Computational Methods in Systems Biology

134 E. Klinger and J. Hasenauer

ab

True

KDEs

n=3

±2 weighted differential density variation

Parameterθ

n=50 n= 150

Population sizen

0 nt nt+1 4000

Population sizen

0

ECV

ECV(nt)

0.08

Density variation

Bootstrap

Control

Interpolation

Extrapolation

Fig. 1. Adaptive population size selection. (a)True density and kernel density
estimates (KDEs) on populations of sizen, sampled from the true density. The weighted
differential density variation is computed by calculating the pointwise variation (at each
parameterθ) and weighting it by the true density at this point. The density variation is
the integrated weighted differential density variation and is higher for smaller popula-
tion sizesn.(b)Fit of the density variationECVparametrized asECV(n;α, β)=αn−β
on bootstrapped populations (black points) of tentative sizesn∗t,q(see ( 1 )) and corre-
sponding estimated variationsECV(n∗t,q)(see( 2 )). Bootstrapped populations are drawn
from the KDE of populationt. The population sizent+1of the subsequent population
t+ 1 is selected to match the target variationECV, correcting the current variation
ECV(nt). Control estimates were directly obtained from populations of varying sizes
nof the underlying true, unimodal normal distribution. The bandwidth was selected
according to the Silverman rule (Sect.2.2, global bandwidth).

n∗t,q=nt/ 3     +(q−1)nt/ 10   ,q∈{ 1 ,...,Q},Q= max{q|n∗t,q≤ 2 nt}. (1)

To estimate the variation for each tentative population sizen∗t,q, a bootstrapped
populationPt,q,bof sizen∗t,qis drawn from the densityK=KDE(Pt)foreach
bootstrap repetitionb∈{ 1 ,...,B}(usuallyB≈10). Next, a density estimate,
Kt,q,b=KDE(Pt,q,b) is calculated on each of the bootstrapped populationsPt,q,b.
The density variationECVis then defined for each tentative population sizen∗t,q
according to

ECV(n∗t,q)=

∑nt

i=1

wiCV

(

{Kt,q,b(θi)}Bb=1

)

, (2)

with the coefficient of variation CV given by

CV({xb}Bb=1)=

Std

(

{xb}Bb=1

)

Mean

(

{(xb}Bb=1

),

computed from mean Mean and (biased) standard deviation Std:

Mean({xb}Bb=1)=

1

B

∑B

b=1

xb,Std({xb}Bb=1)=

√

√

√

√^1

B

∑B

b=1

(

xb−Mean

(

{xb}Bb=1

)) 2

.