Computational Methods in Systems Biology

An R Package to Assess Transfer Entropies via Permutation Tests 287

It is inspired by the method proposed in [ 8 ] and facilitates a shuffling within
each time series to destroy potential correlations.
Each time series is randomized by shuffling the order of the data points.
Neither the scale of measurements nor the overall composition, meaning the
probabilitiesp(xn) of occurrence of individual measurements, of the vector itself
is altered. Therefore the overall entropy of times series does not change, but the
information transfer between both vectors and the temporal information flow of
each vector is disrupted. Eventually, signatures of causality are being destroyed.
The number of necessary repetitions in the permutation test depends strongly
on the sample size of the used data. Since the shuffling runs are independent from
each other (data parallelism), the randomization can be easily parallelized and
therefore optimized for several CPU cores (Fig. 1 ).

Fig. 1.Schematic of the null model. The sequential

time series are shuffled to remove any causal rela-

tion between them. The entropy of each time series

is kept constant, since the procedure only random-

izes the order of the series, but not their respective

composition.

To test for statistical sig-
nificance we perform the
permutation test and com-
pute from the repeatedly
computed TEs the so called
Z-score for the informa-
tion transfer ofTEY→Xand
TEX→Yindividually.

ZTE=

TE−TEs

σ(TEs)

(3)

With TEs being the
mean Transfer Entropy of
all shuffle runs andσ(TEs)
the respective standard devi-
ation. The resulting score gives the distance of the Transfer Entropy for the
original, raw data in units of standard deviation from the mean of the Trans-
fer Entropy for the shuffled data.Zcan be converted to, e.g.,p-values under
Null-Hypothesis testing under applicable distributions of the test statistics TE.

1.3 Illustrative Application: Coupling of Clocks

In Fig. 2 we show a more involved model of two oscillators, that could implement,
e.g., intracellular clocks. We have four speciesx,y,u,andvwhose concentrations
form four time series that we analyze via the tools implemented in ourRpackage.

2 Results

Here, we show how to detect the coupling of molecular species in our toy model of
Fig. 2. We created time series for the four variablesu, v, x, y, addingN(μ, σrel=
5%) relative noise to account for sampling and measurement errors.
Applyingget.tewe obtain the results in Fig. 4. In Fig. 5 we show the results
for varying the couplingkvybetween the two oscillatorsx, yandu, v. In Fig. 5 (a)