animatics

Model-based clustering for BSS
usage mining,
a case study with the velib’ system of Paris

Etienne Côme

15/10/2012

Outline

Bike Sharing Systems (BSS)

What is fun with BSS ?

Relatively new systems
Rapidly diffusing (EU and US nowadays, Hangzhou, ...)
Important sucesses
Abundant usage data
In interesting and original forms :
Origins / Destinations + timestamp
Real-time stations balances
Interesting and new problematics

Etienne Côme (IFSTTAR) Model-based clustering for BSS usage mining 15/10/2012 2 / 75

Outline

Outline
1 Introduction
Problematics
Usage data : trips records
Velib’ in few numbers and pictures
Tools and approach
2 Stations clustering using temporal usage proﬁles
Data representation : count time series
Generative model : naive Poisson mixture
Analysis of the results on the Velib’ dataset
3 Latent Dirichlet Allocation (LDA) for trips activity recognition
Data representation : dynamical O/D matrices
Generative model under LDA
Analysis of the results on the Velib’ dataset


Introduction Problematics

Problematics

Operational objectives

Planning new systems : position, size of the stations
Quality of service : bikes re-dispatch,...
...

Mining objectives

Building predictive model of usage
Finding spatio-temporal patterns
Better understanding of the usages
...


Introduction Usage data : trips records

Raw data
Trips data

departure time stamp
departure station
arrival time stamp
arrival station
type of subscription
! Will be converted in contingency tables (i.e. tensors of counts)

Data sources
! Velib’, 2 month
Open data : Barclays (Londre), Boston, ...


Introduction Velib’ in few numbers and pictures

in few numbers

BSS size :

1200 stations
≈ 40000 slots
≈ 16000 bikes
≈ 100 000 trips/day
27% trips = day subscription
73% trips = year subscription



Global behavior

100 000 140 000 Day subscription
free use limit

120 000
Year subscription
80 000
free use limit

100 000

60 000
80 000
Trips

60 000
40 000

40 000

20 000
20 000

0 0

0 5 10 0 20 40 60 80 100
Distances (Km) Duration (min)

F IG . 1: Histograms of trips lengths and durations



Temporal effects
35 000

Subscription :
30 000 Short

Long

25 000
Trips

20 000

15 000

10 000

5 000

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Time

F IG . 2: Number of Trips / hour (short / long subscriptions)



Temporal effects

7 500

Average number of trips

5 000

2 500

0
0 2 4 6 8 10 12 14 16 18 20 22
Hours

F IG . 3: Number of trips in week day / en week-end



Spatial effects

F IG . 4: Incoming trips map [6h,7h] for week days



Spatial effects

24

20
Mean activity / hour

16

12

8

4

2 4 6 8 10
Distance from the center ("Les Halles") in Km

F IG . 5: Stations activities / distance to "Les Halles"


Introduction Tools and approach

Approach, exploratory data analysis

General methodologie

Use clustering algorithms to ﬁnd interesting patterns in the data
Confront the found clusters to the city geography and sociology
⇒ Extract important factors that inﬂuence BSS system behavior.

2 developments :

1 Find clusters of stations with similar temporal usage pattern
2 Find latent activities that govern the BSS system dynamics



Tools, model based clustering

General methodologie
Imagine a data generation process
⇒ which include non-observed or latent variables
Latent variables can be discrete or continuous

Examples of latent variables

Species for ﬂowers
Topics for texts
Communities for graph vertices
...



Generative approach
Clustering
Model-based clustering :
1 Draw the cluster of sample (i)
2 Depending on the cluster draw the observed values of (i)

0.05

0.04

0.03
f(x)

0.02

0.01

0
-80 -60 -40 -20 0 20 40
x

F IG . 6: Example of 1D Gaussian mixture model



Data generation process
Graphical model representation

1. Draw the cluster of sample (i)

Zi ∼ M(1, π)
⇒ π prior proportions of the clusters.



Data generation process
Graphical model representation

2. Depending on the cluster draw the observed values of (i)

p(x|Zik = 1) = f (x; θ k ), ∀k ∈ {1, . . . , K }.
⇒ f can be tuned to exploit speciﬁcities of the problem.



Model based clustering framework

Task and tools
Inferring the parameters :
⇒ EM algorithm or Variational EM for complex models




Task and tools
Finding the clustering
⇒ Byproducts of EM




Task and tools
Finding the clustering
⇒ Byproducts of EM
Fixing the number of clusters
⇒ Model selection criterion : BIC, AIC, ICL, perplexity.


Stations clustering using temporal usage proﬁles

Stations clustering using
temporal usage proﬁles




Objectives :

Find groups of stations with similar temporal usage proﬁles
Temporal usage proﬁles = incoming, outgoing activity / hour
Taking into account the week-days /week-end discrepancy
With a model for counts data
Cross the results with possible explanatory variables :
population, employments, amenities, ...


Stations clustering using temporal usage proﬁles Data representation : count time series

Data representation : count time series
Observed data :
out
Xsdt : # of bikes taken at station s during day d at hour t
in
Xsdt : # of bikes returned at station s during day d at hour t
in in out out
Xsd = (Xsd1 , . . . , Xsd24 , Xsd1 , . . . , Xsd24 )
⇒ X tensor of size N × D × T .
⇒ temporal behavior / stations.

Variables
Xsd (observed) : # of bike leaving/coming
Zs (latent) : cluster of station s
Wd (observed) : cluster of days (week / week-end)


Stations clustering using temporal usage proﬁles Generative model : naive Poisson mixture

Generative model : naive Poisson mixture

F IG . 7: Graphical model representation

Parameters, Θ

αs = stations attractivity effects
π = (π1 , . . . , πK ) cluster proportions
λ = (λklt ) temporal proﬁles of the clusters


Generative model
Naive Poisson mixture

Zs ∼ M(1, π)
Xsd1 ⊥ . . . ⊥ XsdT
⊥ ⊥ | {Zsk = 1, Wdl = 1}
Xsdt |{Zsk = 1, Wdl = 1} ∼ P(αs λklt )

Constraints

Dl λklt = DT , ∀k ∈ {1, . . . , K },
l,t

with Dl number of day in cluster l.



Parameters estimation, likelihood

Marginal likelihood
 

L(Θ; X) = log  πk p(Xsdt ; αs λklt )Wdl  (1)
s k d,t,l

Completed likelihood
 

Lc(Θ; X, Z) = Zsk log πk p(Xsdt ; αs λklt )Wdl  (2)
s,k d,t,l

where Z is unknown.



EM algorithm
⇒ Straightforward solution for parameters estimation EM :

E step
Conditional expectation of Lc given the current parameters
 

E[Lc(Θ, x, Z)|x, Θ(q) ] = tsk log πk p(xsdt ; αs λklt )Wdl  (3)
s,k d,t,l

with tsk the posteriori probabilities :
(q) (q) (q)
πk d,t,l p(xsdt ; αs λklt )Wdl
tsk = (q) (q) (q)
(4)
k πk d,t,l p(xsdt ; αs λklt )Wdl



EM algorithm
⇒ Straightforward solution for parameters estimation EM :

M step
Maximization of the lower bound with respect to the parameters
1
αs : mean station activity αs =
ˆ DT d,t Xsdt ,
1
πk : proportion of cluster k , πk =
ˆ N s tsk
λklt : activity of time frame t for cluster k , for week day or during
the week-end (day cluster l)

ˆ 1
λklt = tsk Wdl Xsdt (5)
s tsk αs d Wdl
s,d


Stations clustering using temporal usage profiles Analysis of the results on the Velib’ dataset

Results

Setting

One month of data (September)
Number of clusters (K=8) set manually
⇒ good trade off between interpretability and fit of the clustering

Outputs

Zs : station s clusters
λk : temporal profile of cluster k
αs : stations s attractivity



Railway stations
Week Week-end
5

4

Departures
3

2

1
Activity

0
5

4

3

Arrivals
2

1

0
0 5 10 15 20 0 5 10 15 20
Hours



Railway stations



Parks
Week Week-end
5

4

Departures
3

2

1
Activity

0
5

4

3

Arrivals
2

1

0
0 5 10 15 20 0 5 10 15 20
Hours



Parks



Spare time, night
Week Week-end
5

4

Departures
3

2

1
Activity

0
5

4

3

Arrivals
2

1

0
0 5 10 15 20 0 5 10 15 20
Hours



Spare time, night



Spare time, night and week-end
Week Week-end
5

4

Departures
3

2

1
Activity

0
5

4

3

Arrivals
2

1

0
0 5 10 15 20 0 5 10 15 20
Hours



Spare time, night and week-end



Housing
Week Week-end
5

4

Departures
3

2

1
Activity

0
5

4

3

Arrivals
2

1

0
0 5 10 15 20 0 5 10 15 20
Hours


Housing Inhabitants / ha
0
200
400
600
800
1 000
1 200


Employment (1)
Week Week-end
5

4

Departures
3

2

1
Activity

0
5

4

3

Arrivals
2

1

0
0 5 10 15 20 0 5 10 15 20
Hours



Employment (2)
Week Week-end
5

4

Departures
3

2

1
Activity

0
5

4

3

Arrivals
2

1

0
0 5 10 15 20 0 5 10 15 20
Hours


Employment (1 and 2)

Jobs / ha
0
500
1 000
1 500
2 000


Mixed usage
Week Week-end
5

4

Departures
3

2

1
Activity

0
5

4

3

Arrivals
2

1

0
0 5 10 15 20 0 5 10 15 20
Hours



Mixed usage



Crossing with population/employments/services rates

hab/ha emp/ha serv/ha com/ha
162 237 4.2 3.7
Spare time (1) 367 189 6.3 4.4
Spare time (2) 261 322 7.7 6.9
Parks 172 90 2 1.7
Railway stations 209 206 2.4 1.8
Housing 375 108 3.8 2.7
Employment (1) 138 409 4.5 2.8
Employment (2) 157 456 5.7 5.6
Mixed usage 301 163 3.8 2.8
TAB . 1: Mean of each cluster with respect to population, employment,
services and shops densities . Sources "Recensement 2008", "Base
permanente des équipements", Insee.



Conclusion on stations clustering
Discussion on the model
Model adapted to counts
Scaling factors for stations important
Stations described by incoming and outgoing ﬂow dynamics
Taking into account week-day week-end differences

Discussion on the results
Clusters are interpretable
Population, employment and amenities densities are highly
explanatory for the clusters
Temporal proﬁles are also interpretable and informative


Latent Dirichlet Allocation (LDA) for trips activity recognition

Latent Dirichlet Allocation
(LDA),
for trips activity recognition


Latent Dirichlet Allocation (LDA) for trips activity recognition

Objectives

Decompose, the trips into interpretable clusters
⇒ look for stationarities and change points in the OD dynamics
LDA with documents = small bags of successive trips

Analyse the found clusters with respect to their :

Temporal positions, cycles
Spatial distribution of ﬂows
Spatial distribution of incoming / outgoing ﬂows per stations


Latent Dirichlet Allocation (LDA) for trips activity recognition Data representation : dynamical O/D matrices

Data representation : dynamical O/D matrices

Observed data :
Xijt : # of bikes that were
1 taken at station i
2 returned at station j
3 at time t
t ∈ {1, . . . , Nt } :
i, j ∈ {1, . . . , Ns } : set of stations
⇒ Xijt tensor of dimension Ns × Ns × Nt .
⇒ taking into account spatial and temporal BSS behavior


Latent Dirichlet Allocation (LDA) for trips activity recognition Generative model under LDA

LDA, background
LDA = Latent Dirichlet Allocation
Bayesian mixture for discrete data
⇒ originally to ﬁnd topics in text corpus
Each document (bag of words) is a mixture of topics
Each topic has its own words probabilities vector

F IG . 8: Graphical model representation of LDA.



LDA for dynamical O/D matrices analysis

Hypothesis :

Local stationarity of BSS behaviour / OD
Cyclostationarity : week, day

Small bags of successive trips ≈ stationarity of OD
⇒ Documents (bags of words) = bags of successive trips (5000)

, with :

Words = Origin/Destination couples
Topics = Latent activities



LDA, for dynamical O/D matrices analysis

For each activity a, draw an O/D matrices generator :

Λa ∼ D(β)

For each "bag of trips" t ∈ {1, . . . , Nt } :

1 Draw the activities proportions : πt ∼ D(α)
2 For each trips of the bag t :
Draw its activity A : A ∼ M(1, πt )
Draw an O/D couple D using activity A generator :
D ∼ M(1, ΛA )


Latent Dirichlet Allocation (LDA) for trips activity recognition Analysis of the results on the Velib’ dataset

Fixing the number of activities
perplexity analysis

Perplexity = f( likelihood of test data )
Clear drop off at K=5

165000
q q
q

160000
perplexity

155000
q

q q
q q
q q q q
q
q

4 8 12
K

F IG . 9: Perplexity on the September dataset with respect to the number of
latent activities.



Temporal results : πt

9000
trips / hour

6000

3000

0
avril 11 avril 18 avril 25

F IG . 10: Temporal evolution of πt

Remarks :
Cyclostationarity clearly visible (even holidays)
Low mixture between the latent activities
Interpretable temporal clusters : Home ↔ Work, Lunch,...



Spatial results : Λa as ﬂows

F IG . 11: Latent activity "House→Work commute", ﬂows (blue for f=10/10 000)




F IG . 12: Latent activity "Lunch", ﬂows (blue for f=10/10 000)




F IG . 13: Latent activity "Work→House commute", ﬂows (blue for f=10/10 000)




F IG . 14: Latent activity "Evening", ﬂows (blue for f=10/10 000)




F IG . 15: Latent activity "Spare time", ﬂows (blue for f=10/10 000)



Incoming / Outgoing specificities, question :
Which stations have an increased in/out-degree for a latent activity a ?

a
Introduce stations incoming specificities ISs and outgoing
specificities OSsa :

a a a ag g
ISs = log(pins /pins ), OSs = log(pouts /pouts ), (6)
a a
with pins , pouts the probabilities that a trips end/start in station s
for activity a :
a
pins = Λa , pouts =
js
a
Λa ,
sj
j j
g g
and pins , pouts the global probabilities that a trips end/start in
station s :
g j,t Xjst g j,t Xsjt
pins = , pouts = .
i,j,t Xijt i,j,t Xijt



Spatial results : incoming speciﬁcities

F IG . 16: Latent activity "House→Work commute", stations incoming
speciﬁcity


Spatial results : outgoing speciﬁcities

F IG . 17: Latent activity "House→Work commute", stations outgoing speciﬁcity



Expected bike balance, question :
Positive/negative bike balance of stations for a latent activity a ?

The O/D matrix D follow a multinomial law of parameter Ndep
(number of trips) and Λa :
D ∼ M(Ndep , Λa ),
The bike balance Bs for a station s is thus given by :
Incoming bikes Outgoing bikes

Bs = Djs − Dsj
j j

And the expectation of the balance vector B is thus equal to :
E[B] = Ndep (Λa )t − Λa v, (7)
with v = (1, . . . , 1)t .


Spatial results : expected bike balance
Balance
-30
-20
-10
0
10
20
30

F IG . 18: Latent activity "House→Work commute", stations expected balances
with Ndep = 10 000


"Lunch", incoming speciﬁcity

F IG . 19: Stations incoming speciﬁcity



"Lunch", outgoing speciﬁcity

F IG . 20: Stations outgoing speciﬁcity



"Lunch", balance
Balance
-30
-20
-10
0
10
20
30

F IG . 21: Stations expected balances with Ndep = 10 000



"Work→House commute", incoming speciﬁcity




"Work→House commute", outgoing speciﬁcity




"Work→House commute", balance
Balance
-30
-20
-10
0
10
20
30




"Evening" incoming speciﬁcity




"Evening", outgoing speciﬁcity




"Evening", balance
Balance
-30
-20
-10
0
10
20
30




"Spare time", incoming speciﬁcity




"Spare time", outgoing speciﬁcity




"Spare time", balance
Balance
-30
-20
-10
0
10
20
30




Conclusion on LDA for activities recognition

Interpretable latent activities
Give good picture of city "pulse" and geography
Better understanding of the system behaviour
Strong evidence of cyclostationarity
Week-day / Week-end pattern


Thanks for your attention

@comeetie, etienne.come@ifsttar.fr

Ifsttar
Centre de Marne-la-Vallée
Batiment le “Descartes 2”
2, rue de la Butte Verte F-93166 Noisy le Grand cedex

Mél. etienne.come@ifsttar.fr
Tél. +33 (0)1 45 92 56 57

Site : www.ifsttar.fr


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