Data-driven characterization of multidirectional pedestrian trac

Transcription

1 Data-driven characterization of multidirectional pedestrian trac Marija Nikoli Michel Bierlaire Flurin Hänseler TGF 5 Delft University of Technology, the Netherlands October 8, 5 / 5

2 Outline. Motivation and objective. Related research 3. Methodology. Empirical analysis 5. Conclusion and future work / 5

3 3 / 5

4 Background - context Understanding and predicting of pedestrian trac Convenience and safety for pedestrians LOS indicators: based on data or/and models of pedestrian dynamics Trac indicators Density: the number of pedestrians present in an area at a certain time instance [#ped/m ] Flow: the number of pedestrians passing a line segment in a unit of time [#ped/ms] Velocity: the average of the velocities of pedestrians present in an area at a certain time instance / passing a line segment in a unit of time [m/s] / 5

5 Characterization based on Edie's denitions General formulation Density: k(v ) = Flow: q(v ) = N t i i dx dy dt N d i i dx dy dt N d i i t i i Velocity: v(v ) = q(v ) k(v ) = [van Wageningen-Kessels et al., ], [Saberi and Mahmassani, ] 5 / 5

6 Characterization based on Edie's denitions Limit conditions Photo: [Saberi and Mahmassani, ] / 5

7 Characterization based on Edie's denitions Results depend on size, shape and the placement of a measurement unit May be highly sensitive to minor changes Arbitrary aggregation May generate noise in the data [Openshaw, 983] May lead to loss of heterogeneity across space Density indicator 7 / 5

8 Characterization based on Edie's denitions Arbitrary aggregation May lead to loss of heterogeneity across pedestrians Does not comply with multi-directional nature of pedestrian ows Extreme case: velocity and ow vectors cancel out when equally sized streams of pedestrians walk with the same speed but in the opposite directions Velocity and ow indicators 8 / 5

9 Voronoi-based spatial discretization A personal region A i is asigned to each pedestrian i Each point p = (x, y) in the personal region is closer to i positioned at p i = (x i, y i ) than to any other pedestrian, with respect of the Euclidean distance A i = {p d E (p, p i ) d E (p, p j ), j} Pedestrian ows: [Steen and Seyfried, ] 9 / 5

10 Data-driven discretization framework Pedestrian trajectories The trajectory of pedestrian i is a curve in space and time p i = (x i, y i, t i ) 3D Voronoi diagrams associated with trajectories Each trajectory Γ i is associated with a 3D Voronoi 'tube' V i A point p = (x, y, t) belongs to the set V i if d(p, Γ i ) d(p, Γ j ), j d(p, Γ i ) = min{d (p, p i ) p i Γ i } d (p, p i ) - spatio-temporal assignment rule / 5

11 Data-driven discretization framework Sample of points The trajectory is described as a nite collection of triplets p is = (x is, y is, t s ),t s = [t, t,..., t f ] 3D Voronoi diagrams associated with the points p is = (x is, y is, t s ) Each point p is is associated with a 3D Voronoi cell V is A point p = (x, y, t) belongs to the set V is if d (p, p is ) d (p, p js ), j d (p, p is ) - spatio-temporal assignment rule / 5

12 Spatio-temporal assignment rules Naive distance d N (p, p i ) = { (p pi ) T (p p i ), t =, otherwise Distance To Interaction (DTI) { (p pi ) T (p p i ), t = d DTI (p, p i ) = (p p i (t i + t)) v i (t i ) ( v i (t i )), otherwise p = (x, y, t), p i = (x i, y i, t i ), t = t t i / 5

13 Spatio-temporal assignment rules Time-Transform distance (TT) d TT (p, p i ) = (x x i ) + (y y i ) + α(t t i ) α is a conversion constant expressed in meters per second Mahalanobis distance d M (p, p i ) = (p p i ) T M i (p p i ) M i - symmetric, positive-denite matrix that denes how distances are measured from the perspective of pedestrian i p = (x, y, t), p i = (x i, y i, t i ) 3 / 5

14 Voronoi-based trac indicators The set of all points in V i corresponding to a specic time t V i (t) = {(x, y, t) V i } [m ] Density indicator k i = V i (t) / 5

15 Voronoi-based trac indicators The set of all points in V i corresponding to a given location x and y V i (x) = {(x, y, t) V i } [ms] V i (y) = {(x, y, t) V i } [ms] Flow indicator Velocity indicator q i = ( V i (x) V i (y) v i = q i k i ) 5 / 5

16 Empirical analysis Scenarios Synthetic pedestrian trajectories Voronoi-based characterization for trajectories with d N, d TT, d M, d DTI / 5

17 Properties of 3D Voronoi-based characterization! Discretization is performed at the level of an individual: suitable for multi-directional ow composition Reproduces dierent simulated settings with uniform and non-uniform movement Preserves heterogeneity across pedestrians and space Discretization is adjusted to the reality of the ow: leads to smooth transitions in measured trac characteristics 7 / 5

18 Properties of the 3D Voronoi-based characterization! Discretization is performed at the level of an individual: suitable for multi-directional ow composition! Reproduces dierent simulated settings with uniform and non-uniform movement Preserves heterogeneity across pedestrians and space Discretization is adjusted to the reality of the ow: leads to smooth transitions in measured trac characteristics 7 / 5

19 Unidirectional non-uniform straight-line movement

20 Bidirectional non-uniform straight-line movement

21 Properties of 3D Voronoi-based characterization! Discretization is performed at the level of an individual: suitable for multi-directional ow composition! Reproduces dierent simulated settings with uniform and non-uniform movement! Preserves heterogeneity across pedestrians and space Discretization is adjusted to the reality of the ow: leads to smooth transitions in measured trac characteristics 8 / 5

22 3D Voronoi-based characterization - density map 9 / 5

23 Properties of 3D Voronoi-based characterization! Discretization is performed at the level of an individual: suitable for multi-directional ow composition! Reproduces dierent simulated settings with uniform and non-uniform movement! Preserves heterogeneity across pedestrians and space! Discretization is adjusted to the reality of the ow: leads to smooth transitions in measured trac characteristics / 5

24 3D Voronoi-based characterization vs. grid-based method Density sequences / 5

25 Robustness to the sampling rate Sample of points from synthetic pedestrian trajectories obtained using dierent sampling rate Voronoi-based characterization for points with d N, d TT, d M, d DTI Numerical analysis Statistics of interests: distribution of k and v errors for randomly sampled points / 5

26 Unidirectional uniform straight-line movement Density indicator Speed indicator rate:.5s rate:.5s rate:s rate:s rate:.5s rate:.5s

27 Unidirectional non-uniform straight-line movement Density indicator Speed indicator rate:.5s rate:.5s rate:s rate:s rate:.5s rate:.5s

28 Unidirectional non-uniform zig-zag movement Density indicator Speed indicator rate:.5s rate:.5s rate:s rate:s rate:.5s rate:.5s

29 Bidirectional uniform straight-line movement Density indicator Speed indicator rate:.5s rate:.5s rate:s rate:s rate:.5s rate:.5s

30 Bidirectional non-uniform straight-line movement Density indicator Speed indicator rate:.5s rate:.5s rate:s rate:s rate:.5s rate:.5s

31 Bidirectional non-uniform zig-zag movement Density indicator Speed indicator rate:.5s rate:.5s rate:s rate:s rate:.5s rate:.5s

32 Crossing uniform straight-line movement Density indicator Speed indicator rate:.5s rate:.5s rate:s rate:s rate:.5s rate:.5s

33 Conclusions Pedestrian-oriented ow characterization: Edie's denitions adapted through a data-driven discretization Suitable for multi-directional ow composition Reproduces dierent simulated settings with uniform and non-uniform movement Preserves heterogeneity across pedestrians and space Leads to smooth transitions in measured trac characteristics Sampled data: 3D Voronoi diagrams with Time-Transform distance perform the best Reproduces dierent simulated settings Robust with respect to the sampling rate 3 / 5

34 Future directions Additional spatio-temporal assignment rules will be tested More numerical analysis based on synthetic and real-world data (case study: Lausanne train station) Stream-based denitions of indicators and their interaction [Nikoli and Bierlaire, ] / 5

35 Thank you for your attention Marija Nikoli Transport and Mobility Laboratory EPFL - École Polytechnique Fédérale de Lausanne marija.nikolic@ep.ch 5 / 5

36 Mahalanobis distance Directions of interest p is = (x is, y is, t s ), v i (t s ) = t (s+) t s x i(s+) x is y i(s+) y is d (t s ) = v i (t s) v i (t, d s) (t s ) = dx (t s ) d (t s ) = dy (t s ), d (t s ) T d (t s ) =, d (t s ) = d 3 (t s ) = t (s+) t s, d 3 (t s ) = t (s+) t s 5 / 5

37 Mahalanobis distance Change of coordinates S (t s, δ) = p is + (t (s+) t s )v i (t s ) + δd (t s ) S (t s, δ) = p is (t (s+) t s )v i (t s ) δd (t s ) S 3 (t s, δ) = p is + δd (t s ) S (t s, δ) = p is δd (t s ) S 5 (t s, δ) = p is + δd 3 (t s ) S (t s, δ) = p is δd 3 (t s ) d M = (S j (t s, δ) p is ) T M is (S j (t s, δ) p is ) = δ, j =,.., 5 / 5