Computational Methods in Systems Biology

A Scheme for Adaptive Selection of Population Sizes 135

For the inter- and extrapolation ofECV a functional approximationfis used.
The functional form offis motivated by the scaling of the KDE mean squared
error as function of the population sizen[ 3 ]. Silverman [ 21 ] showed that the
mean squared error of KDEs decreases withαn−β, depending on the proper-
ties of the distribution and the choice of the selected KDE. In this study, this
functional form is employed onECV. The parametersαandβof the function
f(n;α, β)=αn−βare fitted to the points{(n∗t,q,ECV(n∗t,q))}Qq=1(Fig. 1 b, black
points) with non-linear least squares and the Levenberg-Marquardt algorithm
[ 11 , 15 ], yielding the optimized parametersαtandβtas well as the correspond-
ing curve (Fig. 1 b, interpolation and extrapolation). Finally, the sizent+1of the
subsequent population is selected such that the target variation is expected to
be achieved:nt+1= round(f−^1 (ECV;αt,βt)). In the case of multiple models,
this scheme is performed on the joint parameter space. The pseudo-code is pro-
vided in Algorithms 3 and 4. There, Fit denotes fitting the functionf, Round
rounding to the nearest integer, Sample(K, n∗) drawingn∗samples fromKand
n∗qcorresponds ton∗t,qof ( 1 ).

Algorithm 3.AdaptPopulationSize

Input:P,KDE

Output:n

N∗←[n∗ 1 ,...,n∗Q]

C∗←[EstimateCV(n∗,KDE,P)forn∗inN∗]

f←Fit(N∗,C∗)

n←Round(f−^1 (ECV))

Algorithm 4.EstimateCV

Input:n∗,KDE,P

Output:cv

K←KDE(P)

K∗←[KDE(Sample(K, n∗))forbin{ 1 , ..., B}]

cv←

∑

(w,θ)∈PwCV([K

′(θ)forK′inK∗])

The population size is selected before the sampling of a population starts, instead
of being continuously re-evaluated during sampling (after acceptance of each
particle), to avoid potential bias towards distributions yielding lower variation
for the same population size.

3 Results

To assess the proposed adaptation scheme, we applied it to problems with known
true parameters and to problems of high practical relevance. We first assessed
the appropriateness of the functional approximation, then examined the example
of an analytical model with a multimodal posterior. We next applied the scheme
to model selection of Markov Jump Process models and finally investigated a
multiscale tumor growth model.