Let 
denote the estimated setup time. Let 
denote the setup time observed from our experiments. In an ideal
analysis we must always observe an equality between the two
times. Obviously there are some errors in the experiments and in the
formula itself (due to neglecting some low-order terms). We ran a
multiple linear regression on our experiments where
is minimized over all the experiments. This
is equivalent to finding the best constants for 
formula (again over all the experiments)
where A and B are the constants and 
term is obtained from the theoretical complexities given in O
notation in previous sections. The setup has a constant A as we need
to represent file open and close times spent in the experiments. After
the regression we obtained A=1.16 and B=0.0000177. Hence,
.
We did a similar analysis for the selection time. Let 
denote the estimated selection time. Let 
denote the
selection time observed from our experiments. Therefore, we can get
the formula 
. Selection does not have a
constant like A as we had in setup so A=0. The
regression produced B=0.00000121 and C=0.00000116. Hence,
.
Figure 5: A subset of experiments (25 experiments out of 3600): The
starfield size is 
pixels, the range slider size is 200
pixels, the point size is 
pixels, and the jump size is 
pixels (which forms the average case for our
experiments).
Finally, let 
denote the estimated querying time. Let
denote the querying time observed from our experiments.
Again using the theoretical complexities we can get 
. Due to initialization routines we
found usage of A appropriate in this case (eventually it turned out
to be a small value). Unfortunately, the analysis for querying is not
trivial, as we have to find an estimate for u in terms of
screen size, point size, and etc. We saw that the u term is
directly proportional to the jump size and the number of records
needed to be updated on the starfield. Since we paint more than one
pixel per point (in general) the formula counts the number of pixels
that are updated. So our estimate for u is 
where 
is the number of pixels to be painted per
each point and j is the jump size. Hence, becomes 
for our
case. Similarly, the regression produced A=0.00528, B=0.0000157,
and C=0.000000263. Hence, 
.