Generalized Least-squares Method Research Articles

Using GLS to Generate Forecasts in Regression Models with Auto-correlated Disturbances with simulation and Palestinian Market Index Data

This paper involves an important statistical problem concerning forecasting in regression models in time series processes. It is well known that the most famous method of estimating and forecasting is the Ordinary Least Squares (OLS). OLS may be not the optimal in this context. So over the years many specialized estimation techniques have been developed, for example Generalized Least Squares (GLS). We are comparing the forecasting based on some estimators with the prediction using the GLS estimate. This comparison will be used by what is known as measures of forecast accuracy. We conduct an extensive computer simulation time series data, to make comparison among these methods. The similar forecasting criteria were developed and evaluated for the real data set on daily closing price in the Palestinian market index (Alquds Index). The data consists of 164 monthly observations and obtained from the website of the Palestine Stock Exchange. The main finding is that, for forecasting purposes there is not much gained in trying to identifying the exact order and form of the auto-correlated disturbances by using GLS estimation method. In addition, we noticed that the accuracy of forecasting using GLS method does not differ substantially than the other methods as Maximum Likelihood Estimation (MLE), Minimize Conditional Sum of Squares (CSS) and the combination of these two methods. Moreover, for parameter estimation, the GLS is nearly as efficient as the exact parameter estimation. On the other hand, the Ordinary Least Squares (OLS) method performs much less efficient than the other estimation methods and producing poor forecasting accuracy.

분해능 모델 제한자를 사용하는 자력탐사자료의 2차원 역산

새로운 2차원 자력탐사자료 역산 방법을 개발하였다. 중,자력탐사와 같은 포텐셜 자료의 역산에서 가장 문제가 되는 점은 비유일해 문제이다. 일반적으로 자력탐사 자료의 역산은 모델변수의 수가 자료의 수보다 훨씬 많은 불충분 문제이며, 이는 비유일해 문제를 더욱 심화시키게 된다. 일반적인 최소자승법을 자력탐사자료의 역산에 적용하게 되면, 이 상대가 지표면에 집중되는 결과를 초래한다. 본 연구에서는 이러한 비유일해 문제를 극복하기 위하여 모델분해능에 근거한 새로운 모델제한자를 제안하였다. 이 모델제한자는 분해능이 높은 모델변수에는 큰 제한을 가하고, 작은 모델변수에는 약한 제한을 가하게 된다. 따라서 분해능이 낮은 심부의 모델변수도 효과적으로 추정할 수 있다. 개발된 역산 알고리듬을 이용하여, 전형적인 모델에 대한 이론자료의 역산에 적용하였다. 또한 옥천대에서 얻어진 항공자력탐사자료 역산에 적용하였다. We developed a method for inverting magnetic data to image 2D susceptibility models. The major difficulty in the inversion of the potential data is the nonuniqueness. Furthermore, generally the number of inversion blocks are greater than the number of the magnetic data available, and thus the magnetic inversion leads to under-determined problem, which aggravates the nonuniqueness. When the magnetic data were inverted by the general least-squares method, the anomalous susceptibility would be concentrated near the surface in the inverted section. To overcome this nonuniqueness problem, we propose a new resolution model constraint that is calculated from the parameter resolution. The model constraint imposes large penalty on the model parameter with good resolution, on the other hand small penalty on the model parameter with poor resolution. Thus, the deep-seated model parameter, generally having poor resolution, can be effectively resolved. The developed inversion algorithm is applied to the inversion of the synthetic data for typical models of magnetic anomalies and is tested on real airborne data obtained at the Okcheon belt of Korea.

A slice‐wise latent structure regression method for the analysis of functional magnetic resonance imaging data

ABSTRACTWe introduce a novel slice‐wise latent structure regression (LSR) method for the analysis of functional magnetic resonance imaging (fMRI) data instead of the conventional voxel‐wise generalized least squares (GLS) method. LSR method is designed for application to data sets from slices where fMRI responses (voxels Y* of a slice) are highly correlated with the design matrix X*. Also, we compared the performances of LSR, principal component regression (PCR), and GLS methods in terms of model parameters using experimental fMRI data. The LSR method exhibits an enhanced predictive ability and model coefficients as compared to the PCR and GLS methods. © 2013 Wiley Periodicals, Inc. Concepts Magn Reson Part A 42A: 130‐139, 2013.

Performance of the generalized least-squares method for the Gumbel distribution and its application to annual maximum wind speeds

The Gumbel model is often used to fit annual maximum wind speed or wind velocity pressure. The commonly used fitting methods include the method of moments, the method of maximum likelihood, the method of L-moments, and the Lieblein BLUE (i.e., generalized least-squares method (GLSM)). Previously, the coefficients of the estimators for the latter method have not been available for large sample size, and the relative performance of the GLSM to other fitting methods such as the method of L-moments is unknown. In this study, we evaluate these coefficients for a sample size up to 100, and identify trends in the calculated coefficients. The relative performance of commonly used fitting methods for the Gumbel distribution, including the GLSM, is evaluated in terms of efficiency, bias, and root-mean square error. We illustrate their application and impact on the estimated return period values of the annual maximum wind speed for 14 locations in Canada.

Step detection in single-molecule real time trajectories embedded in correlated noise.

Single-molecule real time trajectories are embedded in high noise. To extract kinetic or dynamic information of the molecules from these trajectories often requires idealization of the data in steps and dwells. One major premise behind the existing single-molecule data analysis algorithms is the Gaussian ‘white’ noise, which displays no correlation in time and whose amplitude is independent on data sampling frequency. This so-called ‘white’ noise is widely assumed but its validity has not been critically evaluated. We show that correlated noise exists in single-molecule real time trajectories collected from optical tweezers. The assumption of white noise during analysis of these data can lead to serious over- or underestimation of the number of steps depending on the algorithms employed. We present a statistical method that quantitatively evaluates the structure of the underlying noise, takes the noise structure into account, and identifies steps and dwells in a single-molecule trajectory. Unlike existing data analysis algorithms, this method uses Generalized Least Squares (GLS) to detect steps and dwells. Under the GLS framework, the optimal number of steps is chosen using model selection criteria such as Bayesian Information Criterion (BIC). Comparison with existing step detection algorithms showed that this GLS method can detect step locations with highest accuracy in the presence of correlated noise. Because this method is automated, and directly works with high bandwidth data without pre-filtering or assumption of Gaussian noise, it may be broadly useful for analysis of single-molecule real time trajectories.

Combination of multichannel single‐voxel MRS signals using generalized least squares

To propose using the generalized least square (GLS) algorithm for combining multichannel single-voxel magnetic resonance spectroscopy (MRS) signals. Phantom and in vivo brain MRS experiments on a 7 T scanner equipped with a 32-channel receiver coil, as well as Monte Carlo simulations, were performed to compare the coefficient of variation (CV) of the GLS method with those of two recently reported spectral combination methods. Compared to the two existing methods, the GLS method significantly reduced CV values for the simulation, phantom, and in vivo experiments. The GLS method can lead to improved precision of peak quantification.

Robust sensor bias estimation for ill-conditioned scenarios

Sensor bias estimation is an inherent problem in multi-sensor data fusion systems. Classical methods such as the Generalized Least Squares (GLS) method can have numerical problems with ill-conditioned sets which are common in practical applications. This paper describes an azimuth-GLS method that provides a solution to the ill-conditioning problem while maintaining reasonable accuracy compared with the classical GLS method. The mean square error is given for both methods as a criterion to determine when to use this azimuth-GLS method. Furthermore, the separation boundary between the azimuth-GLS favorable region and that of the GLS method is explicitly plotted. Extensive simulations show that the azimuth-GLS approach is preferable in most scenarios.

Thermoviscoelastic state of multiply connected anisotropic plates

A method to solve a thermoviscoelastic problem for multiply connected anisotropic plates is proposed. The perturbation method is used to reduce the problem to a recurrent sequence of thermoelastic problems that are solved using generalized complex potentials. By expanding the unknown functions into Faber–Laurent series and satisfying the boundary conditions using the generalized least-squares method, the problem for a plate with elliptic rigid inclusions is reduced to a system of algebraic equations. The thermostressed state of a plate with one or two inclusions is numerically analyzed depending on time and geometry

A Balanced System of U.S. Industry Accounts and Distribution of the Aggregate Statistical Discrepancy by Industry

This article describes and illustrates a generalized least squares (GLS) method that systematically incorporates all available information on the reliability of initial data in the reconciliation of a large disaggregated system of national accounts. The GLS method is applied to reconciling the 1997 U.S. Input-Output and Gross Domestic Product (GDP)-by-industry accounts with benchmarked GDP estimated from expenditures. The GLS procedure produced a balanced system of industry accounts and distributed the aggregate statistical discrepancy by industry according to the estimated relative reliabilities of initial estimates. The study demonstrates the empirical feasibility and computational efficiency of the GLS method for large accounts reconciliation.

A New Formulation of the Unified Monte Carlo Approach (UMC-B) and Cross-Section Evaluation for the Dosimetry Reaction 55Mn(n,γ)56Mn

Abstract Two relatively new approaches to neutron cross section data evaluation are described. They are known collectively as Unified Monte Carlo (versions UMC-G and UMC-B). Comparisons are made between these two methods, as well as with the well-known generalized least-squares (GLSQ) technique, through the use of simple, hypothetical (toy) examples. These new Monte Carlo methods are based on stochastic sampling of probability functions that are constructed with the use of theoretical and experimental data by applying the principle of maximum entropy. No further assumptions are involved in either UMC-G or UMC-B. However, the GLSQ procedure requires the linearization of non-linear terms, such as those that occur when cross section ratio data are included in an evaluation. It is shown that these two stochastic techniques yield results that agree well with each other, and with the GLSQ method, when linear data are involved, or when the perturbations due to data discrepancies and nonlinearity effects are small. Otherwise, there can be noticeable differences. The present investigation also demonstrates, as observed in earlier work, that the least-squares approach breaks down when these conditions are not satisfied. This paper also presents an actual evaluation of the 55Mn(n,γ)56Mn neutron dosimetry reaction cross section in the energy range from 100 keV to 20 MeV, which was performed using both GLSQ and UMC-G approaches.

System of Covariance Data Evaluation at China Nuclear Data Centre-COVAC

To determine the uncertainty of neutron cross sections for modern nuclear data application, a new evaluation system, COVAC is being developed at China Nuclear Data Centre (CNDC). In COVAC, experimental data were firstly pre-analyzed and handled via processing tools. The sensitivities of theoretical model parameters were analyzed by a special designed subsystem-SEMAW which offers a platform to obtain parameter sensitivities for different nuclear reaction codes. In addition the generalized least-squares method (GLS) and Bayesian method were both contained in COVAC to generate the recommended covariance. The experimental and theoretical uncertain information can be included in the covariance by each of the two methods. The outputs are formatted in ENDF-6. So far, COVAC is suitable to generate the covariance files for structure and fission nuclei in the fast neutron energy range, and n+Ti was taken to illuminate the whole procedure.

Comparative Methods as a Statistical Fix: The Dangers of Ignoring an Evolutionary Model

Abstract Comparative methods are widely used in ecology and evolution. The most frequently used comparative methods are based on an explicit evolutionary model. However, recent approaches have been popularized that are without an evolutionary basis or an underlying null model. Here we highlight the limitations of such techniques in comparative analyses by using simulations to compare two commonly used comparative methods with and without evolutionary basis, respectively: generalized least squares (GLS) and phylogenetic eigenvector regression (PVR). We find that GLS methods are more efficient at estimating model parameters and produce lower variance in parameter estimates, lower phylogenetic signal in residuals, and lower Type I error rates than PVR methods. These results can very likely be generalized to eigenvector methods that control for space and both space and phylogeny. We highlight that GLS methods can be adapted in numerous ways and that the variance structure used in these models can be flexibly optimized to each data set.

Analysis of covariance with pre-treatment measurements in randomized trials under the cases that covariances and post-treatment variances differ between groups

When primary endpoints of randomized trials are continuous variables, the analysis of covariance (ANCOVA) with pre-treatment measurements as a covariate is often used to compare two treatment groups. In the ANCOVA, equal slopes (coefficients of pre-treatment measurements) and equal residual variances are commonly assumed. However, random allocation guarantees only equal variances of pre-treatment measurements. Unequal covariances and variances of post-treatment measurements indicate unequal slopes and, usually, unequal residual variances. For non-normal data with unequal covariances and variances of post-treatment measurements, it is known that the ANCOVA with equal slopes and equal variances using an ordinary least-squares method provides an asymptotically normal estimator for the treatment effect. However, the asymptotic variance of the estimator differs from the variance estimated from a standard formula, and its property is unclear. Furthermore, the asymptotic properties of the ANCOVA with equal slopes and unequal variances using a generalized least-squares method are unclear. In this paper, we consider non-normal data with unequal covariances and variances of post-treatment measurements, and examine the asymptotic properties of the ANCOVA with equal slopes using the variance estimated from a standard formula. Analytically, we show that the actual type I error rate, thus the coverage, of the ANCOVA with equal variances is asymptotically at a nominal level under equal sample sizes. That of the ANCOVA with unequal variances using a generalized least-squares method is asymptotically at a nominal level, even under unequal sample sizes. In conclusion, the ANCOVA with equal slopes can be asymptotically justified under random allocation.

Total Least-Squares Algorithm for Time of Arrival Based Wireless Sensor Networks Location

Location of a source is of considerable interest in wireless sensor networks, and it can be estimated from passive measurements of the arrival times. A novel algorithm for source location by utilizing the time of arrival (TOA) measurements of a signal received at spatially separated sensors is proposed. The algorithm is based on total least-squares (TLS) method, which is a generalized least-squares method to solve an overdetermined set of equations whose coefficients are noisy, and gives an explicit solution. Comparisons of performance with standard least-squares method are made, and Monte Carlo simulations are performed. Simulation results indicate that the proposed TLS algorithm gives better results than LS algorithm.

On Testing Linear Hypothesis in a Nested Error Regression Model

Consider the problem of testing the linear hypothesis on regression coefficients in the nested error regression model. The standard F-test statistic based on the ordinary least squares (OLS) estimator has the serious shortcoming that its type I error rates (sizes) are much larger than nominal significance levels, because the covariance matrix of data is not the identity but has the intraclass correlation structure. One of methods for fixing the problem is to consider an F-test statistic based on the generalized least squares (GLS) estimator, and the resulting GLS F-test performs well in controlling the sizes. However, numerical investigations show that the sizes remain still slightly larger than nominal levels. In this article, we derive two test procedures: One is an exact test based on the within analysis of variance, and the other is a testing procedure based on the asymptotic correction of the GLS method. It is numerically shown that both procedures are superior to the GLS F-test in controlling the sizes and that the latter test is more powerful than the exact test.

Misfit functional for recovering data in 2D ElectroCardioGraphy problems

This paper focuses on the inverse ElectroCardioGraphy problem which consists in recovering the electrical potential and flux on the heart from electrical potential on the body surface. An energy-like error functional introduced in Andrieux et al. [3] and a generalized least-squares method are performed in the context of this ill-posed problem of missing boundary data recovering. Numerical experiments with 2D domains highlight the efficiency of the proposed methods as well as their robustness in the model context.

Robust Method for Sensitivity Analysis in Simulation Model/a Comparison Study

Kleijnen proposed using Ordinary Least Squares method combining with experimental design to estimate polynomial regression metamodels, but I/O data violates some classical assumptions of OLS as the correlation between output which due to common random numbers and Heterogeneous variances which caused by using different factor combinations. Thus Kleijnen and David referred to using repeated OLS (OLSR) or Generalized Least Squares (GLS) as a robust methods instead of OLS. In this study we compare these two methods using a simulation model M/M/1 which represented by a Queuing model in the repair and maintenance fields. We validated the estimated first order polynomial regression meta-model using adjusted R2 and Relative average absolute error, Our results demonstrate that As a result, OLSR is more efficient and more validation than GLS method.

Data boundary fitting using a generalized least-squares method

In many astronomical problems one often needs to determine the upper and/or lower boundary of a given data set. An automatic and objective approach consists in fitting the data using a generalised least-squares method, where the function to be minimized is defined to handle asymmetrically the data at both sides of the boundary. In order to minimise the cost function, a numerical approach, based on the popular downhill simplex method, is employed. The procedure is valid for any numerically computable function. Simple polynomials provide good boundaries in common situations. For data exhibiting a complex behaviour, the use of adaptive splines gives excellent results. Since the described method is sensitive to extreme data points, the simultaneous introduction of error weighting and the flexibility of allowing some points to fall outside of the fitted frontier, supplies the parameters that help to tune the boundary fitting depending on the nature of the considered problem. Two simple examples are presented, namely the estimation of spectra pseudo-continuum and the segregation of scattered data into ranges. The normalisation of the data ranges prior to the fitting computation typically reduces both the numerical errors and the number of iterations required during the iterative minimisation procedure.

An Investigation of the Performance of the Unified Monte Carlo Method of Neutron Cross Section Data Evaluation

The Unified Monte Carlo method (UMC) has been suggested to avoid certain limitations and approximations inherent to the well-known Generalized Least Squares (GLS) method of nuclear data evaluation. This contribution reports on an investigation of the performance of the UMC method in comparison with the GLS method. This is accomplished by applying both methods to simple examples with few input values that were selected to explore various features of the evaluation process that impact upon the quality of an evaluation. Among the issues explored are: i) convergence of UMC results with the number of Monte Carlo histories and the ranges of sampled values; ii) a comparison of Monte Carlo sampling using the Metropolis scheme and a brute force approach; iii) the effects of large data discrepancies; iv) the effects of large data uncertainties; v) the effects of strong or weak model or experimental data correlations; and vi) the impact of ratio data and integral data. Comparisons are also made of the evaluated results for these examples when the input values are first transformed to comparable logarithmic values prior to performing the evaluation. Some general conclusions that are applicable to more realistic evaluation exercises are offered.

Diallel crossing in Pinus cembra: IV. Age trends in genetic parameters and genetic gain for growth and branching traits.

This paper reports results from a complete 10 x 10 diallel carried out in a natural population of Swiss stone pine (Pinus cembra L.) from the southern Carpathian Mountains. At age six, after nursery testing, the material was field planted on one site, using a completely randomized block design with 100 families, four replicates and 15 tree row-plots per replication, spaced 2.5 x 2.5m. Total and annual height growth, root collar diameter, number of branches per whorl and survival were assessed at successive ages between ages eight and 14 after seed. In addition, several traits that were assessed during the nursery test were used in correlation and some other analyses. Plot means of the measured traits were analyzed using the general least-squares method by means of the computer DIALL programme prepared by Schaffer and Usanis (1969). Across the field testing periods, significant (p<0.05) and highly significant (p<0.01; p<0.001) differences occurred in total height growth and root collar diameter for general and specific combining ability as well for maternalinteraction effects. These results suggest that the traits are controlled by nuclear (additive and non-additive) and by nuclear x extra-nuclear gene interactions. In an ascendant trend, the additive variance, as a percent of the total genetic variance, ranged from 35% at age eight to 66% at age 14 for total height growth, while that for root collar diameter trend varied less between 16% and 34%. In a descendant trend, the dominance ratios s2SCA/ s2GCA for total height growth ranged from 0.9 at age eight to 0.3 at age 14, suggesting that the additive variance should be used in the breeding programme. Parents with significant general combining effects for all but one trait were found. For total height growth, the narrow-sense family mean heritability estimates varied in an ascendant trend between 0.45 and 0.65 while the narrow- sense individual tree heritability varied irregularly from year to year between 0.31 and 0.37. By selecting the best 20 families and the best 20% of individuals within families, a genetic gain in total height growth of 9.7% and 10.9%, respectively, could be achieved at age 14. The improvement of growth and branching by using both family and individual selections could be applied. The very high age-age and trait-trait genetic correlations suggest that both early and indirect selection could be applied effectively.