This article presents a new estimation method for the parameters of a time series model. We consider here composite Gaussian processes that are the sum of independent Gaussian processes which, in turn, explain an important aspect of the time series, as is the case in engineering and natural sciences. The proposed estimation method offers an alternative to classical estimation based on the likelihood, that is straightforward to implement and often the only feasible estimation method with complex models. The estimator furnishes results as the optimization of a criterion based on a standardized distance between the sample wavelet variances (WV) estimates and the model-based WV. Indeed, the WV provides a decomposition of the variance process through different scales, so that they contain the information about different features of the stochastic model. We derive the asymptotic properties of the proposed estimator for inference and perform a simulation study to compare our estimator to the MLE and the LSE with different models. We also set sufficient conditions on composite models for our estimator to be consistent, that are easy to verify. We use the new estimator to estimate the stochastic error's parameters of the sum of three first order Gauss-Markov processes by means of a sample of over 800,000 issued from gyroscopes that compose inertial navigation systems. Supplementary materials for this article are available online.

-In this article, we investigate two different algorithms for the integration of GPS with redundant MEMS-IMUs. Firstly, the inertial measurements are combined in the observation space to generate a synthetic set of data which is then integrated with GPS by the standard algorithms. In the second approach, the method of strapdown navigation needs to be adapted in order to account for the redundant measurements. Both methods are evaluated in experiments where redundant MEMSIMUs are fixed in different geometries: orthogonallyredundant and skew-redundant IMUs. For the latter configuration, the performance improvement using a synthetic IMU is shown to be 30% on the average. The extended mechanization approach provides slightly better results (about 45% improvement) as the systematic errors of the individual sensors are considered separately rather than their fusion when forming compound measurements. The maximum errors are shown to be reduced even by a factor of 2.

This research studies the reduction and the estimation of the noise level within a redundant configuration of low-cost (MEMS-type) inertial measurement units (IMUs). Firstly, independent observations between units and sensors are assumed and the theoretical decrease in the system noise level is analyzed in an experiment with four MEMS-IMU triads. Then, more complex scenarios are presented in which the noise level can vary in time and for each sensor. A statistical method employed for studying the volatility of financial markets (GARCH) is adapted and tested for the usage with inertial data. This paper demonstrates experimentally and through simulations the benefit of direct noise estimation in redundant IMU setups.

Abstract-The integration of observations issued from a satellite-based system (GNSS) with an Inertial Navigation System (INS) is usually performed through a Bayesian filter such as the Extended Kalman Filter (EKF). The task of designing the navigation EKF is strongly related to the inertial sensor error modeling problem. Accelerometers and gyroscopes may be corrupted by random errors of complex spectral structure. Consequently, identifying correct error-state parameters in the INS/GNSS EKF becomes difficult when several stochastic processes are superposed. In such situations, classical approaches like the Allan variance or PSD analysis fail due to the difficulty of separating the error-processes in the spectral domain. For this purpose, we propose applying a recently developed estimator based on the Generalized Method of Wavelet Moments (GMWM) which was proven to be consistent and asymptotically normally distributed. The GMWM estimator matches theoretical and sample based wavelet variances, and can be computed using the method of indirect inference. This article mainly focuses on the implementation aspects related to the GMWM, and its integration within a general navigation filter calibration procedure. Regarding this, we apply the GMWM on error signals issued from MEMS-based inertial sensors by building and estimating composite stochastic processes for which classical methods cannot be used. In a first stage, we validate the resulting models using (Allan) variance and PSD analyses and then, in a second stage, study the impact of the resulting navigation filter design in terms of final positioning accuracy. We demonstrate that the GMWM-based calibration framework enables to estimate complex stochastic models in a few seconds only, and that these models are relevant in terms of the resulting navigation accuracy.

Along the ever increasing data size and model complexity, an important challenge frequently encountered in constructing new estimators or in implementing a classical one such as the maximum likelihood estimator, is the computational aspect of the estimation procedure. To carry out estimation, approximate methods such as pseudo-likelihood functions or approximated estimating equations are increasingly used in practice as these methods are typically easier to implement numerically although they can lead to inconsistent and/or biased estimators.In this context, we extend and provide refinements on the known bias 1 correction properties of two simulation based methods, respectively indirect inference and bootstrap, each with two alternatives. These results allow one to build a framework defining simulation based estimators that can be implemented for complex models. Indeed, based on a biased or even inconsistent estimator, several simulation based methods can be used to define new estimators that are both consistent and with reduced finite sample bias. This framework includes the classical method of indirect inference for bias correction without requiring specification of an auxiliary model. We demonstrate the equivalence between one version of the indirect inference and the iterative bootstrap, both correct sample biases up to the order n −3 .The iterative method can be thought of as a computationally efficient algorithm to solve the optimization problem of the indirect inference.Our results provide different tools to correct the asymptotic as well as finite sample biases of estimators and give insight on which method should be applied for the problem at hand. The usefulness of the proposed approach is illustrated with the estimation of robust income distributions and generalized linear latent variable models.