Databases and Systems

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query on that band. As the maps in GDB increasingly come to span multiple levels of
resolution all the way down to sequence features, errors such as these become more
serious.
Nonlinear Alignment
There are a number of reasons why points in the dispersion plot do not fall on the line
y=x. These can include measurement error in mapping, mistakes made in mapping or
data entry, as in the outliers in figure 2, which reflect dramatic disagreements
between two maps as to the position of a locus. More importantly, distances in one
map may have a nonlinear relationship to distances in another. For example, genetic
and physical distances typically have a non-linear relationship because recombination
occurs more frequently near the middle of the chromosomal arms, causing a high
ratio of genetic to physical distance in those regions, but a low ratio in the
pericentromeric regions where recombination is inhibited. This nonlinear relationship
between different types of map distances means that the regression line will be a
more accurate transformation in some regions than in others.
One solution to the problem of nonlinearity is to use a nonlinear transformation
function which warps the maps into better correspondence. The best transformation
function would be the curve defined by the common markers, which we can
approximate by a piece-wise linear function. However the transformation must be
monotonic (nondecreasing) if it is to preserve marker orders when transforming a
map into universal coordinates, and we have seen how order discrepancies between
maps can introduce outliers in the plots. To remove these we select a maximal subset
of the common markers that are order-consistent between the two maps. Such a
subset is called a longest monotonic chain. A piecewise-linear function is then
defined over that set of points. Figure 5 shows the dispersion for a set of maps
aligned with the longest chain method as compared with linear regression. The
reduction in dispersion relative to the linear transformation is apparent.
Special care must be taken at the ends of maps, where there may be loci that occur
beyond the ends of the piecewise linear function. The function must be extrapolated
to handle such points. Two obvious solutions are to use the regression line slope or
the slope of the last linear segment; we have found the former to be more robust.
Although longest chain piecewise linear alignment is clearly an improvement over
linear regression, it can probably be further improved on. The algorithm chooses
arbitrarily among equally long longest chains; it should be possible to somehow
average among all possible longest chains. Also it might be worth considering other
measures of length besides number of points, such as distance covered, or composite
measures that include both number of points and distance spanned