Model Evaluation Based on the Distribution of Estimated Absolute Prediction Error

The construction of a reliable, practically useful prediction rule for future response is heavily dependent on the "adequacy" of the fitted regression model. In this article, we consider the absolute prediction error, the expected value of the absolute difference between the future and predicted responses, as the model evaluation criterion. This prediction error is easier to interpret than the average squared error and is equivalent to the mis-classification error for the binary outcome. We show that the distributions of the apparent error and its cross-validation counterparts are approximately normal even under a misspecified fitted model. When the prediction rule is "unsmooth", the variance of the above normal distribution can be estimated well via a perturbation-resampling method. We also show how to approximate the distribution of the difference of the estimated prediction errors from two competing models. With two real examples, we demonstrate that the resulting interval estimates for prediction errors provide much more information about model adequacy than the point estimates alone.

Inference on Survival Data with Covariate Measurement Error - An Imputation-based Approach

We propose a new method for fitting proportional hazards models with error-prone covariates. Regression coefficients are estimated by solving an estimating equation that is the average of the partial likelihood scores based on imputed true covariates. For the purpose of imputation, a linear spline model is assumed on the baseline hazard. We discuss consistency and asymptotic normality of the resulting estimators, and propose a stochastic approximation scheme to obtain the estimates. The algorithm is easy to implement, and reduces to the ordinary Cox partial likelihood approach when the measurement error has a degenerative distribution. Simulations indicate high efficiency and robustness. We consider the special case where error-prone replicates are available on the unobserved true covariates. As expected, increasing the number of replicate for the unobserved covariates increases efficiency and reduces bias. We illustrate the practical utility of the proposed method with an Eastern Cooperative Oncology Group clinical trial where a genetic marker, c-myc expression level, is subject to measurement error.

Fading pdf of free-space optical communication system with pointing error

A free-space optical (FSO) laser communication system with perfect fast-tracking experiences random power fading due to atmospheric turbulence. For a FSO communication system without fast-tracking or with imperfect fast-tracking, the fading probability density function (pdf) is also affected by the pointing error. In this thesis, the overall fading pdfs of FSO communication system with pointing errors are calculated using an analytical method based on the fast-tracked on-axis and off-axis fading pdfs and the fast-tracked beam profile of a turbulence channel. The overall fading pdf is firstly studied for the FSO communication system with collimated laser beam. Large-scale numerical wave-optics simulations are performed to verify the analytically calculated fading pdf with collimated beam under various turbulence channels and pointing errors. The calculated overall fading pdfs are almost identical to the directly simulated fading pdfs. The calculated overall fading pdfs are also compared with the gamma-gamma (GG) and the log-normal (LN) fading pdf models. They fit better than both the GG and LN fading pdf models under different receiver aperture sizes in all the studied cases. Further, the analytical method is expanded to the FSO communication system with beam diverging angle case. It is shown that the gamma pdf model is still valid for the fast-tracked on-axis and off-axis fading pdfs with point-like receiver aperture when the laser beam is propagated with beam diverging angle. Large-scale numerical wave-optics simulations prove that the analytically calculated fading pdfs perfectly fit the overall fading pdfs for both focused and diverged beam cases. The influence of the fast-tracked on-axis and off-axis fading pdfs, the fast-tracked beam profile, and the pointing error on the overall fading pdf is also discussed. At last, the analytical method is compared with the previous heuristic fading pdf models proposed since 1970s. Although some of previously proposed fading pdf models provide close fit to the experiment and simulation data, these close fits only exist under particular conditions. Only analytical method shows accurate fit to the directly simulated fading pdfs under different turbulence strength, propagation distances, receiver aperture sizes and pointing errors.

Numerical solutions of elliptic inverse problems via the equation error method

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method.

Direct MR arthrography at 1.5 and 3.0 T: signal dependence on gadolinium and iodine concentrations--phantom study

PURPOSE: To prospectively quantify in vitro the influence of gadopentetate dimeglumine and ioversol on the magnetic resonance (MR) imaging signal observed with a variety of musculoskeletal pulse sequences to predict optimum gadolinium concentrations for direct MR arthrography at 1.5 and 3.0 T. MATERIALS AND METHODS: In an in vitro study, T1 and T2 relaxation times of three dilution series of gadopentetate dimeglumine (concentration, 0-20.0 mmol gadolinium per liter) at ioversol concentrations with iodine concentration of 0, 236.4, and 1182 mmol iodine per liter (corresponding to 0, 30, and 150 mg of iodine per milliliter) were measured at 1.5 and 3.0 T. The relaxation rate dependence on concentrations of gadolinium and iodine was analytically modeled, and continuous profiles of signal versus gadolinium concentration were calculated for 10 pulse sequences used in current musculoskeletal imaging. After fitting to experimental discrete profiles, maximum signal-to-noise ratio (SNR), gadolinium concentration with maximum SNR, and range of gadolinium concentration with 90% of maximum SNR were derived. The overall influence of field strength and iodine concentration on these parameters was assessed by using t tests. The deviation of simulated from experimental signal-response profiles was assessed with the autocorrelation of the residuals. RESULTS: The model reproduced relaxation rates of 0.37-38.24 sec(-1), with a mean error of 4.5%. Calculated SNR profiles matched the discrete experimental profiles, with autocorrelation of the residuals divided by the mean of less than 5.0. Admixture of ioversol consistently reduced T1 and T2, narrowed optimum gadolinium concentration ranges (P = .004-.006), and reduced maximum SNR (P < .001 to not significant). Optimum gadolinium concentration was 0.7-3.4 mmol/L at both field strengths. At 3.0 T, maximum SNR was up to 75% higher than at 1.5 T. CONCLUSION: Admixture of ioversol to gadopentetate dimeglumine solutions results in a consistent additional relaxation enhancement, which can be analytically modeled to allow a near-quantitative a priori optimized match of contrast media concentrations and imaging protocol for a broad variety of pulse sequences.