An information-theoretic approach to infer the underlying interaction domain among elements from finite length trajectories in a noisy environment

Abstract

Transfer entropy in information theory was recently demonstrated [Basak et al., Phys. Rev. E 102, 012404 (2020)] to enable us to elucidate the interaction domain among interacting elements solely from an ensemble of trajectories. Therefore, only pairs of elements whose distances are shorter than some distance variable, termed cutoff distance, are taken into account in the computation of transfer entropies. The prediction performance in capturing the underlying interaction domain is subject to the noise level exerted on the elements and the sufficiency of statistics of the interaction events. In this paper, the dependence of the prediction performance is scrutinized systematically on noise level and the length of trajectories by using a modified Vicsek model. The larger the noise level and the shorter the time length of trajectories, the more the derivative of average transfer entropy fluctuates, which makes the identification of the interaction domain in terms of the position of global minimum of the derivative of average transfer entropy difficult. A measure to quantify the degree of strong convexity at the coarse-grained level is proposed. It is shown that the convexity score scheme can identify the interaction distance fairly well even while the position of the global minimum of the derivative of average transfer entropy does not. We also derive an analytical model to explain the relationship between the interaction domain and the change in transfer entropy that supports our cutoff distance technique to elucidate the underlying interaction domain from trajectories.

Transfer entropy dependent on distance among agents in quantifying leader-follower relationships

“…7]) and no interaction exists outside it. Then, the analytical expression of average TE, < TE > λ as a function of is easily acquired, showing < TE > λ remains the same for ≤ that yields <TE> = 0 [71]. In VM, as seen in Fig.…”

“…In recent studies it has been elucidated that informationtheoretic measures are successful in identifying leaders and followers [9,16,33,34,67,[70][71][72]. However, these measures have disadvantages in the cases where there are multiple interacting variables or when the amount of available data is limited.…”

“…We proposed information-theoretic schemes to infer the domain of interaction using the trajectories of the agents. The schemes are based on a quantity termed as the 'cutoff distance 𝜆', which is defined as a predefined maximum distance up to which the interaction between agents is considered for the estimation of TE [9,71]. In fact, for a pre-defined cutoff distance 𝜆, the TE between two agents is computed as follows: only if the distance between two agents at time 𝑡 is less than or equal to 𝜆, their time series of the two agents are used to estimate probability distributions at that time instance [9].…”

“…However, it was found that for shorter length (𝑇 = 12,000 − 25,000), is the minimum. It should be noted that, those 𝑇 for which 𝑅 ̂(𝑀, 𝑇) was not chosen uniquely due to the degeneracy of 𝜅, or 𝑅 ̂(𝑀, 𝑇) was undefined due to the absence of strongly convex part of 𝑑<TE> 𝜆 𝑑𝜆 , were excluded from the computation of optimal 𝑀 [71].…”

“…It was shown using a relative error diagram that the convexity score scheme identifies the interaction radius satisfactorily, (at moderate noise levels) even when data length is short for those data lengths 𝑇 where the global minimum scheme has high relative error [Fig. 6] [71]. Also, when the data length 𝑇 is very short (≈ 6,000), the convexity score fails to identify the interaction radius for large 𝑅 [Fig.…”