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Concept# Estimation of covariance matrices

Summary

In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in Rp×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. In addition, if the random variable has a normal distribution, the sample covariance matrix has a Wishart distribution and a slightly differently scaled version of it is the maximum likelihood estimate. Cases involving missing data, heteroscedasticity, or autocorrelated residuals require deep

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During the last twenty years, Random matrix theory (RMT) has produced numerous results that allow a better understanding of large random matrices. These advances have enabled interesting applications in the domain of communication. Although this theory can contribute to many other domains such as brain imaging or genetic research, its has been rarely applied. The main barrier to the adoption of RMT may be the lack of concrete statistical results from probabilistic Random matrix theory. Indeed, direct generalisation of classical multivariate theory to high dimensional assumptions is often difficult and the proposed procedures often assume strong hypotheses on the data matrix such as normality or overly restrictive independence conditions on the data.
This thesis proposes a statistical procedure for testing the equality of two independent estimated covariance matrices when the number of potentially dependent data vectors is large and proportional to the size of the vectors corresponding to the number of observed variables. Although the existing theory builds a very good intuition of the behaviour of these matrices, it does not provide enough results to build a satisfactory test for both the power and the robustness. Hence, inspired by spike models, we define the residual spikes and prove many theorems describing the behaviour of many statistics using eigenvectors and eigenvalues in very general cases. For example in the two central theorems of this thesis, the Invariant Angle Theorem and the Invariant Dot Product Theorem.
Using numerous generalisations of the theory, this thesis finally proposes a description of the behaviour of a statistic under a null hypothesis. This statistic allows the user to test the equality of two populations, but also other null hypotheses such as the independence of two sets of variables. Finally, the robustness of the procedure is demonstrated for different classes of models and criteria for evaluating robustness are proposed to the reader.
Therefore, the major contribution of this thesis is to propose a methodology both easy to apply and having good properties. Secondly, a large number of theoretical results are demonstrated and could be easily used to build other applications.

This paper presents a new concept for simultaneous modeling and adjusting of raw inertial observations with optical and (if available) GNSS data streams. The presented post-mission procedure of dynamic networks allows treating dynamic (e.g. inertial) and static (e.g. optical) raw observations with a spatial-temporal complexity that cannot be expressed in the traditional form of optimal filtering/smoothing. The theory is supported by a simulation scenario of terrestrial mobile mapping where sections of trajectory lacking GNSS coverage are visited several times and the optical observations (ranges and angles) are optimally combined, by using the presented approach, with angular and specific force observations of an onboard IMU. This simulation reveals that the parameter and covariance estimation via dynamic networks is i) equal to that obtained by the conventional INS/GNSS (if available) integration via filtering/optimal smoothing; and, ii) largely superior to the smoother when positioning states are conditioned across different times thanks to optical observations.