Neural Collective Graphical Models
for Estimating Spatio-temporal Population Flow from Aggregated Data
Tomoharu Iwata and Hitoshi Shimizu
NTT Communication Science Laboratories
Kyoto, Japan

Abstract
We propose a probabilistic model for estimating population
flow, which is defined as populations of the transition between areas over time, given aggregated spatio-temporal population data. Since there is no information about individual
trajectories in the aggregated data, it is not straightforward
to estimate population flow. With the proposed method, we
utilize a collective graphical model with which we can learn
individual transition models from the aggregated data by analytically marginalizing the individual locations. Learning aspatio-temporal collective graphical model only from the aggregated data is an ill-posed problem since the number of parameters to be estimated exceeds the number of observations.
The proposed method reduces the effective number of parameters by modeling the transition probabilities with a neuralnetwork that takes the locations of the origin and the destination areas and the time of day as inputs. By this modeling, we can automatically learn nonlinear spatio-temporal
relationships flexibly among transitions, locations, and times.
With four real-world population data sets in Japan and China,
we demonstrate that the proposed method can estimate thetransition population more accurately than existing methods.

1

Introduction

Analyzing people flow is critical in a wide variety of applications, such as location-based advertisements (Dhar andVarshney 2011), marketing (Kuo, Chi, and Kao 2002), urban
development (Ashworth and Voogd 1990), transportation
system planning (Koopmans 1949), and evacuation guidance (Yi and Özdamar 2007). Using such sensor devices as
GPS, WiFi, Bluetooth, and infrared beacons, we can obtaintrajectories for each person. However, trajectory data are often aggregated to protect privacy. For example, mobile spatial statistics (Terada, Nagata, and Kobayashi 2013) are thehourly aggregated spatio-temporal population data of 500meter grid squares that are generated based on operational
data from mobile terminal networks so that individuals cannot be tracked.
In this paper, we propose a probabilistic model for estimating people flow given the aggregated data, which we callthe neural collective graphical models. Here the aggregateddata are the populations over time for each area, and the people flow is the transition populations between areas. Fig. 1
Copyright c 2019, Association for the Advancement of Artificial
Intelligence (www.aaai.org). All rights reserved.

shows an example of the input aggregated spatio-temporal
population data and the output people flow. In our task, thenumber of parameters to be estimated, which correspond to
the people flow over time, is (T − 1)LM , where T is thenumber of time points, L is the number of areas, and M
is the number of neighboring areas, where we assume thatpeople move to the neighboring areas. On the other hand,the number of observations, which correspond to the spatiotemporal population data, is T L, which is much smaller than
the number of parameters. Learning the population flow only
from the aggregated data is an ill-posed problem.
To make the problem tractable, the proposed method assumes that the transition probabilities at different locations
and time points are correlated to each other. For example, thetransition probabilities at closely located areas or those at thesame time of day resemble each other. Since the correlationis unknown, we automatically learn it with neural networks
from the given aggregated data. The neural networks enable
us to flexibly extract nonlinear spatio-temporal dependence.
In the following discussion, we use the locations of the center of the origin and destination areas and the time of day
as the input of a neural network. However, it can take otherauxiliary information as input, such as the day of the week
and the area’s weather. We can reduce the effective numberof parameters using a neural network whose parameter size
is less than (T −1)LM , where the neural network parametersize is constant with respect to the number of areas and time
points.
Since no transition population data are given in ourtask, standard supervised learning methods are inapplicable because they require observations of target variables
for training. The proposed model uses the framework ofcollective graphical models (Sheldon and Dietterich 2011;
Sheldon et al. 2013) to estimate the population flow, where
the locations of individual persons are analytically marginalized out, and a probabilistic model is investigated about theirsufficient statistics. Sufficient statistics, which correspondto the transition populations between areas, are modeled bymultinomial distributions, where the transition probability isdefined by the neural network described above. By maximizing the likelihood, we simultaneously estimate the transition
populations as well as the neural network parameters withgradient-based optimization methods.
The remainder of this paper is organized as follows. Sec-(a) Input
SymbolTLzt``0(b) Output

yt`
N`
x`
θt``0
f (·; φ)Figure 1: Example of input aggregated spatio-temporal population data and output population flow: (a) Population foreach area over time and longitude and latitude of each area
are given. (b) Our task estimates transition populations between areas over time.
tion 2 reviews related work. In Section 3, we formulate ourtask and propose a probabilistic model for estimating the
population flow from the aggregated spatio-temporal population data. Section 4 demonstrates the effectiveness of our
proposed model with experiments using real-world aggregated population data. Finally, we present concluding remarks and discuss future work in Section 5.

2

Related Work

Collective graphical models have been successfully used ina wide variety of applications, including modeling bird migration (Sheldon, Elmohamed, and Kozen 2007; Nguyen etal. 2016), information diffusion (Kumar, Sheldon, and Srivastava 2013), and contingency tables (Sheldon and Dietterich 2011). To the best of our knowledge, our work is thefirst attempt to automatically learn dependence among theparameters in collective graphical models with neural networks. Although this paper focuses on population flow, ourframework is applicable to other applications of collective
graphical models, where auxiliary information is available
to effectively estimate parameters.
A number of methods have been proposed for estimating
population flow from aggregated population data based oncollective graphical models (Kumar, Sheldon, and Srivastava 2013; Du, Kumar, and Varakantham 2014; Sun, Sheldon, and Kumar 2015; Iwata et al. 2017; Tanaka et al. 2018;
Akagi et al. 2018). The collective flow diffusion model (Kumar, Sheldon, and Srivastava 2013) assumed static transition probabilities that do not change over time. Iwata etal. (2017) modeled the temporal dependence of transition
probabilities by clustering time points using mixture models. Akagi et al. (2018) modeled spatial dependence using
the distance between the origins and destinations. However,these methods do not consider spatio-temporal dependence.
In addition, they model it in a relatively simple parametric

Table 1: Notation
Description
number of time points
number of areas
population who moves from area ` to area `0
at time t
population in area ` at time t
set of neighboring areas of area `longitude and latitude of area `transition probability that a person at area `0moves to area
P ` at time t,θt``0 ≥ 0, `0 ∈N` θt``0 = 1
neural network with parameter φmodel. On the other hand, our proposed method can flexibly
model nonlinear spatio-temporal dependence using neuralnetworks.
Great interest is being shown in developing methods forpredicting spatio-temporal populations (Zhang, Zheng, andQi 2017; Hoang, Zheng, and Singh 2016; Zhang et al. 2016;2018; Li et al. 2018; Cheng et al. 2017; Xie et al. 2010;
Yu et al. 2016; Yao et al. 2018). For example, Zhang etal. (2018) proposed a deep learning method for forecasting
future populations using spatio-temporal dependence as well
as external conditions, such as weather and events. However,these methods do not estimate the transition populations between areas, which is our task. Xu et al. (2017) proposed amethod to recover individual trajectories given aggregatedpopulation data and transition probabilities. This method requires the calculation of the transition probabilities in advance. On the other hand, the proposed method does not
require transition probabilities, because both the transition
probabilities and the transition populations are simultaneously estimated by taking the aggregated data as inputs.

3

Proposed Method

Task
Suppose that we are given spatio-temporal population dataT
Y = {{yt` }L`=1 }t=1 , where yt` is the population in area` ∈ {1, 2, · · · , L} at time t ∈ {1, 2, · · · , T }. For simplicity,
we assume that each area is a grid cell, but the proposed
method is applicable to any forms of disjoint areas. Eacharea is associated with location information x` ∈ R2 , e.g.,
the latitude and longitude of the area’s center. Our task isto estimate the population flow between areas for each timeT −1point Z = {{zt` }L`=1 }t=1 , zt` = {zt``0 }`0 ∈N` , where zt``0is the number of people who move from area ` to area `0 attime t and N` is the set of neighbors of area `. Table 1 shows
our notation.

ModelFirst, we consider a probabilistic model for individuals. Each
person moves independently based on transition probability
θt``0 , which is the probability that a P
person in area ` moves
to area `0 at time t, θt``0 ≥ 0, and `0 ∈N` θt``0 = 1. Let
stn ∈ {1, · · · , L} be the location of person n at time t. Thenthe probabilistic generative process of individual locations
T
S = {{stn }Nn=1 }t=1 , where N is the total population, is asfollows:
1) For each time point t = 1, · · · , T − 12) For each person n = 1, · · · , N3) st+1,n ∼ Categorical(θtstn ).
Here, Categorical(θ) represents a categorical distribution
with event probability θ, and θt` = {θt``0 }`0 ∈N` . Given individual locations S, area population Y and transition popPN
ulation Z are calculated by yt` =n=1 I(stn = `), andPN
zt``0 = n=1 I(stn = ` ∧ st+1,n = `0 ), respectively, whereI(·) is the indicator function, i.e., I(A) = 1 if A is true andI(A) = 0 otherwise. Therefore, we can estimate individual
locations S by maximizing the likelihood of observed area
population Y and then estimate transition population Z using estimated individual locations Ŝ. However, estimating
individual locations S is too expensive to compute when thenumber of people is large.
To overcome this problem, we utilized the framework of
the collective graphical models, where individual locations
S are analytically marginalized out and a probabilistic modelof their sufficient statistics, i.e., transition population Z, isobtained. Due to this marginalization, we do not need to explicitly estimate individual behaviors. In particular, the following is the generative process of transition population Zof given area population Y:
1) For each time point t = 1, · · · , T − 12) For each area ` = 1, · · · , L3) zt` ∼ Multinomial(θt` , yt` ).
Here multinomial(θ, y) is the multinomial distribution with
event probability θ and number of trials y. The multinomial
distribution of transition population zt` is as follows:
Y z 0
yt` !
t``
θt``
0 .
0
`0 ∈N` zt`` ! 0p(zt` |θt` , yt` ) = Q

(1)

` ∈N`

The following two relationships exist between area population Y and transition population Z:
X
X
yt` =
zt``0 , yt+1,` =zt`0 ` ,(2)`0 ∈N``0 ∈N`

where the first equation indicates that the sum of the transition populations from an area equals the population in thearea, and the second equation indicates that the sum of thetransition populations to an area equals the population in thearea at the next time point.
The number of parameters in transition probabilities Θ =T{{θt` }L`=1 }t=1 is O(T LM ), where M is the number ofneighbors. On the other hand, since the number of observations Y is O(T L), the transition probabilities cannot be determined when only these constraints are used. The proposed
method imposes more constraints on the transition populations by incorporating spatio-temporal correlation amongtransition probabilities Θ. We model the transition probabilities with neural network f (·), which takes time t, location

Figure 2: Example of proposed model with its four areas: A,
B, C and D, and location information xA , i.e., latitude andlongitude, is associated to each area. The transition probability between areas θtAB at time t is calculated by nonlinear
function f , which is modeled by a neural network with parameter φ, taking time t, origin location xA , and destination
location xB as input.
of origin area x` , and location of destination area x`0 as the
input, as follows:θt``0 = f (t, x` , x`0 ; φ),(3)where φ is the parameters of the neural network. By thismodeling, related transition probabilities, e.g., those withclose time points, near locations and/or similar directions,are constrained to have similar values to each other. The nonlinear correlation is automatically extracted from the given
data by learning the neural network parameters. Fig. 2 shows
an example of the proposed model.
We use the following input vector:
ut``0 = [τ̃ (t), x̃` , x̃`0 − x̃` ],(4)where τ (t) represents the time of day of time t, x̃ indicatesthe normalized value of x to a range of −0.5 to 0.5, [a, b]
indicates the concatenation of a and b, and ut``0 ∈ R5 . Herethe transition probability depends on time of day τ (t), origin location x` , and the direction from the origin to destination x`0 −x` . Note that our framework can straightforwardly
incorporate other auxiliary information for modeling transition probabilities, such as the day of the week and the area’s
weather, by concatenating the information to input vectorut``0 .
We transform the input vector into a transition probability by the following three-layered, feed-forward neural network:ht``0 = tanh(W1 ut``0 + b1 ),(5)
θt``0 = softmax(w2 ht``0 + b2 ),(6)
where ht``0 ∈ RH is the hidden unit of the neural network,
W1 ∈ RH×5 , w2 ∈ RH , b1 ∈ RH , b2 ∈ R are the weights
and bias parameters to be estimated, H is the number of hidden units, and φ = {W1 , w2 , b1 , b2 }. The proposed model
can handle different numbers of neighbors across different
locations since the number of neural network parameters
does not depend on the number of neighbors. The softmax
function in Eq. (6) outputs the transition probability normalized over the neighbors.

Estimation
We estimate transition population Z and neural network parameters φ based on the maximum likelihood with regularizers for the constraints. Using Eqs. (1) and (3), the log likelihood is given by
L(Z, φ) =

T
−1 X
L
X

X

log p(zt` |θt` , yt` )(7)

t=1 `=1 `0 ∈N`

∝

T
−1 X
L
X

X

− log zt``0 ! + zt``0 log f (t, x` , x`0 ; φ)t=1 `=1 `0 ∈N`

≈

T
−1 X
L
X

X



zt``0 1 − log zt``0 + log f (t, x` , x`0 ; φ)t=1 `=1 `0 ∈N`
0≡ L (Z, φ),(8)where Stirling’s approximation, log z! ≈ z log z − z, is usedin the third line. We incorporate the constraints in Eq. (2) as
regularizers and give the objective function to be maximized
as follows:
T
−1 X
L
X
X
G(Z, φ) = L0 (Z, φ) − λk yt` −
zt``0 k2
`0 ∈N`

t=1 `=1

−λ

L
T
−1 X
X

k yt+1,` −X

zt`0 ` k2 ,(9)`0 ∈N`

t=1 `=1

where the second and third terms correspond to the soft constraints in Eq. (2) and λ > 0 is the hyperparameter. By using the regularizers, we can estimate neural network parameters φ as well as transition population Z with a gradientbased optimization method, since the objective function isdifferentiable with respective to the parameters. Here we
relax transition population zt``0 to take non-negative realvalues instead of non-negative integers and parameterize it
0
by zt``
0 = log zt``0 through which the non-negative constraint becomes unnecessary. Algorithm 1 shows the estimation procedure of the proposed model.
Hyperparameter λ is tuned using the following prediction
error of the area population at the next time step:
E=

L
T X
X

k ŷt` − yt` k2 .

(10)

t=2 `=1

The predictive value of area population ŷt` is calculated using the transition probability and the population at the previous time step as follows:Xŷt` =
θ̂t−1,`0 ` yt−1,`0 ,(11)`0 ∈N`

where θ̂t``0 = f (t, x` , x`0 ; φ̂) is the transition probability
obtained by estimated neural network parameters φ̂.

4

Experiments

Data
We evaluated the proposed method using four real-world
population data sets: Tokyo, Osaka, Nagoya, and Beijing (Iwata et al. 2017). Fig. 3 shows examples of them.

Algorithm 1 Estimation procedure for the proposed neuralcollective graphical model
Require: Spatio-temporal population data Y, location inL
formation {x` }L`=1 , neighbor information {N` }`=1 , hyperparameter λ;Ensure: Population flow Z, estimated neural network parameters φ̂;Initialize neural network parameters φ;repeatCalculate transition probability θt` by neural networkf (t, x` , x`0 ; φ) in (3) for t = 1 to T − 1, ` = 1 to L;
Calculate the objective function G(Z, φ) in (9) and its
gradient with respect to the population flow Z and neural network parameters φ;Update the population flow Z and neural network parameters φ using the gradient;until End condition is satisfied

The Tokyo, Osaka, and Nagoya data are spatio-temporal
aggregated population data in the areas of these three cities
in Japan. They were generated from individual trajectorydata and interpolated from geotagged tweets using railway
and road information (Sekimoto et al. 2011)1 . The unit timewas 30 minutes, the grid size was 10 km×10 km, and thenumber of grids was 16×14. The Tokyo data contained the
population data on July 1 and 7, October 7 and 13, and December 16 and 22 of 2013, where the total population ofeach day was 6,432, 9,166, 6,822, 10,134, 6,646, 10,338,
respectively. The Osaka data contained the population dataon August 8 and 11, September 16 and 22, and December
24 and 29 of 2013, where the total population of each day
was 2,256, 3,034, 2,999, 3,569, 2,487, 3,480, respectively.
The Nagoya data contained the population data on July 22
and 28, September 16 and 22, and December 24 and 29 of2013, where the total population of each day was 929, 1,332,
1,148, 1,460, 975, 1,570, respectively.
The Beijing data are spatio-temporal aggregated population data in Beijing in China that were generated from TDrive trajectory data (Yuan et al. 2010; 2011), which contained the trajectories of 10,357 taxis from February 3 to
7, 2008. The unit time was 15 minutes, the grid size was 2
km×2 km, and the number of grids was 20×16.
With all of the data sets, the neighbors were the surrounding eight cells with the addition of the cell itself.

Measurement
For the evaluation measurement, we used the following normalized absolute error:
PL PT −1 P∗0`0 ∈N` |zt``0 − ẑt`` |,(12)NAE = `=1 t=1
PL PT −1`=1
t=1 yt`1

We used the following sources: SNS-based People Flow Data,
Nightley, Inc., Shibasaki & Sekimoto Laboratory, the University ofTokyo, Micro Geo Data Forum, People Flow project, and Center
for Spatial Information Science at the University of Tokyo, http://nightley.jp/archives/1954

(a) Tokyo

0:00

9:00

(b) Osaka

18:00

0:00

(c) Nagoya

0:00

9:00

9:00

18:00

(d) Beijing

18:00

0:00

9:00

18:00

Figure 3: Aggregated spatio-temporal population data: (a) Tokyo, (b) Osaka, (c) Nagoya, and (d) Beijing data sets. Darker redcolors represent higher populations in each area. A green line represents a main road.
which is the sum of the absolute errors between true tran∗
sition population zt``
0 and its estimation ẑt``0 for each time
step and for each neighbor area pair divided by the sum of
the total number of populations yt` . Note that we did not use∗
true transition populations zt``
0 for training because they areused only for evaluation.

Comparing methods
We compared the proposed method with the following four
methods: VCGMM, ICGM, CGM, and STAY. The STAY
method, which assumes that all people remain at the current
area without moving to other areas, estimates the transition
populations by zt``0 = yt` if ` = `0 , zt``0 = 0 otherwise.
The CGM method is the collective graphical model, whichassumes that the transition probability does not change overtime θt``0 = θ``0 . The ICGM method is the inhomogeneoustransition probability collective graphical model, which assumes that transition probabilities θt``0 are different acrossdifferent time points. VCGMM is the variational collective
graphical mixture models (Iwata et al. 2017), which cluster
the time of day using mixture models. With the proposed
method, we used ten hidden units and maximized the objective function by ADAM (Kingma and Ba 2014).
The number of parameters to be estimated with the proposed method is O(H), where H is the number of hidden units. Those with VCGMM, ICGM and CGM are
O(KLM ), O(T LM ) and O(LM ), respectively, where K
is the number of clusters, L is the number of locations, M is
the number of neighbors, and T is the number of time points.

Results
Table 2 shows the normalized absolute error averaged over
all the time points and all the area pairs. The proposed neural

Table 2: Normalized absolute errors on population flow estimation averaged over all time points
Proposed VCGMM ICGM CGM STAY
Tokyo
0.148
0.167 0.176 0.208 0.192
Osaka
0.186
0.250 0.265 0.280 0.272
Nagoya0.227
0.250 0.281 0.291 0.269
Beijing
0.408
0.470 0.500 0.479 0.532

CGM achieved the lowest error among all of the data sets.
Fig. 4 shows the normalized absolute error for each time ofday. The error with the proposed method was lower than theother methods, especially in the daytime, when the transition
populations are large. This result indicates that the proposed
method effectively learns the transition probabilities using
spatio-temporal dependencies with neural networks. The error with the STAY method was high in the daytime since itcannot model transitions to different areas. The error with
the CGM method was high at night since it assumes thatthe transition probability in the daytime and at night are thesame. VCGMM achieved lower error than ICGM because
it reduces the effective number of parameters by clustering
the time points. However, since VCGMM cannot utilize location information nor flexibly model the temporal dependence, its performance was worse than the proposed method,which can model the nonlinear spatio-temporal dependence
using both location and time information. With the proposed
method, the error from 4:00 a.m. to 7:00 a.m. was relatively
high, because the transition probability drastically changes
in this period, and its switching was not learned properly.
Interesting future work will extend our framework to modelsuch switching transition probabilities using mixture of ex-

(a) Tokyo(b) Osaka(c) Nagoya(d) Beijing

Figure 4: Normalized absolute errors on people flow estimation over time of day.
perts (Jacobs et al. 1991).
Figure 5 shows the transition populations estimated by
the proposed method. Reasonable transitions were estimatedwith all of the data sets. There were few transitions around2:00 a.m. Around 8:00 a.m., people commuted to the city
centers from the suburbs. Around 2:00 p.m., many people
moved around, especially in the city centers. Around 8:00
p.m., people left the city centers and returned to their homes
in the suburbs.

5

Conclusion

We proposed neural collective graphical models for estimating spatio-temporal population flow given aggregated population data. With our proposed model, nonlinear spatiotemporal dependence on transition probabilities are automatically learned from the given data using neural networks. We experimentally confirmed that the proposedmodel achieved higher performance on transition population estimation than the existing methods. Although our results are encouraging, we must extend our approach in a
number of directions. First, we want to apply our frame-

work to other collective graphical model applications, suchas modeling contingency tables and information diffusion.
Our framework, which models multinomial distribution parameters with neural networks, is straightforwardly applicable to them. Second, we plan to evaluate our proposedmodel with higher order Markov transition models that contain more parameters to be estimated. Our model would
more effectively reduce the number of parameters with models that have many parameters. Third, we want to investigate
using different types of neural networks, such as convolutional neural networks for incorporating images, and recurrent neural networks for forecasting future transitions.

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