by increasing the window size, at the cost of being
slightly less robust to aliasing errors.
GVAD+LOCAL: GVAD-0.10 followed by the local
search procedure described previously.
GVAD+KALMAN: Same as above, but followed by for-ward-backward message passing in a graphical model
similar to the one in figure 9 where the wrapped-normal likelihoods are approximated a priori by Gaussian
potentials, so the model has the structure of a Kalman
EP: Same as above, but followed by additional forward
and backward passes using the EP message updates,
with the approximate local search procedure used
EP+HESS: Same as above, but MATLAB’s numerical
optimizer and the more accurate five-term approximation to the infinite sum in the wrapped normal
density are used in place of the local-search procedure.
Additional details about the experimental setup
and the algorithms can be found in the conference
paper (Sheldon et al. 2013).
Figure 10 shows the average RMSE and 95 percent
confidence interval attained by each algorithm on
the 142 test scans. All comparisons are highly signifi-
cant (p < 10–7; paired t-test) except GVAD+LOCAL versus
GVAD+KALMAN and EP versus EP+HESS. It is clear from
the results that each of the improvements we proposed
makes a substantial improvement to performance.
GVAD-0.10 is much better than GVAD-0.01, highlight-
ing the importance of our extension to reduce variance
by increasing the window size. By adding the simple
dealias-and-refit local search procedure, GVAD+LOCAL
performs much better than GVAD alone. Finally, by per-
forming approximate Bayesian inference with the more
accurate wrapped normal likelihood, EP performs sig-
nificantly better than GVAD+LOCAL. The lesser per-
formance of GVAD+KALMAN indicates that the graphi-
cal model structure alone does not explain the better
performance of EP: the approach of approximating the
wrapped normal likelihood in the context of the current
posterior is important. Finally the performance of
EP+HESS shows that EP loses no accuracy by using the
approximate local search in the Laplace approximation.
By examining individual scans (not shown), we
observed that the better models were generally in close
agreement about the direction of travel, but the GVAD-
based models seemed to underestimate speed, which is
likely due to not being completely robust to aliasing
˚N 6 9˚N 7 2˚N 8 1˚N 7 8˚N 7 5˚N 6 9˚N 7 2˚N
Figure 11. Reflectivity, Velocity, Average Bird Density, and Direction and Speed Data
from 12 Radar Stations on 20 August 2010, Two Hours after Local Sunset.
From left to right: (a) reflectivity, (b) velocity. Bubbles represent average bird density proportional to the number of birds aloft, whereas arrows represent an average direction of movement and length of the arrow represents average speed. This night shows a pattern
typical of light to moderate bird migration.