Volume 9, Issue 2

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Updating a Geographic Database

Mathematical Background

For the covariance matrix, we do not take the actual covariance of this particular measurement, but start from a standardized covariance matrix whose coefficients depend only on the distances between points. This has the advantage that the warping does not depend on the particular measurement. This choice is allowed because we know that the measurements are sufficiently reliable since they have passed a standardized test. In fact, the specifications on the covariance in this test are reflected in the new matrix.

The matrix is constructed in several steps. Let us say that the vicinity of two coordinate values and is measured by

for some fixed positive number . If would be negative, we set it equal to 0. The standardized covariance matrix for two points and has the form

This tells us that if two points are close, then the -coordinates and the -coordinates are close, but there is a priori no cross relation between the -coordinate and the -coordinate. For two sets of points and the matrix is just made from blocks by taking as -th block the covariance matrix of and :

The above is the standard covariance matrix in the local coordinates of the measurement. During the warping we use the coordinate system of the database, so the matrix is affected by the similarity transformation and the result is . This is worked out in [2], where you may find more details.