Polynomial Regression Coefficients Research Articles

PNS161 HEALTH COST ANALYSIS: WEIGHTINESS OF THE INTERACTION BETWEEN POPULATION SEX AND AGE

Predicting total health plan paid costs by gender and age of the population is fundamental for strategic planning of health care systems. In this study we determined the polynomial regression coefficients of healthcare costs by gender and age in a health care plan with statewide coverage.

Automation of recognition of chemicals using electronic sensor technology based on neural network data processing

The composition of the initial substance was determined using an electronic sensor “electronic nose”, consisting of 8 sensors, to which air was supplied with a syringe with alcohol vapor containing various kinds of impurities. The signal from the sensors was recorded with a sampling frequency of 1 s for 120 s. The output of the device was presented in two different interpretations - in the form of curves obtained from each sensor, or the areas under the curves. The purpose of the work is to build a recognition system for 11 impurities and water in the starting material. The composition of the initial substance was determined using an “electronic nose”, which allows one to obtain 120 values from each of 8 sensors in the form of curves or the values of the areas under the curves. A large number of classes (12), the dynamic presentation of the source data information make it advisable to build a pattern recognition system based on a neural network - a multilayer perceptron trained on the basis of the error back propagation algorithm. When training the network, existing samples are used, indicating which class they belong to. The properties of each analyte are represented as a vector of 120 values of 8 attributes defining one of 12 classes. To reduce the dimensionality of the input data of the neural network, the authors proposed the use of convolution of the available information without significant loss of information capacity of signs by constructing 8 polynomial regressions of the 19th order that describe the curves from each of the 8 sensors of the “electronic nose”. The input matrix obtained as a result of convolution consisted of 20 polynomial regression coefficients of each of 8 curves for 12 classes under consideration. A two-layer neural network with 43 neurons and a sigmoidal activation function in the hidden layer and 12 neurons and a linear activation function in the output layer was constructed. As a result of network training, 2 classification errors were obtained, which allows us to use the approach proposed by the authors to build a recognition system based on preliminary convolution of data dynamically obtained from the “electronic nose”

Optimal designs for estimating individual coefficients in polynomial regression with no intercept

We identify optimal designs for estimating individual coefficients in a polynomial regression with no intercept. Here the regression functions do not form a Chebyshev system such that the seminal results of Studden (1968) characterizing c-optimal designs are not applicable.

Milk yield prediction between Bunaji Holstein Friesian cross bred and Holstein Frisian cows using linear, cubic and quadratic models

The aim of this study was to compare three different models (Linear, cubic and quadratic) to find best model for predicting milk yield. Data originated from the monthly milk yields records of 251 Bunaji Holstein Friesian crossed and Holstein Friesian cows from 2010 to 2015. The daily milk yield data were regressed against time (day of lactation) for individual cow, using the procedure of SAS, (2002). The resulting polynomial regression coefficients (linear, quadratic and cubic) were then subjected to variance of analysis. All models provided an acceptable level of accuracy in predicting milk yield for Bunaji Holstein Friesian crossed and Holstein Friesian cows, but cubic model is observed to be the most suitable with (R2) values of (0.659, 0.582, 0.810 and 0.621) followed by quadratic model (0.447, 0.516, 0.614 and 0.605) while linear model has the least R2 values (0.02, 0.496, 0.548 and 0.309) in all the study farms.

UAV image registration based on the weighted total-least-squares

In optical image registration,the polynomial regression model generally supposes that the reference control points( RCPs) used as the coefficient matrix is error- free. However,the actual RCPs often inevitably contain errors and RCPs residual errors between different images are not the same. The general least squares method( LS) only considers the error in the observation vector whereas the total least squares method( TLS) takes the errors of both the observation vector and the coefficient matrix into account and assumes that they have the same residual error. In view of this situation,this paper introduces a more reasonable weighted total least squares method( WTLS) for polynomial regression coefficients estimation. Experiments show that the WTLS can estimate the parameters better and significantly improve the image registration accuracy.

Ultra-Wideband Source Localization Using a Particle-Swarm-Optimized Capon Estimator from a Frequency-Dependent Channel Modeling Viewpoint

We introduce a realistic frequency-dependent channel model for ultra-wideband (UWB) communication systems and develop a generalized broadband Capon spatial spectrum estimator for localization of multiple incoherently distributed scattering clusters. The proposed estimator is able to address the three crucial features of practical UWB impulse propagation: presence of local scattering for multiple incoherently distributed clusters, wideband array signals, and frequency-dependent dispersive effects. The particle-swarm optimization, which is a recently invented high-performance optimizer based on the movement and intelligence of swarms, is then implemented to perform a multidimensional parameter search to jointly estimate the source central angles, the polynomial regression coefficients for angle spreads, and the frequency-dependence of various clusters. Numerical experiments are also carried out to examine the performance of the algorithm under various environments and model mismatches.

Optimal designs for estimating individual coefficients in polynomial regression—a functional approach

In this paper the optimal design problem for the estimation of the individual coefficients in a polynomial regression on an arbitrary interval [a,b] (−∞<a<b<∞) is considered. Recently, Sahm (Ann. Statist. 2000, accepted for publication) demonstrated that the optimal design is one of four types depending on the symmetry parameter s ∗=(a+b)/(a−b) and the specific coefficient which has to be estimated. In the same paper the optimal design was identified explicitly in three cases. It is the basic purpose of the present paper to study the remaining open fourth case. It will be proved that in this case the support points and weights are real analytic functions of the boundary points of the design space. This result is used to provide a Taylor expansion for the weights and support points as functions of the parameters a and b, which can easily be used for the numerical calculation of the optimal designs in all cases, which were not treated by Sahm (Ann. Statist. 2000, accepted for publication).

Extension of the Carter-Yang polynomial growth curve model to allow unique times of measurement for subjects

A PC program extending the procedure due to Carter and Yang (Commun Stat: Theory Methods, 8 (1986) 2507–2526) to allow unique times of measurement for subjects is described, illustrated and made available. Given longitudinal observations on each of N subjects comprising a single group, this program determines the lowest degree polynomial in time adequate to fit the average growth curve (AGC); estimates this curve and provides confidence bands for the AGC, and confidence intervals for the corresponding polynomial regression coefficients; and so-called prediction intervals which, with a given level of confidence, will contain the growth curve of a ‘new’ subject from the same population of which the N subjects constitute a random sample. Two kinds of missing data are accommodated. First, in the context of studies planned so that subjects will be measured at identical times and, second, in unstructured studies where subjects may present with their own, unique times of measurement.

PC program for estimating polynomial growth, velocity and acceleration curves when subjects may have missing data

A stand-alone, menu-driven PC program, written in GAUSS386i, for estimating polynomial growth, velocity, and acceleration curves from longitudinal data is described, illustrated and made available to interested readers. Missing data are accommodated: we assume that the study is planned so that individuals will have common times of measurement, but allow some of the sequences to be incomplete. The degrees, D i , adequate to fit the growth profiles of the N individuals are determined and the corresponding polynomial regression coefficients are calculated and can be saved in ASCII files which may then be imported into a statistical computing package for further analysis. Examples of the use of the program are provided.

A PC program for performing multigroup longitudinal comparisons using the Potthoff-Roy analysis and orthogonal polynomials

A PC-program performing the Potthoff-Roy (PR) multigroup ( G-sample) analysis of longitudinal data is described and illustrated. This program and the underlying statistical model are useful in the comparison of several longitudinal samples. Applications include the study of growth, development, adaptation, aging, and treatment effects (in short, any phenomenon in which the passage of time is important) for which serial data are available. Specifically, this method fits polynomials to the average growth curves in the samples, and tests hypotheses concerning the curves themselves and the individual coefficients of the polynomials. The program features the utilization of orthogonal polynomial regression coefficients (OPRCs) and is written in GAUSS, a relatively inexpensive yet comprehensive matrix programming language. It is documented that using OPRCs to comprise the within-individual or time design matrix has several advantages over the more usual choice of the successive-powers-of- t form of this matrix and an example of one important such advantage is provided. GAUSS was employed to make the program readily-accessible (i.e., executable code) to biomedical investigators. The GAUSS compiler is not required to run this program. Information regarding the availability of the program is provided in the Appendix.

PC program for obtaining orthogonal polynomial regression coefficients for use in longitudinal data analysis.

Much of longitudinal data analysis begins with dimensionality reduction, i.e., the replacement of the T observations x1 , x2 , …, xT on an individual taken at times t1 , t2 , …, tT (not necessarily equally spaced) by a smaller number, P, of parameters which are then used to describe and compare growth processes. We focus on the class of polynomial growth curve models for one-sample data matrices in which the P regression coefficients are estimated by an equation of the form \documentclass{article}\pagestyle{empty}\begin{document}$ \hat \tau = ({\rm W'W}) $\end{document} 1 W'x and consider the choice of the design matrix W. The case in favor of using orthogonal polynomials to comprise the elements of W and provide a PC program, written in GAUSS, for obtaining them is presented. This program can be used instead of existing tables of orthogonal polynomials in the case of equally spaced time points, and to avoid laborious hand-computation to obtain them when the time points are not equally spaced. The program also computes the corresponding orthogonal polynomial regression coefficients \documentclass{article}\pagestyle{empty}\begin{document}$ \hat \alpha = (\Phi '\Phi)^{ - 1} \Phi '{\rm x} $\end{document}, where Φ consists of orthogonal polynomials, which may then be input into other programs for subsequent analysis, e.g., to compare the growth profiles of several groups of individuals. Examples of the use of the program are given. Information on obtaining a copy of the program is provided in Appendix A. © 1992 Wiley-Liss, Inc.

A monte carlo comparison of tests of parallelism of the ponses over time of two populations vith non-uniform observation times

Repeated measures data collected at random observation times are quite common in clinical studies and are often difficult to analyze. A Monte Carlo comparison of four analysis procedures with respect to significance level and power is presented. The basic procedures compared are successive difference analyses and three procedures using the data as summarized in the estimated quadratic polynomial regression coefficients for each subject. These three procedures are (1) Hotelling's T-square, (2) Multivariate Multisample Rank Sum Test (MMRST) and (3) Multivariate Multisample Median Test (MMMT). For the variety of dispersion structures, sample sizes and treatement groups simulated the MMRST and successive difference analysis were the most satisfactory.

A potential method of correcting intracavitary left ventricular filling pressures for the effects of positive end-expiratory airway pressure.

Based on the observation that positive end-expiratory airway pressure (PEEP) causes comparable increments in intrapericardial and right-sided intracardiac pressures, we hypothesized that intracavitary left ventricular filling pressures measured in the presence of PEEP can be corrected for increased intrathoracic pressure by subtracting the effects of PEEP on intracavitary right ventricular filling pressures. Ventricular function curves (aortic blood flow vs intracavitary left ventricular end-diastolic pressure [LVEDP]) were generated with and without 15 cm of water of PEEP in eight dogs. All curves were shifted to the right by PEEP (i.e., intracavitary LVEDP was higher for any submaximal level of aortic blood flow). However, when pressures measured in the presence of PEEP were "corrected" by subtracting the corresponding increment in intracavitary right ventricular end-diastolic pressure caused by PEEP at each level of ventricular filling, control and corrected PEEP data points appeared to fall on the same curve in five dogs, and differed only slightly in three dogs. Mean control and corrected PEEP curves derived by averaging polynomial regression coefficients for each condition differed significantly from uncorrected PEEP curves (p less than .05), but not from each other. Analogous curves based on mean left atrial pressure were corrected equally well by subtracting the effects of PEEP on mean right atrial pressure. We conclude that the increments in intracavitary right heart filling pressures caused by PEEP can be used to correct intracavitary left heart filling pressures for the effects of PEEP on intrathoracic pressure.

An analysis of the factors determining yields in crosses between semi-dwarf and taller wheat varieties

SUMMARYThe growth and development of two semi-dwarf winter wheat varieties, of two taller varieties, and of random selections from two crosses between semi-dwarf and taller varieties were compared over 2 years. Changes with time of crop dry weight/m2, leaf area index and ear size were expressed as orthogonal polynomial regression coefficients. Significant correlations were obtained between yield, the polynomial coefficients and growth measurements relating to number of tillers and flag leaf area duration, but many of these variables were significantly correlated with each other. Multiple regression equations were therefore calculated expressing yield in terms of the variables. These showed flag leaf area duration and rate of ear growth to be the most important characters determining crop yield. The greatest reduction in total yield variance was obtained by the simultaneous consideration of characters relating to photosynthetic capacity and ear development. The significance of this finding in relation to the relative importance of ‘apos;source’ and ‘sink’ in determining grain yield are discussed.

Varietal differences in growth parameters of wheat and their importance in determining yield

The growth and development of five winterwheat varieties and of random selections from hybrids between them were compared. Successive sampling demonstrated varietal differences in orthogonal polynomial regression coefficients fitted to data for tiller number, shoot weight and leaf area. The correlation coefficients between these parameters and yield were statistically significant, but when polynomial regression equations of yield on them were determined, only the terms relating to shoot weight and leaf area duration after anthesis were significant.The heritabilities of these parameters in each of 2 years were determined from measurements of genetic and environmental variance. The results obtained were compared each year with estimates of yield derived from parameters determined in the other year. The correlation between successive estimates of yield was found to be closer than that of of yield was found to be closer than that of yield with any of the growth parameters.Separate analyses of crosses involving each of the parental varieties showed seasonal and varietal differences in the usefulness of the growth parameters for predicting yield.The value of this approach to the choice of parents is discussed in view of these findings.

Some properties of optimal spacing in polynomial estimation

It is shown that the choice of x values that minimizes the generalized variance of the estimates of the coefficients in polynomial regression also minimizes the maximum of the generalized variance of the estimates of an arbitrary number of ordinates on the regression curve. This optimum spacing is also optimal with respect to the estimation of the coefficients of the first derivative of the polynomial regression function.