140 E. Klinger and J. Hasenauer
mean reference posterior. This KS distance increased asECVincreased (Fig. 3 f).
The number of required particles decreased however substantially for increasing
ECV(Fig. 3 g), resulting in a decrease of the computation time.
3.4 Multiscale Models
The parameter estimation problems considered in the previous sections possessed
up to two unknown parameters and their computational complexity was com-
paratively low. To assess if the method is also applicable to higher-dimensional
parameter spaces and computationally more demanding problems, we considered
a multiscale model for tumor growth on a two-dimensional plane, as described in
[ 9 ]. This model possessed seven unknown parameters, which we estimated from
artificial data generated by drawing 100 independent samples (Fig. 4 a) from the
model at the reference parameters as given in [ 9 ]. The artificial datasdatawas
obtained from these samples via averaging. We imposed on each parameter a
prior which was uniform in the log 10 domain with upper and lower bounds given
in [ 9 ]. We also used the distance function from [ 9 ]. For ABC-SMC we employed
a KDE with local bandwidth considering for each particle only the 20% nearest
neighbors, measured in Euclidean distance (see Sect.2.2).
The populations slowly contracted around the true (reference) parameters
and clustered already for generationt= 13 around them (Fig. 4 b). The last
generationt= 40 showed that the posterior converged to the true parameters
(Fig. 4 b). To evaluate the quality of the posterior approximation, we drew 100
samples from the maximum a posteriori (MAP) parameters. We found that the
distances (to the observed data) of samples from the MAP parameters were
comparable to the distances of samples from the true parameters (Fig. 4 c). This
indicated that the population size adaptation method, paired with a local pro-
posal distribution, worked successfully for the investigated multiscale model. We
then examined the adaptation of the population sizes (Fig. 4 d). The acceptance
threshold (Fig. 4 d 1 ) and the effective population size (Fig. 4 d 2 ) decreased over
the generations. Surprisingly, the population size increased instead (Fig. 4 d 2 ).
The estimation ofECV took in this example a fraction of about 2. 08 · 10 −^5 of
the total computation time and was therefore negligible.
4 Discussion
In many systems biology applications, model developers face the parameteri-
zation of complex computational models. While ABC-SMC algorithms are well
suited for this task, the need of manually tuning population sizes – a task which
requires substantial experience – limits their applicability. In this manuscript, we
proposed a method to adaptively select population sizes based on the uncertainty
of kernel density estimates. Our method complements existing methods for the
adaptive choice of perturbation kernels [ 6 ], acceptance thresholds [ 20 ], and sum-
mary statistics [ 13 , 17 ]. We illustrated the method’s applicability to parameter