large-scale nonparametric estimation of vehicle travel time distributions

DataModel and Fitting

Experimental Results

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Large-Scale Nonparametric Estimation

of Vehicle Travel Time Distributions

Rikiya Takahashi, Takayuki Osogami,and Tetsuro Morimura

{rikiya,osogami,tetsuro}@jp.ibm.com

IBM Research - Tokyo

Rikiya Takahashi, Takayuki Osogami, and Tetsuro Morimura Large-Scale Nonparametric Estimation of Vehicle Travel Time Distributions



.. Route recommendation and traffic simulation

Which route (e.g. A or B) is chosen by a car driver?Route recommendation

Which route should you select?

Traffic simulation

Which route do you select?

Dijkstra for minimizing expected travel-time is inflexible because of

Risk unawareness Variability oftravel-time is notconsidered.

Unrealistic homogeneity Everyonetakes the same route.




.. Example of risk-sensitive route choice: ICTE

Instead of its mean, evaluate Iterated Conditional TailExpectation (ICTE) (Osogami, 2011) of travel-time.

Quantiles of travel-time distribution are utilized.The value q of CTE q can be different among drivers.




.. Agenda

What we need: probability density function (p.d.f.) oftravel-time for every link of a road network.

Main proposal: data-mining algorithm to interpolatingp.d.f. for every link.

...1 Summary of real data

...2 Model and how to fit it

...3 Experimental prediction performance




.. Our road network and travel-time samplesWe have a road network and probe-car dataset as

1.2M intersections and 3.3M links in Greater Tokyo Area.3.1M travel-time samples by totally 58,584 taxis.Data sparseness especially in suburban or rural regions.

Figure: Heatmaps based on the total number of travel-timesamples in 24 hours for each link. The green, yellow or red pointsare located on the links that have at least 1, 10, or 100 samples,respectively.Rikiya Takahashi, Takayuki Osogami, and Tetsuro Morimura Large-Scale Nonparametric Estimation of Vehicle Travel Time Distributions



.. Distribution of relative travel-time

Histogram of the relative travel-time yy =(actual travel-time)/(travel time by legal speed limit)Modes of P(y) are about from 0 to 2.

0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

Link ID=’1539993’#samples=84

x=(actual time)/(standard time)

14:00-14:59

0 2 4 6 8 10

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16:00-16:59

0 2 4 6 8 10

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18:00-18:59

0 2 4 6 8 10

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1.0



20:00-20:59

0 2 4 6 8 10

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1.0



22:00-22:59Link ID=’1539993’

0 2 4 6 8 10

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8:00-8:59

0 2 4 6 8 10

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9:00-9:59

0 2 4 6 8 10

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10:00-10:59

0 2 4 6 8 10

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15:00-15:59

0 2 4 6 8 10

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16:00-16:59Link ID=’1049171’

Figure: Histograms of the relative travel-timeRikiya Takahashi, Takayuki Osogami, and Tetsuro Morimura Large-Scale Nonparametric Estimation of Vehicle Travel Time Distributions



.. Issues we must solve

Scalability The sizes of the road network and travel-timesamples are large.

Data sparseness Travel-time samples are limited or missing insuburban links.

Non-Gaussianity Distribution of travel-time is not Gaussian.Multi-modality or heavy tails could happen.

Least-square (L2-loss) regression is inflexible.

Assumption for solving: connected links have similardistributions of vehicle velocities, depending on the requiredhops.




.. Conditional density estimator of relative travel-time

Conditional p.d.f. of the relative travel-time y

fe(y) =λ0ϕ0(y)+

∑mi=1 λiK (e, eπ[i ])ϕi(y)

λ0+∑m

i=1 λiK (e, eπ[i ]),

EΦ,{eπ[1], · · · , eπ[m]} : subset of E

Φ,{ϕ0, ϕ1, · · · , ϕm} : set of basis density functions

K (·, ·) : similarity function between links

λ,(λ0, λ1, · · · , λm)T : vector of link importance

The link-independent terms λ0 and ϕ0(·) are introduced forhandling the case ∀i ∈ {1, · · · ,m},K (e, eπ[i ])≡0.




.. 3 steps in estimating the parameters

fe(y) =λ0ϕ0(y)+

∑mi=1 λiK (e, eπ[i ])ϕi(y)

λ0+∑m

i=1 λiK (e, eπ[i ])

A) Basis function Φ,{ϕ0, ϕ1, · · · , ϕm} Mixture of gamma orlog-normal distributions using convex clustering.

B) Link similarity K (·, ·) Sparse diffusion kernel on alink-connectivity graph.

C) Link importance λ,(λ0, λ1, · · · , λm)T Kullback-LeiblerImportance Estimation Procedure (KLIEP)(Sugiyama et al., 2008).

Stability of fitting: each component can be fitted with eitherconvex optimization or simple matrix multiplication.




.. A) Fitting nonparametric basis density functionsAt most L mixtures of gamma or log-normal distributions

ϕi(y) =L∑

`=1

θi`ψ`(y)

Figure: Sliding windowsfor fitting ψ1,· · ·, ψL

Optimize mixture weights as

maxθi

∑y∈Yi

log

[L∑

`=1

θi`ψ`(y)

].

Convex w.r.t. θi ,(θi1, · · · , θiL)T

Fast convergence with SequentialMinimal Optimization (SMO)(Takahashi, 2011)




.. B) Link connectivity graph and its LaplacianAdjacency matrix A=(aij ; ei =(ui , vi), ej =(uj , vj)) as

aij =

{12+

∆T (ei )∆(ej )

2‖∆(ei )‖‖∆(ej )‖if ui =vj ∪ vi =uj

0 otherwise.

Values of {aij} when the widearrow represents ei .

∆(e),xv− xu for e=(u, v) and xu, xv ∈R2 : location

D=diag(∑|E |

j=1 a1j , · · · ,∑|E |

j=1 a|E |

)H=D−1/2 (A−D)D−1/2 : negative normalized Laplacian




.. B) Sparse diffusion kernel as link similarity

The diffusion kernel exp(βH) (Kondor and Lafferty, 2002)is dense and computationally infeasible, while H is sparse.

Assume that traffic does not diffuse broadly in short time.

Then β is small and an approximate kernel matrix is

K (β, p)=

(I+

β

pH

)p

=

p∑q=0

p!βq

q!(p−q)!pqHq,

where p is a resolution hyperparameter in discretization.

The (i , j)-th element of the matrix K (β, p) gives thesimilarity value between the edges ei and ej .




.. C) Optimize the link importance with SMO

The vector of link importance λ is optimized with KLIEP as

maxλ

∑e∈E+

∑y∈Y[e]

log

[λ0ϕ0(y)+

m∑i=1

λiK (e, eπ[i ])ϕi(y)

]

s.t.∑e∈E+

∑y∈Y[e]

[λ0+

m∑i=1

λiK (e, eπ[i ])

]= n.

Convex optimization

Equivalent objective to that of convex clustering, with avariable transformation

Also can be accelerated with SMO




.. Experimental setting

10-fold likelihood cross-validation to evaluate predictiveperformances.

Evaluate performances independently for 24 hourlydatasets.

Hyperparameters are also chosen with validations.

L=100 and p=8 (fixed)r ∈{1, 1.5, 2, · · · , 3} and β∈{1, 2, 3, 4, 5}.

Compare with parametric regression methods assumingsingle log-normal distribution.




.. Time dependent size of the data

Table: The numbers, N, of travel-time samples, and the numbers,|E+|, of links that have at least one sample for each time slot.

hour N |E+| hour N |E+|0:00- 273,168 69,126 12:00- 129,148 41,569

1:00- 185,567 53,018 13:00- 133,987 40,083

2:00- 109,662 38,994 14:00- 128,288 37,594

3:00- 49,821 25,620 15:00- 130,971 36,980

4:00- 22,501 15,484 16:00- 134,056 37,794

5:00- 24,433 16,189 17:00- 174,748 43,074

6:00- 23,868 16,579 18:00- 196,978 45,676

7:00- 62,753 30,025 19:00- 162,816 41,468

8:00- 149,906 47,400 20:00- 149,438 42,592

9:00- 154,597 47,067 21:00- 169,125 47,856

10:00- 131,383 42,445 22:00- 169,956 49,328

11:00- 111,664 37,080 23:00- 165,835 47,297Rikiya Takahashi, Takayuki Osogami, and Tetsuro Morimura Large-Scale Nonparametric Estimation of Vehicle Travel Time Distributions



.. Experimental predictive performances

Nonparametric CDEs outperform for all of the datasets.

-1.8

-1.6

-1.4

-1.2

-1

-0.8

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0

0 3 6 9 12 15 18 21

avg.

test

-set

log-

likel

ihoo

d

hour (index of the dataset)

Euclid-kNNNadaraya-Watson

CDE(Gamma)CDE(LogNormal)CDE(MixGamma)

CDE(MixLogNormal)

Figure: Average test-set log-likelihood for each hourly dataset,based on the 10-fold likelihood cross-validations.




.. Links having complex distributions

ge(y): single exponential-family approximation of fe(y)based on moment matchingCauchy-Schwarz (CS) divergence (Prıncipe, 2010)

CS(f , g |e)=− log

∫yfe(y)ge(y)dy√∫

yf 2e (y)dy

∫yg 2e (y)dy

.

12:00- 18:00- 0:00-

Figure: Links having top-1% highest CS divergence scores.Rikiya Takahashi, Takayuki Osogami, and Tetsuro Morimura Large-Scale Nonparametric Estimation of Vehicle Travel Time Distributions



.. Conclusion and future directions

A novel nonparametric estimator of travel-timedistributions conditioned on the link of a road network.

A) Basis density functions by mixture of gamma orlog-normal distributionsB) Sparse diffusion kernel as link similarityC) Optimizing link importance with KLIEP and SMO

Future directions

Interpolate p.d.f.s also in time domain, as well as thespatial domainIncorporate correlation among linksEstimate each driver’s preference for realistic simulation


large-scale nonparametric estimation of vehicle travel time distributions

Automotive

traveltime sampleswe

actual traveltimetravel

link id

relative traveltime

quantiles of travel

takayuki osogami

fitting experimental

total number of travel