Traffic Speed Data Investigation with

Hierarchical Modeling

Tomonari MASADA

Nagasaki University

masada@nagasaki-u.ac.jp

Real-Time Traffic Speed Data | NYC Open Datahttps://data.cityofnewyork.us/Transportation/Real-Time-Traffic-Speed-Data/xsat-x5sa

Traffic speed measurements at 128 streets

(Regrettably, no longer maintained)

Problem 1

• Traffic speed data show a clear

periodicity at one day period.

• However, many different traffic speed

distribution patterns can be observed

also within each period.

Solution 1 [Masada+ 14]

• We take intuition from topic models

in text mining.

–The data set of each day should be

modeled as a mixture of many

different speed distributions.

Latent Dirichlet Allocation (LDA) [Blei+ 03]

• LDA achieves a word token level clustering.

• Not a document level clustering

• Each document is modeled as a mixture of

many different word probability distributions.

topic <-> word probability distribution

document <-> topic probability distribution

v3

v1

v3

v2

v2

v1 v2 v3 v4

t3φ31

φ32

φ33

φ34

v1 v2 v3 v4

t2φ21

φ22φ23φ24

v1 v2 v3 v4

t1

φ11

φ12φ13

φ14

θj1 θj2

θj3

An important difference

• Words are discrete entities.

– LDA uses multinomial distribution for modeling

per-topic word distribution.

• Speeds (in mph) are continuous entities.

– Our model uses gamma distribution.

gamma distribution

Comparison with LDA

• word token

<-> speed measurement (in mph)

• topic (multinomial)

<-> topic (gamma)

• document

<-> document (24 hrs from midnight)

Full joint distribution

• We estimated parameters by a variational

Bayesian inference. [Masada+ 14]

Problem 2

• Traffic speed data may show a similarity

at the same time point of day.

• Traffic speed data may show a similarity

for the streets whose locations are close

to one another.

Solution 2 [Masada+ FDSE15]

• We use metadata in topic models.

–time points

–geographic locations

TRINH = TRaffic speed INvestigation

with Hierarchical modeling

• Make topic probabilities dependent on

time points and on locations

– probability that the speed measured by the sensor

s at the time point t is assigned to the topic k

𝜃𝑑𝑡𝑘 ≡exp(𝑚𝑑𝑘 + 𝜆𝑘𝑠 + 𝜏𝑘𝑡)

𝑘′ exp(𝑚𝑑𝑘′ + 𝜆𝑘′𝑠 + 𝜏𝑘′𝑡)

Parameters

• 𝑚𝑑𝑘

– How often the document d provides the topic k

• 𝜆𝑘𝑠

– How often the sensor s provides the topic k

• 𝜏𝑘𝑡

– How often the time point t (of day) provides the

topic k

Priors for parameters ("hierarchical")

• 𝑚𝑑𝑘

–K Gaussian priors

• 𝜆𝑘𝑠

–K Gaussian process priors

• 𝜏𝑘𝑡

–K Gaussian process priors

Full joint distribution

Inference by MCMC

• Sample from the posterior distribution

– Slice sampling for topic probability

parameters 𝑚𝑑𝑘, 𝜆𝑘𝑠, and 𝜏𝑘𝑡

–Metropolis-Hastings for hyperparameters

Context dependency

Observations of the same mph

are assigned to different topics.

Context dependency

On May 27, this topic is dominant. On May 28, this

topic is dominant.

Comparison experiment

• Log likelihood per measurement

– Larger is better.

• Data

–May 27 ~ June 16, 2013 (three weeks)

• Data files were downloaded every minute.

–20% measurements for testing

Prior as regularization

Too strong?

What we achieved

• We obtained an MCMC for a topic model

whose topic probabilities are defined by

combining multiple factors.

• And the factors are correlated via Gaussian.

– Our model can also be applied to other types of

metadata indicating intrinsic similarity of data.

Summary

• We proposed a topic model for traffic data analysis.

• Sensor locations and measurement timestamps

affects topic assignment.

• TRINH achieves better likelihood in earlier iterations.

• However, TRINH gives worse likelihood in later

iterations.

Future work

• Control the strength of regularization

– e.g. by weighting the factors.

𝜃𝑑𝑡𝑘 ≡exp(𝑚𝑑𝑘 + 𝜆𝑘𝑠 + 𝜏𝑘𝑡)

𝑘′ exp(𝑚𝑑𝑘′ + 𝜆𝑘′𝑠 + 𝜏𝑘′𝑡)

• Look for other data sets

– Location information should be more relevant.