Prediction of Driving Behavior through Probabilistic Inference

Transcription

1 Held n Torremolnos, Malaga (SPAIN) 8-10 September 2003 Predcton of Drvng Behavor through Probablstc Inference Toru Kumaga*, Yasuo Sakaguch**, Masayuk Okuwa*** and Motoyuk Akamatsu* *Natonal Insttute of Advanced Industral Scence and Technology, Tsukuba, Japan **Res. Inst. of Human Engneerng for Qualty Lfe ***Toyota Central Res. & Dev. Labs. Inc. Abstract: Drvng assstance systems are essental technologes to avod traffc accdents, reduce traffc jams, and solve envronmental problems. Not only observable behavoral data, but also unobservable nferred values should be consdered to realze advanced drvng assstance systems that are adaptable to ndvdual drvers and stuatons. For ths purpose, Bayesan networks, whch are the most consstent nference approach, have been appled for estmaton of unobservable physcal values and nternal states ntroduced for convenence s sake. Nevertheless, only a few reports have addressed predcton of future states of drvng behavor. Ths paper proposes predctng drvng behavor n the near future through a smple dynamc Bayesan network, whch s a hdden Markov model or a swtchng lnear dynamc system. The proposed predctors were examned wth real data. We focused on predcton of the future stop probablty at an ntersecton because t s one of the most mportant maneuvers for safety to avod collson wth other traffc elements (.e. other vehcles and pedestrans) at an ntersecton. Both the HMM and the swtchng lnear dynamc system worked well as stop probablty predctors. The HMM represented the temporal structure of human drvng behavor. Keywords: dynamc Bayesan network, hdden Markov model, swtchng lnear dynamc system, drvng behavor predcton, drvng assstance system, mult-step ahead predcton 1. Introducton Drvng assstance systems are essental technologes for avodng traffc accdents, reducng traffc jams, and solvng envronmental problems. These assstance systems, whose typcal feature s a warnng system, produce a decson based on physcal parameters such as headway dstance and vehcle speed. However, ndvdual drvng behavor depends on myrad varables ncludng ndvdual drvng characterstcs, envronmental condtons, drvng ntenton, and so on. Addtonally, some physcal parameters mght not be measured accurately. Not only observable behavoral data, but also unobservable nferred values should be consdered to realze advanced drvng assstance systems that can adapt to ndvdual drvers and stuatons. For ths purpose, Bayesan networks, whch are the most consstent nferental approaches, have been appled for estmaton of unobservable physcal values and nternal states ntroduced for convenence sake. For example, one study [1] nferred a probablstc dstrbuton of brake onset tme to cross lne from varous evdence, such as weather condton, methodcal drvng style scores, accelerator pedal release tmng, and so on. Dynamc Bayesan networks, whch nclude well-known hdden Markov models, have also attracted many researchers. The model used n another study [2] provded a decson-makng model for an autonomous vehcle of a smple smulaton envronment through a dynamc probablstc network. Another study [3] used a hdden Markov model for modelng and recognzng drvng maneuvers at a tactcal level. Dynamc Bayesan networks have also been appled for general behavor modelng: one study [4] appled a swtchng lnear dynamc system for modelng and recognzng human locomoton. Another [5] appled a swtchng Kalman flter for modelng and recognzng smulated drvng behavor. Nevertheless, only a few nvestgatons have addressed predcton of future states of drvng behavor. Most dynamc Bayesan network applcatons have recognzed temporal nformaton or nferred current states, but have not predcted future states. Although one [1] estmated probablstc dstrbutons of future events, t was based on the statc relaton between predefned varables: t used no temporal nformaton. 117

2 Held n Torremolnos, Malaga (SPAIN) 8-10 September 2003 Ths study s ntended to predct drvng behavor n the near future through a smple dynamc Bayesan network, whch s a hdden Markov model or a swtchng lnear dynamc system. The proposed predctors were examned wth real data. We focused on the predcton of the stop probablty at an ntersecton durng a drver s sde turn because that s a very mportant maneuver for safety. It avods collson wth other traffc elements (.e., other vehcles and pedestrans) at an ntersecton. The remander of ths paper wll be organzed as follows. Secton 2 ntroduces dynamc Bayesan networks and predcton algorthms. Secton 3 descrbes measurement of actual vehcle data n the real road envronment. Secton 4 descrbes the learnng procedure of dynamc Bayesan networks. Secton 5 explans stop-probablty predcton. Fnally, Secton 6 concludes wth a dscusson of the method used n ths paper. 2. Dynamc Bayesan networks 2.1 Model structure In ths study, we denoted dynamc Bayesan networks as where ( ) = a, j t δ j t +1 y t ( )δ t ( ) = f s(t) ( y(t 1) ) t s dscrete tme, s t ( ), (1) ( ) s the dscrete state at tme t, ( ) s the probablty of state j at tme t,.e. Pr s( t) = δ j t a, j s the state transton probablty from state to j, ( ) s the observable value vector at tme t y t and f When we assume that f When we assume that f 2.2 Predcton algorthm Gven observaton y t straghtforward ways: (1) Hdden Markov model ( ) = δ ( ), ( ) s the functon that decdes observaton values at state. ( ) ~ N µ,σ ( ) ~ N µ + w y t 1 δ j T + n ( ), (1) s a Gaussan hdden Markov model. ( ( ),Σ ), (1) s a swtchng lnear dynamc system. ( ) { t = 1...T } and nferred δ T ( ) = a, j δ ( T + n 1) ( ) ~ δ ( T + n) y T + n (2) Swtchng lnear dynamc system n = 1... ( ) = a, j δ ( T + n 1) δ j T + n y T + n ( ), the predcton s performed n the followng { } N ( µ,σ ) ; and (2) { } n = 1... ( ) ~ δ ( T + n) N ( µ + w y( T + n 1),Σ ). (3) In ths study, we approxmate (3) to (4) because (3) s computatonally expensve. 118

3 Held n Torremolnos, Malaga (SPAIN) 8-10 September 2003 ( ) = a, j δ ( T + n 1) δ j T + n y T + n n = 1... ( ) ~ δ ( T + n) N µ + w y T + n ( ) ~ N µ + w ( ) = arg max J T + n 1 y ( T ) = y( T ) { } ( y ( T + n 1),Σ ) ( y J ( T +n 1) ( T + n 1 ),Σ ) j ( ) a j, δ j T + n 1 (4) Note that y ( T + n) { n = 1... } s always a normal dstrbuton. 3. Data preparaton We evaluated the proposed predctor usng actual data n a real road envronment. We developed a vehcle equpped wth sensng devces to measure drvng behavor [6]. Sensng devces ncluded those for drver's operatonal behavor, such as steerng wheel operaton, and those for the vehcle condton, such as vehcle speed. Ths study used vehcle speed and pedal strokes of the acceleraton and brake pedals. Pedal sensors attached to the pedals detected the pedal strokes. The speed sgnal was obtaned from the front wheel speed sensor. We resampled the data at 15 Hz after measurements; the samplng rate was 30 Hz for sensor sgnals. We measured the drver s sde turn behavor (.e., rght turn behavor n Japan) 33 tmes at an ntersecton n a suburb of Tsukuba, Japan (Fg. 1). One testee drove the car. We removed portons of 20 [Km/h] or hgher speeds from records. The car stopped once or twce n 16 of 33 cases because the roadway beyond was blocked wth traffc or the traffc sgnal (see the top row of Fgs. 4 and 7). In other cases, the car passed the ntersecton wthout stoppng. We assumed 16 of 33 records to be learnng data. The remanng 17 records were assumed to be test data. 4. Learnng and nference The Baum-Welch algorthm and the Vterb algorthm are the most wdely used learnng and nference algorthms for HMMs. Learnng and nference algorthms for swtchng lnear dynamc systems are gven as the extenson of those of HMMs [9]. In ths study, we used The Bayes Net Toolbox for Matlab [10] for learnng and nference. We gave the speed of the vehcle and the pedal stroke to a dynamc Bayesan network as observable data. The pedal stroke was gven as the subtracton of the stroke of the brake pedal from the stroke of the accelerator pedal (hereafter called the pedal stroke). Szes of the HMM and the swtchng lnear dynamc system were determned by the number of states, Q. The ncrease of Q contrbuted to mprovement of the stop-predcton accuracy. However, performance was not so senstve to Q when Q was somewhat larger. In ths study, we chose Q = 15 for the HMM and Q = 11 for the swtchng lnear dynamc system. Fgure 2 shows topology of the traned HMM. Each f ( ) was plotted on the plane of the speed and the pedal stroke. An ellpse shows Σ. Arrows show major state transtons and ther drectons. The process of deceleratng, stoppng and acceleratng was clearly obtaned n the model. Fgure 3 shows the topology of the traned swtchng lnear dynamc system. Each f ( ) was plotted on the plane of w 11 and w 12. Here, w 11 s the weght between the speed at tme t and tme (t 1) ; w 12 s the weght between the speed at tme t and the pedal stroke at tme (t 1). Arrows show major state transtons and ther drectons. It s dffcult to nterpret topology of the swtchng lnear dynamc system because nodes have many varables and output values depend on nput values. However, t was at least readly plausble that w 12 s greater than 0 and w 11 s nearly equal to

4 Held n Torremolnos, Malaga (SPAIN) 8-10 September Stop probablty predcton We appled the above HMM to estmate the future stop-probablty. We used the followng functon to estmate the stop probablty. stop T (T p ) = Pr(speed(T p ) < Const speed y(t) { t = 1...T }) Pr(pedal _ stroke(t p ) < Const pedal y(t) { t = 1...T }) (5) Here, stop T (T p ) s the stop probablty at tmet p predcted at a predcton pont T. We chose Const speed = 0.5 and Const pedal = 5. Fgure 4 shows an example of the stop probablty predcton at some predcton pont. We denoted the actual stop tme as T stop. The vehcle stopped at around tme 7 s: T stop 7[s]. Predcton of the stop probablty at the actual stop tme, stop T (T stop ), approached 1 as the predcton pont approached the actual stop tme (see also Fg. 6). At around predcton tme 21 s, the predcted stop probablty of the near future rose wthout an actual stop because the speed was suffcently low and the deceleraton rate was hgh. At tme 22 s, the brake pedal was released and the accelerator pedal was depressed. Thereby the stop probablty decreased to zero. The predcted stop dd not actually occur when t was canceled n mdcourse: detectng the change n the pedal stroke, the predctor sgnaled the cancelng of the stop maneuver. Fgure 5 shows the relatonshp between the predcted stop probablty, stop T (T p ), and the actual stop rate. In an deal predctor, a predcted stop probablty s equal to an actual stop rate. Fgure 5 shows that the stop probablty was nearly equal to the actual stop rate even though predcted values were somewhat larger than actual rates. Fgure 6 shows the average value of stop T (T stop ) at every tme to the stop, (T stop T ). Ths fgure shows agan that the stop probablty approached 1 when predcton pont T approached actual stop tme T stop. Ths fact corresponds to ntuton. Along wth the HMM, the swtchng lnear dynamc system also worked well to predct the stop probablty. Fgures 7 9 show the predcton results. In addton to the stop probablty predcton, fg. 7 shows the mean value of predcted vehcle speed. 6. Dscusson The proposed predctor produced the future stop probablty gven current and past observable data. Both the HMM and the swtchng lnear dynamc system worked well. In almost all cases, the proposed system predcted the possblty of a future stop several seconds before ts occurrence, as shown n Fgs. 4 9; the vehcle dd not stop when the predctor dd not predct t, as seen n Fgs. 5 and 8. Ths result s very sgnfcant to consder the applcaton of the proposed system to drvng assstance systems. The predcted stop-probablty consdered changes of the drver s ntenton. The probablty was smaller than 1 even durng the typcal stop maneuver (see Fgs. 6 and 9) because the drver mght change the ntenton and cancel the maneuver before the predcted stop. Therefore, the predcted stop-probablty was nearly equal to the actual stop rate, as seen n Fgs. 5 and 8. Varous factors cause changes of the drver s ntenton. They were modeled n the state transton probabltes. In ths study, predcton was done by sequental nference through dynamc Bayesan networks. In contrast, many conventonal studes have used dynamc Bayesan networks as a classfer. A textbook example s a strategy that prepares HMMs of the same number as recognton objects and then recognzes them by comparng ther lkelhoods. Ths strategy s applcable when only a part of a recognton object 120

5 Held n Torremolnos, Malaga (SPAIN) 8-10 September 2003 s observable. Gven current and past data, HMMs could recognze ongong behavor. However, for a recognton problem of behavor, t mght be dffcult to prepare a categorzed learnng data set for supervsed learnng of such dynamc Bayesan networks. Classfyng human behavor nto dscrete categores s complcated because behavor s dependent on ndvduals and stuatons. Moreover, t s mpossble to determne when an acton starts and ends. These engender the dffculty of behavor defnton. Ths study avoded ths problem by replacng the defnton of the whole sequence of behavor wth behavor results as defned n (5). The easest way to construct a stop-probablty predctor s to prepare a statc table that descrbes the relatonshp between observable data and the frequency of future stops. However, several reasons recommend the dynamc approach. For example, the statc approach requres more tranng data because of long-range predctons. In the prelmnary study of the stop predcton problem of ths paper, a smple probablstc table could not work well n some cases;.e. t could not determne the probablty because of the lack of learnng data. In general, t s dffcult to forecast future states precsely through dynamc Bayesan networks because nference addresses tme slces for whch no observatons have been gven (e.g. [7],[8]). However, the predctor of the present study showed good performance for the specfc temporal structure of human drvng behavor as seen n Fg. 2. That s, the drver behaved accordng to certan habts whle changng ntenton. Ths structure s also an mportant fndng of ths study. The HMM and the swtchng lnear dynamc system descrbed behavor n a qute dfferent manner. The HMM could be a good proflng tool of the drver s behavor because t vsualzes behavor as seen n Fg. 2. On the other hand, the swtchng dynamc system could descrbe behavor wth a smaller number of states than the HMM because a sngle lnear dynamc system descrbes the relaton of observable data between tme t and t 1 ( ). References [1] Y. Sakaguch, M. Okuwa, Ken chro Takguch, and M. Akamatsu, Measurng and modelng of drver for detectng unusual behavor for drvng assstance, (to appear n Proceedngs of 18th Internatonal Conference on Enhanced Safety Vehcles, 2003) [2] J. Forbes, T. Huang, K. Kanazawa, and S. Russell, The BATmoble: Towards a Bayesan Automated Tax, The 1995 Internatonal Jont Conference on AI (1995). [3] N. Olver and A. Pentland, Graphcal Models for Drver Behavor Recognton n a Smart Car, IEEE Intl. Conference on Intellgent Vehcles (2000). [4] Vladmr Pavlovc, James M. Rehg, Tat-Jen Cham, and K. P. Murphy, A dynamcal Bayesan Network Approach to Fgure Trackng Usng Learned Dynamc Models, Internatonal Conference on Computer Vson (1999). [5] A. Pentland and A. Lu, Modelng and Predcton of Human Behavor, Neural Computaton, 11, pp (1999). [6] M. Akamatsu, Measurng Drvng Behavor, detectng unusual behavor for drvng assstance, SICE Annual Conference n Osaka (2002). [7] U. Kjærulff, A computatonal scheme for reasonng n dynamc probablstc networks, Proceedngs of the Eghth Conference on Uncertanty n Artfcal Intellgence, Morgan Kaufmann Publshers, San Mateo, Calforna, pp (1992). [8] P. Dagum, A. Galper, E. Horvtz and A. Sever, Uncertan Reasonng and Forecastng, Internatonal Journal of Forecastng, 11, pp (1995). [9] K. P. Murphy, Dynamc Bayesan Networks: Representaton, Inference and Learnng, dssertaton, PhD thess, U.C. Berkeley, Dept. Comp. Sc. (2002). [10] K. P. Murphy, The Bayes Net Toolbox for Matlab, Computng Scence and Statstcs: Proceedngs of Interface, 33 (2001). 121

6 Held n Torremolnos, Malaga (SPAIN) 8-10 September 2003 Hanamuro rver To Sakura To central Tsukuba To Tuchura 500m To Arakawaok Fg. 1 An ntersecton n a Tsukuba suburb Fg. 2 Topology of the HMM Fg. 3 Topology of the swtchng lnear dynamc system Fg. 4 Stop probablty through the HMM Fg. 5 Predcted stop probablty through the HMM and the actual stop rate Fg. 6 Averaged predcted stop probablty through the HMM 122

7 Held n Torremolnos, Malaga (SPAIN) 8-10 September 2003 Fg. 7 Stop probablty through the swtchng lnear dynamc system Fg. 8 Predcted stop probablty through the swtchng lnear dynamc system and the actual stop rate Fg. 9 Averaged predcted stop probablty through the swtchng lnear dynamc system 123