{{x}}$$ pollution episodes

We present a method to forecast pollution episodes with a bivariate response. The method simultaneously estimates the concentrations of two pollutants, using historical data. It is based on a location–scale model where the means and the standard deviations are approximated by kernel smoothers in additive models, while the variance–covariance matrix is obtained from the residuals of the previous models. The method provides not only an estimation of the concentration of both pollutants over time but also uncertainty regions covering a specific percentage of the data. The suitability of the model was tested with both simulated and real data (specifically $$\hbox {SO}_2$$ and $$\hbox {NO}_x$$ concentrations from a coal-fired power station). The results have proved highly satisfactory in both cases. The percentage of data covered by the uncertainty region, its area and a new loss function, a variant of the pinball loss function, were used as metrics to evaluate the performance of the model.

A PRECIPITAÇÃO E O CRITÉRIO DE ESCOLHA DE ANOS-PADRÃO

A proposta deste estudo é desenvolver com base no elemento precipitação, as etapas para a eleição de anos-padrão ao estado do Ceará, considerando a aplicação do método de Tavares (1976). Como critério de escolha, fez-se uso também da espacialização geoestatística dos dados por meio do estimador Kernel Smoothing. A base de dados utilizada é composta de 184 postos pluviométricos da Fundação Cearense de Meteorologia e Recursos Hídricos (FUNCEME), e compreende uma escala temporal de 20 anos, entre 1991-2010. Os dados foram organizados conforme os critérios adotados por Tavares (1976), sendo estes validados por meio das ferramentas de validação cruzada e validação/previsão. Feito a validação, os dados foram quantificados em histogramas e espacializados a partir do estimador Kernel Smoothing. Pela quantificação dos histogramas e análises dos mapas, foram selecionados como anos-padrão para o estado do Ceará: 2005, como ano seco, 2004, como ano habitual e 2009, como ano chuvoso.

L&T bags EPC contract for NTPC's CO2-to-methanol demo plant

Accurate information of path followed by vehicles at several real-life scenarios is essential for modeling and testing the optimal path for autonomous vehicles, analyzing the lateral dynamics such as lane-changing and overtaking, vehicle dynamics at the horizontal curves, intersections, and roundabouts. Recently, the availability of such data is plenty, thanks to the advancements in drone and computer-vision technologies. Nevertheless, extracting useful information from the observed data requires certain pre-processing that can handle the noises added at several stages of the data collection. The adequacy of the existing off-the-shelf smoothing techniques (e.g., Moving average, Savitzky–Golay, Kalman filter) for denoising the trajectory data of the modern era is certainly a question. The motivation for the present study is twofold. Firstly, the challenges of contemporary trajectory data collection and associated error patterns are relatively unknown, and thus an appropriate denoising technique needs to be developed. Secondly, many of the existing smoothing techniques have chosen invariant parameters across all the vehicle paths, even though the error patterns are dissimilar across the vehicles and, in the longitudinal and lateral directions. Thus, an adaptive smoothing needs to be applied for each observed vehicle path and its components. The present study proposes an adaptive, and data-driven path smoothing technique that can be universally applied to the video-based path data obtained from any camera platform (either drone or fixed camera setup). The proposed reconstruction framework works in three-stages; in the first stage, the data is prepared by resampling any missing data. Also, any outlier in the observed path data is identified and removed at this stage. In the second stage, a Recursively Ensembled Low-Pass (RELP) filter is proposed to handle the ’heavily tailed’ noise found in the video-processing based trajectory data, particularly for drone videos. A robust adaptive Gaussian kernel smoothing is applied in the third stage to have a localized reconstruction. For Kernel smoothing, the smoothing parameters such as the optimal bandwidth and polynomial order are estimated using the proposed grid-search algorithm. The parameter estimation process ensures that the bias and variance are perfectly traded-off to achieve a smooth vehicle path. The performance evaluation of the proposed method shows an internally and physically consistent trajectory reconstruction.

Vehicle path reconstruction using Recursively Ensembled Low-pass filter (RELP) and adaptive tri-cubic kernel smoother

Accurate information of path followed by vehicles at several real-life scenarios is essential for modeling and testing the optimal path for autonomous vehicles, analyzing the lateral dynamics such as lane-changing and overtaking, vehicle dynamics at the horizontal curves, intersections, and roundabouts. Recently, the availability of such data is plenty, thanks to the advancements in drone and computer-vision technologies. Nevertheless, extracting useful information from the observed data requires certain pre-processing that can handle the noises added at several stages of the data collection. The adequacy of the existing off-the-shelf smoothing techniques (e.g., Moving average, Savitzky–Golay, Kalman filter) for denoising the trajectory data of the modern era is certainly a question. The motivation for the present study is twofold. Firstly, the challenges of contemporary trajectory data collection and associated error patterns are relatively unknown, and thus an appropriate denoising technique needs to be developed. Secondly, many of the existing smoothing techniques have chosen invariant parameters across all the vehicle paths, even though the error patterns are dissimilar across the vehicles and, in the longitudinal and lateral directions. Thus, an adaptive smoothing needs to be applied for each observed vehicle path and its components. The present study proposes an adaptive, and data-driven path smoothing technique that can be universally applied to the video-based path data obtained from any camera platform (either drone or fixed camera setup). The proposed reconstruction framework works in three-stages; in the first stage, the data is prepared by resampling any missing data. Also, any outlier in the observed path data is identified and removed at this stage. In the second stage, a Recursively Ensembled Low-Pass (RELP) filter is proposed to handle the ’heavily tailed’ noise found in the video-processing based trajectory data, particularly for drone videos. A robust adaptive Gaussian kernel smoothing is applied in the third stage to have a localized reconstruction. For Kernel smoothing, the smoothing parameters such as the optimal bandwidth and polynomial order are estimated using the proposed grid-search algorithm. The parameter estimation process ensures that the bias and variance are perfectly traded-off to achieve a smooth vehicle path. The performance evaluation of the proposed method shows an internally and physically consistent trajectory reconstruction.

Diffusive Limit of Non-Markovian Quantum Jumps.

We solve two long-standing problems for stochastic descriptions of open quantum system dynamics. First, we find the classical stochastic processes corresponding to non-Markovian quantum state diffusion and non-Markovian quantum jumps in projective Hilbert space. Second, we show that the diffusive limit of non-Markovian quantum jumps can be taken on the projective Hilbert space in such a way that it coincides with non-Markovian quantum state diffusion. However, the very same limit taken on the Hilbert space leads to a completely new diffusive unraveling, which we call non-Markovian quantum diffusion. Further, we expand the applicability of non-Markovian quantum jumps and non-Markovian quantum diffusion by using a kernel smoothing technique allowing a significant simplification in their use. Lastly, we demonstrate the applicability of our results by studying a driven dissipative two level atom in a non-Markovian regime using all of the three methods.

Inference in time series models using smoothed-clustered standard errors

This paper proposes a long run variance estimator for conducting inference in time series regression models that combines the nonparametric approach with a cluster approach. The basic idea is to divide the time periods into non-overlapping clusters. The long run variance estimator is constructed by first aggregating within clusters and then kernel smoothing across clusters or applying the nonparametric series method to the clusters with Type II discrete cosine transform. We develop an asymptotic theory for test statistics based on these “smoothed-clustered” long run variance estimators. We derive asymptotic results holding the number of clusters fixed and also treating the number of clusters as increasing with the sample size. For the kernel smoothing approach, these two asymptotic limits are different whereas for the cosine series approach, the two limits are the same. When clustering before kernel smoothing, we find that the “fixed-number-of-clusters” asymptotic approximation works well whether the number of clusters is small or large. Finite sample simulations suggest that the naive i.i.d. bootstrap mimics the fixed-number-of-clusters critical values. The simulations also suggest that clustering before kernel smoothing can reduce over-rejections caused by strong serial correlation at a cost of power. When there is a natural way of clustering, clustering can reduce over-rejection problems and achieve small gains in power for the kernel approach. In contrast, the cosine series approach does not benefit from clustering.

Nonparametric Range-Based Double Smoothing Spot Volatility Estimation for Diffusion Models

We consider nonparametric spot volatility estimation for diffusion models with discrete high frequency observations. Our estimator is carried out in two steps. First, using the local average of the range-based variance, we propose a crude estimator of the spot volatility. Second, we use usual nonparametric kernel smoothing to reconstruct the volatility function from the crude estimator. By inference, we find such a double smoothing operation can effectively reduce the estimation error.

Functional lagged regression with sparse noisy observations

A functional (lagged) time series regression model involves the regression of scalar response time series on a time series of regressors that consists of a sequence of random functions. In practice, the underlying regressor curve time series are not always directly accessible, but are latent processes observed (sampled) only at discrete measurement locations. In this article, we consider the so‐called sparse observation scenario where only a relatively small number of measurement locations have been observed, possibly different for each curve. The measurements can be further contaminated by additive measurement error. A spectral approach to the estimation of the model dynamics is considered. The spectral density of the regressor time series and the cross‐spectral density between the regressors and response time series are estimated by kernel smoothing methods from the sparse observations. The impulse response regression coefficients of the lagged regression model are then estimated by means of ridge regression (Tikhonov regularization) or principal component analysis (PCA) regression (spectral truncation). The latent functional time series are then recovered by means of prediction, conditioning on all the observed data. The performance and implementation of our methods are illustrated by means of a simulation study and the analysis of meteorological data.

An Enhanced pix2pix Dehazing Network with Guided Filter Layer

In this paper, we propose an enhanced pix2pix dehazing network, which generates clear images without relying on a physical scattering model. This network is a generative adversarial network (GAN) which combines multiple guided filter layers. First, the input of hazy images is smoothed to obtain high-frequency features according to different smoothing kernels of the guided filter layer. Then, these features are embedded in higher dimensions of the network and connected with the output of the generator’s encoder. Finally, Visual Geometry Group (VGG) features are introduced to serve as a loss function to improve the quality of the texture information restoration and generate better hazy-free images. We conduct experiments on NYU-Depth, I-HAZE and O-HAZE datasets. The enhanced pix2pix dehazing network we propose produces increases of 1.22 dB in the Peak Signal-to-Noise Ratio (PSNR) and 0.01 in the Structural Similarity Index Metric (SSIM) compared with a second successful comparison method using the indoor test dataset. Extensive experiments demonstrate that the proposed method has good performance for image dehazing.

Effects of different smoothing on global and regional resting functional connectivity.

Spatial smoothing is an essential pre-processing step in the process of analysing functional magnetic resonance imaging (fMRI) data, both during an experimental task or during resting-state fMRI (rsfMRI). The main benefit of this spatial smoothing step is to artificially increase the signal-to-noise ratio of the fMRI signal. Previous fMRI studies have investigated the impact of spatial smoothing on task fMRI data, while rsfMRI studies usually apply the same analytical process used for the task data. However, this study investigates changes in different rsfMRI analyses, such as ROI-to-ROI, seed-to-voxels and ICA analyses. Nineteen healthy volunteers were scanned using rsfMRI with three applied smoothing kernels: 0 mm, 4 mm and 8 mm. Appropriate statistical comparisons were made. The findings showed that spatial smoothing has a greater effect on rsfMRI data when analysed using seed-to-voxel-based analysis. The effect was less pronounced when analysing data using ROI-ROI or ICA analyses. The results demonstrated that even when analysing the data without the application of spatial smoothing, the results were significant compared with data analysed using a typical smoothing kernel. However, data analysed with lower-smoothing kernels produced greater negative correlations, particularly with the ICA analysis. The results suggest that a medium smoothing kernel (around 4 mm) may be preferable, as it is comparable with the 8 mm kernel in all of the analyses performed. It is also recommended that the researchers consider analysing the data using two different smoothing kernels, as this will help to confirm the significance of the results and avoid overestimating the findings.

Estimating the Nonparametric Regression Function by Using Padé Approximation Based on Total Least Squares

In this paper, we propose a Padé-type approximation based on truncated total least squares (P – TTLS) and compare it with three commonly used smoothing methods: Penalized spline, Kernel smoothing and smoothing spline methods that have become very powerful smoothing techniques in the nonparametric regression setting. We consider the nonparametric regression model, and discuss how to estimate smooth regression function g where we are unsure of the underlying functional form of g. The Padé approximation provides a linear model with multi-collinearities and errors in all its variables. The P – TTLS method is primarily designed to address these issues, especially for solving error-contaminated systems and ill-conditioned problems. To demonstrate the ability of the method, we conduct Monte Carlo simulations under different conditions and employ a real data example. The outcomes of the experiments show that the fitted curve solved by P – TTLS is superior to and more stable than the benchmarked penalized spline (B – PS), Kernel smoothing (KS) and smoothing spline (SS) techniques.

Fixed design local polynomial smoothing and bandwidth selection for right censored data

The local polynomial smoothing of the Kaplan–Meier estimate for fixed designs is explored and analyzed. The first benefit, in comparison to classical convolution kernel smoothing, is the development of boundary aware estimates of the distribution function, its derivatives and integrated derivative products of any arbitrary order. The advancements proceed by developing asymptotic mean integrated square error optimal solve-the-equation plug-in bandwidth selectors for the estimates of the distribution function and its derivatives, and as a byproduct, a mean square error optimal bandwidth rule for integrated derivative products. The asymptotic properties of all methodological contributions are quantified analytically and discussed in detail. Three real data analyses illustrate the benefits of the proposed methodology in practice. Finally, numerical evidence is provided on the finite sample performance of the proposed technique with reference to benchmark estimates.

Wavelet Estimation of Semi-parametric Regression Model

The semi-parametric regression model combines parametric and nonparametric regression. However, non-parametric estimation may provide flexible solutions to the problems suffers by the regression model, but the problem of dimensionality that this estimator suffers, which occurs due to the increasing number of explanatory variables, still remain, this, in turn, may reduce the accuracy of the estimation process. Estimate the non-parametric part of the semi-parametric models that can be studied using conventional non-parametric methods such as the Spline Smoothing and Kernel Smoothing. However, there are other non-parametric methods that can be used, therefore, in this paper, the semi-parametric regression model was estimated by employing the wavelet estimate for the soft threshold, according to the "Speckman" method, and then comparing it with the two methods, Nadaraya-Watson and Local Linear, through the implementation of simulation experiments that included different sample sizes and threshold values. The parametric part estimation of the partially linear model according to the least-squares method was not identical to those estimates using the Speckman method, that is because the least-squares method was not appropriate for the uneven nature of the number of weekly work hours. Simulation experiments have demonstrated the efficiency of the wavelet estimation method and its superiority over other methods. The above estimation methods were applied to real data related to the study of the production value for the public industrial sector in Iraq, and some factors affect it, such as the value of industrial supplies, the total wages of workers, and the number of workers.

Particle filter-based data assimilation technique for the evaluation of transport of pollutants in small rivers

The discharge of pollutants into water resources has caused many undesirable effects on human health and the environment. In this context, meta-heuristic methods, such as the Simulated Annealing algorithm, allow for only offline estimation of the concentration and dispersion coefficient of pollutant. However, online data assimilation is a very important for decision support in real time. In this regard, the present paper focuses on the use of the so-called Particle Filter as an online observer. Our studies were performed through the algorithms Sampling Importance Resampling (SIR) and Liu and West (LW), using tracer experimental observations from a small river in Brazil. Such approach allowed for tracking satisfactorily the pollutant concentration, longitudinal dispersion coefficient, flow velocity, pollutant load released and distance from the pollution source. The results can be enhanced ever more through the LW filter, which reduces the impoverishment of the samples as well as the computational time due to the use of a smoothing kernel.

Optimal bandwidth selection for a kernel density with a location-scale property

This paper proposes a novel estimator of the bandwidth in kernel density estimation, built from an adaptive combination of the strengths of classical kernel smoothing and the desirable properties inherent in adjusted kernel smoothing as studied under a variety location and scale family of functions. The optimal bandwidth exists as a function of density functionals derived as the minimizer of the mean square error. We provide a substantial computational assessment of the performance of our proposed estimator on both simulated and real data, and it is very encouraging to notice that our proposed method exhibits a lower variance than its previous counterparts, and also yields an overall smaller mean squared error.

Estimating Yield Curves by Kernel Smoothing Methods

We introduce a new method for the estimation of discount functions, yield curves and forward curves from government issued coupon bonds. Our approachis nonparametric and does not assume a particular functional form for thediscount function although we do show how to impose various restrictions inthe estimation. Our method is based on kernel smoothing and is defined asthe minimum of some localized population moment condition. The solution tothe sample problem is not explicit and our estimation procedure isiterative, rather like the backfitting method of estimating additivenonparametric models. We establish the asymptotic normality of our methodsusing the asymptotic representation of our estimator as an infinite serieswith declining coefficients. The rate of convergence is standard for onedimensional nonparametric regression.

Dynamic model of photovoltaic module temperature as a function of atmospheric conditions

Abstract. The temperature of photovoltaic modules is modelled as a dynamic function of ambient temperature, shortwave and longwave irradiance and wind speed, in order to allow for a more accurate characterisation of their efficiency. A simple dynamic thermal model is developed by extending an existing parametric steady-state model using an exponential smoothing kernel to include the effect of the heat capacity of the system. The four parameters of the model are fitted to measured data from three photovoltaic systems in the Allgäu region in Germany using non-linear optimisation. The dynamic model reduces the root-mean-square error between measured and modelled module temperature to 1.58 K on average, compared to 3.03 K for the steady-state model, whereas the maximum instantaneous error is reduced from 20.02 to 6.58 K.