Theoretical insight into diagnosing observation error correlations using observation-minus-background and observation-minus-analysis statistics

To improve the quantity and impact of observations used in data assimilation it is necessary to take into account the full, potentially correlated, observation error statistics. A number of methods for estimating correlated observation errors exist, but a popular method is a diagnostic that makes use of statistical averages of observation-minus-background and observation-minus-analysis residuals. The accuracy of the results it yields is unknown as the diagnostic is sensitive to the difference between the exact background and exact observation error covariances and those that are chosen for use within the assimilation. It has often been stated in the literature that the results using this diagnostic are only valid when the background and observation error correlation length scales are well separated. Here we develop new theory relating to the diagnostic. For observations on a 1D periodic domain we are able to the show the effect of changes in the assumed error statistics used in the assimilation on the estimated observation error covariance matrix. We also provide bounds for the estimated observation error variance and eigenvalues of the estimated observation error correlation matrix. We demonstrate that it is still possible to obtain useful results from the diagnostic when the background and observation error length scales are similar. In general, our results suggest that when correlated observation errors are treated as uncorrelated in the assimilation, the diagnostic will underestimate the correlation length scale. We support our theoretical results with simple illustrative examples. These results have potential use for interpreting the derived covariances estimated using an operational system.

Evaluation of positioning error-induced pixel shifts on satellite linear push-broom imagery

Georeferencing is one of the major tasks of satellite-borne remote sensing. Compared to traditional indirect methods, direct georeferencing through a Global Positioning System/inertial navigation system requires fewer and simpler steps to obtain exterior orientation parameters of remotely sensed images. However, the pixel shift caused by geographic positioning error, which is generally derived from boresight angle as well as terrain topography variation, can have a great impact on the precision of georeferencing. The distribution of pixel shifts introduced by the positioning error on a satellite linear push-broom image is quantitatively analyzed. We use the variation of the object space coordinate to simulate different kinds of positioning errors and terrain topography. Then a total differential method was applied to establish a rigorous sensor model in order to mathematically obtain the relationship between pixel shift and positioning error. Finally, two simulation experiments are conducted using the imaging parameters of Chang’ E-1 satellite to evaluate two different kinds of positioning errors. The experimental results have shown that with the experimental parameters, the maximum pixel shift could reach 1.74 pixels. The proposed approach can be extended to a generic application for imaging error modeling in remote sensing with terrain variation.

An Ensemble Smoother with Error Estimates

A smoother introduced earlier by van Leeuwen and Evensen is applied to a problem in which real obser vations are used in an area with strongly nonlinear dynamics. The derivation is new , but it resembles an earlier derivation by van Leeuwen and Evensen. Again a Bayesian view is taken in which the prior probability density of the model and the probability density of the obser vations are combined to for m a posterior density . The mean and the covariance of this density give the variance-minimizing model evolution and its errors. The assumption is made that the prior probability density is a Gaussian, leading to a linear update equation. Critical evaluation shows when the assumption is justified. This also sheds light on why Kalman filters, in which the same ap- proximation is made, work for nonlinear models. By reference to the derivation, the impact of model and obser vational biases on the equations is discussed, and it is shown that Bayes’ s for mulation can still be used. A practical advantage of the ensemble smoother is that no adjoint equations have to be integrated and that error estimates are easily obtained. The present application shows that for process studies a smoother will give superior results compared to a filter , not only owing to the smooth transitions at obser vation points, but also because the origin of features can be followed back in time. Also its preference over a strong-constraint method is highlighted. Further more, it is argued that the proposed smoother is more efficient than gradient descent methods or than the representer method when error estimates are taken into account

Data Assimilation and Inverse Methods in Terms of a Probabilistic Formulation

The weak-constraint inverse for nonlinear dynamical models is discussed and derived in terms of a probabilistic formulation. The well-known result that for Gaussian error statistics the minimum of the weak-constraint inverse is equal to the maximum-likelihood estimate is rederived. Then several methods based on ensemble statistics that can be used to find the smoother (as opposed to the filter) solution are introduced and compared to traditional methods. A strong point of the new methods is that they avoid the integration of adjoint equations, which is a complex task for real oceanographic or atmospheric applications. they also avoid iterative searches in a Hilbert space, and error estimates can be obtained without much additional computational effort. the feasibility of the new methods is illustrated in a two-layer quasigeostrophic model.

DETECTABILITY AND ERROR ESTIMATION IN ORBITAL FITS OF RESONANT EXTRASOLAR PLANETS

We estimate the conditions for detectability of two planets in a 2/1 mean-motion resonance from radial velocity data, as a function of their masses, number of observations and the signal-to-noise ratio. Even for a data set of the order of 100 observations and standard deviations of the order of a few meters per second, we find that Jovian-size resonant planets are difficult to detect if the masses of the planets differ by a factor larger than similar to 4. This is consistent with the present population of real exosystems in the 2/1 commensurability, most of which have resonant pairs with similar minimum masses, and could indicate that many other resonant systems exist, but are currently beyond the detectability limit. Furthermore, we analyze the error distribution in masses and orbital elements of orbital fits from synthetic data sets for resonant planets in the 2/1 commensurability. For various mass ratios and number of data points we find that the eccentricity of the outer planet is systematically overestimated, although the inner planet`s eccentricity suffers a much smaller effect. If the initial conditions correspond to small-amplitude oscillations around stable apsidal corotation resonances, the amplitudes estimated from the orbital fits are biased toward larger amplitudes, in accordance to results found in real resonant extrasolar systems.

A Survey on Graphical Methods for Classification Predictive Performance Evaluation

Predictive performance evaluation is a fundamental issue in design, development, and deployment of classification systems. As predictive performance evaluation is a multidimensional problem, single scalar summaries such as error rate, although quite convenient due to its simplicity, can seldom evaluate all the aspects that a complete and reliable evaluation must consider. Due to this, various graphical performance evaluation methods are increasingly drawing the attention of machine learning, data mining, and pattern recognition communities. The main advantage of these types of methods resides in their ability to depict the trade-offs between evaluation aspects in a multidimensional space rather than reducing these aspects to an arbitrarily chosen (and often biased) single scalar measure. Furthermore, to appropriately select a suitable graphical method for a given task, it is crucial to identify its strengths and weaknesses. This paper surveys various graphical methods often used for predictive performance evaluation. By presenting these methods in the same framework, we hope this paper may shed some light on deciding which methods are more suitable to use in different situations.

Methods of Estimation in Multiple Linear Regression: Application to Clinical Data

Nesse artigo, tem-se o interesse em avaliar diferentes estratégias de estimação de parâmetros para um modelo de regressão linear múltipla. Para a estimação dos parâmetros do modelo foram utilizados dados de um ensaio clínico em que o interesse foi verificar se o ensaio mecânico da propriedade de força máxima (EM-FM) está associada com a massa femoral, com o diâmetro femoral e com o grupo experimental de ratas ovariectomizadas da raça Rattus norvegicus albinus, variedade Wistar. Para a estimação dos parâmetros do modelo serão comparadas três metodologias: a metodologia clássica, baseada no método dos mínimos quadrados; a metodologia Bayesiana, baseada no teorema de Bayes; e o método Bootstrap, baseado em processos de reamostragem.

Bayesian analysis of skew-t multivariate null intercept measurement error model

The multivariate skew-t distribution (J Multivar Anal 79:93-113, 2001; J R Stat Soc, Ser B 65:367-389, 2003; Statistics 37:359-363, 2003) includes the Student t, skew-Cauchy and Cauchy distributions as special cases and the normal and skew-normal ones as limiting cases. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis of repeated measures, pretest/post-test data, under multivariate null intercept measurement error model (J Biopharm Stat 13(4):763-771, 2003) where the random errors and the unobserved value of the covariate (latent variable) follows a Student t and skew-t distribution, respectively. The results and methods are numerically illustrated with an example in the field of dentistry.

Bayesian analysis for a skew extension of the multivariate null intercept measurement error model

Skew-normal distribution is a class of distributions that includes the normal distributions as a special case. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis in a multivariate, null intercept, measurement error model [R. Aoki, H. Bolfarine, J.A. Achcar, and D. Leao Pinto Jr, Bayesian analysis of a multivariate null intercept error-in -variables regression model, J. Biopharm. Stat. 13(4) (2003b), pp. 763-771] where the unobserved value of the covariate (latent variable) follows a skew-normal distribution. The results and methods are applied to a real dental clinical trial presented in [A. Hadgu and G. Koch, Application of generalized estimating equations to a dental randomized clinical trial, J. Biopharm. Stat. 9 (1999), pp. 161-178].

Proximal methods for nonlinear programming: double regularization and inexact subproblems

This paper describes the first phase of a project attempting to construct an efficient general-purpose nonlinear optimizer using an augmented Lagrangian outer loop with a relative error criterion, and an inner loop employing a state-of-the art conjugate gradient solver. The outer loop can also employ double regularized proximal kernels, a fairly recent theoretical development that leads to fully smooth subproblems. We first enhance the existing theory to show that our approach is globally convergent in both the primal and dual spaces when applied to convex problems. We then present an extensive computational evaluation using the CUTE test set, showing that some aspects of our approach are promising, but some are not. These conclusions in turn lead to additional computational experiments suggesting where to next focus our theoretical and computational efforts.

Inference and local influence assessment in skew-normal null intercept measurement error model

In this article, we discuss inferential aspects of the measurement error regression models with null intercepts when the unknown quantity x (latent variable) follows a skew normal distribution. We examine first the maximum-likelihood approach to estimation via the EM algorithm by exploring statistical properties of the model considered. Then, the marginal likelihood, the score function and the observed information matrix of the observed quantities are presented allowing direct inference implementation. In order to discuss some diagnostics techniques in this type of models, we derive the appropriate matrices to assessing the local influence on the parameter estimates under different perturbation schemes. The results and methods developed in this paper are illustrated considering part of a real data set used by Hadgu and Koch [1999, Application of generalized estimating equations to a dental randomized clinical trial. Journal of Biopharmaceutical Statistics, 9, 161-178].

Semiparametric Bayesian measurement error modeling

This work presents a Bayesian semiparametric approach for dealing with regression models where the covariate is measured with error. Given that (1) the error normality assumption is very restrictive, and (2) assuming a specific elliptical distribution for errors (Student-t for example), may be somewhat presumptuous; there is need for more flexible methods, in terms of assuming only symmetry of errors (admitting unknown kurtosis). In this sense, the main advantage of this extended Bayesian approach is the possibility of considering generalizations of the elliptical family of models by using Dirichlet process priors in dependent and independent situations. Conditional posterior distributions are implemented, allowing the use of Markov Chain Monte Carlo (MCMC), to generate the posterior distributions. An interesting result shown is that the Dirichlet process prior is not updated in the case of the dependent elliptical model. Furthermore, an analysis of a real data set is reported to illustrate the usefulness of our approach, in dealing with outliers. Finally, semiparametric proposed models and parametric normal model are compared, graphically with the posterior distribution density of the coefficients. (C) 2009 Elsevier Inc. All rights reserved.

Multivariate measurement error models based on scale mixtures of the skew-normal distribution

Scale mixtures of the skew-normal (SMSN) distribution is a class of asymmetric thick-tailed distributions that includes the skew-normal (SN) distribution as a special case. The main advantage of these classes of distributions is that they are easy to simulate and have a nice hierarchical representation facilitating easy implementation of the expectation-maximization algorithm for the maximum-likelihood estimation. In this paper, we assume an SMSN distribution for the unobserved value of the covariates and a symmetric scale mixtures of the normal distribution for the error term of the model. This provides a robust alternative to parameter estimation in multivariate measurement error models. Specific distributions examined include univariate and multivariate versions of the SN, skew-t, skew-slash and skew-contaminated normal distributions. The results and methods are applied to a real data set.

Influence Diagnostics for a Skew Extension of the Grubbs Measurement Error Model

Influence diagnostics methods are extended in this article to the Grubbs model when the unknown quantity x (latent variable) follows a skew-normal distribution. Diagnostic measures are derived from the case-deletion approach and the local influence approach under several perturbation schemes. The observed information matrix to the postulated model and Delta matrices to the corresponding perturbed models are derived. Results obtained for one real data set are reported, illustrating the usefulness of the proposed methodology.

A robust Bayesian approach to null intercept measurement error model with application to dental data

Measurement error models often arise in epidemiological and clinical research. Usually, in this set up it is assumed that the latent variable has a normal distribution. However, the normality assumption may not be always correct. Skew-normal/independent distribution is a class of asymmetric thick-tailed distributions which includes the Skew-normal distribution as a special case. In this paper, we explore the use of skew-normal/independent distribution as a robust alternative to null intercept measurement error model under a Bayesian paradigm. We assume that the random errors and the unobserved value of the covariate (latent variable) follows jointly a skew-normal/independent distribution, providing an appealing robust alternative to the routine use of symmetric normal distribution in this type of model. Specific distributions examined include univariate and multivariate versions of the skew-normal distribution, the skew-t distributions, the skew-slash distributions and the skew contaminated normal distributions. The methods developed is illustrated using a real data set from a dental clinical trial. (C) 2008 Elsevier B.V. All rights reserved.