Least-Squares Variance Component Estimation. Theory and GPS Applications

Abstract

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 (LS-VCE) 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, respectivel...

Improving radar interferometry for monitoring fault-related surface deformation

“…The VCE process employs the arcs of first order network that have no common PS and are distributed uniformly over the scene. Once these arcs are unwrapped in time and the parameters of interest estimated, the variance is calculated from the least squares residuals ê = Φ − Φ, (Teunissen, 1988;Amiri-Simkooei, 2007), where Φ and Φ are the observed and modeled phases, respectively. In this method, the variance covariance matrix of the observations QΦ is decomposed using the cofactor matrix Q k :…”

“…The estimator of the variance vector of N interferograms σ = [σ1, ..., σN ] T is given by (Amiri-Simkooei, 2007;Kampes, 2006)…”

Functional and stochastic modelling of satellite gravity data

“…It is shown in section 4.3 that under normality this estimator has minimum variance, (restricted) maximum likelihood and can be derived in a weighted least-squares sense. The derivation of the Least-Squares Variance Component Estimation (LSVCE) was done in a recent PhD study within the DEOS Department [Amiri-Simkooei, 2007]. A much faster estimator, which converges to the same results as MINQUE (under normality), but is biased in the iterations, is the Iterative Restricted Maximum Likelihood Estimator (IREML); see section 4.4.…”

“…In Sjöberg [1984] a non-negative alternative to MINQUE is suggested. However, negative variance components indicate either a low redundancy, an improperly designed variance component model or badly chosen a-priori variance components [Amiri-Simkooei, 2007]. Therefore, these estimators will not be addressed.…”

“…As we can write the vector of variance components as a vector of unknowns in the Gauss-Markov model and estimate them in a least-squares sense, we can test the estimation of the variance components for signiÀcance and derive the equations for linearization and non-negative estimation [Amiri-Simkooei, 2007]. In this section, we will shortly address these topics.…”