Local Polynomial Estimation of Heteroscedasticity in a Multivariate Linear Regression Model and Its Applications in Economics

Multivariate local polynomial fitting is applied to the multivariate linear heteroscedastic regression model. Firstly, the local polynomial fitting is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. One noteworthy feature of our approach is that we avoid the testing for heteroscedasticity by improving the traditional two-stage method. Due to non-parametric technique of local polynomial estimation, it is unnecessary to know the form of heteroscedastic function. Therefore, we can improve the estimation precision, when the heteroscedastic function is unknown. Furthermore, we verify that the regression coefficients is asymptotic normal based on numerical simulations and normal Q-Q plots of residuals. Finally, the simulation results and the local polynomial estimation of real data indicate that our approach is surely effective in finite-sample situations.     

Estimation of Individual Growth Curves by Empirically Weighted Least Squares

The simultaneous estimation of individual growth curves and a mean growth curve is accomplished by weighted least squares. A polynomial curve is fitted for each individual and the polynomial parameters are linear functions of parameters corresponding to covariates. A simple, computationally efficient variance-covariance estimator is derived. The resultant estimate is used in the weighted least squares estimation. The results are compared to empirical Bayes estimation.     

Estimation from Censored Medical Cost Data

This paper applies the inverse probability weighted least‐squares method to predict total medical cost in the presence of censored data. Since survival time and medical costs may be subject to right censoring and therefore are not always observable, the ordinary least‐squares approach cannot be used to assess the effects of explanatory variables. We demonstrate how inverse probability weighted least‐squares estimation provides consistent asymptotic normal coefficients with easily computable standard errors. In addition, to assess the effect of censoring on coefficients, we develop a test comparing ordinary least‐squares and inverse probability weighted least‐squares estimators. We demonstrate the methods developed by applying them to the estimation of cancer costs using Medicare claims data. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Penalized local polynomial regression for spatial data

When performing spatial regression analysis in environmental data applications, spatial heterogeneity in the regression coefficients is often observed. Spatially varying coefficient models, including geographically weighted regression and spline models, are standard tools for quantifying such heterogeneity. In this paper, we propose a spatially varying coefficient model that represents the spatially varying parameters as a mixture of local polynomials at selected locations. The local polynomial parameters have attractive interpretations, indicating various types of spatial heterogeneity. Instead of estimating the spatially varying regression coefficients directly, we develop a penalized least squares regression procedure for the local polynomial parameter estimation, which both shrinks the parameter estimation and penalizes the differences among parameters that are associated with neighboring locations. We develop confidence intervals for the varying regression coefficients and prediction intervals for the response. We apply the proposed method to characterize the spatially varying association between particulate matter concentrations ( ${\text{PM}}_{2.5}$) and pollutant gases related to the secondary aerosol formulation in China. The identified regression coefficients show distinct spatial patterns for nitrogen dioxide, sulfur dioxide, and carbon monoxide during different seasons.     

Photometric elisa calibration for antibody assay

Summary The calibration of a quantitative ELISA assay for monoclonal antibodies based on a photometric measurement may be carried out using a cubic polynomial function of the logarithm of the antibody concentration, The fit is good and the error structure is well suited to the least squares fit method. The antibody concentrations are obtained from the roots of the calibration equation calculated by the cubic formulae.     

A contribution to optimal control of fed-batch biochemical processes

The problem of looking for high efficient modern control strategies in fermentation technology is very urgent, nowdays. Particular attention should be paid to the processes in fed-batch mode. Both, optimal feedforward and feedback control approaches are suggested. A contribution is considered to have been made in the feedback control where continuous and discrete versions are treated as well. The control laws are carried out by a variation calculus problem and a polynomial pole placement synthesis solution, respectively. All the algorithms result in an optimal substrate feed rate profile. On the basis of recursive least squares identification of the model coefficients an adaptive discrete-time control strategy is proposed. Some satisfying simulation results are dealt with.     

Numerical Results on Growth Functions

To given values for the weights of twelve rats and the averages of some of them there are fitted different smooth functions by the method of least squares. Piecewise polynomial approximation before and after the turning point suggests via a differential equation an approximation by exponential functions. The composition of these exponential functions leads to an implicit function, which can approximately be substituted by an inverses polynomial. Contrary to the usual growth functions, here the mass of the animals is unbounded, what is adequate to ad libitum feeding supposing immortality. The numerical results are laid down in ten tables and they are provided with comments.     

Missing value estimation for DNA microarray gene expression data: local least squares imputation

MOTIVATION: Gene expression data often contain missing expression values. Effective missing value estimation methods are needed since many algorithms for gene expression data analysis require a complete matrix of gene array values. In this paper, imputation methods based on the least squares formulation are proposed to estimate missing values in the gene expression data, which exploit local similarity structures in the data as well as least squares optimization process. RESULTS: The proposed local least squares imputation method (LLSimpute) represents a target gene that has missing values as a linear combination of similar genes. The similar genes are chosen by k-nearest neighbors or k coherent genes that have large absolute values of Pearson correlation coefficients. Non-parametric missing values estimation method of LLSimpute are designed by introducing an automatic k-value estimator. In our experiments, the proposed LLSimpute method shows competitive results when compared with other imputation methods for missing value estimation on various datasets and percentages of missing values in the data. AVAILABILITY: The software is available at http://www.cs.umn.edu/~hskim/tools.html CONTACT: hpark@cs.umn.edu     

Nonparametrically Weighted Least Squares Estimation in Heteroscedastic Linear Regression

The weights used in iterative weighted least squares (IWLS) regression are usually estimated parametrically using a working model for the error variance. When the variance function is misspecified, the IWLS estimates of the regression coefficients β are still asymptotically consistent but there is some loss in efficiency. Since second moments can be quite hard to model, it makes sense to estimate the error variances nonparametrically and to employ weights inversely proportional to the estimated variances in computing the WLS estimate for β. Surprisingly, this approach had not received much attention in the literature. The aim of this note is to demonstrate that such a procedure can be implemented easily in S-plus using standard functions with default options making it suitable for routine applications. The particular smoothing method that we use is local polynomial regression applied to the logarithm of the squared residuals but other smoothers can be tried as well. The proposed procedure is applied to data on the use of two different assay methods for a hormone. Efficiency calculations based on the estimated model show that the nonparametric IWLS estimates are more efficient than the parametric IWLS estimates based on three different plausible working models for the variance function. The proposed estimators also perform well in a simulation study using both parametric and nonparametric variance functions as well as normal and gamma errors.     

A hybrid least squares and principal component analysis algorithm for Raman spectroscopy

Raman spectroscopy is a powerful technique for detecting and quantifying analytes in chemical mixtures. A critical part of Raman spectroscopy is the use of a computer algorithm to analyze the measured Raman spectra. The most commonly used algorithm is the classical least squares method, which is popular due to its speed and ease of implementation. However, it is sensitive to inaccuracies or variations in the reference spectra of the analytes (compounds of interest) and the background. Many algorithms, primarily multivariate calibration methods, have been proposed that increase robustness to such variations. In this study, we propose a novel method that improves robustness even further by explicitly modeling variations in both the background and analyte signals. More specifically, it extends the classical least squares model by allowing the declared reference spectra to vary in accordance with the principal components obtained from training sets of spectra measured in prior characterization experiments. The amount of variation allowed is constrained by the eigenvalues of this principal component analysis. We compare the novel algorithm to the least squares method with a low-order polynomial residual model, as well as a state-of-the-art hybrid linear analysis method. The latter is a multivariate calibration method designed specifically to improve robustness to background variability in cases where training spectra of the background, as well as the mean spectrum of the analyte, are available. We demonstrate the novel algorithm's superior performance by comparing quantitative error metrics generated by each method. The experiments consider both simulated data and experimental data acquired from in vitro solutions of Raman-enhanced gold-silica nanoparticles.     

Varying coefficient model with unknown within-subject covariance for analysis of tumor growth curves

SUMMARY: In this article we develop a nonparametric estimation procedure for the varying coefficient model when the within-subject covariance is unknown. Extending the idea of iterative reweighted least squares to the functional setting, we iterate between estimating the coefficients conditional on the covariance and estimating the functional covariance conditional on the coefficients. Smoothing splines for correlated errors are used to estimate the functional coefficients with smoothing parameters selected via the generalized maximum likelihood. The covariance is nonparametrically estimated using a penalized estimator with smoothing parameters chosen via a Kullback-Leibler criterion. Empirical properties of the proposed method are demonstrated in simulations and the method is applied to the data collected from an ovarian tumor study in mice to analyze the effects of different chemotherapy treatments on the volumes of two classes of tumors.     

Estimating drug shelf-life with random batches.

The problem of assessing the shelf-lives of drug products in the market is considered. We focus on the situation where the drug characteristic of interest has a linear relationship over time and batch-to-batch variation is present. Two methods for assessing shelf-lives are proposed based on the weighted least squares method under a regression model with random coefficients. An application to some stability data from the pharmaceutical industry is presented.     

A two-stage model for multiple time series data of counts

We propose a two-stage model for time series data of counts from multiple locations. This method fits first-stage model(s) using the technique of iteratively weighted filtered least squares (IWFLS) to obtain location-specific intercepts and slopes, with possible lagged effects via polynomial distributed lag modeling. These slopes and/or intercepts are then taken to a second-stage mixed-effects meta-regression model in order to stabilize results from various locations. The representation of the models from the stages into a combined mixed-effects model, issues of inference and choices of the parameters in modeling the lag structure are discussed. We illustrate this proposed model via detailed analysis on the effect of air pollution on school absenteeism based on data from the Southern California Children's Health Study.     

Robust framework for PET image reconstruction incorporating system and measurement uncertainties

In Positron Emission Tomography (PET), an optimal estimate of the radioactivity concentration is obtained from the measured emission data under certain criteria. So far, all the well-known statistical reconstruction algorithms require exactly known system probability matrix a priori, and the quality of such system model largely determines the quality of the reconstructed images. In this paper, we propose an algorithm for PET image reconstruction for the real world case where the PET system model is subject to uncertainties. The method counts PET reconstruction as a regularization problem and the image estimation is achieved by means of an uncertainty weighted least squares framework. The performance of our work is evaluated with the Shepp-Logan simulated and real phantom data, which demonstrates significant improvements in image quality over the least squares reconstruction efforts.     

Pattern Recognition Analysis of Metabolites in Escherichia coli Based on ESI-Orbitrap Mass Spectrometry

To achieve rapid detection of carbapenem-resistant Escherichia coli strains, a pattern recognition method based on electrospray ionization Orbitrap mass spectrometry (ESI-Orbitrap MS) was used for the analysis of drug-resistant, and sensitive strains of metabolites were analyzed. Results of five clustering methods applied to analytical data of metabolites were evaluated using iso-phenotypic coefficients. The effectiveness of three methods, principal component analysis (PCA), partial least squares discriminant analysis (PLS-DA) and orthogonal partial least squares discriminant analysis (OPLS-DA), was compared. Univariate statistics such as t-test and fold change were also used to examine the screened differential information. Both PLS-DA and OPLS-DA could achieve rapid identification of strain classes, and OPLS-DA was more powerful in screening 96 significantly different ions. This work is expected to be useful for rapid and accurate identification of strains.     

The use of weighted quadratic regression for the study of latencies of the P100 component of the visual evoked potential

The aim of this study was to determine the range of normal values of the latency of the P100 component of the VEP from a control population. Weighted least squares polynomial regression analysis showed that a quadratic curve best described the relationship between age and latency and separate curves, along with normal ranges, are presented for males and females. We would recommend the use of this statistical method for all determinations of the normal range of P100 latencies.     

Simulation of Diffusive Transport of Radionuclides in the Soil of a Radioactive Waste Disposal Site

The diffusive transport of ¹³⁷Cs, ⁹⁰Sr, and ⁶⁰Co in the clay of a radioactive waste disposal site at PINSTECH was studied to assess the safety of the underlying permeable zone against the release of these radionuclides from buried waste containers in the clay. Diffusion coefficients of these radionuclides were estimated by reservoir to sediment diffusion method via their stable counterparts in a laboratory experiment. A curve-fitting procedure was applied on the measured concentration-time profiles of the reservoir using the one-dimensional solute transport equation with a nonlinear least squares technique. Distribution coefficients were determined in laboratory batch experiments. Diffusive transport simulations were performed with the estimated values of diffusion coefficients and distribution coefficients using the one-dimensional solute transport equation describing Fickian diffusion, equilibrium adsorption, and radioactive decay. The transport simulation results showed that ¹³⁷Cs, ⁹⁰Sr, and ⁶⁰Co will transport distances of 4.33, 3.77, and 1.51 meters, respectively, in the clay before their activity concentrations will drop to clearance levels set by the International Atomic Energy Agency (IAEA), below which the waste is treated as non-radioactive. This showed that concentrations more than clearance levels will not be able to transport to the permeable zone at a minimum depth of seven meters from the ground surface if the waste containers are disposed in a trench below which a clay layer with a thickness of 4.33 meters or more exists.     

Computing Projections into Cones Generated by a Matrix

A method is described for computing orthogonal projections into finitely generated cones. This method can be used to solve nonnegative least squares approximation problems, to find the multivariate onesided Maximum Likelihood estimator and also determines the most stringent somewhere most powerful test of Schaafsma. The gist of the procedure is the unconstrained maximization of a numerically simple function. This function has a global maximum and allows an uncomplicated maximum search since local maxima do not exist. The maximum can be obtained after a finite number of iterations.     

Identification and Severity Determination of Wheat Stripe Rust and Wheat Leaf Rust Based on Hyperspectral Data Acquired Using a Black-Paper-Based Measuring Method

It is important to implement detection and assessment of plant diseases based on remotely sensed data for disease monitoring and control. Hyperspectral data of healthy leaves, leaves in incubation period and leaves in diseased period of wheat stripe rust and wheat leaf rust were collected under in-field conditions using a black-paper-based measuring method developed in this study. After data preprocessing, the models to identify the diseases were built using distinguished partial least squares (DPLS) and support vector machine (SVM), and the disease severity inversion models of stripe rust and the disease severity inversion models of leaf rust were built using quantitative partial least squares (QPLS) and support vector regression (SVR). All the models were validated by using leave-one-out cross validation and external validation. The diseases could be discriminated using both distinguished partial least squares and support vector machine with the accuracies of more than 99%. For each wheat rust, disease severity levels were accurately retrieved using both the optimal QPLS models and the optimal SVR models with the coefficients of determination (R²) of more than 0.90 and the root mean square errors (RMSE) of less than 0.15. The results demonstrated that identification and severity evaluation of stripe rust and leaf rust at the leaf level could be implemented based on the hyperspectral data acquired using the developed method. A scientific basis was provided for implementing disease monitoring by using aerial and space remote sensing technologies.