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Data-driven generation of spatio-temporal routines in human mobility

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Human mobility modelling is of fundamental importance in a wide range of applications, such as the developing of protocols for mobile ad hoc networks or for what-if analysis and simulation in urban ecosystems. Current generative models generally fail in accurately reproducing the individuals' recurrent daily schedules and at the same time in accounting for the possibility that individuals may break the routine and modify their habits during periods of unpredictability of variable duration. In this article we present DITRAS (DIary-based TRAjectory Simulator), a framework to simulate the spatio-temporal patterns of human mobility in a realistic way. DITRAS operates in two steps: the generation of a mobility diary and the translation of the mobility diary into a mobility trajectory. The mobility diary is constructed by a Markov model which captures the tendency of individuals to follow or break their routine. The mobility trajectory is produced by a model based on the concept of preferential exploration and preferential return. We compare DITRAS with real mobility data and synthetic data produced by other spatio-temporal mobility models and show that it reproduces the statistical properties of real trajectories in an accurate way.
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Data Min Knowl Disc (2018) 32:787–829
https://doi.org/10.1007/s10618-017-0548-4
Data-driven generation of spatio-temporal routines in
human mobility
Luca Pappalardo1,2·Filippo Simini3
Received: 16 July 2016 / Accepted: 5 December 2017 / Published online: 27 December 2017
© The Author(s) 2017. This article is an open access publication
Abstract The generation of realistic spatio-temporal trajectories of human mobility
is of fundamental importance in a wide range of applications, such as the develop-
ing of protocols for mobile ad-hoc networks or what-if analysis in urban ecosystems.
Current generative algorithms fail in accurately reproducing the individuals’ recurrent
schedules and at the same time in accounting for the possibility that individuals may
break the routine during periods of variable duration. In this article we present Ditras
(DIary-based TRAjectory Simulator), a framework to simulate the spatio-temporal
patterns of human mobility. Ditras operates in two steps: the generation of a mobility
diary and the translation of the mobility diary into a mobility trajectory. We propose a
data-driven algorithm which constructs a diary generator from real data, capturing the
tendency of individuals to follow or break their routine. We also propose a trajectory
generator based on the concept of preferential exploration and preferential return. We
instantiate Ditras with the proposed diary and trajectory generators and compare
the resulting algorithm with real data and synthetic data produced by other genera-
tive algorithms, built by instantiating Ditras with several combinations of diary and
trajectory generators. We show that the proposed algorithm reproduces the statistical
Responsible editor: Johannes Fürnkranz.
BLuca Pappalardo
lpappalardo@di.unipi.it; luca.pappalardo@isti.cnr.it
Filippo Simini
f.simini@bristol.ac.uk
1Institute of Information Sciences and Technologies, National Research Council, Pisa, Italy
2Department of Computer Science, University of Pisa, Pisa, Italy
3Department of Engineering Mathematics, University of Bristol, Bristol, UK
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788 L. Pappalardo, F. Simini
properties of real trajectories in the most accurate way, making a step forward the
understanding of the origin of the spatio-temporal patterns of human mobility.
Keywords Data science ·Human mobility ·Complex systems ·Mathematical
modelling ·Big data ·Spatiotemporal data ·Human dynamics ·Urban dynamics ·
Mobile phone data ·GPS data ·Smart cities
1 Introduction
Understanding the complex mechanisms governing human mobility is of fundamental
importance in different contexts, from public health (Colizza et al. 2007; Lenormand
et al. 2015) to official statistics (Marchetti et al. 2015; Pappalardo et al. 2016b), urban
planning (Wang et al. 2012; De Nadai et al. 2016) and transportation engineering
(Janssens 2013). In particular, human mobility modelling has attracted a lot of interest
in recent years for two main reasons. On one side, it is crucial in the performance
analysis of networking protocols such as mobile ad hoc networks, where the displace-
ments of network users are exploited to route and deliver the messages (Karamshuk
et al. 2011;Hessetal.2015). On the other side human mobility modelling is crucial
for urban simulation and what-if analysis (Meloni et al. 2011; Kopp et al. 2014), e.g.,
simulating changes in urban mobility after the construction of a new infrastructure
or when traumatic events occur like epidemic diffusion, terrorist attacks or interna-
tional events. In both scenarios the developing of generative algorithms that reproduce
human mobility patterns in an accurate way is fundamental to design more efficient and
suitable protocols, as well as to design smarter and more sustainable infrastructures,
economies, services and cities (Batty et al. 2012; Kitchin 2013).
Clearly, the first step in human mobility modelling is to understand how people
move. The availability of big mobility data, such as massive traces from GPS devices
(Pappalardo et al. 2013b), mobile phone networks (González et al. 2008) and social
media records (Spinsanti et al. 2013), offers nowadays the possibility to observe human
movements at large scales and in great detail (Barbosa-Filho et al. 2017). Many studies
relied on this opportunity to provide a series of novel insights on the quantitative
spatio-temporal patterns characterizing human mobility. These studies observe that
human mobility is characterized by a stunning heterogeneity of travel patterns, i.e.,
a heavy tail distribution in trip distances (Brockmann et al. 2006; González et al.
2008) and the characteristic distance traveled by individuals, the so-called radius of
gyration (González et al. 2008; Pappalardo et al. 2015b). Moreover human mobility is
characterized by a high degree of predictability (Eagle and Pentland 2009; Song et al.
2010b), a strong tendency to spend most of the time in a few locations (Song et al.
2010a), and a propensity to visit specific locations at specific times (Jiang et al. 2012;
Rinzivillo et al. 2014).
Building upon the above findings, many generative algorithms of human mobility
have been proposed which try to reproduce the characteristic properties of human
mobility trajectories (Karamshuk et al. 2011; Barbosa-Filho et al. 2017). The goal
of generative algorithms of human mobility is to create a population of agents whose
mobility patterns are statistically indistinguishable from those of real individuals. Typ-
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Data-driven generation of spatio-temporal routines 789
ically each generative algorithm focuses on just a few properties of human mobility.
A class of algorithms aims to realistically represent spatial properties: they are mainly
concerned with reproducing the trip distance distribution (Brockmann et al. 2006;
González et al. 2008) or the visitation frequency to a set of preferred locations (Song
et al. 2010a; Pappalardo et al. 2015b). Another class of algorithms focus on the accu-
rate representation of the time-varying behavior of individuals, relying on detailed
schedules of human activities (Jiang et al. 2012; Rinzivillo et al. 2014). However, the
major challenge for generative algorithms lies in the creation of realistic temporal
patterns, in which various temporal statistics observed empirically are simultaneously
reproduced, including the number and sequence of visited locations together with the
time and duration of the visits. In particular, the biggest hurdle consists in the simulta-
neous description of an individual’s routine and sporadic mobility patterns. Currently
there is no algorithm able to reproduce the individuals’ recurrent or quasi-periodic
daily schedules, and at the same time to allow for the possibility that individuals may
break the routine and modify their habits during periods of unpredictability of variable
duration.
In this work we present Ditras (DIary-based TRAjectory Simulator), a framework
to simulate the spatio-temporal patterns of human mobility. The key idea of Ditras is
to separate the temporal characteristics of human mobility from its spatial character-
istics. In order to do that, Ditras operates in two steps. First, it generates a mobility
diary using a diary generator. A mobility diary captures the temporal patterns of human
mobility by specifying the arrival time and the time spent in each location visited by
the individual. A diary generator is an algorithm which generates a mobility diary for
an individual given a diary length. In this paper we propose a data-driven algorithm
called Mobility Diary Learner (MDL) which is able to infer from real mobility data a
diary generator, MD, represented as a Markov model. The Markov model captures the
propensity of individuals to follow quasi-periodic daily schedules as well as to break
the routine and modify their mobility habits.
Second, Ditras transforms the mobility diary into a mobility trajectory by using
proper mechanisms for the exploration of locations on the mobility space, so capturing
the spatial patterns of human movements. The trajectory generator we propose, d-EPR,
is based on previous research by the authors (Pappalardo et al. 2015b,2016a) and
embeds mechanisms to explore new locations and return to already visited locations.
The exploration phase takes into account both the distance between locations and
their relevance on the mobility space, though taking into account the underlying urban
structure and the distribution of population density.
We instantiate Ditras with the proposed diary and trajectory generators and com-
pare it with nation-wide mobile phone data, region-wide GPS vehicular data and
synthetic trajectories produced by other generative algorithms on a set of nine dif-
ferent standard mobility measures. We show that d-EPR MD, a generative algorithm
created by combining diary generator MD with trajectory generator d-EPR, simu-
lates the spatio-temporal properties of human mobility in a realistic manner, typically
reproducing the mobility patterns of real individuals better than the other considered
algorithms. Moreover, we show that the distribution of standard mobility measures
can be accurately reproduced only by modelling both the spatial and the temporal
aspects of human mobility. In other words, the spatial mechanisms and the temporal
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790 L. Pappalardo, F. Simini
mechanisms have to be modeled together by proper diary and trajectory generators in
order to reproduce the observed human mobility patterns in an accurate way. The gen-
erative algorithm we propose, d-EPR MD, captures both the spatial and the temporal
dimensions of human mobility and is a useful tool to develop more reliable protocols
for ad hoc networks as well as to perform realistic simulation and what-if scenarios
in urban contexts. In summary this paper provides the following novel contributions:
the modeling framework Ditras which allows for the combinations of different
spatial and temporal mechanisms of human mobility and whose code is freely
available (https://github.com/jonpappalord/DITRAS);
the data-driven algorithm MDL to construct from real mobility data a diary gen-
erator (MD) which is realistic in reproducing the temporal patterns of human
mobility;
a comparison of existing algorithms as well as algorithms resulting from novel
combinations of temporal and spatial mechanisms, on a set of nine mobility mea-
sures and two large-scale mobility datasets.
Our modeling framework goes towards a comprehensive approach which combines a
network science perspective and a data mining perspective to improve the accuracy
and the realism of human mobility models.
This paper is organized as follows. Section 2revises the relevant literature on human
mobility modelling. In Sect. 3we present the structure of the Ditras framework.
Section 4describes the first step of Ditras, the generation of the mobility diary,
and in Sect. 4.1 we describe the mobility diary learner MDL and the Markov model.
Section 5describes the second step of Ditras, the generation of the mobility trajectory,
and in Sect. 5.1 we propose a trajectory generator called d-EPR. Section 6shows the
comparison between an instantiation of Ditras with the proposed diary and trajectory
generators with real trajectory data and the trajectories produced by other generative
algorithms. In Sect. 6.4 we discuss the obtained results and, finally, Sect. 7concludes
the paper.
2 Related work
All the main studies in human mobility document a stunning heterogeneity of human
travel patterns that coexists with a high degree of predictability: individuals exhibit a
broad spectrum of mobility ranges while repeating daily schedules dictated by routine
(Giannotti et al. 2013). Brockmann et al. study the scaling laws of human mobility by
observing the circulation of bank notes in United States, finding that travel distances
of bank notes follow a power-law behavior (Brockmann et al. 2006). González et al.
analyze a nation-wide mobile phone dataset and find a large heterogeneity in human
mobility ranges (González et al. 2008): (i) travel distances of individuals follow a
power-law behavior, confirming the results by Brockmann et al.; (ii) the radius of
gyration of individuals, i.e., their characteristic traveled distance, follows a power-
law behavior with an exponential cutoff. Song et al. observe on mobile phone data
that individuals are characterized by a power-law behavior in waiting times, i.e., the
time between a displacement and the next displacement by an individual (Song et al.
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Data-driven generation of spatio-temporal routines 791
2010a). Pappalardo et al. find the same mobility patterns on a dataset storing the
GPS traces of 150,000 private vehicles traveling during one month in Tuscany, Italy
(Pappalardo et al. 2013b). Song et al. study the entropy of individuals’ movements and
find a high predictability in human mobility, with a distribution of users’ predictability
peaked at approximately 93% and having a lower cutoff at 80% (Song et al. 2010b).
Pappalardo et al. analyze mobile phone data and GPS tracks from private vehicles and
discover that individuals split into two profiles, returners and explorers, with distinct
mobility and geographical patterns (Pappalardo et al. 2015b). Several studies focus
on the prediction of the kind of activity associated to individuals’ trips on the only
basis of the observed displacements (Liao et al. 2007; Jiang et al. 2012; Rinzivillo
et al. 2014), and to discover geographic borders according to recurrent trips of private
vehicles (Rinzivillo et al. 2012; Thiemann et al. 2010), or to predict the formation of
social ties (Cho et al. 2011; Wang et al. 2011). Other works demonstrate the connection
between human mobility and social networks, highlighting that friendships and other
types of social relations are significant drivers of human movements (Brown et al.
2013b;Hristovaetal.2016; Wang et al. 2011; Volkovich et al. 2012; Brown et al.
2013a; Hossmann et al. 2011a,b).
How to combine the discovered patterns to create a generative algorithm that repro-
duces the salient aspects of human mobility is an open task. This task is particularly
challenging because generative algorithms should be as simple, scalable and flexible
as possible, since they are generally purposed to large-scale simulation and what-if
analysis. In the literature many generative algorithms have been proposed so far to
model individual human mobility patterns (Karamshuk et al. 2011; Barbosa-Filho
et al. 2017).
Some algorithms try to reproduce the heterogeneity of individual human mobility
and simulate how individuals visits locations. ORBIT (Ghosh et al. 2005) is an exam-
ple of such algorithms. It splits into two phases: (i) at the beginning of the simulation it
generates a predefined set of locations on a bi-dimensional space; (ii) then every syn-
thetic individual selects a subset of these locations and moves between them according
to a Markov chain. In the Markov chain every state represents a specific location in the
scenario and proper probability of transitions guarantee a realistic distribution of loca-
tion frequencies. SLAW (Self-similar Least-Action Walk) produces mobility traces
having specific statistical features observed on human mobility data, namely power-
law waiting times and travel distances with a heavy-tail distribution (Lee et al. 2012,
2009). In a first step SLAW generates a set of locations on a bi-dimensional space so
that the distance among them features a heavy-tailed distribution. Then, a synthetic
individual starts a trip by randomly choosing a location as starting point and making
movement decisions based on the LATP (Least-Action Trip Planning) algorithm. In
LATP every location has a probability to be chosen as next location that decreases with
the power-law of the distance to the synthetic individual’s current location. SLAW is
used in several studies of networking and human mobility modelling and is the base
for other generative algorithms for human mobility, such as SMOOTH (Munjal et al.
2011), MSLAW (Schwamborn and Aschenbruck 2013) and TP (Solmaz et al. 2015,
2012).
Small World In Motion (SWIM) is based on the concept of location preference
(Kosta et al. 2010). First, each synthetic individual is assigned to a home location,
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792 L. Pappalardo, F. Simini
which is chosen uniformly at random on a bi-dimensional space. Then the synthetic
individual selects a destination for the next move depending of the weight of each
location, which grows with the popularity of the location and decreases with the dis-
tance from the home location. The popularity of a location depends on a collective
preference calculated as the number of other people encountered the last time the
synthetic individual visited the location. Another category of generative algorithms
combine notions about the sociality of individuals with mobility patterns to define
socio-mobility models, demonstrating how they can be exploited to design more real-
istic protocols for ad hoc and opportunistic networks (Borrel et al. 2009; Yang et al.
2010; Fischer et al. 2010; Boldrini and Passarella 2010; Musolesi and Mascolo 2007).
In contrast with many generative algorithms of human mobility, the Exploration
and Preferential Return (EPR) model does not fix in advance the number of visited
locations on a bi-dimensional space but let them emerge spontaneously (Song et al.
2010a). The model exploits two basic mechanisms that together describe human mobil-
ity: exploration and preferential return. Exploration is a random walk process with a
truncated power-law jump size distribution (Song et al. 2010a). Preferential return
reproduces the propensity of humans to return to the locations they visited frequently
before (González et al. 2008). A synthetic individual in the model selects between
these two mechanisms: with a given probability the synthetic individual returns to
one of the previously visited places, with the preference for a location proportional
to the frequency of the individual’s previous visits. With complementary probability
the synthetic individual moves to a new location, whose distance from the current one
is chosen from the truncated power-law distribution of travel distances as measured
on empirical data (González et al. 2008). The probability to explore decreases as the
number of visited locations increases and, as a result, the model has a warmup period
of greedy exploration, while in the long run individuals mainly move around a set
of previously visited places. Recently the EPR model has been improved in different
directions, such as by adding information about the recency of location visits during
the preferential return step (Barbosa et al. 2015), or adding a preferential exploration
step to account for the collective preference for locations and the returners and explor-
ers dichotomy, as the authors of this paper have done in previous research by defining
the d-EPR model (Pappalardo et al. 2015b,2016a). It is worth noting that although
the algorithms described above are able to reproduce accurately the heterogeneity of
mobility patterns, none of them can reproduce realistic temporal patterns of human
movements.
Recent research on human mobility show that individuals are characterized by a
high regularity and the tendency to come back to the same few locations over and over
at specific times (González et al. 2008; Pappalardo et al. 2013b). Temporal models
focus on these temporal patterns and try to reproduce accurately human daily activ-
ities, schedules and regularities. Zheng et al. (Zheng et al. 2010) use data from a
national survey in the US to extract realistic distribution of address type, activity type,
visiting time and population heterogeneity in terms of occupation. They first describe
streets and avenues on a bi-dimensional space as horizontal and vertical lines with
random length, and then use the Dijkstra’s algorithm to find the shortest path between
two activities taking into account different speed limits assigned to each street. WDM
(Working Day Movement) distinguishes between inter-building and intra-building
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Data-driven generation of spatio-temporal routines 793
movements (Ekman et al. 2008). It consists of several submodels to describe mobility
in home, office, evening and different transportation means. For example a home model
reproduces a sojourn in a particular point of a home location while an office model
reproduces a star-like trajectory pattern around the desk of an individual at specific
coordinates inside an office building. Although Zheng et al.’s algorithm and WDM
provide an extremely thorough representations of human movements in particular sce-
narios, they suffer two main drawbacks: (i) they represent specific scenarios and their
applicability to other scenarios is not guaranteed; (ii) they are too complex for ana-
lytical tractability; (iii) they generally fail in capturing some global mobility patterns
observed in individual human mobility, e.g., the distribution of radius of gyration. A
recent study (McInerney et al. 2013) proposes methods to identify and predict depar-
tures from routine in individual mobility using information-theoretic metrics, such
as the instantaneous entropy, and developing a Bayesian framework that explicitly
models the tendency of individuals to break from routine.
Position of our work. From the literature it clearly emerges that existing generative
algorithms for human mobility are not able to accurately capture at the same time the
heterogeneity of human travel patterns and the temporal regularity of human move-
ments. On the one hand exploration models accurately reproduce the heterogeneity
of human mobility but do not account for regularities in human temporal patterns.
On the other hand temporal models accurately reproduce human mobility schedules
paying the price in complexity, but fail in capturing some important global mobility
patterns observed in human mobility. In this paper we try to fill this gap and propose
d-EPR MD, a scalable generative algorithm that creates synthetic individual trajecto-
ries able to capture both the heterogeneity of human mobility and the regularity of
human movements. Despite its great flexibility, d-EPR MD is to a large extent analyti-
cally tractable and several statistics about the visits to routine and non-routine locations
can be derived mathematically. In fact, since the temporal mechanism of d-EPR MD is
based on a Markov chain, using standard results in probability theory one can compute
various quantities, including the probability to go between any two states in a given
number of steps, the average number of visits to a state before visiting another state,
the average time to go from one state to another and the probability to visit one state
before another. Moreover the spatial mechanism of d-EPR MD is based on the EPR
model for which various analytical results, such as the distributions of the radii of
gyration and of the location frequencies, have been derived (Song et al. 2010a). The
data-driven algorithm MDL (Mobility Diary Learner), is another novel contribution
of this paper. MDL infers from real mobility data a diary generator for realistic mobil-
ity diaries. It is highly adaptive and can be applied to different geographic areas and
different types of mobility data.
The modelling framework we propose, Ditras, can generate synthetic mobility
trajectories and can be easily integrated in transportation forecast models to infer trip
demand. Our approach has some similarity with activity-based models (Bellemans
et al. 2010), as they both aim to estimate trip demand by reproducing realistic indi-
vidual temporal patterns, however there are important differences between the two
approaches. In fact, while the goal of activity-based models is to produce detailed
agendas filled with activities performed by the agents and are calibrated on surveys
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794 L. Pappalardo, F. Simini
with a limited number of participants, our framework produces mobility diaries con-
taining the time and duration of the visits in the various locations without explicitly
specifying the type of activity performed there, and is calibrated on a large population
of mobile phone users.
A recent paper introduces TimeGeo, a modelling framework to generate a popula-
tion of synthetic agents with realistic spatio-temporal trajectories (Yang et al. 2016).
Similarly to the modelling framework presented here, TimeGeo combines a Markov
model to generate temporal patterns with the correct periodicity and duration of visits,
with a model to reproduce spatial patterns with the characteristic number of visits and
distribution of distances. Albeit having similar aims, there are important differences
between our modelling approach and TimeGeo’s. In fact, while TimeGeo proposes a
parsimonious model which is based on few tunable parameters and is to some extent
analytically tractable, the approach proposed in this paper is markedly data driven
and parameter-free, with a greater level of complexity which ensures the necessary
flexibility to reproduce realistic temporal patterns.
3 The DITRAS modelling framework
Ditras is a modelling framework to simulate the spatio-temporal patterns of human
mobility in a realistic way.1The key idea of Ditras is to separate the temporal char-
acteristics of human mobility from its spatial characteristics. For this reason, Ditras
consists of two main phases (Fig. 1): first, it generates a mobility diary which captures
the temporal patterns of human mobility; second it transforms the mobility diary into a
sampled mobility trajectory which captures the spatial patterns of human movements.
In this section we define the main concepts which constitute the mechanism of Ditras.
The output of a Ditras simulation is a sampled mobility trajectory for a synthetic
individual. A mobility trajectory describes the movement of an object as a sequence
of time-stamped locations. The location is described by two coordinates, usually a
latitude-longitude pair or ordinary Cartesian coordinates, as formally stated by the
following definition:
Definition 1 (Mobility trajectory) A mobility trajectory is a sequence of triples T=
(x1,y1,t1),...,(xn,yn,tn), where ti(i=1,...,n)is a timestamp, 1i<nti<
ti+1and xi,yiare coordinates on a bi-dimensional space.
For modelling purposes it is convenient to define a sampled mobility trajectory, S(t),
which can be obtained by sampling the mobility trajectory at regular time intervals of
length tseconds:
Definition 2 (Sampled mobility trajectory) A sampled mobility trajectory is a
sequence S(t)=l1,...,lN, where li(i=1,...,N)is the geographic location
where the individual spent the majority of time during time slot i, i.e., between (i1)t
and ti seconds from the first observation. Nis the total number of time slots considered.
A location liis described by coordinates on a bi-dimensional space.
1The Python code of Ditras is freely available for download on a public GitHub repository: https://github.
com/jonpappalord/DITRAS
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Data-driven generation of spatio-temporal routines 795
Fig. 1 Outline of the DITRAS framework. Ditras combines two probabilistic models: a diary generator
(e.g., MD(t)) and trajectory generator (e.g., d-EPR). The diary generator uses a typical diary W(t)to produce
a mobility diary D. The mobility diary Dis the input of the trajectory generator together with a weighted
spatial tessellation of the territory L.FromDand Lthe trajectory generator produces a sampled mobility
trajectory S
To generate a sampled mobility trajectory Ditras exploits two probabilistic models:
a diary generator and a trajectory generator (see Fig. 1). In this paper we propose as
diary generator MD(t), a Markov model responsible for reproducing realistic temporal
mobility patterns, such as the distribution of the number of trips per day and the
tendency of individuals to change location at specific hours of the day (González
et al. 2008; Jiang et al. 2012). Essentially, MD(t)captures the tendency of individuals
to follow or break a temporal routine at specific times. As trajectory generator we
propose the d-EPR generative model (Pappalardo et al. 2015b,2016a), which is able
to reproduce realistic spatial mobility patterns, such as the heavy-tail distributions of
trip distances (Brockmann et al. 2006; González et al. 2008; Pappalardo et al. 2013b)
and radii of gyration (González et al. 2008; Pappalardo et al. 2013b,2015b), as well
as the characteristic visitation patterns, such as the uneven distribution of time spent
in the various locations (Song et al. 2010a; Pappalardo et al. 2013b). d-EPR embeds
a mechanism to choose a location to visit on a bi-dimensional space given the current
location, the spatial distances between locations and the relevance of each location.
Figure 1provides an outline of Ditras and Algorithm 1describes its pseudocode.
Ditras is composed of two main steps. During the first step, the diary generator builds
a mobility diary Dof Ntime slots, each of duration t. The operation of this step is
described in detail in Sect. 4. During the second step, Ditras uses the trajectory gen-
erator and a given spatial tessellation Lto transform the mobility diary into a sampled
mobility trajectory. We describe in detail the second step of Ditras in Sect. 5.Note
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796 L. Pappalardo, F. Simini
The DITRAS framework
input :L={(l1,r1),...,(ln,rn)}, weighted spatial tessellation
G, diary generator
N, length of trajectory to generate
W, typical diary
output:S=(x1,y1,t1),...,(xn,yn,tn), sampled mobility trajectory of length N
1D=generateMobilityDiary(G,N)// use the diary generator DG to create a
mobility diary Dof length N
2S=generateMobilityTrajectory(D,L,W)// scan the mobility diary Dand create
a sample mobility trajectory Sof length N
3return S
1Function generateMobilityTrajectory(D, L, W )
2S=newList()
3t=1
4Wm=assignLocationsTo(W)// assign a physical location to every abstract
location in typical diary W
5while d<lengt h(D)do
6// scan the mobility diary D
7if D[d]=|then
8// when it sees a separator ‘|’
9d=d+1
10 continue
11 end
12 if D[d]=0then
13 // the individual follows the routine (i.e., she visits a typical
location)
14 S.append((Wm[t],t))
15 t=t+1
16 end
17 else
18 // the individual breaks the routine
19 l=TG(S,P)// call the trajectory generator TG to obtain the next
location to visit
20 S.append((l,t))
21 t=t+1
22 j=d+1
23 while D[d]=D[j]do
24 // stay in location luntil the next separator appears
25 S.append((l,t))
26 t=t+1
27 j=j+1
28 end
29 d=j1
30 end
31 d=d+1
32 end
33 return S
Algorithm 1: The algorithm describing the Ditras framework. Python code is
freely available at https://github.com/jonpappalord/DITRAS.
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Data-driven generation of spatio-temporal routines 797
that the two-step process described above is a general framework common to many
generative models of human mobility, which are often composed by two sequential
parts, the first generating temporal patterns and the second determining the spatial
trajectory. However, in some models the division between the temporal and the spatial
mechanisms is present but not explicitly acknowledged.
In Sect. 6we will instantiate Ditras by using MD(t)and d-EPR and compare it
with other generative models obtained combining diary generators (first step) with
trajectory generators (second step).
4 Step 1: Generation of mobility diary
A diary generator G produces a mobility diary, D(t), containing the sequence of trips
made by a synthetic individual during a time period divided in time slots of tseconds.
For example, G(3600)and G(60)produce mobility diaries with temporal resolutions of
one hour and one minute, respectively. In Sect. 4.1 we illustrate a data-driven algorithm
to construct a diary generator, MD(t), using real mobility trajectory data such as mobile
phone data.
To separate the temporal patterns from the spatial ones, we define the abstract
mobility trajectory, A(t), which contains the time ordered list of the “abstract loca-
tions” visited by a synthetic individual during a period divided in time slots of t
seconds. An abstract location uniquely identifies a place where the individual is sta-
tionary, like home or the workplace, but it does not contain any information on the
specific geographic position of the location (i.e., its coordinates). The abstract mobility
trajectory is thus equivalent to the sampled mobility trajectory where the geographic
locations, lk, are substituted by placeholders, ak, called abstract locations:
Definition 3 (Abstract mobility trajectory) An abstract mobility trajectory is a
sequence A(t)=a1,...,aN, where ai(i=1,...,N)is the abstract location where
the individual spent the majority of time during time slot i, i.e., between (i1)tand
it seconds from the first observation.
The mobility diary, D(t), is generated with respect to a typical mobility diary, W(t),
which represents the individual’s routine. W(t)is a sequence of time slots of duration t
seconds and specifies the typical and most likely abstract location the individual visits
in every time slot. Here we consider the simplest choice of typical mobility diary, in
which the most likely location where a synthetic individual can be found at any time is
her home location. It is possible to relax this simplifying assumption and estimate an
individual’s typical mobility diary from the data by computing her mobility regularity,
which is the time series of the most visited location in each time slot (Song et al. 2010b).
Computing the weekly mobility regularity of individuals on real large-scale mobile
phone data and GPS vehicular data and performing a clustering of their typical diaries
we find that there is one dominant cluster containing 90% of the individuals and
whose representative typical diary has a single location (see “Appendix A”). This result
supports the validity of the simplifying assumption to consider one typical diary with a
single location for all agents. The proposed generative model does not change if there
are two or more typical mobility diaries which have more than one typical location.
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798 L. Pappalardo, F. Simini
When a synthetic individual is generated it can be randomly assigned to one of the
typical diaries in proportion to the overall frequency of the various diaries among real
users. Then, the rest of the algorithm remains the same.
Definition 4 (Typical mobility diary) A typical mobility diary is a sequence W(t)=
w1,...,w
Nwhere wk=wk=1,...,Ndenotes the home location of the
synthetic individual. Nis the total number of time slots considered.
The mobility diary, D(t), specifies whether an individual’s abstract mobility trajec-
tory, A(t), follows her typical mobility diary, W(t), or not. In particular, for every time
slot i,D(t)(i)can assume two values:
D(t)(i)=1ifA(t)(i)=W(t)(i), meaning that the individual visits the abstract
location W(t)(i)following her routine, i.e., she is at home;
D(t)(i)=0if A(t)(i)= W(t)(i), meaning that the individual visits a location
other than the abstract location W(t)(i)being out of her routine.
Definition 5 (Mobility Diary) A mobility diary is a sequence D(t)of time slots of
duration tseconds generated by the regular language L=(1+|(0+|)), where 1 at
time slot iindicates that the individual visits the abstract location in her typical diary at
time i,W(t)(i), and 0 indicates a visit to a location different from the abstract location
W(t)(i). The symbol “|” indicates a transition or trip between two different abstract
locations.
An example of mobility diary generated by language Lis D(t)=11|00|0|1.
The first two entries indicate that A(t)(1)=W(t)(1)and A(t)(2)=W(t)(2), i.e., the
individual follows her routine and she is at home. Next, the third, fourth and fifth entries
indicate that A(t)(3)= W(t)(3),A(t)(4)= W(t)(4)and A(t)(5)= W(t)(5), i.e., the
individual breaks the routine and visits a non-typical location for two consecutive
time slots, then she visits a different non-typical location for one time slot. Finally,
the last time slot indicates that A(t)(6)=W(t)(6), the individual follows the routine
and returns home. We assume that the travel time between any two locations is of
negligible duration.
4.1 Mobility diary learner (MDL)
In this section we propose diary generator MD(t)and illustrate MDL (Mobility Diary
Learner), a data-driven algorithm to compute MD from the abstract mobility trajec-
tories of a set of real individuals (Algorithm 2). We use a Markov model to describe
the probability that an individual follows her routine and visits a typical location
at the usual time, or she breaks the routine and visits another location. First, MDL
translates mobility trajectory data of real individuals into abstract mobility trajecto-
ries (Sect. 4.1.1). Second, it uses the obtained abstract trajectory data to compute the
transition probabilities of the Markov model MD(t)(Sect. 4.1.2).
4.1.1 Mobility trajectory data
The construction of MD(t)is based on mobility trajectory data of real individuals.
We assume that raw mobility trajectory data describing the movements of a set of
123
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Data-driven generation of spatio-temporal routines 799
MDL (Mobility Diary Learner)
input :D={T1,...,Tn}, dataset of real trajectories of nagents
t, time slot length
output:G, a Markov chain
1G=emptyMarkovChain()
2forall the i∈{1,...,n}do
3Ai=createTimeSeries(Ti)// create abstract trajectory of i
4G=updateMarkovChain(Ai)// update the Markov chain using Ai
5end
6return G
1Function updateMarkovChain(A, G)
2slot =0
3while slot <len(A)1do
4h=slot%24 // hour of the day
5nexth=(h+1)%24 // next hour of the day
6loch=A[sl ot]// abstract location at the slot
7loch+1=A[sl ot +1]// abstract location at next slot
8if loch== 1then
9if loch+1== 1then
10 // Case 1: lochis typical and loch+1is typical
11 G[(h,1), (nexth,1)]=G[(h,1), (nexth,1)]+1
12 end
13 else
14 // Case 2: lochis typical and loch+1is not typical
15 τ=1
16 for j=slot +2to len (A)do
17 loc2h=A[j]
18 if loc2h== loc2h+1then
19 τ=τ+1
20 end
21 else
22 break
23 end
24 end
25 hτ=(h+τ)%24
26 G[(h,1), (hτ,0)]=G[(h,1), (hτ,0)]+1
27 slot =j1
28 end
29 end
30 else
31 if loch+1== 1then
32 // Case 3: lochis not typical and loch+1is typical
33 G[(h,0), (nexth,1)]=G[(h,0), (nexth,1)]+1
34 end
35 else
36 // Case 4: both lochand loch+1are not typical
37 τ=1
38 for j=slot +2to len (A)do
39 loc2h=A[j]
40 if loc2h== loc2h+1then
41 τ=τ+1
42 end
43 else
44 break
45 end
46 hτ=(h+τ)%24
47 G[(h,0), (hτ,0)]=G[(h,0), (hτ,0)]+1
48 slot =j1
49 end
50 end
51 end
52 slot =slot +1
53 end
54 G=normalizeMarkovChain(G)
55 return G
Algorithm 2: Algorithm for the construction of the MD generator.
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800 L. Pappalardo, F. Simini
individuals are in the form (u1,x1,y1,t1),...,(un,xn,yn,tn)where uiindicates
the individual who visits location (xi,yi)at time ti,1i<nti<ti+1.
Mobility trajectory data can be obtained from various sources (e.g., mobile phones,
GPS devices, geosocial networks) and describe the movements of individuals on a
territory. Since the purpose of MD(t)is to capture the temporal patterns regardless
the geographic position of locations, we translate raw mobility trajectory data into
abstract mobility trajectories (see definition in Section 3).
Starting from the raw trajectory data, we assign an abstract location to every time
slot in an individual’s abstract mobility trajectory A(t)according to the following
method. If the individual visits just one location during time slot i, we assign that
location to i. If the individual visits multiple locations during slot i, we choose the
most frequent location in i, i.e., the location where the individual spends most of
the time during the time slot. If there are multiple locations with the same visitation
frequency in time slot i, we choose the location with the highest overall frequency. If
there is no information in the abstract trajectory data about the location visited in time
slot i(e.g., no calls during the time slot in the case of mobile phone data), we assume
no movement and choose the location assigned to time slot i1.
To clarify the method let us consider the following example. A mobile phone user
has the following hourly time series of calls: [A,A,,,B,(C,C,B,B)], where
A,B,Care placeholders for different cell phone towers (i.e., abstract locations). Here
the symbol indicates that there is no information in the data about the location
visited during the 1-hour time slot, while all the locations in round brackets are visited
during the same time slot. Using the method described above, the abstract mobility
trajectory of the individual becomes A(3600)=A,A,A,A,B,Bbecause: (i) the
two symbols in the third and fourth time slots are substituted by location Aassuming
no movement with respect to the second time slot; (ii) the location assigned to the last
time slot is Bsince Cand Bhave the same visitation frequency in (C,C,B,B)but
f(B)> f(C), i.e., Bhas the highest overall visitation frequency.
It is worth noting that the choice of the duration of the time slot, t, is crucial and
depends on the specific kind of mobility trajectory data used. GPS data from private
vehicles, for example, generally provide accurate information about the location of
the vehicle every few seconds. In this scenario, a time slot duration of one minute can
be a reasonable choice. In contrast when dealing with mobile phone data a time slot
duration of an hour or half an hour is a more reliable choice, since the majority of
individuals have a low call frequency during the day (Pappalardo et al. 2015b).
4.1.2 Markov model transition probabilities
Let Au=a(u)
0,...,a(u)
n1and Wu=w(u)
0,...,w
(u)
n1be the abstract mobility
trajectory and the typical mobility diary of individual uU, where Uis the set of all
individuals in the data – we omit the superscript (t)for clarity. Elements a(u)
hAu
and w(u)
hWudenote the abstract and the typical locations visited by individual uat
time slot hwith h=0,...,N1.
A state in the Markov model MD is a tuple of two elements s=(h,R). The state’s
first element, h, is the time slot of the time series denoted by an integer between 0
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Data-driven generation of spatio-temporal routines 801
Tab l e 1 Formulae to compute the transition probabilities of the Markov chain MD from abstract mobility
trajectories
Transition, ssFrequency, MDss
(h,1)(h+1,1)uUaAuδu
h(au
h+1(a)
uUaAuδu
h(a)
(h,1)(h+τ,0)uUaAuδu
h(a)[1δu
h+1(a)]τ1
i=1ˆ
δu
h+i(a)[1ˆ
δu
h+τ(a)]
uUaAuδu
h(a)
(h,0)(h+1,1)uUaAu[1δu
h(a)]δu
h+1(a)
uUaAu[1δu
h(a)]
(h,0)(h+τ,0)dD[1δu
h(a)][1δu
h+1(a)][1ˆ
δu
h(a)]τ1
i=1ˆ
δu
h+i(a)[1ˆ
δu
h+τ(a)]
uUaAu[1δu
h(a)]
and N1. The state’s second element, R, is a boolean variable that is 1 (True) if at
time slot hthe individual is in her typical location, w(u)
h, and 0 (False) otherwise –
just like in the mobility diary. In total there are N×2=2Npossible states in the
model. The transition matrix, MD, is a 2N×2Nstochastic matrix whose element
MDsscorresponds to the conditional probability of a transition from state sto state s,
MDssp(s|s). The normalization condition imposes that the sum over all elements
of any row sis equal to 1, sMDss=1,s. We consider two types of transitions,
ss, depending on whether in state sthe individual is in typical location or not:
if the individual is in the typical location at time slot h, i.e., s=(h,1), then
she can either go to the next typical location at time slot h+1, s=(h,1)
s=(h+1,1), or go to a non-typical location and stay there for τtime slots,
s=(h,1)s=(h+τ,0);
if instead the individual is not in the typical location at time slot h, i.e., s=(h,0),
then she can either go to the typical location at time slot h+1, s=(h,0)s=
(h+1,1), or go to a different non-typical location and stay there for τtime slots,
s=(h,0)s=(h+τ,0).
The formulae to compute the empirical frequencies for the four types of transitions
are shown in Table 1. In the table, δu
x(a)=δ(a(u)
x,w
(u)
x),ˆ
δu
x(a)=δ(a(u)
x,a(u)
x+1),
where δ(i,j)=1ifi=jand 0 otherwise, is the Kronecker delta. By convention,
the product τ1
i=1... is equal to 1 if τ=1.
5 Step 2: Generation of sampled mobility trajectory
Starting from the mobility diary D(t), the sampled mobility trajectory S(t)is generated
to describe the movement of a synthetic individual between a set of discrete locations
called weighted spatial tessellation. A weighted spatial tessellation is a partition of a bi-
dimensional space into locations each having a weight corresponding to its relevance.
Definition 6 (Weighted spatial tessellation) A weighted spatial tessellation is a set of
tuples L={(l1,r1), . . . , (lm,rm)}, where rjN(j=1,...,m)is the relevance of a
location and the ljare a set of non-overlapping polygons that cover the bi-dimensional
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802 L. Pappalardo, F. Simini
space where individuals can move. The location of each polygon is identified by the
coordinates of its centroid, (xj,yj).
The weighted spatial tessellation indicates the possible physical locations on a
finite bi-dimensional space a synthetic individual can visit during the simulation. The
relevance of a location measures its popularity among real individuals: locations of
high relevance are the ones most frequently visited by the individuals (Pappalardo
et al. 2015b,2016a). The relevance is introduced to generate synthetic trajectories
that take into account the underlying urban structure. An example of weighted spatial
tessellation is the one defined by a set of mobile phone towers, where the relevance of a
tower can be estimated as the number of calls performed by mobile phone users during
a period of observation, and the polygons correspond to the regions obtained from the
Voronoi partition induced by the towers. If information about location relevance is
not available to the user of the simulator, the distribution of population can be used
to estimate the relevance of the locations. For example, the websites http://sedac.
ciesin.columbia.edu/ and http://www.worldpop.org.uk/ provide a fine-grained spatial
tessellation for the entire globe, together with an estimate of population density in
every location.
First, Ditras assigns to every abstract location in the typical mobility diary W(t)a
physical location on the weighted spatial tessellation L, creating W(t)
m, a typical mobil-
ity diary where each abstract location has a specific geographic position (Algorithm 1,
line 4, procedure assignLocationsTo). The geographic position of an abstract
location is chosen according to the distribution of location relevance specified in the
spatial tessellation, i.e., the more relevant a location is the more likely it is chosen as
a geographic position of an abstract location. This choice ensures the generation of
synthetic data with a realistic distribution of locations across the territory (Pappalardo
et al. 2016a). Next, Ditras scans D(t)to assign a physical location to every entry. For
every entry D(t)(i)D(t)we have two possible scenarios:
D(t)(i)=1, the entry indicates a visit to a typical location, i.e., the abstract location
in W(t)(i)(Algorithm 1, line 12). In this scenario the synthetic individual visits
location l=W(t)
m(i)which is added to the sampled trajectory at time slot i, i.e.
S(t)(i)=W(t)
m(i)(Algorithm 1, lines 14);
D(t)(i)=0, the entry indicates a visit to a non-typical location (Algorithm 1,
line 17). In this second scenario Ditras calls the trajectory generator to choose a
location lto visit, where l= W(t)
m(i)(Algorithm 1, lines 19). The chosen location
lis added to the sampled mobility trajectory ktimes, where kis the number of
consecutive 0 characters before the next separator character ‘|’ appears in D(t),
i.e., the total number of time slots spent in location l(Algorithm 1, lines 23-27).
Example of trajectory generation To clarify how the second step of Ditras works
let us consider the following example. A synthetic individual is assigned a mobility
diary D(t)=1|00|1and the chosen typical diary is W(t)=w, w, w, w, where w
denotes the individual’s home. To generate a synthetic sampled mobility trajectory S,
Ditras operates as follows. First, Ditras assigns a physical location to the individual’s
home w, generating W(t)
m=(x1,y1), (x1,y1), (x1,y1), (x1,y1).Next,Ditras starts
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Data-driven generation of spatio-temporal routines 803
from the first entry D(t)(1). Since D(t)(1)=1 the synthetic individual is at home.
Therefore, tuple (x1,y1,1)is added to trajectory S.Next,Ditras processes the second
entry D(t)(2), sees a separator and then proceeds to entry D(t)(3). Since D(t)(3)=0,
the synthetic individual is not at home in the third time slot. Hence, Ditras calls a
trajectory generator (e.g., d-EPR) which chooses to visit physical location (x2,y2).
Ditras hence adds the tuples (x2,y2,2)and (x2,y2,3)to trajectory S, since there two
0 characters until the next separator in D(t). The last entry D(t)(6)=1 indicates that
the synthetic individual returns home in the fourth time slot. So, Ditras adds tuple
(x1,y1,4)to trajectory S. At the end of the execution, the sampled mobility trajectory
generated by Ditras is S=(x1,y1,1), (x2,y2,2), (x2,y2,3), (x1,y1,4).
5.1 The d-EPR model
As trajectory generator we propose the d-EPR individual mobility model (Pappalardo
et al. 2015b,2016a) that assigns a location on the bi-dimensional space to an entry in
mobility diary D(t).Thed-EPR (density-Exploration and Preferential Return) is based
on the evidence that an individual is more likely to visit relevant locations than non-
relevant locations (Pappalardoet al. 2015b,2016a). For this reason d-EPR incorporates
two competing mechanisms, one driven by an individual force (preferential return)
and the other driven by a collective force (preferential exploration). The intuition
underlying the model can be easily understood: when an individual returns, she is
attracted to previously visited places with a force that depends on the relevance of such
places at an individual level. In contrast, when an individual explores she is attracted to
new places with a force that depends on the relevanceof such places at a collective level.
In the preferential exploration phase a synthetic individual selects a new location to
visit depending on both its distance from the current location, as well as its relevance
measured as the collective location’s relevance in the bi-dimensional space. In the
model, hence, the synthetic individual follows a personal preference when returning
and a collective preference when exploring. The d-EPR uses the gravity model (Zipf
1946; Jung et al. 2008; Lenormand et al. 2016) to assign the probability of a trip
between any two locations in L, which automatically constrains individuals within
a territory’s boundaries. The usage of the gravity model is justified by the accuracy
of the gravity model to estimate origin-destination matrices even at the country level
(Erlander and Stewart 1990; Wilson 1969; Simini et al. 2012; Balcan et al. 2009;
Lenormand et al. 2016).
Algorithm 3describes how d-EPR assigns a location on the bi-dimensional space
defined by a spatial tessellation Lfor an entry in mobility diary D(t).Thed-EPR takes
in input two variables: (i) the current sampled mobility trajectory of the synthetic
individual S=(x1,y1,t1), . . . , (xn,yn,tn); (ii) a probability matrix Pindicating,
for every pair of locations i,jL,i= jthe probability of moving from ito j.Every
probability pij is computed as:
pij =1
Z
rirj
d2
ij
,
123
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804 L. Pappalardo, F. Simini
The d-EPR model
input :S=(x1,y1,t1),...,(xn,yn,tn), the current sample mobility trajectory of the synthetic individual
P, the gravity-probability matrix
output:j, the next location to visit
ρ=0.6, γ=0.21 // distributions’ constants (Pappalardo et al. 2015b, 2016a;
Song et al. 2010a)
1N=|set (S)|// number of distinct visited locations
2i=last(S)// the current location of the synthetic individual
3pnew=getReturnProbability() // generate a probability to return or explore
4if pnewρNγthen
5j=PreferentialExploration(i,P)// explore a new location
77 return j
8end
9else
10 j=PreferentialReturn(S)// return to a previously visited location
1212 return j
13 end
1Function PreferentialExploration(i)
2j=weightedRandom(P[i])// choose jaccording to prob.s in P[i]
44 return j
1Function PreferentialReturn(S)
2j=weightedRandom(S)// choose jaccording to visitation frequency of
locations in S
44 return j
Algorithm 3: The psuedo-code of the d-EPR trajectory generator. The function
weightedRandom randomly chooses an element in a vector according to its
probability.
where ri(j)is the relevance of location i(j)as specified in the weighted spatial tes-
sellation L,dij is the geographic distance between iand j, and Z=i,j=ipij is a
normalization constant. The matrix Pis computed before the execution of the Ditras
model by using the spatial tessellation L.
With probability pnew=ρNγwhere Nis the number of distinct locations in
Sand ρ=0.6, γ=0.21 are constants (Pappalardo et al. 2015b,2016a; Song
et al. 2010a), the individual chooses to explore a new location (Algorithm 3, line 5),
otherwise she returns to a previously visited location (Algorithm 3, line 10). If the
individual explores and is in location i, the new location j= iis selected according to
the probability pij P(Algorithm 3, function PreferentialExploration).
If the individual returns to a previously visited location, it is chosen with probability
proportional to the number of her previous visits to that location (Algorithm 3, function
preferentialReturn). The d-EPR model hence returns the chosen location j.
It is worth highlighting the difference between typical locations and preferred loca-
tions. Typical locations indicate places where individuals repeatedly return as part
of their mobility routine. Examples of typical locations are home and work loca-
tions, where individuals regularly return in their everyday routine. Besides typical
123
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Data-driven generation of spatio-temporal routines 805
locations, individuals can also return to preferred locations, i.e., places which are not
part of a schematic routine but where people return occasionally, such as cinemas or
restaurants. The preferential return mechanism of d-EPR models the existence of such
preferred locations, allowing the agents to return to previously visited locations with
a probability depending of the past visitation frequency.
6 Results
In this section we show the results of simulation experiments where we instantiate
Ditras by using d-EPR as trajectory generator and MD(t)as diary generator. We
construct MD(t)from nation-wide mobile phone data covering a period of three month
using MDL. We refer to the spatio-temporal model as d-EPR (CDR)
MD and use it to generate
sampled mobility trajectories of 10,000 agents. We compare the resulting sampled
mobility trajectories with:
the trajectories of 10,000 mobile phone users whose mobility is tracked during 3
months in a European country;
the sampled mobility trajectories produced by other 8 spatio-temporal mobility
models created through the Ditras framework by combining different diary and
trajectory generators, whose parameters are fitted on the mobile phone data.
Similarly we instantiate Ditras by using d-EPR and MD(t)constructed on GPS
vehicular tracks covering a period of one month. We refer to the spatio-temporal model
as d-EPR (GPS)
MD. We use this model to generate sample mobility trajectories of 10,000
agents and compare the resulting sample mobility trajectories with:
the trajectories of 10,000 private vehicles whose mobility is tracked through on-
board GPS devices during 4 weeks in Tuscany;
the sampled mobility trajectories produced by other 8 spatio-temporal mobility
models created through the Ditras framework by combining different diary and
trajectory generators, whose parameters are fitted on the GPS vehicular data.
In Sect. 6.1 and in Sect. 6.2 we describe respectively the mobile phone data and
the GPS vehicular data we use in our experiments to describe the mobility of real
individuals and the pre-processing operations we carry out on the data. In Sect. 6.3
we provide a comparison on a set of spatio-temporal mobility patterns of d-EPR(CDR)
MD’s
trajectories, mobile phone data’s trajectories, and the trajectories produced by the
other models. These simulations are performed by using a weighted spatial tessellation
induced by the mobile phone towers. Analogously, we provide a comparison on a set of
spatio-temporal mobility patterns of d-EPR(GPS)
MD’s trajectories, GPS data’s trajectories,
and the trajectories produced by the other models. These simulations are performed
by using a weighted spatial tessellation induced by the census cells in Tuscany. All
the simulations are performed using a time slot duration t=3600s =1h.
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806 L. Pappalardo, F. Simini
6.1 CDR data
We have access to a set of Call Detail Records (CDRs) gathered by a European carrier
for billing and operational purposes. The dataset records all the calls made during 11
weeks by 1 million anonymized mobile phone users. CDRs collect geographical,
temporal and interaction information on mobile phone use and show an enormous
potential to empirically investigate the structure and dynamics of human mobility on a
society wide scale (Reades et al. 2007; Hidalgo and Rodriguez-Sickert 2008; González
et al. 2008; Jiang et al. 2012; Calabrese et al. 2011; Pappalardo et al. 2015b,a). Each
time an individual makes a call the mobile phone operator registers the connection
between the caller and the callee, the duration of the call and the coordinates of
the phone tower communicating with the phone, allowing to reconstruct the user’s
approximate position. Table 2illustrates an example of the structure of CDRs.
CDRs have been extensively used in literature to study different aspects of human
mobility, due to several advantages: they provide a means of sampling user locations
at large population scales; they can be retrieved for different countries and geographic
scales given their worldwide diffusion; they provide an objective concept of location,
i.e., the phone tower. Nevertheless, CDR data suffer different types of bias (Ranjan
et al. 2012; Iovan et al. 2013), such as: (i) the position of an individual is known at
the granularity level of phone towers; (ii) the position of an individual is known only
when she makes a phone call; (iii) phone calls are sparse in time, i.e., the time between
consecutive calls follows a heavy tail distribution (González et al. 2008; Barabási
2005). In other words, since individuals are inactive most of their time, CDRs allow
to reconstruct only a subset of an individual’s mobility. Several works in literature
study the bias in CDRs by comparing the mobility patterns observed on CDRs to the
same patterns observed on GPS data (Pappalardo et al. 2013b,2015b,2013a,c)or
handover data (data capturing the location of mobile phone users recorded every hour
Tab l e 2 Example of call detail
records (CDRs) Timestamp Tower Caller Callee
(a)
2007/09/10 23:34 36 4F80460 4F80331
2007/10/10 01:12 36 2B01359 9H80125
2007/10/10 01:43 38 2B19935 6W1199
.
.
.
.
.
.
.
.
.
.
.
.
Tower Latitude Longitude
(b)
36 49.54 3.64
37 48.28 1.258
38 48.22 1.52
.
.
.
.
.
.
.
.
.
Every time a user makes a call, a
record is created with timestamp,
the phone tower serving the call,
the caller identifier and the
callee identifier (a). For each
tower, the latitude and longitude
coordinates are available to map
the tower on the territory (b)
123
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Data-driven generation of spatio-temporal routines 807
or so) (González et al. 2008). The studies agree that the bias in CDRs does not affect
significantly the study of human mobility patterns.
Data preprocessing In order to cope with sparsity in time of CDRs and focus
on individuals with reliable call statistics, we carry out some preprocessing steps.
Firstly, for each individual uwe discard all the locations with a visitation frequency
f=ni/N0.005, where niis the number of calls performed by uin location
iand Nthe total number of calls performed by uduring the period of observation
(Schneider et al. 2013; Pappalardo et al. 2015b). This condition checks whether the
location is relevant with respect to the specific call volume of the individual. Since it is
meaningless to analyze the mobility of individuals who do not move, all the individuals
with only one location after the previous filter are discarded. We select only active
individuals with a call frequency threshold of f=N/(hd)0.5 calls per hour,
where Nis the total number of calls made by u,h=24 is the hours in a day and
d=77 the days in our period of observation. Starting from 1 millions users, the
filtering results in 50,000 active mobile phone users.
Weighted spatial tessellation The weighted spatial tessellation Lweuseinthe
experiments is defined by the mobile phone towers in the CDR data. The relevance
of a phone tower is estimated as the total number of calls served by that tower by the
50,000 active mobile phone users during the 3 months. Every location’s position on
the space is identified by the latitude and longitude coordinates of a phone tower.
6.2 GPS data
The GPS dataset stores information of approximately 9.8 Million different trips from
159,000 private vehicles tracked during one month (May 2011) which passed through
Tuscany (central Italy). The GPS traces are provided by Octo Telematics Italia Srl,2
a company that provides a data collection service for insurance companies. The GPS
device is embedded in the private vehicles’ engine and automatically turns on when the
vehicle starts. The sequence of GPS points that the device transmits every 30 seconds
to the server via a GPRS connection forms the global trajectory of a vehicle. When
the vehicle stops no points are logged nor sent.
We exploit these stops to split the global trajectory into several sub-trajectories,
corresponding to the trips performed by the vehicle. Clearly, the vehicle may have
stops of different duration, corresponding to different activities. To ignore small stops
like gas stations, traffic lights, bring and get activities and so on, we choose a stop
duration threshold of at least 20 minutes: if the time interval between two consecutive
observations of the vehicle is larger than 20 minutes, the first observation is considered
as the end of a trip and the second observation is considered as the start of another trip.
We also performed the extraction of the trips by using different stop duration thresholds
(5, 10, 15, 20, 30, 40 minutes), without finding significant differences in the sample
of short trips and in the statistical analysis we present in the paper. Since GPS data
do not provide explicit information about visited locations, we assign each origin and
destination point of the obtained sub-trajectories to the corresponding census cell,
2http://www.octotelematics.com/.
123
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808 L. Pappalardo, F. Simini
according to the information provided by the Italian National Institute of Statistics
(ISTAT).3We hence obtain a data format similar to CDR data, where we describe
the movements of a vehicle by the time-ordered list of census cells where the vehicle
stopped. We filter the data by discarding all the vehicles with only one visited location
or with less than one trip per day on average during the period of observation. This
filtering results in a dataset of 46,121 vehicles.
Weighted spatial tessellation The weighted spatial tessellation Lweuseinthe
experiments is defined by the census cells in Tuscany. The relevance of a location is
estimated as the total number of stops in the corresponding cell by the 159,000 private
vehicles during the month of observation. Every location’s position on the space is
identified by the latitude and longitude coordinates of the census cell.
6.3 Models comparison and validation
We use the Ditras framework to build 18 models (9 models fitted on CDRs and 9
models fitted on GPS data) which use different combinations for the diary generator
and the trajectory generator. In particular, we consider three diary generators – MD, RD
and WT – and three trajectory generators – d-EPR, SWIM and LATP. For every model
we simulate the mobility of 10,000 agents for a period of N=1,848 hours (3 months)
and N=744 hours (1 month) for models fitted on CDRs and GPS data respectively.
Tables 3and 4show the ability of every model in reproducing a set of characteristic
statistical distributions derived from the CDR and the GPS data respectively, quantified
by two measures: (i) the Root Mean Square Error, RMSE(y,ˆ
y)=n
i=1(ˆyiyi)2
nwhere
ˆyiˆ
yindicates a point of the synthetic distribution ˆ
y,yiythe corresponding point in
the empirical distribution yand nthe number of observations; (ii) the Kullback-Leibler
divergence, KL(y||ˆ
y)=H(y,ˆ
y)H(y), where H(y,ˆ
y)is the cross entropy between
the real distribution and the empirical distribution and H(y)is the entropy of the real
distribution. Here we use the notation TG DG to specify that trajectory generator TG
is used in combination with diary generator DG. For example, d-EPR MD indicates
the model using diary generator MD in combination with trajectory generator d-EPR.
Notation TG{DG1,..., DGk}indicates the set of models {TG DG1,…,TGDGk}. Similarly,
notation {TG1,…,TG
k}DG indicates the set of models {TG1
DG,…,TG
k
DG}.
Diary generators In the Random Diary (RD) generator a synthetic individual is in
perpetuum motion: in every time slot of the simulation she chooses a new location to
visit. We use RD to highlight the difference between the diary generator we propose,
MD (Sect. 4.1), and the temporal patterns of a non-realistic diary generator.
In the Waiting Time (WT) diary generator a synthetic individual chooses a waiting
time Δtbetween a trip and the next one from the empirical distribution Pt)
Δt1βexpΔt, with β=0.8 and τ=17 hours as measured on CDR data (Song
et al. 2010a). WT is the temporal mechanism usually used in combination with mobility
models like EPR (Song et al. 2010a) and SWIM (Kosta et al. 2010). It reproduces in
a realistic way the distribution of the time between two consecutive trips (Song et al.
3www.istat.it.
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Data-driven generation of spatio-temporal routines 809
Tab l e 3 Error of fit between CDR data and synthetic data
CDR Δrr
gSunc TDΔtV N f(L)
MD
d-EPR .0001 .0026 .9643 .0061 .0659 .0014 2.6E5.0218 .0122
.0006 .0247 29.34 .0101 .0682 .1915 .0016 .5449 .1200
SWIM .0005 – 3.6069 .0062 .0683 .0029 5.6E5– .0669
.0067 60.97 .0101 .0808 .4996 .0451 1.2892
LATP .0001 .0061 3.2236 .0062 .0684 .0027 6.3E5– .0625
.0008 .3223 258.46 .0101 .0802 .3282 .0600 .9353
RD
d-EPR .0004 .0027 1.1745 .0232 .2098 .0024 4.1E5.0235 .0521
.0029 .0161 20.8015 .197 4.3558 .2048 .0191 1.1773 .3876
SWIM .0041 – .0232 – .0033 7.2E5– .0947
.1501 .1974 .3773 .0460 4.4057
LATP .0002 – .0232 – .0033 4.6E5– .0874
.0014 .1974 .6967 .0321 2.2051
WT
d-EPR .0003 .0024 1.1666 .0232 .1790 .0023 4.0E5.0224 .0502
.0019 .0130 20.00 .1970 3.9769 .1946 .0189 1.0395 .3537
SWIM .0033 – .0232 .2036 .0033 1.9E5– .0943
.0601 .1975 4.3806 .1146 .0070 3.9605
LATP .0001 – .0232 .2037 .0033 7.2E5– .0866
.0010 .1975 4.5672 .6322 .0309 2.1015
Best model d-EPR d-EPR d-EPR d-EPR d-EPR d-EPR SWIM d-EPR d-EPR
MD WT MD MD MD MD WT MD MD
Every row iis a model and every column ja mobility measure. A cell (i,j)indicates the RMSE (first
row) and the KL divergence (second row) of a synthetic distribution w.r.t. the real distribution. The best
RMSE values are in italic. Symbol—indicates that the synthetic distribution is not comparable with the
real distribution. We highlight in bold the combination of temporal and spatial model leading to the highest
number of Italic cells
2010a; Pappalardo et al. 2013b) but does not model the circadian rhythm and the
tendency of individuals to be in certain places and specific times.
We construct two diary generators, MD (CDR) and MD (GPS), by applying algorithm
MDL (Sect. 4.1) on CDR data and GPS data respectively. These diary generators are
based on Markov models and can reproduce the circadian rhythm of individuals and
their tendency to follow or break the routine.
Trajectory generators The trajectory generator SWIM (Kosta et al. 2010) is a mod-
elling approach based on location preference. The model initially assigns to each
synthetic individual a home location Lhchosen randomly from the spatial tessellation.
The synthetic individual then selects a destination for the next movements depending
on the weight of each location (Kosta et al. 2010):
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810 L. Pappalardo, F. Simini
Tab l e 4 Error of fit between GPS data and synthetic data
GPS Δrr
gSunc TDΔtV N f(L)
MD
d-EPR .0254 .0148 1.9855 .0053 .1334 .0738 .0123 .0113 .0323
.5346 .2850 156.92 .0156 .2992 .7567 .1415 .0411 .2429
SWIM .0229 – 3.8403 .0054 .1232 .0589 .0123 .0319 .0358
.8970 210.87 .0156 .2634 .7321 .1522 1.6923 .4914
LATP .0258 .0225 3.7636 .0054 .1233 .0655 .0178 .0315 .0324
.5968 .9508 151.35 .0157 .2636 .7148 .4639 1.9085 .3811
RD
d-EPR .0031 .0237 – .0231 .0923 .0349 .0042 .0271 .0560
.0420 .9939 .1906 1.2493 .4221 .0360 3.3216 .5258
SWIM .0274 – – .0231 – .2647 .0102 – .0915
1.6628 .1912 1.4443 .0919 3.6641
LATP .0169 – – .0231 – .1599 .0168 – .0899
.1381 .1912 1.1524 .3609 2.9663
WT
d-EPR .0069 .0223 – .0231 .0923 .0291 .0045 .0270 .0530
.0518 .8217 .1906 1.0593 .4369 .0394 2.132 .4623
SWIM .0180 – .0231 .0923 .1608 .0095 – .0908
.7278 .1912 .9510 1.0941 .0823 3.2346
LATP .0190 – .0231 .0923 .1027 .0166 – .0890
.1840 .1913 1.0398 .9187 .4282 2.6838
Best model d-EPR d-EPR d-EPR d-EPR SWIM d-EPR SWIM d-EPR d-EPR
RD MD MD MD WT WT WT MD MD
Every row iis a model and every column ja mobility measure. A cell (i,j)indicates the RMSE (first
row) and the KL divergence (second row) of a synthetic distribution w.r.t. the real distribution. The best
RMSE values are in italic. Symbol—indicates that the synthetic distribution is not comparable with the
real distribution. We highlight in bold the combination of temporal and spatial model leading to the highest
number of italic cells
w(L)swim =αd(Lh,L)+(1α) r(L), α =0.75 (1)
which grows with the relevance r(L)of the location and decreases with the distance
from the home (Kosta et al. 2010):
d(Lh,L)=1
(1+distance(Lh,L))2.
SWIM tries to model both the preference for short trips and the preference for relevant
locations, though it does not model the preferential return mechanism.
The trajectory generator LATP (Least Action Trip Planning) (Lee et al. 2012,2009)
is a trip planning algorithm used as exploration mechanism in several mobility models,
such as SLAW (Lee et al. 2012,2009), SMOOTH (Munjal et al. 2011), MSLAW
(Schwamborn and Aschenbruck 2013) and TP (Solmaz et al. 2015,2012). In LATP
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Data-driven generation of spatio-temporal routines 811
a synthetic individual selects the next location to visit according to a weight function
(Lee et al. 2012,2009):
w(L)latp =1
distance(c,L)1.5.(2)
LATP only models the preference for short distances and does not consider the rele-
vance of a location nor model the preferential return mechanism.
We compare the synthetic mobility trajectories of the nine models with CDR trajec-
tories and GPS trajectories on the distributions of several measures capturing salient
characteristics of human mobility. Tables 3and 4display the mobility measures we
consider, which are: trip distance Δr(González et al. 2008; Pappalardo et al. 2013b),
radius of gyration rg(González et al. 2008; Pappalardo et al. 2013b,2015b), mobility
entropy Sunc (Song et al. 2010b; Eagle and Pentland 2009; Pappalardo et al. 2016b),
location frequency f(L)(Song et al. 2010a; Hasan et al. 2013; Pappalardo et al.
2013b), visits per location V(Pappalardo et al. 2016a), locations per user N(Pap-
palardo et al. 2016a), trips per hour T(González et al. 2008; Pappalardo et al. 2013b),
time of stays Δt(Song et al. 2010a; Hasan et al. 2013) and trips per day D.
Trip distance The distance of a trip Δris the geographical distance between the
trip’s origin and destination locations. We compute the trip distances for every indi-
vidual and then plot the distribution Pr)of trip distances in Fig. 2a–c (CDR data)
and Fig. 3a–c (GPS data). Figure 2a compares the distribution of trip distance of CDR
data with the distributions produced by d-EPR (CDR)
MD, SWIM (CDR)
MD and LATP(CDR)
MD.We
observe that d-EPR (CDR)
MD and LATP (CDR)
MD are able to reproduce the distribution of Pr)
although slightly overestimating long-distance trips. In contrast SWIM(CDR)
MD cannot
reproduce the shape of the empirical distribution resulting in a RMSE(SWIM(CDR)
MD)
and KL(SWIM (CDR)
MD) higher than the other two models (see Table 3). The shape of
the synthetic distributions do not vary significantly by changing the diary generator
(Fig. 2b–c). In other words, the choice of the diary generator does not affect the ability
of the model to capture the distribution Pr). This is also evident from Table 3
where the RMSEs and the KLs in the first column vary a little by changing the diary
generator. Model d-EPR(CDR)
MD produces the best fit with CDR data, as we notef in Fig. 2c
and Table 3. This suggests that modelling preferential return and location preference is
crucial to reproduce Pr)as well as the preference for short-distance trips. Although
SWIM embeds a preference for short-distance trips (Eq. 1) the distance is chosen with
respect to the home location Lhleading to an underestimation of short-distance trips
(Fig. 2a–c). Figure 3a–c compares the distribution of trip distance of GPS data with
the distributions produced by the generative algorithms. Results on GPS data con-
firm the observations on CDRs: in contrast with SWIM, d-EPR and LATP are able to
reproduce the distribution of Pr), regardless the diary generator. Also in this case,
d-EPR (GPS)
RD is the model generating the most realistic synthetic data (Table 4).
Radius of gyration The radius of gyration rgis the characteristic distance traveled by
an individual during the period of observation (González et al. 2008; Pappalardo et al.
2013b,2015b). In detail, rgcharacterizes the spatial spread of the locations visited by
an individual ufrom the trajectories’ center of mass (i.e., the weighted mean point of
the locations visited by an individual), defined as:
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812 L. Pappalardo, F. Simini
CDR
(a) (b) (c)
(d)(e) (f)
(g) (h) (i)
Fig. 2 Distributions of human mobility patterns (CDR). The figure compares the models and CDR data on
trip distance, radius of gyration and mobility entropy. Plots in (ac) show the distribution of trip distances
Pr)for real data (black squares) and data produced by three trajectory generators (d-EPR, SWIM and
LATP) in combination with the MD generator (a), the RD generator (b) and the WT generator (c). Plots in
(df) show the distribution of radius of gyration rg, while plots in (gi) show the distribution of mobility
entropy Sunc
rg=
iL(u)
pi(lilcm)2,(3)
where liand lcm are the vectors of coordinates of location iand center of mass,
respectively (González et al. 2008; Pappalardo et al. 2013b,2015b), L(u)Lis
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Data-driven generation of spatio-temporal routines 813
(a) (b) (c)
(d) (e) (f)
(g) (h)
GPS
(i)
Fig. 3 Distributions of human mobility patterns (GPS). The figure compares the models and GPS data on
trip distance, radius of gyration and mobility entropy
the set of locations visited by individual u,pi=ni/|L(u)|is the individual’s visita-
tion frequency of location li, equal to the number of visits to lidivided by the total
number of visits to all locations. In Fig. 2a we observe that d-EPR (CDR)
MD is the only
model capable of reproducing the shape of P(rg)of CDR data, though overestimating
the presence of large radii (see Fig. 2d). RMSE(d-EPR (CDR)
MD)forrgis indeed lower
than RMSE(SWIM (CDR)
MD) and RMSE(LATP (CDR)
MD) as shown in Table 3. SWIM (CDR)
MD and
LATP (CDR)
MD cannot reproduce the shape of P(rg)because rgalso depends on the pref-
erential return mechanism (Song et al. 2010a; Pappalardo et al. 2015b) which is not
123
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814 L. Pappalardo, F. Simini
modeled in SWIM and LATP. In a previous work (Pappalardo et al. 2016a)wealso
show that P(rg)depends on the preferential exploration mechanism of d-EPR since a
version of d-EPR without preferential exploration – the s-EPR model – is not able to
reproduce the shape of P(rg). We also observe that while d-EPR (CDR)
{MD,RD,WT}produce
similar distributions of rg, SWIM and LATP produce different distributions of rgwith
different choices of the diary generator (Fig. 2e, f). The shape of P(rg)for GPS data
is slightly different from the same distribution of CDR data, since short radii are less
likely in GPS due to the nature of car travels (Pappalardo et al. 2013c,b,a). Also for
GPS we observe that, in contrast with LATP and SWIM, d-EPR is the only model that
can reproduce the shape of P(rg). In particular d-EPR (GPS)
MD produces the best fitting
with GPS data in terms of both RMSE and KL (Table 4).
Mobility entropy The mobility entropy Sunc of an individual uis defined as the Shan-
non entropy of her visited locations (Song et al. 2010b; Eagle and Pentland 2009;
Pappalardo et al. 2016b):
Sunc(u)=iL(u)pilog(pi)
log |L(u)|,(4)
where piis the probability that individual uvisits location iduring the period of
observation and log |L(u)|is a normalization factor. The mobility entropy of an indi-
vidual quantifies the possibility to predict individual’s future whereabouts. Individuals
having a very regular movement pattern possess a mobility entropy close to zero and
their whereabouts are rather predictable. Conversely, individuals with a high mobility
entropy are less predictable.
We observe that the average Sunc produced by d-EPR (CDR)
MD data equals the average
Sunc=0.61 in CDR data, although d-EPR (CDR)
MD underestimates the variance of distri-
bution P(Sunc)(Fig. 2g). In contrast, SWIM (CDR)
MD and LATP (CDR)
MD largely overestimate
Sunc and underestimate the variance of P(Sunc), resulting in RMSE and KL much
higher than RMSE(d-EPR (CDR)
MD) and KL(d-EPR (CDR)
MD), as shown in Table 3.Thisis
because SWIM and LATP do not model the preferential return mechanism, which
increases the predictability of individuals since they tend to come back to already vis-
ited locations. P(Sunc)is not robust to the choice of diary generator: diary generator
RD and WT make the models to largely overestimate Sunc (Fig. 2h, i). In particular
SWIM (CDR)
{RD,WT}and LATP (CDR)
{RD,WT}produce distributions with ¯
Sunc 1, indicating that
the typical synthetic individual is much more unpredictable than a typical individual
in CDR data. This makes those distributions not comparable with the distribution of
MD models. Hence, distribution P(Sunc )highly depends on both the choice of the
trajectory generator and the choice of the diary generator. We observe similar results
for GPS data, where only {d-EPR, SWIM, LATP} (GPS)
MD can reproduce P(Sunc)in rea-
sonable agreement with real data. All the other models produce distributions that are
not comparable with the entropies of private vehicles (Fig. 3g–i).
Location frequency Another important characteristic of an individual’s mobility is
the probability of visiting a location given the location’s rank. The rank of a location
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Data-driven generation of spatio-temporal routines 815
CDR
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 4 Distributions of human mobility patterns (CDR). The figure compares the models and CDR data on
location frequency, visits per location and locations per users. Plots in (ac) show the distribution of location
frequency f(L)for d-EPR, SWIM and LATP used in combination with MD, RD and WT respectively. Plots
in (df) show the distribution of the number Vof visits per location and plots in (gi) show the distribution
of the number Nof distinct visited locations per user
depends on the number of times the individual visits the locations over the period
of observation. For instance, rank 1 represents the most visited location (generally
home place); rank 2 the second most visited location (e.g., work place) and so on. We
compute the frequency of each of these ranked locations for every individual and plot
the distribution of frequencies f(Li)in Figs. 4a–c (CDR) and 5a–c (GPS). For CDR
123
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816 L. Pappalardo, F. Simini
GPS
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 5 Distributions of human mobility patterns (GPS). The figure compares the models and GPS data on
location frequency, visits per location and locations per users
data, we observe that d-EPR (CDR)
MD reproduces the shape of f(Li)(with RMSE=0.0122
and KL =0.12) better than SWIM (CDR)
MD and LATP (CDR)
MD (which have RMSE =0.0669,
KL =1.2892 and RMSE=0.0626, KL =0.9353 respectively). If we change the diary
generator in the model, d-EPR (CDR)
{RD,WT}underestimate the frequency of the top-ranked
location and slightly overestimate the frequency of the less visited locations with
respect to CDR data (Fig. 4b, c). A reason for this discrepancy is that RD and WT do
not take into account the circadian rhythm of individuals, hence underestimating the
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Data-driven generation of spatio-temporal routines 817
number of returns to the most frequent location (usually the home place). In SWIM (CDR)
MD
and LATP (CDR)
MD , the absence of a preferential return mechanism produce a more uni-
form distribution of location frequencies (Fig. 4b, c), which is further exacerbated for
SWIM (CDR)
{RD,WT}and LATP (CDR)
{RD,WT}. Location frequency f(Li)is another case where
the choice of the diary generator and the choice of the trajectory generator are both
crucial to reproduce the shape of the distribution in an accurate way. Experiments on
GPS data confirm results observed on CDRs (Fig. 5a–c): model d-EPR (GPS)
MD produces
the best fit with real data, while changing either the diary or the trajectory generators
produces worse fits.
Visits per location A useful measure to understand how a set of individuals exploit the
mobility space is the number Vof overall visits per location, i.e., the total number of
visits by all the individuals in every location during the period of observation. For every
dataset, we compute the number of visits for every location of the weighted spatial
tessellation and plot the distribution P(V)in Fig. 6d–f (CDR) and Fig. 7d–f (GPS).
As for CDR data, d-EPR (CDR)
MD produces a P(V)which follows a heavy tail distribution:
the majority of locations have just one visit while a minority of locations have up to
several thousands visits during the 11 weeks. The value of Vof a location depends on
two factors: (i) its relevance in the weighted spatial tessellation; (ii) its position in the
weighted spatial tessellation. The higher the relevance of a location in the weighted
spatial tessellation, the higher is the probability for the location to be visited in the
exploration mechanisms of d-EPR and SWIM. Indeed, from Fig. 6e, f we observe that
d-EPR and SWIM are the models which better fit P(V). In contrast LATP does not
take into account the relevance of a location during the exploration being unable to cap-
ture the shape of P(V). Experiments on GPS data substantially confirm these results
(Fig. 7d–f): d-EPR and SWIM generates the most realistic distributions of P(V).
Locations per user The number Nuof distinct locations visited by an individual during
the period of observation describes the degree of exploration of an individual, i.e., how
the single individuals exploit the mobility space. In Fig. 4g we observe that the MD
models do not capture the shape of P(Nu)in CDR data: the average number of distinct
locations Naccording to d-EPR (CDR)
MD is about twice Nin CDR data, while SWIM (CDR)
MD
and LATP(CDR)
MD produce distributions whose Nis more than ten times Nin CDR data.
By changing diary generator (Fig. 4h, i) the difference with CDR data becomes even
larger: d-EPR (CDR)
{RD,WT}produce a much broader variance of P(Nu), SWIM(CDR)
{RD,WT}and
LATP (CDR)
{RD,WT}predict a number of distinct visited locations very far from CDR data.
These results suggest that the considered models overestimate the degree of explo-
ration of individuals. In the case of d-EPR (CDR)
MD the overestimation may depend on
the distribution of time of stays, as the distribution of time stays Pt)produced by
d-EPR (CDR)
MD overestimates the number of short stay times, leading to a larger total num-
ber of visited locations (Fig. 6g). For GPS data, model d-EPR (GPS)
MD produces a P(N)that
is more realistic than the other models, as it is evident from Fig. 7g and from Table 4.
Trips per hour Human movements follow the circadian rhythm, i.e., they are preva-
lently stationary during the night and move preferably at specific times of the day
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818 L. Pappalardo, F. Simini
CDR
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 6 Distributions of human mobility patterns (CDR). The figure compares the models and CDR data on
trips per hour, trips per day and time of stays. Plots in (ac) show the distribution of the number Tof trips
per hour of the day for d-EPR, SWIM and LATP used in combination with MD, RD and WT respectively.
Plots in (df) show the distribution of the number Dof trips per day, plots in (gi) show the distribution of
time of stays Δt
(González et al. 2008; Pappalardo et al. 2013b). To verify whether the considered
models are able to capture this characteristic of human mobility, we compute the
number of trips Tmade by the individuals at every hour of the period of observation.
Figures 6a–c and 7a–c show how Tdistribute across the 24 hours of the day, for CDRs
and GPS data respectively. We observe that, regardless the trajectory generator used,
123
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Data-driven generation of spatio-temporal routines 819
GPS
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 7 Distributions of human mobility patterns (GPS). The figure compares the models and GPS data on
trips per hour, trips per day and time of stays
diary generator MD produces a distribution of trips per hour very similar to real data
(Figs. 6a and 7a). The mobility diary generator MD proposed in Sect. 4is hence able
to create mobility diaries which reproduce the circadian rhythm of individuals in an
accurate way. In contrast, diary generators RD and WT are not able to capture this
distribution, regardless the trajectory generator used (Figs. 6b, c and 7b, c). This is
because: (i) in RD individuals are always in motion; (ii) WT takes into account the
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820 L. Pappalardo, F. Simini
waiting times but not the preference of individuals to move at specific times of the
day.
Trips per day The number of trips per day Dindicates the tendency of individuals to
travel in their every-day life. For every dataset, we compute the number of trips per
day made by each individual during the period of observation and plot the distribu-
tion P(D)in Fig. 6d–f (CDR) and Fig. 7d–f (GPS). We observe that d-EPR (CDR, GPS)
MD ,
SWIM (CDR, GPS)
MD and LATP (CDR, GPS)
MD are able to capture the shape of P(D)but overestimate
the variance of the distribution (Fig. 6d). The other diary generators, RD and WT, are
not able to reproduce the CDR distribution since the average number Dof trips per
day is much higher than CDR data (Fig. 6e, f). Again, this is because in RD individuals
are always in motion and because WT does not take into account the circadian rhythm
of individuals.
Time of stays The distribution of stay times Δtis another important temporal features
observed in human mobility. Stay time is the amount of time an individual spends at
a particular location. In our experiments we compute the stay time as the number of
hours every individual spends in her visited locations and plot the distribution Pt)
in Fig. 4g–i (CDR) and Fig. 5g–i (GPS). We observe that, for both CDRs and GPS
data, d-EPR (CDR, GPS)
{MD,RD,WT}capture the shape of the distribution while the other models
do not, though overestimating the presence of short time stays.
6.4 Discussion of results
Two main results emerge from our experiments. First, model d-EPR MD produces
sampled mobility trajectories having in general the best fit to both CDR data and GPS
data (i.e., having the lowest RMSE and KL for most of the measures), as evident in
Tables 3and 4. Diary generator MD, indeed, simulates in a realistic way temporal
human mobility patterns such as the distribution of location frequency (Fig. 4a) and
the distribution of trips per hour (Figs. 6a, 7a). This is mainly because MD reproduces
the circadian rhythm of individuals, while RD and WT do not. Moreover, trajectory
generator d-EPR embeds two mobility mechanisms: preferential return and preferen-
tial exploration. The preferential return mechanism – absent in SWIM and LATP –
allows for a realistic simulation of, for example, the distribution of radius of gyration
(Figs. 2d, 3d) and the distribution of stay times (Fig. 6g). The preferential exploration
mechanism, which is modeled by both d-EPR and SWIM but it is absent in LATP,
allows for a realistic description of the territory exploitation by individuals, in terms
of the distribution of the number of visits per location (Figs. 4d, 5d). Also, model
d-EPR MD produces realistic distributions for both CDR and GPS data, suggesting
that it can be used in different simulation scenarios where its parameters are fitted on
different types of data and different spatio-temporal resolutions.
Second interesting result is that the temporal and the spatial mechanisms have dif-
ferent roles in shaping the distribution of standard mobility measures. Some measures,
such as trip distance (Figs. 2a–c, 3a–c), radius of gyration (Figs. 2d–f, 3d–f), visits
per location (Figs. 4d–f, 5d–f) and time of stays (Fig. 2g–i) mainly depend on the
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Data-driven generation of spatio-temporal routines 821
choice of the trajectory generator, i.e., on the spatial mechanism of the model. Indeed,
by changing the underlying diary generator the shape of these distribution, the RMSE
and the KL divergence w.r.t. real data do not change in a significant way. Other mea-
sures, such as trips per hour (Figs. 6a–c, 7a–c) and trips per day (Fig. 6d–f) mainly
depend on the choice of the diary generator, i.e., on the temporal mechanism of the
model. Conversely, both the spatial and the temporal mechanism are determinant in
reproducing the distribution of some other measures like mobility entropy (Figs. 2g–i,
3g–i) and locations per user (Figs. 4g–i, 5g–i). Moreover the right combination of
diary and trajectory generator, d-EPR MD, leads to more accurate fits w.r.t. both CDR
data and GPS data for the majority of measures (Tables 3,4). Human mobility pat-
terns depend on both where people go and when people move: our results show that
to reproduce them in an accurate way we need proper choices for the spatial and the
temporal generative models to use in the Ditras framework.
7 Conclusion and future work
In this paper we propose Ditras, a framework for the generation of individual human
mobility trajectories with realistic spatio-temporal patterns. The framework consists
of two steps: (i) the generation of a mobility diary by using a diary generator; (ii) the
generation of a mobility trajectory by using a trajectory generator. In the paper we
propose a novel diary generator MD together with MDL, a data-driven algorithm to
build it from real mobility data.
We instantiate Ditras by using MD and the state-of-the-art trajectory generator
d-EPR and obtain a novel generative algorithm, d-EPR MD. We use it to generate the
spatio-temporal trajectories of thousands of agents visiting the locations on a large
European country and a region in Italy. The generated sampled mobility trajectories
are compared with CDR data, GPS vehicular data, and the trajectories produced by
other generative algorithms, each obtained by using a different combination of diary
generator and trajectory generator in the Ditras framework. Among the considered
algorithms, d-EPR MD produces the best fit with respect to both CDR data and GPS
data. We also observe that different combinations of diary and trajectory generators
show different abilities to reproduce the distribution of standard mobility measures.
This result highlights the importance of considering both the spatial and temporal
dimensions in human mobility modelling.
The proposed model d-EPR MD has a limited number of parameters to fit. The
generation of the mobility diary is parameter-free as the Markov chain is a non-
parametric model where each element of the transition matrix MD is estimated using
the empirical frequencies observed in the data. The generation of the mobility trajectory
is based on the d-EPR model. The details on how to fit the d-EPR parameters are
explained in detail in (Pappalardo et al. 2015b,2016a). Here, for the two parameters
of the exploration probability pnew, we choose the values ρ=0.6 and γ=0.21 that
have been estimated in previous work (Song et al. 2010a). For the gravity model used
in the exploration phase, we use a power law deterrence function of the distance with
exponent 2, although other types of gravity or intervening opportunities models can
be used. Given that the model is non-parametric or depends on a very small number
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822 L. Pappalardo, F. Simini
of parameters, it does not suffer from training/test issues and its calibration is quite
robust to changes in the size of the training set.
Applications Given its flexibility, Ditras can be used in a wide range of applications.
Here we provide three examples where Ditras and d-EPR MD can be particularly
useful and profitably applied.
In urban science, the generation of what-if scenarios to imagine the new mobility
that could emerge from the construction of new infrastructures requires the generation
of realistic mobility data and hence the presence of an accurate generative algorithm
(Barbosa-Filho et al. 2017; Kopp et al. 2014). d-EPR MD could be used to generate
synthetic data given the tessellation of the territory that emerges from the construction
of the new infrastructure, allowing urban planners and managers to quantify changes
in urban mobility and visualize preferred path that could emerge from the simula-
tion.
Computational epidemiology has attracted particular attention in the last decade, as
the arrival of the 2009 flu pandemic prompted scientists to develop realistic mobility
models to simulate the spread of viruses on a territory (Merler et al. 2013; Ajelli et al.
2010; Venkatramanan et al. 2017). The possibility to use Ditras to combine different
temporal and spatial mechanisms is particularly valuable for this type of studies, as
generative algorithms for individual human mobility are the basic mechanism used in
computational epidemiology to generate synthetic population mimicking at an indi-
vidual level the realistic aspects related to disease propagation.
Opportunistic Networks (OppNets) enable communications in disconnected envi-
ronments in the absence of an end-to-end path between the sender and the receiver.
In OppNets, the mobility of nodes (e.g., mobile devices such as smartphones and
tables) help the delivery of messages by connecting, asynchronously in time, other-
wise disconnected subnetworks. This means that the network protocols responsible
for finding a route between two disconnected devices must embed patterns in human
movements and make prediction of future encounters. Realistic generative algorithms
for human mobility are fundamental for testing the efficiency of OppNets protocol,
as real data about the functioning of the network is obviously not available during the
protocol design (Tomasini et al. 2017). Ditras can be used to instantiate many gen-
erative algorithms and then generate realistic mobility routines to test the efficiency
of a given network protocol for OppNets. Given its accuracy in reproducing human
mobility patterns, d-EPR MD can be used to uncover the characteristics of the network
protocol in real-life, such as the speed of message delivery.
A possible application of Ditras and d-EPR MD in data mining is anomaly detec-
tion. The proposed model can be used to detect individuals with an anomalous mobility
behavior with respect to the typical mobility patterns of the majority of the individuals.
In particular, within our framework an individual is anomalous if her trajectory is not
a likely outcome of the model, i.e., if the probability that the model would generate
such trajectory is below a given threshold. To this end, the log-likelihood of each indi-
vidual’s trajectory can be computed and the individuals can be ranked according to
their log-likelihood values: individuals with a low rank and a very high log-likelihood
values would be the most typical, whereas individuals with the highest ranks and low
log-likelihood values would be the most anomalous.
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Data-driven generation of spatio-temporal routines 823
Improvements The instantiation of Ditras we propose, d-EPR MD, can be further
improved in severaldirections. First, in t his work the construction of the diary generator
MD(t)through the mobility diary learner MDL is based on the simplest possible typical
diary W(t), where the most likely location where a synthetic individual can be found at
any time is her home location. More complex typical diaries can be used specifying, for
example, the typical times where an individual can be found at work, school, friends’
home and so on. Such a composition of W(t)can be constructed by using surveys
or generative algorithms describing the daily schedule of human activities (Rinzivillo
et al. 2014; Jiang et al. 2012; Liao et al. 2007) as a way to enrich an individual’s
trajectory with information about the type of activity associated to a location.
Second, in d-EPR the preference for short-distance trips is embedded in the prefer-
ential exploration phase only. A preference for short-distance trips can be introduced
during the preferential return mechanisms as well, in order to eliminate the overesti-
mation of long-distance trips and long-distance radii observed in Figs. 2a and 2d.
Third, in d-EPR MD we make the simplifying assumption that the travel time is of
negligible duration. This may not be a good assumption especially when the duration of
the time slot is one hour or less. The proposed algorithm can be modified to explicitly
include realistic information on the travel time between locations, which imposes
constraints on the locations that are reachable in a given time window and on the
time that can be spent in a location given the travel time needed to reach the next
location in the mobility diary. Moreover, another interesting improvement can be to
map the sampled mobility trajectories to a road network specifying specific road routes
with specific velocities. This mapping would be of great help, for example, in what-if
analysis where we want to study how human mobility changes with the construction
of a new infrastructure in an urban context.
Finally, there is a large number of studies that demonstrate the connection between
human mobility and social networks (Brown et al. 2013b; Hristova et al. 2016; Wang
et al. 2011; Volkovich et al. 2012; Brown et al. 2013a; Hossmann et al. 2011a,b), as
well as several approaches that include information on social connections in human
mobility models (Borrel et al. 2009; Yang et al. 2010; Fischer et al. 2010; Boldrini
and Passarella 2010; Musolesi and Mascolo 2007). A mechanism to account for the
influence of social connections on human mobility can be introduced in DITRAS
as a third phase, between the mobility diary generation and the sampled trajectory
construction.
We leave these improvements of DITRAS for future work.
Acknowledgements We thank Paolo Cintia, Gianni Barlacchi and Salvatore Rinzivillo for their invaluable
suggestions. This work has been partially funded by the EU under the H2020 Program by project Cimplex
Grant n. 641191. Filippo Simini has been supported by EPSRC First Grant EP/P012906/1.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-
tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
123
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824 L. Pappalardo, F. Simini
Appendix A: Homogeneity of typical mobility diaries
We investigate to what extent the typical mobility diaries of real individuals are homo-
geneous by performing a clustering experiment. For every individual in the GPS dataset
we compute her typical week, i.e. a time series of length 168 hours. Every time slot is
the most frequent location of the individual in that hour of the week. We then apply
the DBSCAN clustering algorithm (Ester et al. 1996) to group the typical weeks in
dense clusters. We use the Levenshtein metric (Navarro 2001) to measure the sim-
ilarity between two typical weeks. DBSCAN takes two input parameters: minPts
and eps (Ester et al. 1996). We set minPts =4 and eps =70. We estimate the
value of these parameters using the procedure suggested in (Tan et al. 2005): (i) we
fix min Pts =4 and compute for every typical week the distance dto its 4th nearest
neighbor; (ii) we sort the typical weeks in increasing order with respect to dand set
eps to the distance corresponding to an elbow in the curve of Fig. 8a. We observe no
significant differences in the clustering results by varying minPts in the range [2,5].
DBSCAN produces two clusters, one of them consisting of 90% of the typical
weeks (Fig. 8b). The silhouette coefficient of the clustering (Rousseeuw 1987), a
measure of how similar a typical diary is to its own cluster compared to other clusters,
is s=0.50 (in general, s∈[1,1]). The typical weeks in the biggest cluster have
typically one or two locations, while the representative typical week (i.e., the medoid
of the cluster) consists of just one location, the most frequent location of the individual
(Fig. 8c, d). This result supports the validity of the simplifying assumption to consider
one typical diary with a single location for all agents.
Appendix B: Computational complexity of d-EPR MD
Learning phase. In the learning phase, two main tasks are performed:
(1) the construction of the MD model by the MDL algorithm (Algorithm 2). The pro-
cedure UpdateMarkovChain has computational complexity O(N), where N
is the number of slots in the period of observation. As we repeat the procedure for
all the nindividuals in the dataset, the computational complexity of Algorithm 2
is O(Nn). When nN, (e.g., when the period of observation is short and the
dataset contains hundreds of thousands of individuals), the factor Nis negligible
and the computational complexity of Algorithm 2 can be approximated to O(n).
(2) the construction of the probability matrix Pin the d-EPR model, which has
complexity O(L2)where Lis the number of locations in the spatial tessellation.
Generation phase. In the generation phase, the generation of the mobility diary with
MD has complexity O(N). The generation of the trajectory from the mobility diary
has complexity O(LNn)(Algorithm 3): in the worst case, for each individual we
assign a spatial location in each time slot, and the assignment of a spatial location
requires a call to procedure weightedRandom which has complexity O(L). When
nN, the computational complexity can be approximated to O(Ln).
The total complexity of the generation phase is hence O(L2+Ln)when the prob-
ability matrix has to be constructed for the first time. In this case, when Lnthe
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Data-driven generation of spatio-temporal routines 825
(a) (b)
(c) (d)
Fig. 8 First row: aTypical weeks sorted by distance to the 4th nearest neighbor, the elbow suggests to
use eps =70; brelative size of the clusters resulting from DBSCAN algorithm with minP ts =4and
eps =70 and their relative size. Second row: cVisualization of a day of the typical weeks of 100 individuals
in the GPS dataset for the first cluster. Every color represents a different abstract location in the typical
diary. dDistribution of abstract location entropy and number of distinct abstract locations of time series of
individuals in cluster 1
computational complexity can be approximated to O(L2). If the probability matrix is
already available or has been already computed, the computational complexity of the
generation phase is O(LNn), which can be approximated to O(Ln)when nN.
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