This iterative approach (defined explicitly below) is applied here to
log-transformed data z, = %n y, and to the untransformed concentrations
Yye
Residuals at sampte collection points are obtained for the fits in
both scales as well as for the "antilog” scale.
For this latter case,
the estimates obtained in log scale are transformed back to the original
scale by taking antilogarithms.
The antilog residual is then the differ-
ence between the observed datum and the antilog estimate (see Table 1).
Our interest in the antilog scale arises from a desire to present results
in untransformed scale while taking advantage of any benefits to be had
by fitting in the log scale.
The log concentration surface is displayed
here as a contour map in untransformed scale by drawing contours for log
concentrations z such that y = exp(z), where y are contours of interest
in the original scale.
In this paper, contours are displayed for values
of y equal to 1, 10, 100, 1,000, and 10,000 uCi/m
of *°9°24°Pu. This
is done by drawing contours for z's equal to {n 1, £n 10,
on the estimated log concentration surface.
&n 100, etc.,
In this paper, we investigate whether iterating on residuals in any or
all of the three scales (untransformed, log~transformed, and antilog)
result in a "better" estimate of the true plutonium concentration surface
than if the estimation routine were applied only once.
The "best"
estimated surface is considered here to be that for which the deviation
between the true and estimated concentration surface at all locations
(not just at sample points) is a minimum.
Since the true surface is
unknown, this investigation includes examining residuals between the
observed and estimated surface at sample collection points.
This analysis
includes plotting and comparing residuals for each iteration, computing
the mean, median, and standard deviations of residuals, and by computing
the proportion of the total variation in the observed data explained by .
the estimated values.
Further insight is gained by computing the linear
Table 1.
Notation for Iterative Procedure on Untransformed, Log-Transformed,
and Antilog Fits to 239-240Py Concentrations in Surface Soil
Untrans formed
Observed
Concentration
Estimated
Concentration
at mth
Iteration
Residual
(R,_)
im
*units of uCi/m?.
x
Vi
“
y;
=
“4
x Ya3
Log
Antilog
24
_
~
Any,
“
Zz,
05
“ *
2» 254
J
*
yi
*
y,;
=
*
exp (z5)
J
“
Ye M4
“
4,7 44
Applicable to top 5 cm of soil.
322
*
y, 7 exp (2)