Trajectory Data Mining

IntroductionUrban Computing with Taxi Trajectories

Where to find my next passenger

Trajectory Data Mining

Shaolin ZamanNafeez Abrar

Bangladesh University of Engineering and Technology

July 9, 2013

Shaolin Nafeez Trajectory Data Mining



OutlineKeyWordWhat is our Data

Outline

1 Urban Computing with Taxi Trajectories

2 Where to find my next passenger





KeyWord I

Urban Planning

Land Use planning

Transportation planning

Infrastructure planning

Ubiquitous Computing

Every person, vehicle, building used as computing component.





KeyWord II

Taxicab

Vehicle for hire with a driver

GPS

space-based satellite navigation system that provides locationand time information





What is our Data

Taxi TrajectorySequence of GPS points having

TimeLatitude and LongitudeState (Occupied/Non-occupied)




Traffic ModelingFlaw DetectionEvalutionFuture Direction

Urban Computing with Taxi Trajectories

Section 2

Urban Computing with Taxi Trajectories





Main Objective

Traffic Modeling

model city-wide trafficconnection between region pairs

Flaw Detection

Detect Flawed Region pairRelationship between the flawed pairs

Real Evalution

Justify the effectiveness of proposed method





Some Definition I

Taxi Trajectory

denoted as Tr

Time-ordered point sequence

Tr = p1 → p2 → · · · → pn

p = (lat, long , t, o)

Region

Map is partitioned by disjoint regions bounded by the majorroads.Denoted as r





Some Definition II

Transition

Denoted as sDirectional transition from one region to another region.

Tr = p1 → p2 → · · · → pn

s : r1 → r2

pi : point of Tr falling in region r1

pj : point of Tr falling in region r2

where i < j





Some Definition III

A transtion, s have the following attributes

Leaving Time, pj .tArriving Time, pi .tTravelled distance, d

d(pi , pj) =∑

i≤k<j

Dist(pk , pk+1)

speed of the transition

v =d(pi , pj)

|pi .t − pj .t|





Some Definition IV

Region Pair

Attribute Notation

A pair of region (r1, r2)

Euclidian Distance CenDist(r1, r2)

Count of Transition |S |Expected Travelled Distance E (D)

Expected Speed E (V )

Ratio between actual travelleddistance and Central distance

θ





Some Definition V

Region Pair

E (V ) =

∑si∈S si .v

|S |

E (D) =

∑si∈S si .d

|S |

θ =E (D)

CenDist(r1, r2)





Some Definition VI

<S ,E (V ), θ>

Represents the connectivity and traffic between two region

θ may be smaller than 1

Big θ means that people have to take a long route

Big S and small E (V ) implies heavy traffic





Architecture of the model





Traffic Modeling

1 Urban Computing with Taxi TrajectoriesTraffic ModelingFlaw DetectionEvalutionFuture Direction

2 Where to find my next passengerProbability CalculationParking Place DetectionTime-Dependent ProbablitiesOnline RecommendationEvaluationFuture Direction





Traffic ModelingMap Partition

Partition the map into disjoint regions based on major roadsegments

Employ Connected Component Labeling (an image segmentmethod)

Why not based on road segments?

Region carries knowledge about people’s living and travel

Flaws represented by regions contribute to both land use andtransportation.





Map Partition

Figure: Heat map of the partitioned regions in Beijing





Traffic ModelingBuilding Region Matrix

Steps:

1 Temporal Partition

2 Transition Construction

3 Build the matrix





Step 1: Temporal Partition

Partition the taxi trajectories according to

Work Day

Rest Day (weekend and public holiday)

Time Work day Rest day

Slot 1 7:00am-10:30am 9:00am-12:30pm

Slot 2 10:30am-4:00pm 12:30pm-7:30pm

Slot 3 4:00pm-7:30pm 7:30pm-9:00am

Slot 4 7:30pm-7:00am





Step 2: Transition construction I

Pick out the trajectories with passengers

Project the trajectories onto the map

Construct transition between two regions





Step 2: Transition construction II

Figure: Transfer a trajectory into transitions

Tr1 = r1 → r2

Tr1 = r1 → r2 → r3





Step 3: Building Region matrix

Formulate matrix, M for each time slotEach item a(i , j) is a tuple representing a region pair whichhas three attributes <S ,E (V ), θ>If working days = x , rest days = y total number of matrix =4x+3y

Figure: Region Matrix





Flaw Detection







Flaw DetectionPrimary Steps of Detecting Flaw

Detect region pairs with big S, small V and big theta.

For each matrix, M, select the region paris with transitionsabove the average.

Find the skyline set from the selected region pairs.





Flaw DetectionWhat is Skyline

A point dominates another point if it is good or better in alldimension and at least better in one dimension

Skyline is the point which is not dominated by any other point.





Flaw DetectionExample of Skyline Detection





Flaw DetectionExample of Skyline

Three kinds of region pairs fall in Skyline:





Flaw DetectionPattern mining from Skyline

1 Formulate skyline graph

2 Mining frequent sub-graph pattern





Flaw DetectionFormulating Skyline Graph





Flaw DetectionMining frequent sub-graph pattern

g = subgraph

G = SkylineGraphcontainingg

G = CollectionofSkylineGraphs

Support(g) =|G |g ⊆ G ,G ∈ G |

numofdays

Support(g1 ⇒ g2) =|g1 ∪ g2|

numofdays

Confidence(g1 ⇒ g2) =|g1 ∪ g2||g1|





Flaw DetectionMining Skyline Graph Example

Figure: Mining frequent skyline patterns





Evalution







EvalutionReal Evaluation

Select some new urban planning (such as new road) andcheck if they reduced the flaw.

Check whether some flaws have been detected by city plannerin future.

[Two dataset of Trajectories of year 2009 and 2010 in Beijing havebeen used for evaluation]





EvalutionEvaluation Result

Some flawed planning occured in 2009 disappeared in 2010because of newly built roads

Number of defected regions have been increased in 2010





Future Direction

Studying geographic features of a region

The purpose of people’s travel i.e shopping, sports, work etc.




Probability CalculationParking Place DetectionTime-Dependent ProbablitiesOnline RecommendationEvaluationFuture Direction


Section 3






Purpose

Improve utilization of taxi

Reduce energy consumption

Recommend a Taxi driver some locations highly probable topick up passenger and maximize profit

Recommend people some locations highly probable to findvacant taxi





How does it fill the purpose

Detect parking places

Devise Probabilistic model to formulate time-dependent taxibehavior

Provide just-in-time recommendation to both taxi driver andpeople seeking taxi





Some Definition I

Road Segment

Denoted as rDirected edge r .dirTwo terminal points : r .s and r .eTravel time : r .t

Route

Sequence of Road segmentsdenoted as

R : r1 → r2 → . . .→ rn where rk+1.s = rk .e (1 <= k < n)

R.s = r1.s and R.e = rn.e





Some Definition II

State.

State Taxi Status

Occupied (O) Taxi is occupied by passenger

Cruising (C) Taxi is travelling without passenger

Parking (P) Taxi is waiting for passenger





Some Definition III

Trip A taxi trip is a sub-trajectory which has a single state(C/O/P)

Figure: Taxi Trajectory and Taxi Trip





System Overview





Probability Calculation







Notations

Let , P : A parking placeR : r1 → r2 → . . .→ rn : A RouteARP : an action that a driver drives along R until finds passengertmax : maximum waiting time at PQuestion:

How likely driver will get passenger

If finds then what is the expected duration

What is the expected duration of next trip





Probability of picking up passenger I

Let, S = an event a driver succeeds in getting passenger if takesaction ARP

t(max) = maximum waiting time at Parking place

S =n+1⋃i=1

Si where i = 1, 2, . . . , n and current timeT0 (1)

Sn+1 = event that driver picks passenger at Parking place P (2)

The probability that taxi gets passenger at road segment ri andtime T0 + ti is:

pi = Pr(C O|ri ,T0 + ti ) (3)

p∗ = Pr(P (0,t(max)] O|T0 + tn) (4)





Probability of picking up passenger II

The probability that a taxi succeeds at picking passenger at P is

Pr(Si ) =

p1 i = 1

Pi∏i−1

j=1(1− pj) i = 2, 3, . . . , n,

p∗∏n

j=1(1− pj) i = n + 1

(5)

Pr(S) = 1− Pr(n+1⋃i=1

Si )

= 1− (1− p∗)n∏

j=1

(1− pj)

(6)





Probability of picking up passenger III

Let random variable T = duration from current time T0 tobeginning of next trip

T = TR + TP{TP = 0, if TR ≤ tn

TR = tn, if TP > 0(7)





Probability of picking up passenger IV

The probability mass function

Pr(TR = ti |S) =Pr(TR = ti , S)

Pr(S)

=

{Pr(Si )Pr(S) i = 1, 2, . . . , n − 1,Pr(Sn)+Pr(Sn+1)

Pr(S) , i = n

(8)





Probability of picking up passenger V

Conditional Expectation of TR is

E [TR |S ] =n∑

i=1

Pr(TR = ti |S)

=1

Pr(S)(

n∑i=1

tiPr(Si ) + tnPr(Sn+1)

(9)





Probability of picking up passenger VI

Let W = event that driver waits at PSo probability of waiting at parking place

Pr(W ) =n∏

j=1

(1− pj)

Now let we break TP i.e (0, tmax) into m buckets.

t0 = 0

4t∗ =tmax

2mt∗j = (2j − 1)4t∗, j = 1, 2, . . . ,m,

(10)





Probability of picking up passenger VII

So, probabiilty that waiting time TP belongs to j − th bucket is

pj∗ = Pr(P t∗j −4t∗,t∗j +4t∗ O|TP > 0,T0 + tn) (11)

p∗ =m∑j=1

pj∗ (12)





Probability of picking up passenger VIII

The conditioanl Probability The conditional expectation of TP is

E [TP |S ] =Pr(W )

Pr(S)

m∑j=1

pj∗t∗j

The conditional Expectation of TR is

E [TR |S ] =1

Pr(S)(

n∑i=1

tiPr(Si ) + tnPr(Sn+1)

The conditional Expectation of T is

E [T |S ] = E [TP |S ] + E [TR |S ]

=

∑ni=1 tiPr(Si ) + tnPr(Sn+1) + Pr(W )

∑mj=1 pj

∗t∗j

Pr(S)

(13)





Probability of travel time of Next Trip I

Let

DN = Distance of next trip if driver takes ARP conditioned S happens

qji = Probability that dj−1 < DN ≤ dj when Si happens

dmax = maximum distance of next trip

i = 1, 2, . . . , n + 1 and current time T0

qji = Pr(dj −4d < DN < dj +4d |Si ,T0 + ti )

Now, if distance is splitted into s buckets as earlier then

d0 = 0

4d∗ =dmax

2sd∗j = (2j − 1)4d∗, j = 1, 2, . . . , s,

(14)Shaolin Nafeez Trajectory Data Mining




Probability of travel time of Next Trip II

The conditional probability that DN is dj is

Pr(DN = dj |S) =n+1∑i=1

Pr(Si )qji

Pr(S)(15)

The conditional expected distance of next trip is

E [DN |S ] =1

Pr(S)

s∑j=1

(dj

n+1∑i=1

Pr(Si )qji )

=1

Pr(S)

n+1∑i=1

Pr(Si )(s∑

j=1

djqji )

(16)





Probability of getting taxi for passenger I

Let Pr(C ; r |t) : Probability of a vacant taxi at road segment r attime t So the suggested road segment for a passenger will be

r = argmaxr∈σPr(C ; r |t)





Parking Place Detection

There are three steps

Candidates detection

Filtering

Parking place clustering





Candidates detection I





Candidates detection II





Filtering I

Why Candidates can be generated due to traffic jam.

How Design supervised model to detect real parking places





Filtering II

Filtering Features

Spatial-Temporal features

Minimum bound ratioAverage DistanceCenter DistanceParking DurationLast Speed

POI feature

Point of interest i.e shopping mall, theaters etc.

Collaborative feature

Historical state of the candidate sets.





Parking Place Clustering

Parking place is detected for each single trajectory

Use density-based clustering method (OPTICS) to discoverthe parking places

This method is used because clustered region may havearbitrary shape





Time-Dependent Probablities







Time-Dependent Probablities

Assumed that probability is stable during interval [t, t +4t]

Partion-and-group approach is developed for computing theprobability

Day is partitioned into K small time intervals of width τ

So k − th interval is

Ik = [(k − 1)τ, kτ ], k = 1, 2, . . . ,K

So we learn the probability of each Ik offline





Probability w.r.t Road Segments I

Conduct road segment clustering to integrate road segmentswith similar features

Let,

r a road segmentr̃ : The cluster r belongs to#k(C ; r̃): The number of trips with state C during Ik on allsegments of cluster#k(O; r̃) : The number of trips with state O during Ik on allsegments of cluster





Probability w.r.t Road Segments II

Probability that there exists taxi cruising on r at time t is

Pr(C ; r |t) =

∑b(t+4t)/τck=bt/τc #k(C (̃r)∑b(t+4t)/τc

k=bt/τc∑

r̃∈r̃ (#k(C ; r̃) + #k(O; r̃))(17)

Pr(C O; r |t) =

∑b(t+4t)/τck=bt/τc #k(C O; (̃r)∑b(t+4t)/τck=bt/τc

∑r̃∈r̃ (#k(C ; r̃))

(18)

Pr(da < DN ≤ db|r , t) =

∑b(t+4t)/τck=bt/τc #k(da, db; r̃)∑b(t+4t)/τck=bt/τc #k(0, dmax ; r̃)

(19)





Probability w.r.t Road Segments I

Pr(P (ta,tb] O|Tp > 0, t) =∑b(t+4t)/τck=bt/τc #k(ta, tb,P O; P)∑b(t+4t)/τc

k=bt/τc (#k(P O; P) + #k(P C ; P))

(20)





Online Recommendation







Taxi Recommender

Retrieve set of parking places

For each parking place, Generate the Route R with minimumPr(SR)

Recompute Pr(S) according to the query time

Rank the parking places

Recommend top-k parking places





Passenger Recommender

Perform query for obtaining region within walking distance

If region has parking places, recommend k nearest parkingplaces

Otherwise recommend road segments with k largest probablity(Pr(C ; r |t)) of having vacent taxi





Evaluation







EvaluationEvaluation Dataset

Road network:

road network of Beijing106,579 road nodes141,380 road segments

Trajectory:

over 12,000 taxisperiod of 110 days.total 20 million trips, among which 46% are occupied trips and53% are non-occupied

use 70 days data to build our system and evaluate the methodusing the rest (40 days) data.





Evaluation IParking Place Detection

Ask three local people to label 1000 parking candidates

Features Precision Recall

Spatial 0.695 0.670

Spatial+POI 0.716 0.696

Spatial+POI+Collaborative 0.725 0.706

Spatial+POI+Collaborative+Temporal 0.909 0.889





Evaluation IIParking Place Detection

Conduct survey of more than 20 users for submitting knownparking places

received 70 Parking places uniformly distributed in Beijing

Recall reaches to 81%





Evaluation IStatistical Learning

Calculate overall time-dependent distribution for both Parkingplaces & road segments.





Evaluation IOnline Recommendation

Extract high-profit drivers

Segment them to O/C/P trips

Before each O/P trip, randomly select 10 points

For each query point top-k parking places are recommended

The recommendation is evaluated based on real case.

Precision of recommendation reaches to 67





Future Direction

Incorporate real-time Traffic information to provide betterroutes toward the parking places





Thank you


Documents

Trajectory Data Mining