Learning interaction rules from multi-animal trajectories via augmented behavioral models

1 Introduction . Extracting the interaction rules of real-world agents from data is a fundamental problem in a variety of scientific and engineering fields . For example , animals , vehicles , and pedestrians observe other ’ s states and execute their actions in complex situations . Discovering the directed interaction rules of such agents from observed data will contribute to the understanding of the principles of biological ∗fujii @ i.nagoya-u.ac.jp 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . agents ’ behaviors . Among methods analyzing directed interactions within multivariate time series , Granger causality ( GC ) [ 23 ] is a practical framework for exploratory analysis [ 49 ] in various fields , such as neuroscience [ 65 ] and economics [ 3 ] ( see Section 2 ) . Recent methodological developments have focused on inferring GC under nonlinear dynamics ( e.g. , [ 73 , 31 , 81 , 55 , 43 ] ) . However , the structure of the generative process in biological multi-agent trajectories , which include navigational and motion processes [ 54 ] regarded as time-varying dynamical systems ( see Section 3.1 ) , is not fully utilized in existing base models of GC including vector autoregressive [ 27 ] and recent neural models [ 73 , 31 , 81 ] . Ignoring the structures of such processes in animal behaviors will lead to interpretational problems and sometimes erroneous assessments of causality . That is , incorporating the structures into the base model for inferring GC , e.g. , augmenting ( inherently ) incomplete behavioral models with interpretable data-driven models ( see Section 3.2 ) , can solve these problems . Furthermore , since data-driven models sometimes detect false causality that is counterintuitive to the user of the analysis , e.g. , introducing architectures and regularization to utilize scientific knowledge ( see Sections 3.2 and 4.2 ) will be effective for a reliable base model of a GC method . In this paper , we propose a framework for learning GC from biological multi-agent trajectories via augmented behavioral models ( ABM ) using interpretable data-driven neural models . We adopt an approach for augmenting incomplete multi-agent behavioral models described by time-varying dynamical systems with neural networks ( see Section 3.2 ) . The ABM leverages theory-based architectures separating navigation and motion processes based on a well-known conceptual behavioral model [ 54 ] , and the theory-guided regularization ( see Section 4.2 ) for interpretable and reliable behavioral modeling . This framework can provide interpretable signs of Granger-causal effects over time , e.g. , when specific others cause the approach or separation . The main contributions of this paper are as follows . ( 1 ) We propose a framework for learning Granger causality via ABM , which can extract interaction rules from real-world multi-agent and multi-dimensional trajectory data . ( 2 ) Methodologically , we realized the theory-guided regularization for reliable biological behavioral modeling for the first time . The theory-guided regularization can leverage scientific knowledge such that “ when this situation occurs , it would be like this ” ( i.e. , domain experts often know an input and output pair of the prediction model ) . Existing methods in Granger causality did not consider the utilization of such knowledge . ( 3 ) Biologically , our methodological contributions lies in the reformulation of a well-known conceptual behavioral model [ 54 ] , which did not have a numerically computable form , such that we can compute and quantitatively evaluate it . ( 4 ) In the experiments , our method achieved better performance than various baselines using synthetic datasets , and obtained new biological insights and verified our method using multiple datasets of mice , birds , bats , and flies . In the remainder of this paper , we describe the background of GC in Section 2 . Next , we formulate our ABM in Section 3 , and the learning and inference methods in Section 4 . 2 Granger Causality . GC [ 23 ] is one of the most popular and practical approaches to infer directed causal relations from observational multivariate time series data . Although the classical GC is defined by linear models , here we introduce a more recent definition of [ 73 ] for non-linear GC . Consider p stationary time-series x = { x1 , ... xp } across timesteps t = { 1 , ... , T } and a non-linear autoregressive function gj , such that xjt+1 = gj ( x 1 ≤t , ... , x p ≤t ) + ε j t , ( 1 ) where xj≤t = ( ... , x j t−1 , x j t ) denotes the present and past of series j and ε j t represents independent noise . We then consider that variable xi does not Granger-cause variable xj , denoted as xi 9 xj , if and only if gj ( · ) is constant in xi≤t . Granger causal relations are equivalent to causal relations in the underlying directed acyclic graph if all relevant variables are observed and no instantaneous ( i.e. , connections between two variables at the same timestep ) connections exist [ 59 ] . Many methods for Granger causal discovery , including vector autoregressive [ 27 ] and recent deep learning-based approaches [ 73 , 31 , 81 ] , can be encapsulated by the following framework . First , we define a function fθ ( e.g. , an multilayer perceptrons ( MLP ) in [ 73 ] , a linear model in [ 27 ] ) , which learns to predict the next time-step of the test sequence x . Then , we fit fθ to x by minimizing some loss ( e.g. , mean squared error ) L : θ ? = argminθ L ( x , fθ ) . Finally , we apply some fixed function h ( e.g. , thresholding ) ( e.g. , [ 45 ] ) to the learned parameters to produce a Granger causal graph estimate for x : Ĝx = h ( θ ? ) . Furthermore , we need to differentiate between positive and negative Granger-causal effects ( e.g , approaching and separating ) . Based on the definition of [ 45 ] , we define the effect sign as follows : if gj ( · ) is increasing in all xi≤t , then we say that variable xi has a positive effect on xj , if gj ( · ) is decreasing in xi≤t , then x i has a negative effect on xj . Note that xi can contribute both positively and negatively to the future of xj at different delays . Overall , the causality measures , however elaborate in construction , are simply statistics estimated from a model [ 71 ] . If the model inadequately represents the system properties of interest , subsequent analyses based on the model will fail to address the question of interest . The inability of the model to represent key features of interest can cause interpretational problems and sometimes erroneous assessments of causality . Therefore , in our case , incorporating the structures of the generative process for animal behaviors ( i.e. , Eq . ( 2 ) ) in a numerically computable form will be required . We thus propose the ABM based on a well-known conceptual model [ 54 ] in biological sciences in the next section . 3 Augmented behavioral model . Our motivation for developing interpretable behavior models is to obtain new insights from the results of Granger causality . In this section , we firstly formulate a well-known conceptual behavioral model [ 54 ] so that it can be computable . Second , we propose ( multi-animal ) ABMs with theorybased architectures based on scientific knowledge . Further , we discuss the relation to the existing explainable neural models [ 2 ] . The diagram of our method is described in Appendix C . 3.1 Formulation of a conceptual behavioral model . In movement ecology , which is a branch of biology concerning the spatial and temporal patterns of behaviors of organisms , a coherent framework [ 54 ] has been conceptualized to explore the causes , mechanisms , and patterns of movement . For example , two alternative structural representations [ 54 ] were proposed to model a new position pt+1 from its current location pt ( for details , see Appendix A ) : the motion-driven case pt+1 = fU ( fM ( Ω , fN ( Φ , rt , wt , pt ) , rt , wt , pt ) ) + εt , and the navigation-driven case pt+1 = fU ( fN ( Φ , fM ( Ω , rt , wt , pt ) , rt , wt , pt ) ) + εt , where wt is the internal state , Ω is the motion capacity , Φ is the navigation capacity , and rt is the environmental factors ( these are conceptual parameters ) . fM , fN , and fU are conceptual functions to represent actions of the motion ( or planning ) , navigation , and movement progression processes , respectively . For efficient learning of the weights in the model ( i.e. , coefficient of Granger causality ) in this paper , we consider a simple case with homogeneous navigation and motion capacities , and internal states . Moreover , to make the contribution of fM , fN , and fU interpretable after training from the data for extracting unknown interaction rules ( and for obtaining scientific new insights ) , one of the simplified processes for agent i is represented by xit+1 = f i U ( f i N ( r i t , x i t ) , f i M ( r i t , x i t ) , r i t , x i t ) + ε i t , ( 2 ) where xi ∈ Rd includes location pi and velocity for the agent i . We here consider ri ∈ R ( p−1 ) dr including p− 1 other agents ’ dr-dimensional information . This formulation does not assume either motion-driven or navigation-driven case . Such behaviors have been conventionally modeled by mathematical equations such as force- and rule-based models ( e.g. , reviewed by [ 77 , 47 ] ) . Recently , these models have become more sophisticated by incorporating the models into hand-crafted functions representing anticipation ( e.g. , [ 29 , 51 ] ) and navigation ( e.g. , [ 8 , 76 ] ) . However , these conventional and recent models are sometimes too simplistic and customized for the specific animals , respectively ; thus it is sometimes difficult to define the dynamics of general biological multi-agent systems ( i.e. , multiple species of animals ) . Therefore , methods for learning parameters and interaction rules of behavioral models are needed . There have been some researches to estimate specific parameters ( and their distributions ) of the interpretable behavior models ( e.g. , [ 84 , 85 , 15 ] ) , and others to model the parameters and rules in purely data-driven manners ( i.e. , sometimes uninterpretable ) only for accurate prediction ( e.g. , [ 16 , 28 ] ) . In the proposed framework , we consider flexible data-driven interpretable models to focus on inferring GC for exploratory analysis from the observed data without specific knowledge of the species and obtaining additional data . Recently , some attempts have been made to explore flexible and interpretable models bridging theorybased and data-driven approaches . For example , a paradigm called theory-guided data science has been proposed [ 30 ] , which leverages the wealth of scientific knowledge for improving the effectiveness of data-driven models in enabling scientific discovery . For example , scientific knowledge can be used as architectures or regularization terms in learning algorithms in physical and biological sciences ( e.g. , [ 61 , 21 ] ) . In biological multi-agent systems , an approach extract interpretable dynamical information based on physics-based knowledge [ 18 ] from multi-agent interaction data , and another approach made a particular module such as observation ( e.g. , [ 19 ] ) interpretable in mostly black-box neural models . However , these data-driven models did not sufficiently utilize the above scientific knowledge of multi-animal interactions . In the next subsection , to make the model ( e.g. , of GC ) flexible and interpretable , we propose an ABM with theory-based architectures .

Summary: The manuscript proposes a hybrid method for learning Granger causality that combines data-driven learning methods and theory-based behavioral models for the goal of modeling multi-agent behavior.   The main motivation is that current models either do not consider the data generation process behind the observed time series or if they do, they are tailored to specific animal species, thus having generalization limitations.  More specifically, a Self-Explainable Neural Network (SENN) -like model is used where scientific knowledge on multi-agent interactions is integrated. In this regard, key contributions of the proposed method include: a theory-guided loss function to regularize training and a motion and navigation function Experiments on synthetic and realistic data show the effectiveness of the proposed method.