Nonstationary Least-Squares Collocation

Abstract

Many geodesists worldwide deal with the modelling of functions to approximate or interpolate measured data. For this purpose, a functional model is usually set up and adjusted by parameters so that it fits as precisely as possible to the data. One of the most important questions is how to select the functions of the model. One method, which has the advantage of being as independent as possible from the choice of functions is provided by least-squares collocation. Here, function values are estimated from the dependencies (covariances) between the observations themselves. Even though, when predicting values between the data, an analytic model, which describes the track of the covariances, still must be specified.
This is where this dissertation starts. After a general introduction and the discussion of the crucial terms (chapter 1) as well as the current state of the field (chapter 2), thus it is firstly shown how the estimate of autoregressive (AR) processes can be used to create a continuation of the sequence of variances, while the resulting function also must satisfy all requirements of a covariance function (chapter 3). The benefit of this method lies in the fact that the estimate of AR processes is data-driven. After finding the most suitable AR process of order p, a continuous continuation of the discrete covariances is to be found which is (1) clearly assigned to the AR process, (2) reproduces the discrete covariances and (3) is positive definite. As the spectrum of these covariance functions differ from the spectrum of the discrete variances, I show how they can be transformed into one another by a multiplication with a dirac comb. It must be mentioned that this thesis is limited to one dimensional time series associated with time.
The results of chapter 3 shows that instead of depending on the coefficients directly, the constructed covariance function depends on the roots of the characteristic polynomial of the AR process. These roots are also used to compute the spectrum of the AR process. As they offer the advantage of the coefficient being variable in time, the approach of using a TVAR process with known root motion guarantees that the roots of the characteristic polynomial of the process stay within the unit circle which implies that the variance of the resulting process stays finite. From this, we derive three methods to estimate TVAR processes, i.e. with linear, quadratic and piecewise linear root motion. Following the transformation from the coefficients of the AR process into the corresponding roots with linear or quadratic motions, only TVAR processes of order one, two and three are estimated directly. Since there is no analytic solution for computing the roots of a polynomial of order five or higher, the TVAR processes of higher order are estimated by a successive calculation using TVAR(1) and TVAR(2) processes.
As its title suggests, this study provides a method to compute the least-squares collocation under the assumption of nonstationary time series. As such, chapter 5 combines the findings of chapter 3 and chapter 4. Therefore the discrete covariances of a TVAR process are derived. In contrast to the discrete covariances of the stationary AR process these discrete covariances not only depend on the lag, but also on the epoch of the first observation as well as on the fact whether the second epoch relative to the first is in the past or in the future. As a result, the continuous continuation of the discrete covariances embodies a function of two parameters; the epoch of the first observation and the difference between the second and the first epoch including the sign of the difference.
In chapter 6, the different theories are tested on discrete measurements. First, sea level anomalies are estimated either by the Gauss-Markov model and trigonometric base functions, or by the Least-squares collocation with a covariance function from AR processes. Second, TVAR processes with predetermined root motions are estimated by GNSS elevation measurements and validated over the roots of stationary AR processes estimated for a moving window. In a final example, functions are estimated by temperature anomalies, wherein the covariances for the least-squares collocation come once from an AR(1) process estimate and another time from a TVAR(1) process estimate.