| Résumé : |
(Auteur) Data processing in geodetic applications often relies on the least-squares method, for which one needs a proper stochastic model of the observables. Such a realistic covariance matrix allows one first to obtain the best (minimum variance) linear unbiased estimator of the unknown parameters; second, to determine a realistic precision description of the unknowns; and, third, along with the distribution of the data, to correctly perform hypothesis testing and assess quality control measures such as reliability. In many practical applications the covariance matrix is only partly known. The covariance matrix is then usually written as an unknown linear combination of known cofactor matrices. The estimation of the unknown (co)variance components is generally referred to as variance component estimation (VCE). In this thesis we study the method of least-squares variance component estimation (LSVCE) and elaborate on theoretical and practical aspects of the method. We show that LS-VCE is a simple, flexible, and attractive VCE-method. The LS-VCE method is simple because it is based on the well-known principle of least-squares. With this method the estimation of the (co)variance components is based on a linear model of observation equations. The method is flexible since it works with a user-defined weight matrix. Different weight matrix classes can be defined which all automatically lead to unbiased estimators of (co)variance components. LS-VCE is attractive since it allows one to apply the existing body of knowledge of least-squares theory to the problem of (co)variance component estimation. With this method, one can 1) obtain measures of discrepancies in the stochastic model, 2) determine the covariance matrix of the (co)variance components, 3) obtain the minimum variance estimator of (co)variance components by choosing the weight matrix as the inverse of the covariance matrix, 4) take the a-priori information on the (co)variance component into account, 5) solve for a nonlinear (co)variance component model, 6) apply the idea of robust estimation to (co)variance components, 7) evaluate the estimability of the (co)variance components, and 8) avoid the problem of obtaining negative variance components. LS-VCE is capable of unifying many of the existing VCE-methods such as MINQUE, BIQUE, and REML, which can be recovered by making appropriate choices for the weight matrix. An important feature of the LS-VCE method is the capability of applying hypothesis testing to the stochastic model, for which we rely on the w-test, v-test, and overall model test. We aim to find an appropriate structure for the stochastic model which includes the relevant noise components into the covariance matrix. The w-test statistic is introduced to see whether or not a certain noise component is likely to be present in the observations, which consequently can be included in the stochastic model. Based on the normal distribution of the original observables we determine the mean and the variance of the w-test statistic, which are zero and one, respectively. The distribution is a linear combination of mutually independent central chi-square distributions each with one degree of freedom. This distribution can be approximated by the standard normal distribution for some special cases. An equivalent expression for the w-test is given by introducing the v-test statistic. The goal is to decrease the number of (co)variance components of the stochastic model by testing the significance of the components. The overall model test is introduced to generally test the appropriateness of a proposed stochastic model. We also apply LS-VCE to real data of two GPS applications. LS-VCE is applied to the GPS geometry-free model. We present the functional and stochastic model of the GPS observables. The variance components of different observation types, satellite elevation dependence of GPS observables’ precision, and correlation between different observation types are estimated by LS-VCE. We show that the precision of the GPS observables clearly depends on the elevation angle of satellites. Also, significant correlation between observation types is found. For the second application we assess the noise characteristics of time series of daily coordinates for permanent GPS stations. We apply LS-VCE to estimate white noise and power-law noise (flicker noise and random walk noise) amplitudes in these time series. The results confirm that the time series are highly time correlated. We also use the w-test statistic to find an appropriate stochastic model of GPS time series. A combination of white noise, autoregressive noise, and flicker noise in general best characterizes the noise in all three position components. Unmodelled periodic effects in the data are then captured by a set of harmonic functions, for which we rely on least-squares harmonic estimation (LS-HE) developed in the same framework as LS-VCE. The results confirm the presence of annual and semiannual signals, as well as other significant periodic patterns in the series. To avoid the biased estimation of the variance components, such sinusoidal signals should be included in the functional part of the model before applying LS-VCE. |
Least-square variance component estimation (2007)
