DDP-GP: A Sequential Bayesian NonparametricMethod for Vehicular Trajectory Clustering

Miao Liu1, Trevor Campbell2, Jonathan How2 and Lawrence Carin1

1Department of Electrical & Computer Engineering, Duke University2Department of Aeronautics & Astronautics, Massachusetts Institute of Technology

Introduction

I Persistent surveillance and tracking missions require. Flexible and reliable motion models for describing and predicting

target trajectory (behavior),. Principled way of characterizing motion pattern uncertainty and

detecting new situation,. Scalable inference algorithms for learning model parameters.

I We propose/extend:. Infinite mixture of Gaussian Processes (GPs) for modeling

vehicular motion patterns [1]. A sequential Bayeisan nonparametric method based on

Dependent Dirichlet Process (DDP) [2]

Sequential Vehicular Trajectory clustering based on DDP-GP

Figure 1: Diagram of proposed sequential inference procedure

The generative model for DP-GP

{θk, πk}∞k=1 ∼ DP(H,α)

{fxk , fyk} ∼ GP(µk,Kk)

{zj}NJ

j=1 ∼ Categorical(π)

{xt, yt, vxt, vyt}Tjt=1 ∼ N ({fxzj, f

yzj}, σf)

I θk = {µk,Kk} is the parameter for motion class (pattern) k,I πk is the categorical weight for class k,I fxk : R2→ R, fyk : R2→ R are functions which take position as an

input and output the velocity components for class k,I {xt, yt, vxt, vyt}

Tjt=1 are trajectory location/velocity pairs for target j.

DDP - A Markov chain of DPs

I Given D ∼ DP(µ) and a set of samples Φ ∼ D, in which φiappears ci times, we have

D|Φ ∼ DP(µ+ c1δφ1+ · · ·+ cnδφn) (1)

I In DDP, a new DP D′ is updated by conditioning on the old DPD′|Φ ∼ DP(ανpν + α0pqν +

∑mk=1 qkckT (φk, · )), (2)

where. pν is the innovation distribution,. pqν is the q-subsampled base distribution,. T (φk, · ) is the transition distribution,. qk is the discount factor for forgetting cluster k, 0 < qk < 1.

I The validity of the operations on DPs is based on the validity of theoperations of the underlying Poisson process (PoissonP) [2].

I PoissonP and DP are intrinsically related by Gamma Process (ΓP). ΓP is a compound PoissonP.. DP is a normalized ΓP.

Results

I Time complexity. DP-GP: O(n2). DDP-GP: O(n)

I Space complexity. DP-GP: O(n). DDP-GP: O(1)

I The influence of qk. qk controls the rate for

sampling previous clusters. Smaller qk results in more

clusters

Figure 2: Performance comparison between DDP-GP and DP-GP

I Inference Results

Figure 3: Row 1-3: mean vector field, row 4-6: variance field

Conclusions

I DDP-GP provides a scalable learning framework for assisting longterm surveillance and tracking tasks.

I DDP-GP generalizes DP-GP in several ways:. Either inheriting previous motion patterns or generating new ones,. Controlling the rate for forgetting old clusters,. Smooth variation when old motion pattern is inherited.

Future work

I Embed DDP-GP into the planning system developed in theAerospace Controls Laboratory (ACL) of MIT.

I Customize DDP-GP for multi-agent planning and learning problems(MILP/MDP/POMDP).

Acknowledgement

Thanks to Bobby Klein in ACL helping collecting and processingthe trajectory data used in the experiment.

References

[1] J. Joseph, F. Doshi-Velez, A. Huang, and N. Roy.A bayesian nonparametric approach to modeling motion patterns.Autonomous Robots, 31(4):383–400, Nov. 2011.

[2] Dahua Lin, Eric Grimson, and John Fisher.Construction of dependent dirichlet processes based on poisson processes.In NIPS, 2010.

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