A synthetic example of indirect causation. A sequence was generated using a simplified A → B → C causal model, and then analysed using standard methods and our proposed method. The generation procedure was deliberately designed as not a perfect match to any of the analysis models, so as not to privilege any of them. We show an example timeline of events, along with the cross-correlation plots, and then the influence strengths recovered using each method summarized as a social network diagram. Arrow thicknesses indicate the relative strength of influence (or transition probability for the Markov model); arrows represent excitatory influences, while flat-headed arrows (the self–self loops in our analysis) represent inhibitory influences. Note that the values recovered by each method are different in kind and have been rescaled separately for each of the network plots. Cross-correlation analysis recovers most of the influences/independences but tends to recover false-positive connections for the indirect link A → C (see the lower-left panel of the cross-correlation plots). A simple Markov model recovers influences without timing information, and in this case also adds a connection from C back to A to take the place of the baseline event calling rate of A. Our proposed method recovers a good match for the network structure as well as timing information. It adds self-inhibitory feedback on B and C to account for the fact that in this test case a call by A leads to no more than one call by B (and likewise for B → C). For further details of this synthetic example, see the electronic supplementary material. (Online version in colour.)
Schematic of processes generating events: (a) homogeneous Poisson process, (b) inhomogeneous Poisson process and (c) inhomogeneous Poisson process in which all changes in rate are due to the external influence of stimulus events. In each panel, the rate parameter for the process (λ) is shown as a filled curve, continuous in time, and an example sequence of events sampled from the process is shown as a sequence of spikes. (Online version in colour.)
Principal components plot of kernels recovered from zf4f data. Arrows connect the kernels for each directed pair, between the two days studied. Ellipses show the 50% and 95% probability regions for Gaussian fits, for the self–self (upper, dots) and for the self–other (lower, plusses) datapoints. These Gaussian ellipses give a visual indication of the groupings evaluated in the MRPP test. The number labels indicate the individuals involved in each influence kernel. (Online version in colour.)
Influence strengths, plotted as a social network. Standard arrows indicate kernels whose peak values are positive (excitatory), while flat-headed arrows indicate kernels whose peak values are negative (inhibitory). In this case, all the self–self arrows looping back are inhibitory. Note that this view emphasizes the magnitudes while suppressing the temporal structure recovered using our model. (Online version in colour.)
Aggregate kernels as in figure 3 but for the dataset of Gill et al. Note that the ‘self–partner’ category was labelled retrospectively, according to the pair bonds that eventually formed. The pairings had typically not yet formed on the first day.
Principal components plots of kernels recovered from each day of the Gill et al. data. Ellipses show the 50% and 95% Gaussian probability regions for the three types of connection: self–self (dots), self–partner (crosses), self–other (plusses). (Online version in colour.)
Interindividual influence strengths for the dataset of Gill et al. considering all call types pooled together and analysed using cross-correlation (a) or GLMpp (b). In each matrix, individuals are arranged so that the cells on the counter-diagonal are the intra-pair influences (e.g. F1 to M1). Self–self influences are omitted for visual clarity (black squares on diagonal). For the GLMpp analysis, PCA kernel magnitudes are plotted, with polarity used to indicate whether each kernel's main peak is excitatory or inhibitory.
Network plot of interindividual influence strengths for the dataset of Gill et al. on day 7, considering all call types pooled together and analysed using cross-correlation (a) or GLMpp (b). See also figure 8. (Online version in colour.)
Predictability (Pearson correlation) of kernel magnitudes, from one day to the next, measured under four different permutations for aligning the two days. The four permutations correspond to four rows of a Latin square, one of which matched individuals across days, another which matched physical cage locations across days and two null permutations having no meaningful interpretation.
self–other (n = 12)
self–self (n = 4)
***p < 0.001.
Within-group agreement for self–partner versus self–other influence kernels, for each day of the Gill et al. data, with p-values.
MRPP agreement (%)
Within-group agreement for each day of the Gill et al. data, to measure the mutual distinctiveness of four specific influence kernel types: Tet–Tet, Tet–Stack, Stack–Tet, Stack–Stack (cf. figure 11).