Polynomial Smoothing Research Articles

Auto-Scaling Provision Basing on Workload Prediction in the Virtualized Data Center

With the development in the Cloud datacenters, the purpose of the efficient resource allocation is to meet the demand of the users instantly with the minimum rent cost. Thus, the elastic resource allocation strategy is usually combined with the prediction technology. This article proposes a novel predict method combination forecast technique, including both exponential smoothing (ES) and auto-regressive and polynomial fitting (PF) model. The aim of combination prediction is to achieve an efficient forecast technique according to the periodic and random feature of the workload and meet the application service level agreement (SLA) with the minimum cost. Moreover, the ES prediction with PSO algorithm gives a fine-grained scaling up and down the resources combining the heuristic algorithm in the future. APWP would solve the periodical or hybrid fluctuation of the workload in the cloud data centers. Finally, experiments improve that the combined prediction model meets the SLA with the better precision accuracy with the minimum renting cost.

Comparing smoothing techniques for extracting MOSFET threshold voltage

Measurement noise acts as a barrier to the accurate calculation of threshold voltage by derivative-based extraction methods. We examined several smoothing techniques and their effects on measurement noise in threshold voltage extracted by the linear extrapolation method and the gm/Id method. We applied these techniques on a set of SiC power MOSFET devices before and after they had undergone accelerated thermal testing. The smoothing methods examined are least-squares polynomial fitting, low-pass filtering using Butterworth and Chebyshev filters, and differential smoothing. These methods are compared by their shift in threshold voltage compared to unsmoothed data and their standard deviation among the measurements of each device. All methods performed poorly at smoothing when extracting via the gm/Id due to the large amounts of high-magnitude subthreshold noise, mostly tending to extract threshold voltages far below the expected value. Differential smoothing provided results closest to the expected value with an absolute shift of 1.2 V on average. The smoothing methods performed better with the LE method. On a failed device, all smoothing methods except polynomial smoothing preserved the large threshold voltage anomaly for four out of five measurements. These results show that choice of smoothing method, if used must be evaluated based on the device and extraction method.

Single-index modal regression via outer product gradients

Most existing methods for single-index models (SIM) were focused on either mean regression or quantile regression, while the former is sensitive to outliers or heavy tailed distributions and the latter may lose efficiency for normally distributed data. Then a robust, efficient and easily implemented estimation procedure for index coefficient in SIM is developed by integrating the ideas of local modal regression and outer product gradients. Under some mild conditions, we establish the asymptotic normality of the proposed estimators. We further discuss the optimal choices of tuning parameters including one common bandwidth for nonparametric polynomial smoothing and another key bandwidth that controls the robustness and efficiency of the estimator, based on the derived theories. A practical modified EM algorithm is also presented for implementation. Finally, some simulation studies and two real data analysis are conducted to confirm the merits and theoretical findings of the novel method.

Application of curve-fitting techniques to develop numerical calibration procedures for a river water quality model

River water quality models are often constrained by a lack of understanding of model structures and complicated estimation procedures for unknown parameters. This paper demonstrates a new calibration strategy by setting up a simple model structure for river water quality. The unknown parameters of RWQM were calibrated through the use of small river water quality data sets. In order to facilitate the calibration procedure, data reconstruction and parameter estimation were performed by the systematic application of cubic smoothing spline, polynomial curve-fitting and nonlinear least squares. The quality of calibrated parameters was estimated by developing a sensitivity ranking system. The variation of model outputs showed a slight difference at a sensitivity index of less than 10% and a significant difference at a sensitivity index of more than 50%. The one-way analysis of variance showed a large p-value of 0.8431, indicating that differences between model data and measured data means are not significant. The calibrated model responses and their statistical envelopes were in good agreement with the river water quality data. A MATLAB GUI platform was developed to perform numerical and graphical analysis, which can be used as a relatively simple but robust calibration tool to support model application and data analysis.

A Potts‐mixture spatiotemporal joint model for combined magnetoencephalography and electroencephalography data

AbstractWe develop a new methodology for determining the location and dynamics of brain activity from combined magnetoencephalography (MEG) and electroencephalography (EEG) data. The resulting inverse problem is ill‐posed and is one of the most difficult problems in neuroimaging data analysis. In our development we propose a solution that combines the data from three different modalities, magnetic resonance imaging (MRI), MEG and EEG, together. We propose a new Bayesian spatial finite mixture model that builds on the mesostate‐space model developed by Daunizeau & Friston [Daunizeau and Friston, NeuroImage 2007; 38, 67–81]. Our new model incorporates two major extensions: (i) We combine EEG and MEG data together and formulate a joint model for dealing with the two modalities simultaneously; (ii) we incorporate the Potts model to represent the spatial dependence in an allocation process that partitions the cortical surface into a small number of latent states termed mesostates. The cortical surface is obtained from MRI. We formulate the new spatiotemporal model and derive an efficient procedure for simultaneous point estimation and model selection based on the iterated conditional modes algorithm combined with local polynomial smoothing. The proposed method results in a novel estimator for the number of mixture components and is able to select active brain regions, which correspond to active variables in a high‐dimensional dynamic linear model. The methodology is investigated using synthetic data and simulation studies and then demonstrated on an application examining the neural response to the perception of scrambled faces. R software implementing the methodology along with several sample datasets are available at the following GitHub repository https://github.com/v2south/PottsMix. The Canadian Journal of Statistics 47: 688–711; 2019 © 2019 Statistical Society of Canada

W.F. Sheppard's Smoothing Method: A Precursor to Local Polynomial Regression

SummaryW.F. Sheppard has been much overlooked in the history of statistics although his work produced significant contributions. He developed a polynomial smoothing method and corrections of moment estimates for grouped data as well as extensive normal probability tables that have been widely used since the 20th century. Sheppard presented his smoothing method for actuaries in a series of publications during the early 20th century. Population data consist of irregularities, and some adjustment or smoothing of the data is often necessary. Simple techniques, such as Spencer's summation formulae involving arithmetic operations and moving averages, were commonly practised by actuaries to smooth out equally spaced data. Sheppard's method, however, is a polynomial smoothing method based on central differences. We will show how Sheppard's smoothing method was a significant milestone in the development of smoothing techniques and a precursor to local polynomial regression.

Efficient nonparametric estimation and inference for the volatility function

ABSTRACTIn this paper we focus on nonparametric analysis of the volatility function for mixing processes. Our approach is based on local polynomial smoothing and supplies several tools which can be used to test a specific parametric model: nonparametric function estimation, nonparametric confidence intervals, and nonparametric test for symmetry. At the same time, it faces the main drawbacks of the nonparametric procedures proposed so far in the literature that are the choice of the bandwidth parameter and the slow convergence rate. Another aim of this paper is to focus on the advantages of an alternative representation for the parametric model in terms of a Nonparametric-ARCH model, to be estimated by local polynomials. We prove the consistency of the proposed method and investigate its empirical performance on synthetic and real datasets.

Estimating functions and derivatives via adaptive penalized splines

Adaptive penalized splines via radial basis are constructed to estimate regression functions and their derivatives. A weight vector based on the range of observations is embedded into the penalty matrix, which remarkably improves the adaptability of the penalized spline smoothing model. Fast computation and comparison with traditional spline models are studied, and the empirical results and simulations show that the new method outperforms smoothing splines, traditional penalized splines and local polynomial smoothing when estimating regression functions and their derivatives, particularly when the observations have inhomogeneous variation.

Estimation and Inference for Generalized Geoadditive Models

In many application areas, data are collected on a count or binary response with spatial covariate information. In this article, we introduce a new class of generalized geoadditive models (GGAMs) for spatial data distributed over complex domains. Through a link function, the proposed GGAM assumes that the mean of the discrete response variable depends on additive univariate functions of explanatory variables and a bivariate function to adjust for the spatial effect. We propose a two-stage approach for estimating and making inferences of the components in the GGAM. In the first stage, the univariate components and the geographical component in the model are approximated via univariate polynomial splines and bivariate penalized splines over triangulation, respectively. In the second stage, local polynomial smoothing is applied to the cleaned univariate data to average out the variation of the first-stage estimators. We investigate the consistency of the proposed estimators and the asymptotic normality of the univariate components. We also establish the simultaneous confidence band for each of the univariate components. The performance of the proposed method is evaluated by two simulation studies. We apply the proposed method to analyze the crash counts data in the Tampa-St. Petersburg urbanized area in Florida. Supplementary materials for this article are available online.

A practical and effective regularized polynomial smoothing (RPS) method for high-gradient strain field measurement in digital image correlation

Digital image correlation (DIC) is an effective tool for studying the deformation of tested materials. However, high-gradient strain field (HSF) measurement remains a challenging issue in DIC, because multifold factors have a significant effect on the first correlation calculation step of the original displacement and the exact strain reconstruction in the second post-processing step. In this study, we focus on the second step. The formerly proposed local Hermite has been proven as an accurate and robust method, yet substantial programming work is involved in constructing a Hermite element. To solve this issue, a regularized polynomial smoothing (RPS) method is proposed, which replaces the Hermite function of the polynomial. The RPS applies the same kernel function as the point-wise least-squares (PLS) method for easy implementation. Tikhonov regularization and the generalized cross-validation (GCV) function are also used to decrease unavoidable noise and weaken the dependence on window size. The experiments demonstrate that, in the matched/overmatched case, Tikhonov regularization in RPS can greatly decrease the degree of morbidity caused by ill-posed problems in the PLS method, thereby ensuring smoothness and stability of the solution. Moreover, the influences of window size and polynomial order of both the RPS and PLS are also closely studied. In conclusion, the proposed RPS provides a practical, robust and accurate method for HSF measurement.

Application of KPCA combined with SVM in Raman spectral discrimination

Raman spectroscopy has been widely used in discriminant analysis. In order to improve the accuracy of Raman spectroscopy discrimination, a model combining kernel principal component analysis (KPCA) and support vector machine (SVM) is proposed. Firstly, the Raman spectral discriminant data is collected, which is subjected to the fifth-order polynomial smoothing and vector normalization preprocessing to eliminate the influence of noise. Then, the collected unbalanced data is oversampled by the Synthetic Minority Over-sampling Technique algorithm, and the KPCA method is used to extract the features of the balanced data. The SVM discriminant model is established by selecting different kernel functions for the extracted features. In order to evaluate the performance of the KPCA-SVM discriminant model, it is compared with the PCA-SVM discriminant model under the same experimental conditions. The experimental results show that the KPCA-SVM discriminant model achieves a discriminative accuracy rate of 93.75%, which is better than that of the PCA-SVM discriminant model (87.5%). This study provides a new idea for improving the discrimination accuracy of Raman spectroscopy.

PROPERTIES OF DOUBLY ROBUST ESTIMATORS WHEN NUISANCE FUNCTIONS ARE ESTIMATED NONPARAMETRICALLY

An estimator of a finite-dimensional parameter is said to be doubly robust (DR) if it imposes parametric specifications on two unknown nuisance functions, but only requires that one of these two specifications is correct in order for the estimator to be consistent for the object of interest. In this article, we study versions of such estimators that use local polynomial smoothing for estimating the nuisance functions. We show that such semiparametric two-step (STS) versions of DR estimators have favorable theoretical and practical properties relative to other commonly used STS estimators. We also show that these gains are not generated by the DR property alone. Instead, it needs to be combined with an orthogonality condition on the estimation residuals from the nonparametric first stage, which we show to be satisfied in a wide range of models.

English

A variational two-level method in the class of methods with an aggressive coarsening and a massive polynomial smoothing is proposed. The method is a modification of the method of Section 5 of Tezaur, Vaněk (2018). Compared to that method, a significantly sharper estimate is proved while requiring only slightly more computational work.

The use of weight adjusted for height rather than body mass index to assess growth trajectory: Results from a population-based cohort.

We compared different growth models parameterizations regarding (i) adjustment of weight-for-height, as denoted by body mass index (BMI); (ii) adjustment for different covariates, ie, age or height; and (iii) the use of different smoothing methods, ie, polynomial, fractional polynomial, or linear splines. A total of 11 459 measurements of weight and height from 719 participants were used, obtained from the EPITeen cohort at 13, 17, and 21years, and extracted from child health books. The individual growth curves were modeled using mixed-effects polynomial, fractional polynomial, and linear splines, and each model parameterization included as covariate age or height. The goodness-of-fit of the model parametrizations was compared using the relative squared error (RSE) and the relative absolute error (RAE). The adjustment of weight-for-height as denoted by BMI was found to be biased, especially for extreme values of height and presented the worst fit indexes from all model parameterizations tested (RSE=12.46%; RAE=22.63%). Regardless of the smoothing method, the weight-for-height retrieved the best fit indexes in comparison to the adjustment for age. With regard to the smoothing methods and comparing weight-for-height model parameterizations, the fractional polynomial model performed better (RSE=0.75%; RAE=5.70%), followed by linear splines (RSE=0.77%; RAE=5.82%), and conventional polynomial (RSE=0.91%; RAE=6.82%). Therefore, growth modeling in pediatric age should be based on the modeling of weight-for-height because the use of BMI leaves residual confounding for height. Regarding the smoothing methods, although differences were relatively small, the fractional polynomials performed better in comparison to conventional polynomials and linear splines.

Option Implied Risk-Neutral Density Estimation: A Robust and Flexible Method

In practice, a reliable and flexible estimation of risk-neutral density from empirical data is a challenging task since it can not be observed directly from the market. In this study, we apply Bernstein polynomial basis to recover the risk-neutral density function from the observed price quotes of European-type option contingent on an underlying asset. More importantly, we perform an extensive simulation study to examine the flexibility and robustness of the proposed method in recovering different shapes of the true risk-neutral density function from noisy option price quotes. Also, we compare the proposed method with other two popular nonparametric methods namely the constrained local linear polynomial smoothing and the smoothed implied volatility smile reported in the literature. Accuracy and stability of the three nonparametric methods are assessed by the root mean integrated square error criterion. The simulation results show that the proposed method is flexible as it exhibits the various shapes of the true risk-neutral density function even when the volatility is high. Moreover, in comparison with the other two methods, the proposed approach is robust and yields more accurate densities even in the presence of noise. Finally, we demonstrate the applicability of the proposed method in recovering a smooth risk-neutral density function from the S&P 500 market index option data.

Brexit: Tracking and disentangling the sentiment towards leaving the EU

On 23 June, 2016, the UK held a referendum to decide whether to stay in the European Union or leave. The uncertainty surrounding the outcome of this referendum had major consequences for public policy, investment decisions, and currency markets. We discuss some of the subtleties involved in smoothing and disentangling poll data in light of the problem of tracking the dynamics of the intention to Brexit, and propose a multivariate singular spectrum analysis method that produces trendlines on the unit simplex. The trendline yield via multivariate singular spectrum analysis is shown to resemble that of local polynomial smoothing, and singular spectrum analysis presents the nice feature of disentangling the dynamics directly into components that can be interpreted as changes in public opinion or sampling error. The merits and disadvantages of some different approaches for obtaining smooth trendlines on the unit simplex are contrasted, in terms of both local polynomial smoothing and multivariate singular spectrum analysis.

Detecting Heteroscedasticity in Non-Parametric Regression Using Weighted Empirical Processes

Summary Heteroscedastic errors can lead to inaccurate statistical conclusions if they are not properly handled. We introduce a test for heteroscedasticity for the non-parametric regression model with multiple covariates. It is based on a suitable residual-based empirical distribution function. The residuals are constructed by using local polynomial smoothing. Our test statistic involves a ‘detection function’ that can verify heteroscedasticity by exploiting just the independence–dependence structure between the detection function and model errors, i.e. we do not require a specific model of the variance function. The procedure is asymptotically distribution free: inferences made from it do not depend on unknown parameters. It is consistent at the parametric (root n) rate of convergence. Our results are extended to the case of missing responses and illustrated with simulations.

Obtaining the Amplitude-Frequency Response of the Retina and Formalization of its Parameters for Using in Diagnostic Systems

We consider two problems in this work. The first is the justification for the possibility of obtaining the amplitude-frequency response (AFR) of the retina by processing rhythmic electroretinograms (RERG). The second problem is to approximate the obtained AFR of the retina in order to obtain additional formalized features of the current state of the retina in the form of the coefficients of the approximating polynomials. When we obtain the AFR of the eye retina, we take into consideration the spectrum of the input testing signal (stimulus). Light stimuli are periodically repeated short rectangular light pulses of five standard frequencies. Due to the fact that the retina is a nonlinear dynamic object, the changes in the AFR of the retina are evaluated and taken into consideration in the obtaining of the frequency characteristics for each frequency of light flashes. For the polynomial approximation of the obtained AFR's of the retina, it is proposed to distinguish two frequency ranges: the low-frequency range (from 0 to 50 Hz) and the high-frequency range (from 50 to 120 Hz). In the low-frequency range it is proposed to smooth the retinal AFR by the second degree polynomial smoothing, and in the high-frequency range - by the first degree polynomial smoothing. The proposed approximation of the frequency response allows to obtain 25 additional features from five experimentally determined AFR's for one person. In this case each AFR is characterized by five coefficients of smoothing polynomials. The results of the work allow us to compare different methods of classification (diagnosis) with using the received features.

Studies of Associated Solutions: Evaluation of Thermodynamic Parameters of Blends of 2-Methylaniline and Substituted Ethanols at Various Temperatures

Densities (ρ), speeds of sound (u), and viscosities (η) are reported for binary mixtures of 2-methylaniline with substituted ethanols (2-phenylethanol, 2-chloroethanol and 2-aminoethanol) over the entire composition range of mole fraction at T = (303.15–318.15) K and at atmospheric pressure, 0.1 MPa. The excess molar volume, excess isentropic compressibility and deviation in viscosity are calculated from the corresponding experimental densities, speeds of sound and viscosities. The excess properties are correlated using the Redlich–Kister polynomial smoothing equation. Excess partial molar volumes and excess partial molar isentropic compressibilities were calculated for all the binary systems throughout the composition range and also at infinite dilution. The variations in these properties with composition, for all the binary mixtures, suggest that loss of dipolar association, difference in size and shape of the component molecules, dipole–dipole interaction and hydrogen bonding between molecules of 2-methylaniline with 2-phenylethanol, 2-chloroethanol and 2-aminoethanol are involved.

Interaction of a shock wave with multiple spheres suspended in different arrangements

In this study, the unsteady drag force, Fd, drag coefficient, Cd, and the relevant dynamic behaviors of waves caused by the interaction between a planar incident shock wave and a multi-sphere model are investigated by using imbedded accelerometers and a high-speed Schlieren system. The shock wave is produced in a horizontal 200 mm inner diameter circular shock tube with a 2000 mm × 200 mm × 200 mm transparent test section. The time history of Cd is obtained based on band-block and low-pass Fast Fourier Transformation filtering combined with Savitzky-Golay polynomial smoothing for the measured acceleration. The effects of shock Mach number, Ms, geometry of multi-sphere model, nondimensional distance between sphere centers, H, and channel blockage are analyzed. We find that all time histories of Cd have a similar double-peak shaped main structure. It is due to wave reflection, diffraction, interference, and convergence at different positions of the spheres. The peak Fd increases, whereas the peak Cd decreases monotonically with increasing Ms. The increase of shock strength due to shock focusing by upstream spheres increases the peak Fd of downstream spheres. Both the increase in sphere number and the decrease in distance between spheres promote wave interference between neighboring spheres. As long as the wave interference times are shorter than the peak times, the peak Fd and Cd are higher compared to a single sphere.