Copula functions are very convenient for modelling multivariate observations. Popular estimation methods are the two-stage MLE and an alternative semi-parametric with empirical cdf for the margins. Unfortunately, they are hastily biased whenever relatively small model deviations occur at the marginal (empirical or parametric) and/or copula levels. In this master thesis we propose three robust estimators that do not share this undesirable feature. The bounded-bias of robust estimators is corrected through indirect inference. By means of a simulation study we show that the robust estimators outperform the popular approaches.
Copula functions are very convenient for modelling multivariate observations. Popular es- timation methods are the two-stage maximum likelihood and an alternative semi-parametric with empirical cumulative distribution functions (cdf) for the margins. Unfortunately, they can be hastily biased whenever relatively small model deviations occur at the marginal (empirical or parametric) and/or copula levels. In this paper we propose three robust estimators that do not share this undesirable feature. Since heavy skewed and heavy tailed parametric marginals are often considered in applications, we also propose a bounded-bias robust estimator that is corrected for consistency by means of indirect inference. In a simulation study we show that the robust estimators outperform the popular approaches.
This research presents methods for detecting and isolating faults in multiple Micro-Electro-Mechanical System (MEMS) Inertial Measurement Unit (IMU) configurations. Traditionally, in the inertial technology, the task Fault Detection and Isolation (FDI) is realized by the parity space method. However, this approach performs poorly with low-cost MEMS-IMUs, although, it provides satisfactory results when applied to tactical or navigation grade IMUs. In this article, we propose a more complex approach to detect outliers that takes into account the shape and size of multivariate data. The proposed method is based on Mahalanobis distances. Such approach has already been successfully applied in other fields of applied multivariate statistics, however, it has never been tested with inertial sensors. As Mahalanobis distances (as well as the parity space method) is very sensitive to the presence of the same outliers this method aims to detect, we propose using its robust version. The performances of the proposed algorithm are evaluated using dynamical experiments with several MEMS-IMUs and a reference signal provided by a tactical-grade IMU run in parallel. The conducted experiment shows that, for example, the percentage of false alarms is approximately ten times lower when using a method based on Mahalanobis distances as compared to that based on the parity space approach.
We present an algorithm for determining the nature of stochastic processes together with its parameters based on the analysis of time series of inertial errors. The algorithm is suitable mainly (but not only) for situations when several stochastic processes are superposed. In such cases, classical approaches based on the analysis of Allan variance or PSD are likely to fail due to the difficulty of separating the underlying error-processes in the spectral domain. The developed alternative is based on the recently proposed method called the Generalized Method of Wavelet Moments (GMWM), whose resulting estimator was proven to be consistent and asymptotically normally distributed. The principle of this method is to match the empirical and model-based wavelet variances (WV). In this study we propose a goodness-of-fit criterion which can be used to determine the suitability of a candidate model and apply it to low-cost inertial sensors. The suggested approach of model selection relies on an unbiased estimate of the distance between the theoretical WV and the empirical WV which would be obtained on an independent sample issued from the stochastic process of interest. Such goodness-of-fit criterion is however “penalized” by the complexity of the model. In some sense, the proposed methodology is a generalization of Mallow’s Cp applied to models estimated by the GMWM. By allowing to rank candidate models, this approach permits to construct an algorithm for automatic model identification and determination. The benefits of this methodology are highlighted by providing practical examples of model selection for two types of MEMS-IMUs, the latter of higher quality.
BACKGROUND: Patients admitted to the emergency room with renal impairment and undergoing a contrast computed tomography (CT) are at high risk of developing contrast nephropathy as emergency precludes sufficient hydration prior to contrast use. The value of an ultra-high dose of intravenous N-acetylcysteine in this setting is unknown. METHODS: From 2008 to 2010, we randomized 120 consecutive patients admitted to the emergency room with an estimated clearance lower than 60 ml/min/1.73 m2 by MDRD (mean GFR 42 ml/min/1.73 m2) to either placebo or 6000 mg N-acetylcysteine iv one hour before contrast CT in addition to iv saline. Serum cystatin C and creatinine were measured one hour prior to and at day 2, 4 and 10 after contrast injection. Nephrotoxicity was defined either as 25% or 44 mumol/l increase in serum creatinine or cystatin C levels compared to baseline values. RESULTS: Contrast nephrotoxicity occurred in 22% of patients who received placebo (13/58) and 27% of patients who received N-acetylcysteine (14/52, p = 0.66). Ultra-high dose intravenous N-acetylcysteine did not alter creatinine or cystatin C levels. No secondary effects were noted within the 2 groups during follow-up. CONCLUSIONS: An ultra-high dose of intravenous N-acetylcysteine is ineffective at preventing nephrotoxicity in patients with renal impairment undergoing emergency contrast CT.Trial registration: The study was registered as Clinical trial (NCT01467154).
This article first demonstrates the inconsistency of the estimator based on the standard Allan Variance (AV) for composite stochastic processes. This result motivates the use of a recently developed estimator, called the Generalized Method of Wavelet Moments (GMWM) estimator. This estimator was previously shown to be consistent and asymptotically normally distributed under the settings of the present research. Moreover, and unlike Maximum Likelihood Estimators (MLE), this method is able to estimate parameters of complex composite stochastic processes. After establishing the link between the AV and GMWM estimators we prove that there always exists a GMWM estimator with smaller asymptotic variance than its AV based counterpart. As the GMWM may be biased in finite samples issued from simple models on which the AV or MLE can still be applied, we present several extensions to the GMWM which significantly enhance its performance. One of these extensions is a finite sample bias correction related to the principle of indirect inference which is very general and can be applied beyond the scope of the GMWM framework. Finally, the theoretical findings are supported by simulation studies that compares the finite sample performance of AV based estimators, MLE and the various different forms of GMWM estimators.
Stochastic modeling is a challenging task for lowcost inertial sensors whose errors can have complex spectral structures. This makes the tuning process of the INS/GNSS Kalman filter often sensitive and difficult. We are currently investigating two approaches for bounding the errors in the mechanization. The first is an improved modeling of stochastic errors through the superposition of several Auto-Regressive (AR) processes. A new algorithm is presented based on the Expectation-Maximization (EM) principle that is able to estimate such complex models. The second approach focuses on redundancy through the use of multiple IMUs which don’t need to be calibrated a priori. We present a synthetic IMU computation in which the residuals are modeled by a single ARMA model. The noise power issued from the residuals is then continuously estimated by a GARCH model, which enables a proper weighting of the individual devices in the synthetic IMU.
Modeling and estimation of gyroscope and accelerometer errors is generally a very challenging task, especially for low-cost inertial MEMS sensors whose systematic errors have complex spectral structures. Consequently, identifying correct error-state parameters in a INS/GNSS Kalman filter/smoother becomes difficult when several processes are superimposed. In such situations, the classical identification approach via Allan Variance (AV) analyses fails due to the difficulty of separating the error-processes in the spectral domain. For this purpose we propose applying a recently developed estimation method, called the Generalized Method of Wavelet Moments (GMWM), that is excepted from such inconveniences. This method uses indirect inference on the parameters using the wavelet variances associated to the observed process. In this article, the GMWM estimator is applied in the context of modeling the behavior of low-cost inertial sensors. Its capability to estimate the parameters of models such as mixtures of GM processes for which no other estimation method succeeds is first demonstrated through simulation studies. The GMWM estimator is also applied on signals issued from a MEMS-based inertial measurement unit, using sums of GM processes as stochastic models. Finally, the benefits of using such models is highlighted by analyzing the quality of the determined trajectory provided by the INS/GNSS Kalman filter, in which artificial GNSS gaps were introduced. During these epochs, inertial navigation operates in coasting mode while GNSS-supported trajectory acts as a reference. As the overall performance of inertial navigation is strongly dependent on the errors corrupting its observations, the benefits of using the more appropriate error models (with respect to simpler ones estimated using classical AV graphical identification technique) are demonstrated by a significant improvement in the trajectory accuracy.
In this paper, we propose a new criterion for selection between nested models. We suppose that the correct model is one (or near one) of the available models and construct a criterion which is based on the Bregman divergence between the out-of-sample prediction of the smaller model and the in-sample prediction of the larger model. This criterion, the prediction divergence criterion (PDC), is different from the ones that are often used like the AIC, BIC, Cp, in that, in a sequential approach, it directly considers the prediction divergence between two models, rather that differences between the former criteria evaluated at two different models. We derive an estimator for the PDC (PDCE) using Efron (2004) approach on parametric covariance penalty method, and for the linear model and smoothing splines, we show that the PDCE on a suitable sequence of nested models that we formalize, selects the correct model with probability 1 as the sample size tends to infinity. In finite samples, we compare the performance of our criterion to the other ones as well as to the lasso, as find that it outperforms the other criteria in terms of prediction error in sparse situations.
