A Stochastic Hybrid Framework for Driver Behavior Modeling Based on Hierarchical Dirichlet Process

This work has been accep ted in 201 8 IEEE 8 8th Vehicular T echnology Conference (VTC201 8-Fall) 27–30 August 2018, Chicago, USA IEEE Co pyright Notice: © 20 18 IEEE. Personal u se of this material i s per mitted. P ermission from IEEE must b e o btained for all other uses, in a ny curren t or future media, including rep rinting/republishi ng this material for advertising or promotional purpo ses, creating new collective works, for r esale or red istribution to servers o r lists, or re use of any cop yrighted co mponent of this work in other works. A Stochast ic Hybrid Framework for Driver Behavior Modeli ng Based on Hierarc hical Dirichlet Process Hossein Nou rkhiz Mahjoub Networked Systems Lab ECE Department University of Central Flo rida Orlando, FL, US hnmahjoub@knights. ucf.edu Behrad Toghi Networked Systems Lab ECE Department University of Central Flo rida Orlando, FL, US toghi@knights.ucf.edu Yaser P. Fall ah Networked Systems Lab ECE Department University of Central Flo rida Orlando, FL, US yaser.fallah@ucf.edu Abstract — Scalability is one of the major issues for real-w orld Vehicle-to-Vehicle netw ork rea lization. To tackle this ch allenge, a stochastic hybrid modeling framework based on a non-parametric Bayesian inf erence method, i. e., hierarchical Dirichlet process (HDP), is investigated i n this p aper. This framework is able to jointly model dri ver/vehicle behavior thr ough fo recasting the vehicle dynamical time-series. This modeling framework could be merged with the notion of model-base d i nformation networking, which is recently proposed in t he ve hicular literatur e, to overcome the scalability challenges in dense vehicular netw orks via broadcasting the behavioral models instead of r aw information dissemination. This modeling approach h as been applied on several scenarios from t he realistic Safety Pilot M odel Deployment (SPMD) driving d ata set and the results show a higher performance of th is m od el in com p arison w ith the zero-h old method as the baseline. Keywords— Model-based c ommunications, vehicular ad-hoc network, non-parametric Bayesian inference, hierarchical Dirichlet process, stochastic hybrid syste ms I. I NTRODUCTION From a macro-system perspective, situationa l awareness is an imperativ e for any distri buted d y namic system c omposed of interactive agents (nodes). Thi s crucial feature, attained b y the virtue of sensory information and communication amongst nodes, allow s individual agents to k eep the track of the overall system b ehavio r and assists them to perform the ne cessary coordinated actions more properly . As an illustrati o n, in a distributed vehicula r safety system each vehi c le ne eds to have a precise understan ding of its proximity within a r ange of at least 300 meters, based on N at ional Highw a y Tr affic Safety Admin istration (NHTSA) technical reports [1]-[2]. This requiremen t is m andatory for c r itic al Coo perative Vehicul ar Safety (CVS) applications to properly detect the dangerous and critical situations and perform appropriate reacti ons in a timely manner. T hese reactions could range from issuing warnin gs to the dr iv er to taking the contro l of the vehi cle in full autonomy case. Inter-vehicl e commun ication is essentia l to improve and extend the situational awarenes s, since the sensory data could be restricted and cover a limited range due to the existence of obstacles or other confining circumstan ces forced by the environm ent, such as dimness, fog, rain, etc. Dedicated Short Range Communication ( DSRC) [3], is o ne of the current primary communication technologies for the vehicular ad-hoc networks (VANETs) and is highly expected to b e m andated by US officials in the near future. T his w ill require all autom otive original equi p ment m anufacturers (OEMs) to depl oy DSRC communication devices in their brand new pr oductio ns. Situational awareness could be attain ed in the DSRC -based VANETs by sharing every a gent’s ( vehicle’ s) raw dy namic information , e.g. GPS (latitude, longitude ), vel ocity, accelerati on, heading , etc., enclosed in s p ecific ap plication layer messages known as Basic Safety Messages (BSMs). The definite BSM content f o rmat is define d by a message set dictionary under the SAE J2735 standa r d [4] . The innovative idea of extracti ng a predictive abstract model for vehicles’ dynamics to be disseminated over the network, which has been initi a lly proposed in [5] as the model-based communication (MBC) methodology and then investigated with more depth in [6], is im mensely inspir ed by consi dering the serious inherent c hanne l utilization constra ints of available VANET commu nications techn ologies, e. g . DSRC. V ario us congestion control methods have been proposed so far in the literature [7]-[15], to mitigate the effect of inefficient DSRC channel utilization. High er communications quality achieved b y more efficient channel utilization in turn increases the overall system performance in different aspects such as b ehavio r of critical safety applications and providing more room to sen d lower priority inform ation, to name a few . Some of these communication st rategies are currently regarded as the state o f the art and nominated as the core of SAE J2945/1 congestion control standard [15], [16]. T hese approaches provide fascinating improvements in channel utilizati on in com parison with baseline DSRC communication policy , i.e. constant frequency broa d cast of raw data BSMs, b y continuously adapting diff erent flexible param e ters such as content, length or disseminati on rate of the information packet to the wireless channel quality . However, there is still a significant improvemen t potenti al by shifting the paradigm from any scheme o f raw data communication to the model-base d inform ation netw orking as proposed in [5], [6]. In addition to the noticeably increased This material is based on w ork supported by the National Science Foundation under CA REER Grant 1664968. efficiency in chann el bandwidth utilization, o ther significan t advantage o f apply ing MBC methodology versus the raw data communication is its capability to substanti a lly increase t he forecasting accuracy over the long predicti on durations . This is due to the flexibili ty of this approach to u pdate the model structure and/ or parameters at the subject agent o n the fly and updating the remote a g ents’ knowledge of the updated model accordingly in a real-time fash ion. Appropriate scalable model derivation strategies capa ble of capturing high level driving patterns, such as what has been adopted and employed in this work, i.e. Switching Linear Dynamical Systems- Hierarchical Dirichlet Process- Hidden Markov Model (SLDS-HD P-HMM), in conjunction with a precise MBC-custom ize d commun ication policy design could profoundly o ut perform the conventional forecastin g schemes at the receiver (remote) agents. I n the conventional approach remote a gents always pr esu me a predefined behavior (or roughly speaking a predefined model) of th e subject agent, e.g. constant velocity /a ccelerat ion models, with no structu ral model updates. How e ver, VANETs are compose d of highly dynam ic agents which need to be traced precisely to realize the aforementione d concept of si tuational awareness. Thus, this weak assumption of the su bj ect agent’s d yn amics, w hich almost neglects the plausible evolution of its behavioral structure over time, results in a notable worse prediction q uality and consequently lower situational awareness level in comparison with the MBC approach. I n this work, adopting the notion of MBC, applicabili ty of Stochastic Hybrid Systems (SHS)-bas ed modeling schemes with und erly ing Markovian Switching Processes (M SP) , specifically SLDS-HMMs, is explored and their forecasting p recision of joint driver-vehicl e b ehavio rs is investigat ed. T he underly ing M SP describes the latent behavioral mo de (state) chang es inferred through observable information provided via Cont roller Area Netw ork (CAN) of the host (subject) agent. Sinc e n ew unf oreseen behav iors might always be revea led by the human driver, the model predictions should th eoretically b e drawn from a n infinit e size sample set in order to support the theoretical infinite structu ral cardinality of the SHS-HMM model. To this end, a non-parametric Bayesian approach has been a dopted and studied in this work, enabling the model to inf initely generate new behavioral modes if n one of the already generated modes could adequately mimic the driver cur rent behavi or. The rest of this paper is organi zed as follows. Mathem atical structure o f the adopted modeling fr amework is present ed and discussed in details in section I I. The forecasting perfo rmance of this model is then analyzed and evaluate d over a set of realis tic driving scenarios and its accuracy gain is presented in section III. Finally , secti on IV conc ludes the paper. II. P ROBL EM S TATEMENT A. Stoch astic Hybrid S ystem (SHS)-Based Model Deriva tion Framework Markovian sw itc hing processes ( MSPs) are capable of capturing the evolution trend of an observation sequ ence if the total effect of the s equence history on the n ext unobserved value is a ssum ed to be e n capsulated in and expressed by some finite- length set of the last observ e d points. More specifically , if ,...} , { 2 1 y y Y  denotes a sequence o f observations with r th order Markovian property, then }) ,..., , { | ( }) ,..., , { | ( 1 ) 1 ( 1 2 1       n r n r n n n n y y y y P y y y y P (1) where P here denotes the conditi onal em ission probability of   given the o bservation his tory sequence. However, f or numerous real world p rocesses, the succ essive values o f an observable para meter d o not directly fulf ill th e Markovian pr operty . I n those cases, they are assumed to b e emitted from some H idden states, each has a certain emission probability distributi o n. In hidden M a rkov m od els [17 ], t he transition probability distributions are also de fine d among the hidden layer states. T he most challenging HMM problem, namely p aram eter estim ation, is infer r ing the most meaningful model parameters of the underlying states, i.e. their e mis sion and transition distributions , f rom the observation sequenc e . This problem has been tackled in the literat ure by different methods, mainly iterativ e likelih ood maximization s chemes us ing expectation m aximization (EM). Although HMM is much more effective than conventiona l Markovian models to comprehend and reveal the actual dependencies among observed sequence el ements, it puts a stri ct limiting assumption on these o bse rvations. More specificall y, HMM assumes that the observations are independently drawn from the emiss ion distributi ons of the hidden stat es. Theref ore, this model assumes observati ons as independent random variables, given the hidden state is known at each moment. This restrictive assumption forces t he model to neglect the temporal dependencies b etw ee n co nsecu tive obs erved values. C ombining the notion of hybrid system modeling with HMM framework is a promising candidate to address this limitation a n d take the temporal eff ects of the sequence his tory on its upcom ing values into account . For in stance, if the c onsecutive drawn observations of an HMM model are assum ed to follow a certain dynamic model, e.g. an autoregressive (AR) model, w ithin each hidden state, the r esultan t Stoch astic Hybrid Sy stem (SHS) framework is able to perceive both inherent Markovian and temporal dependencies of the observatio ns, simu ltaneously. This specific hybrid framew o rk which is realized b y switching a m ong different dynamical behaviors based on the rules d esignat ed b y an HMM model is known as Switch ing Linear Dynam ical Systems (SLDS)-HMM model [18], [ 19]. Details of the a do pt ed framework in this work wh ich has been built upon the SLDS- HMM notion is described in the foll o w ing section. B. Non-Pa rametric Baye sian SHS Fo rmulation of S ystems with Theoretic ally Infinite Behaviora l Modes In order to formulate a stochastic hy b rid sy stem, a SLDS- HMM here, with adaptive structural propert ies to the observations received f ro m the ta r get agent in an online manner, the model should be able to continuously track the agent behavior and add/rem ove the necessary/unn ecessary behavioral modes on the fly. What is meant b y structural pro perties here are the num ber of inferred discrete modes (states), dynam ic behaviors assigned to each mode, etc. Therefore, the model is theoretically e xpect ed to generate infinite number o f m o des, since we do not want t o put any restrict ive assum ptions on the model cardinality (num b er of states) and the dy namic behaviors defined b y d ifferent modes. T o this end, a nonparametric Bayesian approach, b ased on hierarchical Dirichlet p rocess [ 20] is nomin ated and its performance to model a highly dynamic system , i.e. VANET, is investigate d in this work. T his extension of SLDS-HMM, known as SLDS-HDP-HMM, proposed in [19], leverages the properties of D irichlet processes to d efine the transition probability measures among unbounded a nd unknow n number of discrete modes. Very few previous studies , such as [21], have also studied non-parametric appr oaches to study driving behaviors. Ho w ever, th eir stu dies are mainly focused on HDP-HMM models based o n B eta process wh ich is conceptually different from what we have in this work, i.e. SLDS-HDP-HMM model . Any single draw fro m a Dirichlet process with a b ase measure  and concentrati o n p aram eter  , ) , ( H DP  ,is composed o f an infinite number of   s, so that         . Here  specifies the param e ter space of the ba se measure,  . However, since any two different sets of draw s fro m a continuous distributi on are fully disjo int and have definitely no overlap, taking the transition probability measures at d ifferen t points of tim e as independent draws of a Dirichlet process with a co ntin uous base measure for ces the set of inferred states after each observati o n to be com pletely disjo int from all other sets of the alre ady visite d (or draw n) states. In o ther words, with a continuous base measure a visited b ehavio ral mode of the system will never be r evisite d. To handle this issu e , a d iscrete probability m easure with the inf inite capability of generating a new unobserved state w ith non-zero probability, i.e., a d iscrete measure with countably infinite parameter d omain, should be selected as the base measur e . Assigning another Dirichl e t process for this purpose solves the problem a nd results in the hierarchical D irichl et process- HM M stru cture. In this work we have consi dered the sticky version of HDP- HMM [22]. T he sticky property notes anothe r extension to the idea of HDP-HM M and all ows capturing m odal behavior of the system in a more accurate manner. This is crucial to model the system s with persist ent dynamic m odes and prevent th e m odel from rapid f luctuations among inher ently sim ilar mo des. Figure 1. Ge nerative model of hierarchical Dirichlet process Figure 2. Evolution of a HDP-HM M mode structure over time.   is the transition probability me asure of the   state     , and              denotes the hidden state at the t ime   . It should be noted that:                            . Generative model of S LDS-HD P-HMM and its evol ution over tim e are dep icted in Figure 1 and Figure 2, respec tively. In these figures, observat ions, discrete and continu ous states at time instant n t , are d enoted b y n t y , n t z , and n t x , respectiv ely. Discrete st ates are w hat have b een als o refe rred to as behavioral modes s o fa r. These parameter s form a state space m odel w ithin an SLDS struc ture:                       (2)               (3) Where A and e are process matrix an d process nois e paramete rs respectively , derived from a mode-specific distributi on. In addition, C an d w are mode in dependent (could also be mode- specific) parameters represent ing the measurement matrix and observation noise distribution, respect ively. The m ode specif ic distributi ons for A and e are from a parameter space  . Therefore, the transi tion pr obability measures, d en oted as   (in a sum of weighted unit mass functi on form ), could be drawn from a Dirichle t process which its ba se measure has the sam e parameter space  . The weights o f this Dirichlet process are elements of a dra w fro m a Griffith- Engen-McCloskey ( GEM) distributi on . GEM distribut ion, wh ich is a distributi o n over the countably infinite size probability measures (Φs), has a generative model, known as stick-b reaking constructi on, which guarantees that the w eights ( φ k probabilities ) always add up to 1. Utilizing this specific generative m odel all ows d raw ing state transition probabilities without knowing the exact number of states in advance. Since any Dirichlet process is theoretically an infinite size stochastic measure, it requires a practical generativ e model in order to b e realized and implemented. Chinese Restauran t Process (CRP) is a Pólya urn-based process which allows to generate a Dirichlet process without d evel o ping its complete realization. Tracing back, the parameters o f the DP and each transition p robability is computed as:                                 (4)                                    (5)                         ,                              (6)                                                                 (7)             (8) The mentioned stick-breaking model is realized by the means o f a B eta distribution with parameters 1 and  , as depicted in (4), and (5). The complexity of the model generate d by the virtue of a Dirichlet p rocess is driven by and adapted t o the hist ory o f the observations in an online manner. To this end, after receiving each new observat ion element Diri chlet process should decide how to reflect it into the model. This task is performed through the inherent CRP mechanism of the Dirichlet process . T he probability of assigning the most recent data point to any of the already generated stat es is proportional to the number of observation elements wh ich have been assigned to that state so far and is inversely proportional to the whole observation sequence length. T his imitates the well-know n effect o f “rich g et richer” in the model structu re. Generating a new state for the most recent observation is proportional to the CRP parameter and is again invers e ly proportional to th e w hole observati o n sequence l ength. Thus, the probability of genera ting a new state tends to zero when the num ber of observations goes towards infinity . This c ould be in terpreted a s a monotonically incre ase in the accur acy o f the inferred model by the Dirichlet process and approaching towards the complete behavioral model of the system through receiving more a nd more o bservation s. After that the d ecision for the model state card inality is mad e, the prob lem reduces to a nor mal HMM with a kno wn number of modes. “Evaluatio n, Deco ding, and P arameter Esti mation” prob lems o f this HM M no w could be tackled utilizing well- known appro aches in the lite rature, i.e. forward-backwar d algorithm, Viterbi algorithm, and Expectation-Maxi mization method, respectively. III. E VALUATI ON Derived S LDS-HDP-HMM model fro m the prev ious section is applied on an e xtensive drivi ng infor mation set o f selected trip s from a realistic rich driving d ata set provide d by US DOT , namely Sa fety P ilot Mo del Deplo ym ent ( SPMD) data set [23]. SPMD d ata set which is composed of infor mation collected through two different setti ngs of Data Acquisiti on Systems (DAS1 and DAS2) in Ann -Arbor, Michiga n, provides different in-vehicle informatio n lo gged f rom CAN, such as longitudinal velocit y a nd accel eration, yaw r ate, s teering angle, turn flash stat us, etc. , alo ng with t he vehic le p ositioning information over the w hole trip duration. The data are analyzed in this work to realize the number of hidden driving b ehavioral states inferre d b y the model in real driving situations. In addition, the obt ained model is employed to predi ct the future trend of the analyzed time series under imperfect netw o rk conditions . Network imperfection has been modeled through the Packet Error Rate (PER) abstraction, which is a common approach to consider wireless channel conditi on from the applicati on layer p oint of view . The sequence of hidden modes assign ed by the model to a sample inte rval of the longitudinal acceleration is d epict e d in Figure 3 . Figure 4 represents the predicted l ongitudinal speed o f the sam e tri p and its c o mpariso n to the b aseline m odel, which is basically a zero-hold estimation m o del he re. This com p arison is performed under 60% PER rate and evidently demonstrates the forecasting d ominance of our m o del during a notable po rtion of the s cenario. The evaluated Empirical Cumulative Dist ribution Function (ECDF) of the prediction errors over 10 tr ips for both baseline and SLDS-HDP-HMM m odels ar e depicte d in Figure 5, which shows higher prediction precision of o ur model for errors less than 1 m eter. Figure 3. Discrete states (modes) of the lo ngitudinal acceleration inferred by SLDS-HDP -HMM model Figure 4. Prediction accuracy comparison of SL D S-HDP-H MM and Baseline models for l ongitudinal speed under 60% PER Figure 5. ECDF of prediction err ors for lo ngitudinal speed in SLDS- HDP- HMM and Baseline model s under 60% PER IV. C ONCLUSION In this work a non-parametric Bayesian approach is investigat ed to track the join t vehicl e-driver behavior in an online manner. A stochastic hybrid model is designe d b ase d o n a hierarchica l Dirichlet process. The HDP serves as the underlyin g Mar kovian switching process of the model a nd 420 430 440 450 460 470 480 2 4 6 8 Discrete States (Modes) of LongAcceleration Inferred by SLDS-HDP-HMM Model 420 430 440 450 460 470 480 Time (100 miliSeconds) -4 -2 0 2 400 410 420 430 440 450 460 470 480 Time (100 miliSeconds) 4 6 8 10 12 14 16 Actual LongSpeed and its Reconstructed Models at Remote Vehicle Remote Vehicle Actual LongSpeed Remote Vehicle HDP-HMM Model LongSpeed Remote Vehicle Constant Velocity Model LongSpeed 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 ECDF Empirical CDFs of Absolute Error Sequences (LongSpeed) HDP-HMM Model Baseline Model determines the appropriate seq uence o f hidden drivin g behavioral modes (states). The non-param e tric nature of the employed modeling f ramework m akes it possi b le to a d d an unbounded num b er of unforeseen sta tes to th e model on the f ly. 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