findSeTheta: Scale pooled standard errors after polar transformation
Description
After polar transformation of Cartesian intensity values, the
estimated standard errors are no longer useful. This function
normalizes the standard errors depending on the polar
intensity value
Usually called from within createAlleleSet. The standard
errors of the Cartesian intensities A and B are not
meaningful when the polar coordinates theta and
intensity are plotted on Cartesian axes. In a plot of
homoscedastic B vs. A (see
transformChannels), the standard error of each bead-type
is independent of the signal intensities. In a Cartesian plot of
intensity vs. theta, however, bead-types with low
intensity will have a large uncertainty, and the precision of the
points will increase with increasing intensity. This is beacause the
arc-length of the first quadrant semi-circle, which increases with the
distance from origin, gets a constant value of unity as the polar
coordinates are plotted on Cartesian axes. The pooled standard errors
are therefore scaled with the intensity dependent arc-length of the
semi-circle between 0 and 90 degrees.
The arc-lengths by which the standard errors are scaled also depend on
dist and pNorm. The circumference of a circle in
Manhattan geometry, using a Euclidean metric, is
4*sqrt(2)*R, and the circumference of a Euclidean circle is
2*pi*R. It follows that the arc-lengths in the first quadrant
only are sqrt(2)*R and pi*R/2, respectively. The more
general arc-length of a Minkowski geometry circle is estimated by
numerical integration along the the curve of the super-ellipse between
0 to 90 degrees.
#A single standard error value for points of increasing intensityR <- .1:10pooledSE.raw <- 1
pooledSE.theta <- findSeTheta(pooledSE.raw=pooledSE.raw,R=R)
print(pooledSE.theta)