Bayesian estimation of the spatial Durbin error model with an application to voter turnout in the 2004 presidential election

The potential for spatial dependence in models of voter turnout, although plausible from a theoretical perspective, has not been adequately addressed in the literature. Using recent advances in Bayesian computation, we formulate and estimate the previously unutilized spatial Durbin error model and apply this model to the question of whether spillovers and unobserved spatial dependence in voter turnout matters from an empirical perspective. Formal Bayesian model comparison techniques are employed to compare the normal linear model, the spatially lagged X model (SLX), the spatial Durbin model, and the spatial Durbin error model. The results overwhelmingly support the spatial Durbin error model as the appropriate empirical model.

Sparse probability density function estimation using the minimum integrated square error

We develop a new sparse kernel density estimator using a forward constrained regression framework, within which the nonnegative and summing-to-unity constraints of the mixing weights can easily be satisfied. Our main contribution is to derive a recursive algorithm to select significant kernels one at time based on the minimum integrated square error (MISE) criterion for both the selection of kernels and the estimation of mixing weights. The proposed approach is simple to implement and the associated computational cost is very low. Specifically, the complexity of our algorithm is in the order of the number of training data N, which is much lower than the order of N2 offered by the best existing sparse kernel density estimators. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with comparable accuracy to those of the classical Parzen window estimate and other existing sparse kernel density estimators.

Bayesian estimation of regression parameters in elliptical measurement error models

The main object of this paper is to discuss the Bayes estimation of the regression coefficients in the elliptically distributed simple regression model with measurement errors. The posterior distribution for the line parameters is obtained in a closed form, considering the following: the ratio of the error variances is known, informative prior distribution for the error variance, and non-informative prior distributions for the regression coefficients and for the incidental parameters. We proved that the posterior distribution of the regression coefficients has at most two real modes. Situations with a single mode are more likely than those with two modes, especially in large samples. The precision of the modal estimators is studied by deriving the Hessian matrix, which although complicated can be computed numerically. The posterior mean is estimated by using the Gibbs sampling algorithm and approximations by normal distributions. The results are applied to a real data set and connections with results in the literature are reported. (C) 2011 Elsevier B.V. All rights reserved.

Convergence Property of the Measurement Gross Error Correction in Power System State Estimation, Using Geometrical Background

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Slope estimation in a parametric measurement error model: A simulation study

In regression analysis, covariate measurement error occurs in many applications. The error-prone covariates are often referred to as latent variables. In this proposed study, we extended the study of Chan et al. (2008) on recovering latent slope in a simple regression model to that in a multiple regression model. We presented an approach that applied the Monte Carlo method in the Bayesian framework to the parametric regression model with the measurement error in an explanatory variable. The proposed estimator applied the conditional expectation of latent slope given the observed outcome and surrogate variables in the multiple regression models. A simulation study was presented showing that the method produces estimator that is efficient in the multiple regression model, especially when the measurement error variance of surrogate variable is large.^

The estimation of truncation error by tau-estimation revisited

The aim of this paper was to accurately estimate the local truncation error of partial differential equations, that are numerically solved using a finite difference or finite volume approach on structured and unstructured meshes. In this work, we approximated the local truncation error using the @t-estimation procedure, which aims to compare the residuals on a sequence of grids with different spacing. First, we focused the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the truncation error for both finite difference and finite volume approaches on different grid topologies. Then, we extended the analysis to two-dimensional problems: first on linear and non-linear scalar equations and finally on the Euler equations. We demonstrated that this approach yields a highly accurate estimation of the truncation error if some conditions are fulfilled. These conditions are related to the accuracy of the restriction operators, the choice of the boundary conditions, the distortion of the grids and the magnitude of the iteration error.

Clinical validation of an algorithm to correct the error in the keratometric estimation of corneal power for normal eyes

To validate clinically an algorithm for correcting the error in the keratometric estimation of corneal power by using a variable keratometric index of refraction (nk) in a normal healthy population.

Algorithm for Correcting the Keratometric Error in the Estimation of the Corneal Power in Eyes With Previous Myopic Laser Refractive Surgery

Purpose: To calculate theoretically the errors in the estimation of corneal power when using the keratometric index (nk) in eyes that underwent laser refractive surgery for the correction of myopia and to define and validate clinically an algorithm for minimizing such errors. Methods: Differences between corneal power estimation by using the classical nk and by using the Gaussian equation in eyes that underwent laser myopic refractive surgery were simulated and evaluated theoretically. Additionally, an adjusted keratometric index (nkadj) model dependent on r1c was developed for minimizing these differences. The model was validated clinically by retrospectively using the data from 32 myopic eyes [range, −1.00 to −6.00 diopters (D)] that had undergone laser in situ keratomileusis using a solid-state laser platform. The agreement between Gaussian (PGaussc) and adjusted keratometric (Pkadj) corneal powers in such eyes was evaluated. Results: It was found that overestimations of corneal power up to 3.5 D were possible for nk = 1.3375 according to our simulations. The nk value to avoid the keratometric error ranged between 1.2984 and 1.3297. The following nkadj models were obtained: nkadj= −0.0064286r1c + 1.37688 (Gullstrand eye model) and nkadj = −0.0063804r1c + 1.37806 (Le Grand). The mean difference between Pkadj and PGaussc was 0.00 D, with limits of agreement of −0.45 and +0.46 D. This difference correlated significantly with the posterior corneal radius (r = −0.94, P < 0.01). Conclusions: The use of a single nk for estimating the corneal power in eyes that underwent a laser myopic refractive surgery can lead to significant errors. These errors can be minimized by using a variable nk dependent on r1c.

New Approach for Correction of Error Associated With Keratometric Estimation of Corneal Power in Keratoconus

The aim of this study was to obtain the exact value of the keratometric index (nkexact) and to clinically validate a variable keratometric index (nkadj) that minimizes this error. Methods: The nkexact value was determined by obtaining differences (DPc) between keratometric corneal power (Pk) and Gaussian corneal power (PGauss c ) equal to 0. The nkexact was defined as the value associated with an equivalent difference in the magnitude of DPc for extreme values of posterior corneal radius (r2c) for each anterior corneal radius value (r1c). This nkadj was considered for the calculation of the adjusted corneal power (Pkadj). Values of r1c ∈ (4.2, 8.5) mm and r2c ∈ (3.1, 8.2) mm were considered. Differences of True Net Power with PGauss c , Pkadj, and Pk(1.3375) were calculated in a clinical sample of 44 eyes with keratoconus. Results: nkexact ranged from 1.3153 to 1.3396 and nkadj from 1.3190 to 1.3339 depending on the eye model analyzed. All the nkadj values adjusted perfectly to 8 linear algorithms. Differences between Pkadj and PGauss c did not exceed 60.7 D (Diopter). Clinically, nk = 1.3375 was not valid in any case. Pkadj and True Net Power and Pk(1.3375) and Pkadj were statistically different (P , 0.01), whereas no differences were found between PGauss c and Pkadj (P . 0.01). Conclusions: The use of a single value of nk for the calculation of the total corneal power in keratoconus has been shown to be imprecise, leading to inaccuracies in the detection and classification of this corneal condition. Furthermore, our study shows the relevance of corneal thickness in corneal power calculations in keratoconus.

Estimation of the Central Corneal Power in Keratoconus: Theoretical and Clinical Assessment of the Error of the Keratometric Approach

Purpose: The aim of this study was to analyze theoretically the errors in the central corneal power calculation in eyes with keratoconus when a keratometric index (nk) is used and to clinically confirm the errors induced by this approach. Methods: Differences (DPc) between central corneal power estimation with the classical nk (Pk) and with the Gaussian equation (PGauss c ) in eyes with keratoconus were simulated and evaluated theoretically, considering the potential range of variation of the central radius of curvature of the anterior (r1c) and posterior (r2c) corneal surfaces. Further, these differences were also studied in a clinical sample including 44 keratoconic eyes (27 patients, age range: 14–73 years). The clinical agreement between Pk and PGauss c (true net power) obtained with a Scheimpflug photography–based topographer was evaluated in such eyes. Results: For nk = 1.3375, an overestimation was observed in most cases in the theoretical simulations, with DPc ranging from an underestimation of 20.1 diopters (D) (r1c = 7.9 mm and r2c = 8.2 mm) to an overestimation of 4.3 D (r1c = 4.7 mm and r2c = 3.1 mm). Clinically, Pk always overestimated the PGauss c given by the topography system in a range between 0.5 and 2.5 D (P , 0.01). The mean clinical DPc was 1.48 D, with limits of agreement of 0.71 and 2.25 D. A very strong statistically significant correlation was found between DPc and r2c (r = 20.93, P , 0.01). Conclusions: The use of a single value for nk for the calculation of corneal power is imprecise in keratoconus and can lead to significant clinical errors.

Error induced by the estimation of the corneal power and the effective lens position with a rotationally asymmetric refractive multifocal intraocular lens

AIM: To evaluate the prediction error in intraocular lens (IOL) power calculation for a rotationally asymmetric refractive multifocal IOL and the impact on this error of the optimization of the keratometric estimation of the corneal power and the prediction of the effective lens position (ELP). METHODS: Retrospective study including a total of 25 eyes of 13 patients (age, 50 to 83y) with previous cataract surgery with implantation of the Lentis Mplus LS-312 IOL (Oculentis GmbH, Germany). In all cases, an adjusted IOL power (PIOLadj) was calculated based on Gaussian optics using a variable keratometric index value (nkadj) for the estimation of the corneal power (Pkadj) and on a new value for ELP (ELPadj) obtained by multiple regression analysis. This PIOLadj was compared with the IOL power implanted (PIOLReal) and the value proposed by three conventional formulas (Haigis, Hoffer Q and Holladay). RESULTS: PIOLReal was not significantly different than PIOLadj and Holladay IOL power (P>0.05). In the Bland and Altman analysis, PIOLadj showed lower mean difference (-0.07 D) and limits of agreement (of 1.47 and -1.61 D) when compared to PIOLReal than the IOL power value obtained with the Holladay formula. Furthermore, ELPadj was significantly lower than ELP calculated with other conventional formulas (P<0.01) and was found to be dependent on axial length, anterior chamber depth and Pkadj. CONCLUSION: Refractive outcomes after cataract surgery with implantation of the multifocal IOL Lentis Mplus LS-312 can be optimized by minimizing the keratometric error and by estimating ELP using a mathematical expression dependent on anatomical factors.

Positional accommodative intraocular lens power error induced by the estimation of the corneal power and the effective lens position

Purpose: To evaluate the predictability of the refractive correction achieved with a positional accommodating intraocular lenses (IOL) and to develop a potential optimization of it by minimizing the error associated with the keratometric estimation of the corneal power and by developing a predictive formula for the effective lens position (ELP). Materials and Methods: Clinical data from 25 eyes of 14 patients (age range, 52–77 years) and undergoing cataract surgery with implantation of the accommodating IOL Crystalens HD (Bausch and Lomb) were retrospectively reviewed. In all cases, the calculation of an adjusted IOL power (PIOLadj) based on Gaussian optics considering the residual refractive error was done using a variable keratometric index value (nkadj) for corneal power estimation with and without using an estimation algorithm for ELP obtained by multiple regression analysis (ELPadj). PIOLadj was compared to the real IOL power implanted (PIOLReal, calculated with the SRK-T formula) and also to the values estimated by the Haigis, HofferQ, and Holladay I formulas. Results: No statistically significant differences were found between PIOLReal and PIOLadj when ELPadj was used (P = 0.10), with a range of agreement between calculations of 1.23 D. In contrast, PIOLReal was significantly higher when compared to PIOLadj without using ELPadj and also compared to the values estimated by the other formulas. Conclusions: Predictable refractive outcomes can be obtained with the accommodating IOL Crystalens HD using a variable keratometric index for corneal power estimation and by estimating ELP with an algorithm dependent on anatomical factors and age.