A Prediction Divergence Criterion for Model Selection

The problem of model selection is inevitable in an increasingly large number of applications involving partial theoretical knowledge and vast amounts of information, like in medicine, biology or economics. The associated techniques are intended to determine which variables are "important" to "explain a phenomenon under investigation. The terms "important" and "explain" can have very different meanings according to the context and, in fact, model selection can be applied to any situation where one tries to balance variability with complexity. In this paper, we introduce a new class of error measures and of model selection criteria, to which many well know selection criteria belong. Moreover, this class enables us to derive a novel criterion, based on a divergence measure between the predictions produced by two nested models, called the Prediction Divergence Criterion (PDC). Our selection procedure is developed for linear regression models, but has the potential to be extended to other models. We demonstrate that, under some regularity conditions, it is asymptotically loss efficient and can also be consistent. In the linear case, the PDC is a counterpart to Mallow's Cp but with a lower asymptotic probability of overfitting. In a case study and by means of simulations, the PDC is shown to be particularly well suited in "sparse" settings with correlated covariates which we believe to be common in real applications.Comment: 56 page

Construction of dynamically-dependent stochastic error models

Stochastic behavior of an instrument is often analyzed by constructing the Allan (or wavelet) variance signatures from an error signal. For inertial sensors, such a signature is conveniently obtained by recording data at rest. The analysis of this signal will result in noise-parameters adequate to such situation. Nonetheless, the value of the noise parameters may change under dynamics or other kind of external influences like for instance the temperature. In this research we study first the influence of the rotational dynamics on the signal of MEMS gyroscopes and then we show how to link this property to the noise-parameter estimation in a rigorous way by a modified version of the Generalized Method of Wavelet Moments (GMWM) estimator. The results of such analysis can then for instance be used in a Kalman filter, where the noise parameters are adapted according to such predetermined functional relationship between sensor noise and the encountered dynamics of the platform/sensor

Parameter Determination of Sensor Stochastic Models under Covariate Dependency

The proliferation of (low-cost) sensors provokes new challenges in data fusion. This is related to the correctness of stochastic characterization that is a prerequisite for optimal estimation of parameters from redundant observations. Different (statistical) methods were developed to estimate parameters of complex stochastic models. To cite a few, there is the maximum likelihood approach estimated via the so-called EM algorithm as well as a linear regression approach based on the log-log-representation of a quantity called Allan Variance. Nevertheless, all these methods suffer from various limitations ranging from numerical instability and computational inefficiency to statistical inconsistency. The relative recent approach called Generalized Method of Wavelet Moments (GMWM) that makes a use of the Wavelet Variance (WV) quantity of the error signal was proven to estimate stochastic models of considerable complexity in a numerically stable and statistically consistent manner with good computational efficiency. The situation is more challenging when stochastic errors are dependent on external factors (e.g. temperature, pressure, dynamics). This paper presents the essence of mathematical extension of the GMWM estimator that allows handling such a scenario rigorously by taking the external influences into consideration. We present the model of the multivariate stochastic process that composes firstly of the process of interest (signal of a sensor) and secondly of an explanatory process (e.g. environmental variable), where the latter is believed to have an impact on the stochastic properties of the former. Next, we assume that the input is composed of a real-valued “smooth” function dependent on external influence (values of which are perfectly observed) and a zero-mean process that is itself a sum of several independent latent processes. Then we define the covariate-dependent latent process (e.g. change of variance of white noise or auto-regressive process) as a class of piece-wise covariate-dependent latent time series models described by n-parameters. We propose to estimate the underlying vector parameter of interest using a modified version of the GMWM methodology that considers linear approximation of the dependency between noise parameters and the external influence. The intuition behind the new GMWM estimator is to select the parameter values that match the empirical WV on the data with the theoretical WV (i.e. those generated by the model parameters). We briefly demonstrate the asymptotic properties of the estimated parameter vector as well the consistency of the estimator

Automatic Identification and Calibration of Stochastic Parameters in Inertial Sensors

We present an algorithm for determining the nature of stochastic processes and their parameters based on the analysis of time series of inertial errors. The algorithm is suitable mainly (but not only) for situations where several stochastic processes are superposed. The proposed approach is based on a recently developed method called the Generalized Method of Wavelet Moments (GMWM), whose estimator was proven to be consistent and asymptotically normally distributed. This method delivers a global selection criterion based on the wavelet variance that can be used to determine the suitability of a candidate model (compared to other models) and apply it to low-cost inertial sensors. By allowing candidate model ranking, this approach enables us 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. Copyright (C) 2015 Institute of Navigation

An Approach for Observing and Modeling Errors in MEMS-Based Inertial Sensors Under Vehicle Dynamic

This paper studies the error behavior of low-cost inertial sensors in dynamic conditions. After proposing a method for error observations per sensor (i.e., gyroscope or accelerometer) and axes, their properties are estimated via the methodology of generalized method of wavelet moments. The developed model parameters are compared with those obtained under static conditions. Then, an attempt is presented to link the parameters of the established model to the dynamic of the vehicle. It is found that a linear relation explains a large portion of the exhibited variability. These findings suggest that the static methods employed for the calibration of inertial sensors could be improved when exploiting such a relationship