Approximating covariance matrices estimated in multivariate models by estimated auto- and cross-covariances |
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Authors: | K R Koch H Kuhlmann W-D Schuh |
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Institution: | (1) Department of Biology, Wesleyan University, Middletown, CT 06459, USA |
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Abstract: | Quantities like tropospheric zenith delays or station coordinates are repeatedly measured at permanent VLBI or GPS stations
so that time series for the quantities at each station are obtained. The covariances of these quantities can be estimated
in a multivariate linear model. The covariances are needed for computing uncertainties of results derived from these quantities.
The covariance matrix for many permanent stations becomes large, the need for simplifying it may therefore arise under the
condition that the uncertainties of derived results still agree. This is accomplished by assuming that the different time
series of a quantity like the station height for each permanent station can be combined to obtain one time series. The covariance
matrix then follows from the estimates of the auto- and cross-covariance functions of the combined time series. A further
approximation is found, if compactly supported covariance functions are fitted to an estimated autocovariance function in
order to obtain a covariance matrix which is representative of different kinds of measurements. The simplification of a covariance
matrix estimated in a multivariate model is investigated here for the coordinates of points of a grid measured repeatedly
by a laserscanner. The approximations are checked by determining the uncertainty of the sum of distances to the points of
the grid. To obtain a realistic value for this uncertainty, the covariances of the measured coordinates have to be considered.
Three different setups of measurements are analyzed and a covariance matrix is found which is representative for all three
setups. Covariance matrices for the measurements of laserscanners can therefore be determined in advance without estimating
them for each application. |
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