arXiv:1711.04710v2 [cs.LG] 17 Nov 2017

Spatio-Temporal Data Mining: A Survey of Problems and
Methods
GOWTHAM ATLURI∗ , University of Cincinnati
ANUJ KARPATNE∗ , University of Minnesota
VIPIN KUMAR, University of Minnesota
Large volumes of spatio-temporal data are increasingly collected and studied in diverse domains including,
climate science, social sciences, neuroscience, epidemiology, transportation, mobile health, and Earth sciences.
Spatio-temporal data differs from relational data for which computational approaches are developed in the data
mining community for multiple decades, in that both spatial and temporal attributes are available in addition
to the actual measurements/attributes. The presence of these attributes introduces additional challenges that
needs to be dealt with. Approaches for mining spatio-temporal data have been studied for over a decade
in the data mining community. In this article we present a broad survey of this relatively young field of
spatio-temporal data mining. We discuss different types of spatio-temporal data and the relevant data mining
questions that arise in the context of analyzing each of these datasets. Based on the nature of the data mining
problem studied, we classify literature on spatio-temporal data mining into six major categories: clustering,
predictive learning, change detection, frequent pattern mining, anomaly detection, and relationship mining.
We discuss the various forms of spatio-temporal data mining problems in each of these categories.
ACM Reference Format:
Gowtham Atluri, Anuj Karpatne, and Vipin Kumar. 2017. Spatio-Temporal Data Mining: A Survey of Problems
and Methods. ACM Comput. Surv. 1, 1 (November 2017), 37 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn

1

INTRODUCTION

Space and time are ubiquitous aspects of observations in a number of domains, including, climate
science, neuroscience, social sciences, epidemiology, transportation, criminology, and Earth sciences, that are rapidly being transformed by the deluge of data. Since the real-world processes being
studied in these domains are inherently spatio-temporal in nature, a number of data collection
methodologies have been devised to record the spatial and temporal information of every measurement in the data, hereby referred to as spatio-temporal (ST) data. For example, in neuroimaging
data, activity measured from the human brain is stored along with the spatial location from which
the activity was measured and the time at which the measurement was made. Similarly, web-search
requests arriving at Google’s servers have a geographic location and time from which they are
made. Effective analysis of such increasingly prevalent ST data holds great promise for advancing
the state-of-the-art in several scientific disciplines.
A unique quality of ST data that differentiates it from other data studied in classical data mining
literature (e.g., see [Tan et al. 2017]) is the presence of dependencies among measurements induced
∗ Both

authors contributed equally to the paper.

Authors’ addresses: Gowtham Atluri, University of Cincinnati, atlurigm@ucmail.uc.edu; Anuj Karpatne, University of
Minnesota, karpa009@umn.edu; Vipin Kumar, University of Minnesota, kumar001@umn.edu.
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by the spatial and temporal dimensions. For example, many of the widely used data mining methods
are founded on the assumption that data instances are independent and identically distributed (i.i.d).
However, this assumption is violated when dealing with ST data, where instances are structurally
related to each other in the context of space and time and show varying properties in different
spatial regions and time periods. Ignoring these dependencies during data analysis can lead to poor
accuracy and interpretability of results [Eklund et al. 2016].
Apart from limiting the effectiveness of classical data mining algorithms, the presence of spatial
and temporal information also makes it possible to consider novel formulations for analyzing data in
the emerging field of spatio-temporal data mining (STDM). Contrary to traditional data mining that
deals with distinct objects (also referred to as data instances) having well-defined features, in STDM,
one can define objects and features in a variety of ways. One scenario involves treating spatial
locations as objects and using the measurements collected from a spatial location over time to define
the features. For example, in climate science, one of the goals is to group locations that experience
similar climatic phenomenon over time. In this case, locations are treated as instances/objects
and features are defined based on climate variables measured over time [Steinbach et al. 2002].
Another scenario involves treating time points as objects and using measurements collected from
all the spatial locations under consideration to define features. For example, in the application of
discovering patterns of human brain activity from neuroimages, the goal is to identify the time
points at which similar brain activity is observed in the brain. In this case, time points are treated
as objects/instances and features are defined using the observed spatial map of activity [Liu et al.
2013]. There are also scenarios where events are treated as objects and features are defined based
on the spatial and temporal information of events. For example, in the context of discovering crimes
that are committed in close proximity in space and time, incidence of a crime is treated as an
object and the location and time stamp of the crime are treated as features, in addition to other
features such as nature of crime and number of victims involved [Tompson et al. 2015]. Hence, the
coupling of spatial and temporal information in ST data introduces novel problems, challenges, and
opportunities for STDM research, with a broad scope of application in several domains of scientific
and commercial significance.
There exists a vast literature on approaches for mining data that is purely spatial in nature,
spanning multiple decades of research in spatial statistics [Cressie and Wikle 2015], spatial data
mining [Aggarwal 2015; Shekhar et al. 2011, 2008], and spatial database management [Ester et al.
1997; Shekhar and Chawla 2003]. An extensive taxonomy of spatial data types and representations
has been explored in the field of spatial data mining for improving the efficiency and effectiveness
of data mining tasks such as clustering, prediction, anomaly detection, and pattern mining when
dealing with spatial data [Shekhar et al. 2011]. Another related area of research is time series
data mining [Esling and Agon 2012; Keogh and Kasetty 2003; Liao 2005], where approaches for
mining useful information from time-series databases have been explored. Existing research in
STDM includes foundational research in the statistics community [Cressie and Wikle 2015], e.g.,
research on spatio-temporal point processes [Diggle 2013]. Approaches for handling spatial and
temporal information have also been explored in the data mining literature for problems such as
spatio-temporal clustering [Kisilevich et al. 2010] and trajectory pattern mining [Giannotti et al.
2007].
There are a few recent surveys that have reviewed the literature on STDM in certain contexts
from different perspectives. Articles by [Vatsavai et al. 2012] and [Chandola et al. 2015] discuss the
computational issues for STDM algorithms in the era of ‘big-data’ for application domains such as
remote sensing, climate science, and social media analysis. The review by [Cheng et al. 2014] covers
STDM approaches for prediction, clustering, and visualization problems in several applications. An
extensive survey of approaches for mining trajectory data, one of the many types of ST data, is
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presented in [Li 2014; Mamoulis 2009; Zheng 2015]. A survey on STDM by [Shekhar et al. 2015]
provides a semantic categorization of ST data types and pattern families from a database-centric
perspective.
Given the richness of problems and the variety of methods being explored in the rapidly advancing
field of STDM, there is a need for developing an over-arching structure of research in STDM
that highlights the similarities and differences of different problems and methods in diverse ST
applications. This can enable the cross-pollination of ideas across disparate research areas and
application domains, by making it possible to see how a solution developed for a certain problem in
a particular domain (e.g., identifying patterns in climate data) can be useful for solving a different
problem in another domain (e.g., understanding the working of the brain). This can also help
in connecting the traditional data mining community with the challenges and opportunities in
analyzing ST data, thus exposing some of the open questions and motivating future directions of
research in STDM.
This review paper on STDM problems and methods attempts to fulfill this need as follows.
First, it builds a foundation of ST data types and properties that can help in identifying the
relevant problems and methods for any class of ST data encountered in real-world applications.
In particular, we provide a broad taxonomy of the different types of ST data, different ways of
defining and describing ST data instances, and different ways of computing similarity among ST
data instances. Second, it presents a survey of STDM approaches for a number of commonly studied
data mining problems such as clustering, predictive learning, frequent pattern mining, anomaly
detection, change detection, and relationship mining. For every category of problems, we review
the novel issues that arise in dealing with the unique properties of ST data types in classical data
mining frameworks. This paper can be used as a guide by data mining researchers and real-world
practitioners working with ST data, to identify STDM formulations fit for their data and to make
most effective use of STDM research in their problem. In addition, by bridging the gap between
classical data mining literature and the novel aspects of spatio-temporal data, this paper helps in
opening novel possibilities of future research.
The rest of the paper is organized as follows. In Section 2 we review the variety of application
areas where analyzing ST data is important. In Section 3, we discuss the types and characteristics of
ST data, and the different ways of defining instances and similarity measures using ST data types.
Section 4 presents a survey of STDM methods developed for different types of ST data instances
in the context of six major data mining problems, viz., clustering, predictive learning, frequent
pattern mining, anomaly detection, change detection, and relationship mining. Section 5 presents
concluding remarks and discusses future research directions.
2

APPLICATIONS

Large volumes of ST data are collected in several application domains such as social media, healthcare, agriculture, transportation, and climate science. In this section, we briefly describe the different
sources of ST data and the motivation for analyzing ST data in different application domains.
Climate Science: Data pertaining to historic and current atmospheric and oceanic conditions
(e.g., temperature, pressure, wind-flow, and humidity) is collected and studied in climate science
[Karpatne et al. 2013]. In addition to observational data 1 collected from weather stations and
reanalysis data that is gridded in space [Kistler et al. 2001], simulated data generated using climate
models [Voldoire et al. 2013] is also studied in this domain. The purpose in studying this data is
to discover relationships and patterns in climate science that advance our understanding of the
1 https://www.ncdc.noaa.gov/cdo-web/datasets

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Earth’s system and help us better prepare for future adverse conditions by informing adaptation
and mitigation actions in a timely manner.
Neuroscience: Continuous neural activity captured using a variety of technologies such as
Functional Magnetic Resonance Imaging (fMRI), Electroencephalogram (EEG), and Magnetoencephalography (MEG) is studied in neuroscience [Atluri et al. 2016]. Spatial resolution of neural
activity measured using these technologies is quite different from another. For example, neural
activity is measured from millions of locations in fMRI data, while it is only measured from tens of
locations in the case of EEG data. Temporal resolution of the data collected using these technologies
is also quite different. For example, fMRI typically measures activity for every two seconds, while
the temporal resolution of EEG data is is typically 1 millisecond. The purpose in studying this data
is to understand the governing principles of the brain and thereby determine the disruptions to
normal conditions that arise in the case of mental disorders [Atluri et al. 2013, 2015]. Discovering
such disruptions can be useful for designing diagnostic procedures and in developing therapeutic
procedures for patients.
Environmental Science: Studying the data pertaining to the quality of air, water, and environment is one of the objectives of environmental science. While air quality is measured based on the
presence of pollutants such as particles, carbon monoxide, nitrogen dioxide, sulphur dioxide, ozone
etc., water quality is measured based on factors such as dissolved oxygen, conductivity, turbidity,
and pH. Air quality sensors are typically placed on streets or on top of buildings, and water quality
sensors are placed in lakes, rivers, and streams. In addition to air and water quality, data pertaining
to sound pollution is also collected. These environmental data sets are studied to detect changes in
levels of pollution, identify the causal factors that contribute to pollution, and to design effective
policies to reduce the different types of pollution [Thompson et al. 2014].
Precision Agriculture: Multi-band high-resolution (ranging from 0.25m to 1m) areal or remotesensing images of large farms are being collected at regular intervals (e.g., daily to weekly). One
of the purposes of collecting and studying this data is to detect plant diseases [Mahlein 2016]
and understand the impact of several factors such as misapplication of fertilizer, compaction
during planting and weeds on crop yield, as well as their inter-relationships. With the help of this
knowledge, steps can be taken in future crop cycles to mitigate the risks due to the factors that
adversely affect the crop yield.
Epidemiology/ Health care: Electronic health record data that is widely stored in hospitals
provide demographic information pertaining to patients as well diagnosis made on patients at
different time points. This dataset can be represented as a spatio-temporal dataset where each
diagnosis has a spatial location and a time-point associated with it. One can construct such spatiotemporal instances for different types diseases such as cancers and diabetes, as well as for infectious
diseases such as influenza. This data is studied to discover spatio-temporal patterns in different
diseases [Matsubara et al. 2014] and to study the spread of an epidemic. This data is also used
in conjunction with environmental, climate science data sets to discover relationships between
environmental factors and public health [Ryan et al. 2007]. Discovery of such relationships will
allow policy-makers to develop effective policies that will ensure the well-being of the population.
Social media: Users of social media portals such as Twitter and Facebook post their experience
at a given place and time. Each social media post captures the experience of a user at a given place
and time. Using this data one can study collective user experience at a given place for a given
time period [Tang et al. 2014]. One can also capture the spread of epidemics such as Influenza or
ebola based on users’ posts. More recently, there is also increased interest in studying the spread of
social and political movements using social media data [Carney 2016]. In addition, events such as
earthquakes, tsunamis, and fires can also be automatically detected from this data.

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Traffic Dynamics: Large scale taxi pick-up/drop-off data is publicly available for several major
cities across the world [Castro et al. 2013]. This data contains information about each trip made
by customers of the taxi service, including the time and location of pick-up and drop-off, and
GPS locations for each second during the taxi ride. This data can be used to understand how the
population in a city moves spatially as a function of time and also the influence of extraneous
factors such as traffic and weather. In addition, this data can also be studied to explore traffic
dynamics based on the collective movement patterns of the taxis. This will enable transportation
engineers to design effective policies to reduce traffic congestion. In addition, the behavior of taxi
drivers can also be studied using this data so effective practices can be designed to detect abnormal
behavior, increasing likelihood of finding new passengers, and taking optimal routes to arrive at a
destination.
Heliophysics: Heliophysics studies the events that occur in the Sun and their impact on the Solar
System. The publicly available Heliophysics Events Knowledgebase [Hurlburt et al. 2010] provides
various observations that include solar events and their annotations on a daily basis. Examples
of these events include Active region, Emerging flux, Filament, Flare, Sigmoid, and Sunspot. The
time and the location of where these events were observed on the Sun are also provided in the
knowledgebase. The spatial and temporal information along with the different observations are
studied to discover patterns in the solar events [Pillai et al. 2012]. The Heliophysics knowledgebase
also enables the study of the impact of solar events and the Earth’s climate system.
Crime data: Law enforcement agencies store information about reported crimes in many cities
and this information is made publicly available in the spirit of open-data [Tompson et al. 2015]. This
data typically has the type of crime (e.g., arson, assault, burglary, robbery, theft, and vandalism),
as well as the time and location of the crime. Patterns in crime and the effect of law enforcement
policies on the amount of crime in a region can be studied using this data with the goal of reducing
crime.
3 DATA
The presence of space and time introduces a rich diversity of ST data types and representations,
which leads to multiple ways of formulating STDM problems and methods. In this section, we first
describe some of the generic properties of ST data, and then describe the basic types of ST data
available in different applications. Building on this discussion, we describe some of the common
ways of defining and representing ST data instances, and generic methods for computing similarity
among different types of ST instances.
3.1

Properties

There are two generic properties of ST data that introduces challenges as well as opportunities for
classical data mining algorithms, as described in the following.
3.1.1 Auto-correlation. In domains involving ST data, the observations made at nearby locations
and time stamps are not independent but are correlated with each other. This auto-correlation in
ST data sets results in a coherence of spatial observations (e.g., surface temperature values are
consistent at nearby locations) and smoothness in temporal observations (e.g., changes in traffic
activity occurs smoothly over time). As a result, classical data mining algorithms that assume
independence among observations are not well-suited for ST applications, often resulting in poor
performance with salt-and-pepper errors [Jiang et al. 2015]. Further, standard evaluation schemes
such as cross-validation may become invalid in the presence of ST data, because the test error
rate can be contaminated by the training error rate when random sampling approaches are used
to generate training and test sets that are correlated with each other. We also need novel ways
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of evaluating the predictions of STDM methods, because estimates of the location/time of an ST
object (e.g., a crime event) may be useful even if they are not exact but in the close ST vicinity of
ground-truth labels. Hence, there is a need to account for the structure of auto-correlation among
observations while analyzing ST data sets.
3.1.2 Heterogeneity. Another basic assumption that is made by classical data mining formulations is the homogeneity (or stationarity) of instances, which implies that every instance belongs
to the same population and is thus identically distributed. However, ST data sets can show heterogeneity (or non-stationarity) both in space and time in varying ways and levels. For example,
satellite measurements of vegetation at a location on Earth show a cyclical pattern in time due to
the presence of seasonal cycles. Hence, observations made in winter are differently distributed than
the observations made in summer. There can also be inter-annual changes due to regime shifts in
the Earth’s climate, e.g., El Nino phase transitions, which can impact climate patterns to change on
a global scale. As another example, different spatial regions of the brain perform different functions
and hence show varying physiological responses to a stimuli. This heterogeneity in space and time
requires the learning of different models for varying spatio-temporal regions.
3.2

Data Types

There is a variety of ST data types that one can encounter in different real-world applications.
They differ in the way space and time are used in the process of data collection and representation,
and lead to different categories of STDM problem formulations. For this reason, it is important to
establish the type of ST data available in a given application to make the most effective use of STDM
methods. In the following, we describe four common categories of ST data types: (i) event data,
which comprises of discrete events occurring at point locations and times (e.g., incidences of crime
events in the city), (ii) trajectory data, where trajectories of moving bodies are being measured (e.g.,
the patrol route of a police surveillance car), (iii) point reference data [Cressie and Wikle 2015],
where a continuous ST field is being measured at moving ST reference sites (e.g., measurements of
surface temperature collected using weather balloons), and (iv) raster data, where observations of
an ST field is being collected at fixed cells in an ST grid (e.g., fMRI scans of brain activity). While
the first two data types (events and trajectories) record observations of discrete events and objects,
the next two data types (point reference and rasters) capture information of continuous or discrete
ST fields. We discuss the basic properties of these four data types using illustrative examples from
diverse applications. Indeed, if an ST data set is collected in a native data type that is different
from the one we intend to use, in some cases, it is possible to convert from one ST data type to
another, e.g., from point reference data to raster data. We briefly discuss some of the possible ways
of converting an ST data type to other data types, for leveraging the STDM methods developed for
those data types in a particular application.
3.2.1 Event Data. An ST event can generally be characterized by a point location and time,
which denotes where and when the event occurred, respectively. For example, a crime event can be
characterized by the location of the crime along with the time at which the crime activity occurred.
Similarly, a disease outbreak can be represented using the location and time where the patient
was first infected. A collection of ST point events is called as a spatial point pattern [Gatrell et al.
1996] in the spatial statistics literature. Figure 1(a) shows an example of a spatial point pattern
in a two-dimensional Euclidean coordinate system, where (li , ti ) denotes the location and time
point of an event. Note that apart from the location and time information, every ST event may also
contain non-ST variables, known as marked variables, that provide additional information about
every ST event. For example, events can belong to different types such as the type of disease or the
nature of crime, which can be denoted by a categorical marked variable. In Figure 1(a), the marked
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variable over events can take three categorical values: A, B, and C. ST events are quite common in
real-world applications such as criminology (incidence of crime and related events), epidemiology
(disease outbreak events), transportation (road accidents), Earth science (land cover change events
like forest fires and insect disease), and social media (Twitter activity or Google search requests).
While the spatial nature of most ST events can be represented
using a Euclidean coordinate system (where every dimension
B
is equally important), sometimes it is more relevant to explore
(l2, t2)
A
(l1, t1)
alternative representations of this data. For example, accidents
C
on freeways can be considered as events occurring on a spatial
(l3, t3)
road network, where the distance between any two events is
B
measured not by their Euclidean distance but by the shortest
B
(l4, t4)
distance of the road segments connecting the events. Further,
(l5, t5)
C
events may not always be point objects in space but instead
(l7, t7)
A
be characterized by other geometric shapes such as lines and
(l6, t6)
polygons. For example, a forest fire event can be represented
as a spatial polygon that delineates the extent of damage due
(a) Events belonging to three types: A (cirto the fire. Similarly, an event may not have an instantaneous cles), B (squares), and C (triangles).
time point of occurrence but instead be associated with a time
period of appearance, denoting the birth and death of the event.
For example, a music concert happening in the city can be repA
resented using the start and end times of the event. While these
simple extensions of ST events are quite common in real-world
applications, most of the existing STDM methods are tailored
B
for analyzing point ST events occurring in Euclidean spaces.
3.2.2 Trajectory Data. Trajectories denote the paths traced by
C
bodies moving in space over time. Some examples of trajectories
include the route taken by a taxi from the pick-up to the drop-off (b) Trajectories of three moving bodies,
location or the migration patterns of animals traveling for better A, B, and C.
access to food, water, and shelter [Farine et al. 2016]. Trajectory
data is commonly collected by mounting sensors on the moving Fig. 1. Illustrations of event data
bodies that periodically transmits information about the location and trajectory data types.
of the body over time. Figure 1(b) illustrates the trajectories of
three moving bodies, A, B, and C. Apart from measuring the series of locations traversed by every
moving body over time, trajectory data may also contain additional marked variables of the moving
body, such as the heart-rate of a person running on a circuit track or the sequence of messages
transmitted and received by a patrolling police car. Trajectories are common in applications such
as transportation, ecology, and law enforcement.
3.2.3 Point Reference Data. A point reference data consists of measurements of a continuous
ST field such as temperature, vegetation, or population over a set of moving reference points
in space and time. For example, meteorological variables such as temperature and humidity are
commonly measured using weather balloons floating in space, which continuously record weather
observations at point locations in space and time. Another example is that of buoy sensors used for
measuring ocean variables such as sea surface temperature at locations that keep changing over
time. In both these examples, a finite sample of ST reference points is used to represent the behavior
of a continuous spatio-temporal field. Figure 2 shows an example of the spatial distribution of a
continuous ST field at two different time stamps, that are being measured at reference locations
(shown as dots) on the two time stamps. The observations at these discrete reference points can
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Fig. 2. An example of ST reference points (shown as dots) on two different time stamps. The colorbars show
the distribution of the ST field on the two time stamps.

be used to reconstruct the ST field at any arbitrary location and time using data-driven methods
(e.g., smoothing techniques) or physics-based methods (e.g., meteorological reanalysis approaches
[Saha et al. 2010]). Point reference data is also known as geostatistical data in the spatial statistics
literature.
3.2.4 Raster Data. In raster data, measurements of a continuous or discrete ST field are recorded
at fixed locations in space and at fixed points in time. This is in contrast to point reference data where
the ST reference sites may keep changing their location over time and collect recordings on different
time stamps. To formally describe a raster, consider a set of fixed locations, S = {s 1 , s 2 , . . . , sm },
either distributed regularly in space with constant distance between adjacent locations, e.g., pixels
in an image (see Figure 3(a)) or distributed in an irregular spatial pattern, e.g., ground-based sensor
networks (see Figure 3(b)). For every location, we record observations on a fixed set of time stamps,
T = {t 1 , t 2 , . . . , tn }, which can again be regularly spaced with equal delays between consecutive
measurements (see Figure 3(c)) or irregularly spaced (see Figure 3(d)). It is the Cartesian product of
S and T that results in the complete spatio-temporal grid, S × T , where every vertex on the ST
grid, (si , t j ), has a distinct measurement.
ST raster data is quite common in several real-world applications such as remote sensing,
climate science, brain imaging, epidemiology, and demography. Some examples of ST raster data
include measurements collected by ground-based sensors of ST fields such as air quality or weather
information, geo-registered images of the Earth’s surface collected by satellites at regular revisit
times, and fMRI video sequences of brain activity. Note that while some examples of ST raster record
observations at point vertices (e.g., measurements collected by a sensor network), others make
aggregate measurements over the region at every grid cell. For example, demographic information
is often collected at aggregate scales over political divisions such as cities, counties, districts, and
states at annual scale. Another feature of an ST raster data is the resolution of the grid (both
in space and in time) used for collecting measurements. In many applications, one commonly
encounters ST raster data sets at varying resolutions of space and time, collected from different
instruments or sensors. For example, satellite measurements of Earth’s surface may be obtained
via Landsat instruments at 30 meter spatial resolution every 16 days or via MODIS instruments
at 500 meter spatial resolution on a daily scale. As another example, fMRI technology can be
used to measure brain activity at each 1mm x 1mm x 1mm location, whereas EEG technology
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(b) Irregular space.
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Fig. 3. Different aspects of ST grids used for representing a raster data. The set of locations in the ST raster
can either be located regularly or irregularly in space. The set of time stamps can also be either spaced
regularly or irregularly.

measures activity at a selected set of tens of locations. In problems involving ST rasters with varying
resolutions, we often need to convert an ST raster from its native resolution to a finer or coarser
resolution, so that a seamless analysis of all ST rasters can be performed at a common resolution.
Interpolation techniques, also referred to as resampling methods in Geographic Information System
(GIS) literature, are commonly used to convert a raster data to a finer resolution in space or time.
Note that the computational requirements of STDM methods generally increase as we move to
finer resolutions in space and time. A raster data can also be converted to a coarser resolution by
aggregating over collections of ST cells. Aggregation generally helps in removing redundancies
in the observations, especially when there is high spatial and temporal auto-correlation at finer
resolutions. However, it is important to keep in mind that aggressive aggregation of data to coarser
resolutions may result in loss of information about the ST field being measured.
3.2.5 Converting Data Types. Even if an ST data is naturally collected in a particular data type
in a certain application, it is possible to transform it to a different ST type so that the relevant
family of STDM tools are used for their analyses. We provide some examples of inter-conversion
among ST data types in the following. An event data type can be converted to a raster data type
by aggregating the counts of events at every cell of an ST grid. For example, crime events can
be counted at the levels of counties in a city at an hourly scale, thus producing an ST raster of
crime occurrences. In some cases, a raster data can also be converted to ST events by using special
algorithms for event extraction, e.g., techniques to find ST regions with abnormal activity. As an
example, ecosystem disturbance events such as forest fires can be extracted from geo-registered
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satellite images of vegetation cover [Mithal et al. 2011a]. Another common type of conversion
among ST data types is between point reference data and raster data. Observations at ST reference
points can be transformed into an ST raster format by interpolating or aggregating over an ST grid.
Raster data can also be converted to a point reference data where every vertex of the ST grid is
viewed as an ST reference point.
3.3

Data Instances

The basic unit of data that a data mining algorithm operData  Type  
Data  Instance  
ates upon is called a data instance. In classical data mining
settings, a data instance is unambiguously represented as
ST  Events
Points
a set of observed features with optional supervised labels.
However, in the context of ST data, there are multiple
Trajectories
Trajectories
ways of defining instances for a given data type, each
Time  Series
resulting in a different STDM formulation. In this section,
ST  Point  
we review five common categories of ST instances that
Reference
Spatial  
one encounters in STDM problems, namely, points, traMaps
jectories, time series, spatial maps, and ST rasters. These
ST  Raster
ST  Raster
data instances form the canonical building blocks of analysis for a broad range of problems and methods in STDM
that we will discuss in Section 4.
Fig. 4. A mapping showing the different catFigure 4 summarizes the different types of data in- egories of ST data instances that can be constances that can be constructed for each ST data type. structed from ST data types.
While ST events can be naturally represented as point
instances, a trajectory data type can either be represented as a collection of point instances (ordered
list of locations visited by the moving object), a trajectory instance, or as a time-series of spatial
identifiers (e.g., location coordinates). A ST reference data type comprises of point instances, where
every instance is a reference point of the ST field in space and time. There are three different
ways of constructing instances for ST raster data. First, we can define an instance to be the set
of measurements at any location, si , represented as a time series, Ti . Second, we can also define
an instance to be the set of measurements at any time stamp, t j , represented as a spatial map, S j .
A third approach is to consider the entire ST raster (collection of observations over the entire
ST grid) as a single data instance. It is the plurality of ways for defining instances for every ST
data types that results in multiple STDM problems and methods. The choice of the right approach
for constructing ST instances from a ST data type depends on the nature of the question being
investigated, and the family of available STDM methods that can be used.
In the following, we introduce the five categories of ST data instances and discuss some of
the common questions that can be investigated for each category. Since ST data instances can be
summarized using a number of ways that go beyond standard dimensionality reduction techniques
such as principal component analysis, we also describe some of the common ways of representing
and summarizing the features for every instance category.
3.3.1 Points. An ST point can be represented as a tuple containing the spatial and temporal
information of a discrete observation, as well as any additional variables associated with the
observation. ST points are frequently used as data instances in STDM analyses involving event data,
e.g., point events such as the occurrence of crimes at certain locations and times can be treated as
basic instances to group similar instances or to find anomalous instances. ST points are also used
as data instances when dealing with point reference data, where the measurements at ST reference
points are used as instances to estimate the ST field at unseen instances. Additionally, a trajectory
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can also be viewed as an ordered collection of ST points, which are the locations visited by the
moving object.
Some of the common questions that can be asked using ST points as data instances include: how
are ST points clustered in space and time? What are the frequently occurring patterns of ST points?
Can we identify ST points that do not follow the general behavior of other ST points? Can we estimate
a target variable of interest at an ST point that has not been seen during training?
A collection of ST points can be summarized by measures that capture the strength of interaction
(or auto-correlation) among the points. For example, the Ripley’s K function [Dixon 2002] is a
commonly used statistic for describing the amount of attraction or repulsion among spatial locations
beyond its expected value. Spatio-temporal extensions of the Ripley’s K function have also been
explored to measure the strength of interaction among ST point events [Lynch and Moorcroft
2008]. The strength of auto-correlation among ST reference points can also be measured using the
Moran’s I function [Li et al. 2007a], local Moran’s I [Anselin 1995], and its ST extension [Hardisty
and Klippel 2010].
3.3.2 Trajectories. Trajectories are a different class of data instances that can be used in STDM
analyses involving moving bodies. Trajectories can be represented as multi-dimensional sequences
that contain a temporally ordered list of locations visited by the moving object, along with any
other information recorded by the object.
Some of the common questions that can be asked using trajectories as data instances include:
can we cluster a collection of trajectories into a small set of representative groups? Are there frequent
sequences of locations within the trajectories that are traversed by multiple moving bodies?
A common approach for representing and extracting features from trajectories is by the use of
generative models [Gaffney and Smyth 1999], where a parametric model is used to approximate the
behavior of every trajectory. The learned parameters can then used as succinct representations of the
trajectories. A trajectory can also be represented using the frequent sub-sequences of locations that
are visited by the moving body. Techniques for identifying frequent trajectory patterns are discussed
in detail later in Section 4.3.3. Distributed and efficient indexing structures for answering trajectoryrelated similarity queries have been developed in [Al-Naymat et al. 2007; Zeinalipour-Yazti et al.
2006]. Apart from these methods, different schemes such as semantic trajectories [Bogorny et al.
2014; Li 2017], symbolic trajectories [Güting et al. 2015], and spatio-textual trajectories [Damiani
2016] are also recently being explored for representing trajectory data.
3.3.3 Time Series. Time series can be used as data instances in two different scenarios involving
ST data. First, given an ST raster data, we can consider the set of observations at every spatial cell
in the ST grid as a time series that can be used as a data instance in an STDM analysis. Second,
a trajectory data can also be treated as a multi-dimensional time-series data, where the multiple
dimensions correspond to the spatial identifiers (e.g., location coordinates) traversed by the moving
objects over time, and any other variables recorded by the moving object in the course of its
trajectory. While representing spatial identifiers as multiple (and independent) dimensions of a
time-series may not preserve the spatial context among the identifiers, it opens up the vast literature
on time series data mining that can be used for analyzing trajectories in novel ways, as will be
described in Section 4.
Some of the common questions that can be asked using time series as data instances include: can
we identify groups of time-series that show similar temporal activity and are located nearby in space?
Are there some temporal patterns that commonly repeat in a number of time-series? Can we identify
time stamps where the time-series deviate from their normal behavior for a short period of time? Can
we discover time stamps where the time-series show a change in its profile? Can we use time-series as
input features to predict a target variable? Can we predict the value of a time-series at a future time
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stamp using its historical values? Can we find distant groups of spatially contiguous time-series that
are related to each other?
A number of approaches exist for extracting useful features from a collection of time-series.
This include methods that can identify temporally frequent sub-sequences that occur in a majority
of time-series (e.g., temporal motifs [Mueen 2014]) or contain discriminatory information about
a particular time-series class (e.g., shapelets [Hills et al. 2014; Ye and Keogh 2009]). A review of
techniques for identifying such frequent patterns in time-series is presented in [Fu 2011] and [Esling
and Agon 2012].
3.3.4 Spatial Maps. An ST raster data can be viewed as a collection of spatial maps observed at
every time stamp, which can also be used as data instance in analyses involving ST rasters.
Some of the common questions that can be asked using spatial maps as data instances in STDM
analyses include: can we cluster the spatial maps to find groups of time stamps showing similar spatial
activity? Can we identify spatial patterns that are observed in a number of spatial maps? Can we
use spatial maps as input variables to predict a target variable? Can we predict the value at a certain
location using observed values at other locations in the map?
A common approach for extracting features among spatial maps is using image segmentation
techniques [Haralick and Shapiro 1985]. The presence or absence of different types of image
segments can then be used as features to represent spatial maps.
3.3.5 ST Rasters. An ST raster data in its entirety, with measurements spanning the entire set
of locations and time stamps, can also be treated as individual data instances in STDM analyses.
Some of the common questions that can be asked using ST rasters as data instances include: can
we cluster ST rasters into groups that show similar behavior in space and time? Can we find frequent
spatio-temporal behavior that occurs in a number of ST raster data sets? Can we find ST rasters that
show distinctly different behavior than other ST rasters? Can we find timestamps where the behavior of
an ST raster changes over time? Can we use ST rasters as input features for predicting a target variable
of interest? Can we predict the value at a certain location and time stamp in the ST grid using observed
values at other locations and time stamps? Can we find subsets of locations in the ST grid that show
interesting relationships in their temporal activity?
A basic approach for representing ST rasters is using N -way arrays also called as tensors. In
a tensor representation of an ST raster data, some dimensions are used to represent the set of
locations while the remaining dimension is used to represent the set of time stamps available in
the ST grid. For example, precipitation data is represented as a 3-dimensional array where the
first two dimensions capture 2D space and the third dimension captures time. Similarly, fMRI
data is represented as a 4-dimensional array where the first three dimensions capture 3D space
and the third dimension captures time. A tensor representation of an ST raster data can then be
summarized using space-time subspaces that have similar values, which are the equivalent of image
segmentation in ST domains. Existing image segmentation techniques that have been extended to
work with videos to discover moving objects are relevant to address this problem [Haritaoglu et al.
2000; Prati et al. 2003].
One way to summarize or extract features from ST rasters is by using network-based representations, where the nodes correspond to the locations and the edges denote the similarity among
the time-series at locations [Atluri et al. 2016; Feldhoff et al. 2015]. Techniques for computing
similarity among time-series are discussed in detail later in Section 3.4.3. The topological properties of nodes such as degree and different variants of centrality measures can then be used to
characterize the ‘role’ and ‘influence’ of locations. Such properties have been found to be useful in
characterizing nodes in different types of networks such as social, biological, and transportation
networks. For example, the use of network-based properties such as coherence among a set of
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locations [De Martino et al. 2007; Sui et al. 2009] or relationship between distant locations [Lynall
et al. 2010; Pettersson-Yeo et al. 2011] has been explored in previous studies.
3.4

Similarity Among Instances

Determining similarity (or dissimilarity) between data instances is key to many data mining
problems such as clustering, classification, pattern discovery, and relationship mining. Similarity
measures have been extensively studied for data instances in traditional data mining settings, that
are able to deal with varying types and formats of attributes. However, in the presence of space and
time, there is a variety of ways we can define similarity among ST data instances such as points,
trajectories, time-series, spatial maps, and ST rasters. Our ability to capture similarities among
ST data instances using proximity measures underpins the effectiveness of STDM methods that
build upon them. In the following, we describe some of the common ways for computing similarity
among different types of ST data instances.
3.4.1 Point Similarity. Two points are considered close if they lie within the ST neighborhoods of
each other. The ST neighborhood of a point can be defined using a fixed distance threshold in space
and time, e.g., within 1 km radius and 1 hour time difference. Alternatively, the ST neighborhood
of every point can also be defined in terms of a fixed number, k, of closest points. The choice of
the right notion of locality depends on the application context and can be decided by the domain
analyst.
3.4.2 Trajectory Similarity. Similarity among trajectories is often measured in terms of the
co-location frequency, which is the number of times two moving bodies appear spatially close to
one another. Other approaches for measuring similarity among trajectories include subsequence
similarity metrics such as the length of the longest common subsequence, Fréchet distance, dynamic
time warping (DTW), and edit distance [Toohey and Duckham 2015]. Trajectory similarity can also
be computed using feature-based representations such as the frequent trajectory patterns extracted
from the data.
3.4.3 Time Series Similarity. If we consider every time-series as a 1D-array of observations, the
similarity among two time-series can be simply computed using proximity measures such as the
Euclidean distance and the correlation strength, that consider a one-to-one correspondence between
the elements of the two arrays. However, sometimes it is the case that two similar time-series are
not exactly aligned with one another but show the same pattern of activity over time. Measures
such as dynamic time warning (DTW) [Keogh and Ratanamahatana 2005] and Fréchet distance
[Alt and Godau 1995] are able to capture such forms of similarity among time-series. We can also
compute the similarity among two time-series based on their closeness in time-series features
such as temporal motifs and shapelets. When it is expected to observe a certain delay or time lag
between the observations of two time series, a common approach is to translate one of the time
series with a range of candidate values of time lag and then choose the time lag that provides
maximum similarity (e.g., highest absolute correlation).
While most measures of time-series similarity consider the entire time duration into account,
it is possible that the similarity structure among time-series is subject to variation over time.
For example, when a subject’s mental activity is switching between planning their day (i.e., an
executive task) and reminding themselves of a recent meeting (i.e., a memory task), the similarity
structure among the time-series at brain regions could be different. In such cases, a desired pattern
of similarity among time-series may only be exhibited for short periods of time, which need to be
determined from the data. An example of an approach that simultaneously identifies the relevant
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time windows for computing time-series similarity and uses this metric to cluster the time-series
can be found in [Atluri et al. 2014].
3.4.4 Spatial Map Similarity. Two spatial maps can be considered similar if they show similar
values at corresponding locations, which can be captured using standard proximity measures such
as Euclidean distance. However, spatial maps can often suffer from small misalignments due to
geo-registration errors, which can result in misleading distance metrics. Further, it is often more
useful to compute similarity over smaller sub-regions in the map that contain foreground objects
than considering the similarity over the entire map. The Earth’s mover distance (EMD) [Rubner
et al. 2000] is one such metric that is robust to changes in the alignment of images, which is based
on the minimal cost that must be paid to transform one image into the other. Spatial map similarity
can also be computed on the basis of features such as image segments extracted from the data.
3.4.5 ST Raster Similarity. Network-based representations of ST rasters can be used to assess if
two rasters are similar or not. For example, link and node similarity scores [Berg and Lässig 2006;
Li and Yang 2009] can be used to compute the similarity among the corresponding links and nodes
of ST rasters. ST raster similarity can also be computed on the basis of features extracted from their
network representations [Soundarajan et al. 2013].
4
4.1

PROBLEMS AND METHODS
Clustering

Clustering refers to the grouping of instances in a data set that share similar feature values. When
clustering ST data, novel challenges arise due to the spatial and temporal aspects of different types
of ST instances. For example, clustering locations based on their time series of observations in an ST
raster data has to ensure that the discovered clusters are spatially contiguous. Ignoring this spatial
information can lead to fragmented clusters of locations that are difficult to interpret and have
salt-and-pepper errors. Since many clustering algorithms use similarity measures for identifying
groups of similar instances, techniques for computing similarity among ST instances (described in
Section 3.4) will come in handy while clustering ST data. In the following, we describe some of the
common methods for clustering different types of ST data instances.
4.1.1 Clustering Points. There are two different objectives that are of interest when clustering ST
points. One objective involves finding clusters that have an unusually high density of ST points, also
termed as hot-spots. This can be used for finding outbreaks of diseases or social movements, where
there is a dense conglomeration of events both in space and time [Gomide et al. 2011; Kulldorff
et al. 2005]. This problem is also referred to as ‘event detection’ in the literature on mining social
media data [Feng et al. 2015]. The problem of finding hot-spots has been studied in the spatial
statistics literature using methods such as the spatial scan statistic [Kulldorff 1997], which explores
all possible circular shaped regions of different sizes to determine regions where the incidence of
points is significantly higher than expected. Generalization of the scan statistic for ST data, termed
as space-time scan statistics, have been explored for studying disease outbreaks, crime hot-spots,
and events in twitter data [Cheng and Wicks 2014; Kulldorff 2001; Kulldorff et al. 2005]. While
these early works provide a promising framework, open questions pertaining to shapes of regions
[Eftelioglu et al. 2016; Takahashi et al. 2008], background distribution [Tango et al. 2011], and speed
of search [Agarwal et al. 2006] are being investigated. Approaches for clustering spatial objects
such as CLARANS [Ng and Han 2002] can also be extended in the ST domain for finding groups of
objects that are related in both space and time.
The second objective involves finding clusters of ST points that also have similar non-ST attributes.
For example, given a collection of crime events, we may be interested in finding regions in space
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Fig. 5. Mobility profiles discoverd from a user’s GPS traces based on clustering of trips. (Figure taken from
[Trasarti et al. 2011])

and time with similar crime activities. This clustering objective has been studied in the context of
crime data [Eftelioglu et al. 2014], twitter data [Abdelhaq et al. 2013; Chierichetti et al. 2014; Ihler
et al. 2006; Walther and Kaisser 2013; Weng and Lee 2011], geo-tagged photos [Zheng et al. 2012],
traffic accidents [Zheng et al. 2012], and epidemiological data [Glatman-Freedman et al. 2016]. A
number of techniques for clustering ST points are based on the DBSCAN algorithm [Ester et al.
1996], which is a widely used method for finding arbitrarily shaped clusters of spatial points based
on the density of points. ST-DBSCAN [Birant and Kut 2007] is one of the popular extensions of
DBSCAN that defines two separate distances between ST points: one that captures spatial attributes
and another that captures temporal and non-ST attributes. Computing distance based on spatial,
temporal and non-ST attributes separately and having separate thresholds for them provides the
user a flexibility to determine the desired spatial density that is relevant to the problem at hand.
However, several challenges including heterogeneity in space and time, varying densities of clusters,
and sampling bias inherent in the data are yet to be addressed.
4.1.2 Clustering Trajectories. When clustering trajectory data, we are often interested in finding
groups of trajectories that are similar to each other across the entire duration of trajectories.
For example, clustering of trajectories has been used to find groups of hurricanes with similar
trajectories [Lee et al. 2007], groups of taxi traces that follow a similar route [Liu et al. 2010],
or groups of moving objects that follow the same motion in video streams [Gaffney and Smyth
1999]. A review of techniques for clustering trajectories is presented in [Kisilevich et al. 2010]. Two
important aspects of trajectory clustering methods are the choice of distance measure (see Section
3.4.2) and the choice of clustering technique [Morris and Trivedi 2009]. One category of methods for
clustering trajectories involves using mixture modeling approaches [Alon et al. 2003; Chudova et al.
2003; Gaffney and Smyth 1999], e.g., mixtures of regression models where a different regression
model is learned for every cluster of trajectories [Gaffney and Smyth 1999]. A different approach
has been explored in [Trasarti et al. 2011], where a two-step clustering method was proposed to
find mobility profiles of users based on their GPS traces. Figure 5 shows two sets of discovered
profiles (A and B), along with several noisy trips that do not conform to these profiles.
In some trajectory clustering problems, we are interested in finding groups of trajectories that
share similarity in only a short duration of the trajectory. One of the methods developed for this
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problem is a partition-and-group framework called TRACLUS [Lee et al. 2007], which first partitions
each trajectory into smaller line segments based on a minimum description length (MDL) principle,
and then groups line segments based on their similarity using a DBSCAN-based approach.
A related area of research is the identification of “moving clusters” of trajectories, where moving
bodies may join or leave a cluster as it progresses in space over time. This is a common pattern that
is observed in several applications, e.g., migrating flocks of animals or convoys of cars. Algorithms
for detecting moving clusters of trajectories have been developed in [Kalnis et al. 2005] and in
subsequent studies [Dodge et al. 2008; Jeung et al. 2008; Li et al. 2010]. More recently, [Zhang et al.
2016] proposed an iterative framework, GMove, that alternates between two tasks: assigning users
to groups and modeling group-level mobility, using an ensemble of hidden Markov models.
4.1.3 Clustering Time Series. A central objective when clustering time series derived from ST
raster data (using the series of measurements at every location) is to find spatially coherent groups of
locations with similar temporal activity. This problem has been approached from multiple directions.
One direction is to use traditional clustering schemes such as k-means clustering [Mezer et al. 2009],
hierarchical clustering [Goutte et al. 1999], shared nearest neighbor (SNN) clustering [Steinbach
et al. 2003], and normalized-cut spectral clustering [Van Den Heuvel et al. 2008] to cluster time
series (using the time series similarity measures discussed in Section 3.4.3). The clustering of Sea
Level Pressure time series, studied in climate science, from all locations on the Earth’s surface is
shown in Figure 6. Data from 1982 to 1993 was used to find these clusters and Pearson’s correlation
coefficient was used to assess similarity between time series.
Since traditional approaches for time series clustering do not guarantee that the resultant clusters
are spatially contiguous, this is typically addressed
in a post-processing step either by increasing the
number of clusters [Smith et al. 2009] or by separating clusters with non-contiguous sets into multiple
contiguous clusters. Another direction is to directly
incorporate spatial contiguity in the clustering process, e.g., by using ‘region growing’ approaches or
by enforcing contiguity constraints in the clustering
technique. Region growing approaches [Bellec et al.
2006; Heller et al. 2006; Lu et al. 2003b] work by
merging spatially adjacent locations that are highly
similar to each other (defined using similarity mea- Fig. 6. Clusters found using SNN clustering of
sures discussed in Section 3.4.3) into a single clus- Sea Level Pressure data (1982-1993). (Figure taken
from [Steinbach et al. 2002])
ter until a minimum number of clusters has been
achieved or no cluster can be further grown without violating a similarity criterion. While region
growing approaches ensure that every cluster has similar locations, they do not ensure that the
different clusters are dissimilar. On the other hand, clustering approaches that utilize contiguity
constraints [Blumensath et al. 2013; Craddock et al. 2012] do not have this problem as these approaches ensure that locations within a cluster are highly similar to each other than locations that
are in two different clusters.
4.1.4 Clustering Spatial Maps. When clustering spatial maps (sets of measurements from all
locations in an ST raster on different times), the central objective is to find groups of time stamps
that have similar spatial maps, e.g., time stamps with similar maps of brain activity [Liu and Duyn
2013]. Similarity measures among spatial maps, such as those discussed in Section 3.4.4, can help
in discovering meaningful clusters of spatial maps irrespective of data artifacts such as changes
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in alignment and registration among the locations of different maps [Liu et al. 2013]. While these
methods are useful in producing temporally contiguous segments with similar spatial activity (due
to the temporal auto-correlation in the data), in some cases, we may be interested in identifying
non-contiguous groups of time stamps. One way to ensure that a given cluster of spatial maps is
not due to temporal auto-correlation is to evaluate this in a post-processing step or use a smaller k
such that distant time points are grouped into clusters.
4.1.5 Finding Dynamic ST Clusters. A common clustering problem that is of interest when dealing
with ST raster data is to identify sub-regions of space and time that show coherent measurements,
termed as ‘dynamic ST clusters’ [Chen et al. 2015]. The discovery of dynamic ST clusters can help
in detecting phenomena that only influence a subset of locations during a subset of time points. For
example, water bodies that grow and shrink in space over time, e.g., lakes and reservoirs, can be
identified as dynamic ST clusters in remote sensing data, as they appear as coherent observations
in subsets of space and time. Note that a dynamic ST cluster may evolve over time and thus change
its shape, size, and appearance as we progress in time. Hence, while some locations are retained
across consecutive time stamps, the cluster assignments of locations are dynamic and the clusters
can grow and shrink over time. A recent work [Chen et al. 2015] explored a novel approach to
identify dynamic ST clusters in raster data sets for detecting surface water dynamics, which used
an iterative algorithm to first identify the set of ‘core’ locations that are part of a dynamic cluster
across all time stamps, and then growing around the core locations at every time stamps to capture
the dynamic behavior occurring at the boundaries. Similar formulations can be developed for
identifying other types of dynamic clusters in ST raster data, e.g., a moving cluster of locations
such as the evolution of a hurricane.
4.1.6 Clustering ST Rasters. Given a collection of ST raster data sets, possibly collected over
different spatial regions and time periods, it is useful to identify groups of ST raster data sets
considered as individual instances. This has applications in several domains dealing with ST raster
data such as climate science and neuroimaging. For example, in climate science, different climate
models produce simulations of global climate variables as ST rasters, which when clustered sheds
light on the similarity of climate models [Steinhaeuser and Tsonis 2014]. ST rasters can be clustered
by using similarity measures among ST rasters as basic building blocks, which typically involve
extracting features from network representations of ST rasters (discussed in Section 3.4.5). For
example, [Yu et al. 2015b] found groups of ST rasters by first constructing networks from ST
rasters and then grouping the resultant networks using a network module-finding approach from
[Newman 2006]. However, extracting such high higher-order features from ST rasters is non-trivial
and often requires domain expertise.
4.2

Predictive Learning

The basic objective of predictive learning methods is to learn a mapping from the input features (also
called as independent variables) to the output variables (also called as dependent variables) using a
representative training set. In spatio-temporal applications, both the input and output variables
can belong to different types of ST data instances, thus resulting in a variety of predictive learning
problem formulations. In the following, we discuss some of the predictive learning problems that
are commonly encountered in ST applications based on the type of ST data instance used as input
variables.
4.2.1 Time Series. Given an ST raster data, we can consider the problem of predicting an output
variable (continuous or categorical) at every location of the raster using the time-series at every
location as input variables. The use of time-series as input features is quite common in a number of
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classification and regression problems. For example, the temporal dynamics of audio frequencies is
used to classify words or sentences in human speech recognition problems. This can be achieved
by using recurrent neural networks [Graves and Schmidhuber 2009; Mikolov et al. 2010], which are
extensions of artificial neural networks with appropriate skip connections among the neural nodes
to model information delay. Another approach to incorporate the temporal context of time-series
features in classification problems is the use of shapelets [Hills et al. 2014; Ye and Keogh 2009].
Shapelets are time-series subsequences that are discriminative in nature, i.e. their occurrence
is selective to certain classes. While these techniques provide an ability to model the temporal
characteristics of input features, there is a need to develop novel methods that can take into account
the spatial information among the time-series in ST rasters. For example, instead of predicting
the class label at every location using its time-series independently, we can leverage information
about spatial neighborhoods to enforce spatial consistence among the labels at nearby locations.
Variants of recurrent neural networks that include spatial features for spatio-temporal prediction
have been explored in [Jain et al. 2016; Jia et al. 2017a,b]. Latent space models that use topological
as well as temporal attributes of locations have also been developed for real-time traffic prediction
using time-varying information from sensor recordings [Deng et al. 2016]. Time series can also be
constructed from trajectory data, where the objective is to predict the future location of a moving
object (or a group of objects), given their past history of visited locations [Horton et al. 2014; Li
et al. 2016].
4.2.2 Spatial Maps. In this class of predictive problems, a scalar output variable has to be
predicted at a time-step of the ST raster, using the spatial map at the same time-step as input
variables. Some examples of applications that use spatial maps as input features include image
classification and object recognition problems, where a categorical value has to be assigned to every
image or sub-region in an image using the information contained in spatial maps. A classification
approach that has recently gained widespread recognition in the computer vision community
is deep convolutional neural networks (CNN) [Krizhevsky et al. 2012; LeCun and Bengio 1995],
that use the spatial nature of inputs to share model parameters and provide robust generalization
performance. In spatio-temporal applications, a promising research direction is to use the temporal
auto-correlation among consecutive spatial maps to share the parameters of CNN models over time.
Spatio-temporal extensions of CNN based learning frameworks have been developed in [Karpathy
et al. 2014; Taylor et al. 2010]. Extensions of neural networks that use both the spatial and temporal
information of data have also been developed in [Ghosh and Deuser 1995; Stiles and Ghosh 1997],
where the network design was inspired by the biological information of habituation mechanisms
in neuroscience.
4.2.3 ST Rasters. Another class of predictive learning problems is to use the entire information
in an ST raster as input variables to predict a scalar output variable. An example of this is to predict
if a subject has a mental disorder or note based on their fMRI scan, stored as an ST raster. This
has tremendous applications in diagnosing mental disorders which is currently done in a very
subjective manner. Another application that is increasingly becoming popular in the realm of brain
imaging is that of ‘brain reading’ [Norman et al. 2006], where the objective is to determine the
nature of activity (e.g., planning, memorizing, recollecting etc.) based on measured spatio-temporal
activity from the brain, represented as an ST raster.
A naïve approach to this problem could involve representing every ST raster with |S| locations
and |T | time stamps as a vector of size |S| × |T |, and employing traditional classification schemes
on these vector representations, e.g., linear discriminant analysis and support vector machines
[Ku et al. 2008]. Note that this is only possible when the ST grid of every grid is perfectly aligned
with each other, such that their sets of locations and time stamps are identical. This is not usually
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the case with resting state fMRI scans, because the time points in one subject’s scan cannot be
matched with those of another subject. Furthermore, especially in fMRI data, the large number of
spatial locations (typically hundreds of thousands) and time points (typically hundreds) leads to
millions of potential features, where the number of instances are often tens to hundreds, leading to
the phenomena of model overfitting. Hence, it is important to use derived features from ST rasters
that summarize the ST activity of every raster, as described in Section 3.3.5. Alternatively, tensor
learning based approaches [Bahadori et al. 2014; Yu et al. 2015a; Zhou et al. 2013a] provide a way to
reduce the model complexity by making use of the spatial and temporal dependencies among the
input features, thus showing a promise in predictive learning problems involving input ST rasters.
4.2.4 ST Reference Points. A common predictive learning problem in spatio-temporal applications is to predict the response at a certain location and time using observations collected at
other locations and time stamps (often in ST neighborhoods). This is important in a number of
domains, e.g., while estimating an ecological variable over every location and time using remote
sensing observations at nearby locations and time stamps. The problem of land cover classification
(estimating a categorical label at every location and time indicating its propensity to belong to a
land cover type) has been heavily studied in the remote sensing literature for a variety of problems
[DeFries and Chan 2000; Jun and Ghosh 2011; Li et al. 2014b; Vatsavai 2008], e.g., the mapping of
surface water dynamics using multi-spectral remote sensing data [Karpatne et al. 2016b; Khandelwal
et al. 2017]. As another example, the outbreak of influenza at a given location and time can be
predicted based on web searches [Ginsberg et al. 2009] and twitter messages [Culotta 2010] at
neighboring locations and times. There are two classes of methods that are relevant for making
predictions at ST reference points: methods that use the temporal information to predict values
at nearby time points, and methods that use the spatial information to estimate values at nearby
spatial points. We discuss both these classes of methods in the following.
Using Temporal Information: In many domains such as climate and health, estimation (or forecasting) of the future conditions based on present and past conditions is desired. For example, the
sea surface pressure and temperature for the present month can be predicted based on values in
the previous months. Similar problems are studied in predicting closing stock prices at the New
York Stock Exchange and in predicting sales in the manufacturing industry [Montgomery et al.
2015]. Some of the widely used methods for time-series forecasting problems include exponential
smoothing techniques [Gardner 2006], ARIMA models [Box and Jenkins 1976], and state-space
models [Aoki 2013]. Another type of methods for making predictions at time stamps include dynamic Bayesian networks such as hidden Markov models and Kalman filters [Harvey 1990; Rabiner
and Juang 1986], that estimate the most likely sequence of latent values at time stamps using the
temporal auto-correlation structure. Techniques that make predictions in time need to be modified
to include the spatial context in spatio-temporal applications. As an example spatio-temporal
Granger causality models, that use both spatial and temporal information in the regression models,
have been explored to model relationships among ST variables [Lozano et al. 2009b; Luo et al. 2013].
Using Spatial Information: There is a vast body of literature on spatial prediction methods that
take into account the spatial auto-correlation structure in the data to ensure spatially coherent
results. This includes the use of spatial auto-regressive (SAR) models [Kelejian and Prucha 1999],
geographically weighted regression (GWR) models [Brunsdon et al. 1998], and Kriging [Oliver and
Webster 1990] in the spatial statistics literature. Markov random field based approaches that are
naturally suited to handle the spatial auto-correlation in the data have also been widely studied
[Kasetkasem and Varshney 2002; Schroder et al. 1998; Zhao et al. 2007]. There is a promise in
informing such techniques with the temporal nature of ST points in spatio-temporal applications,
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such that both spatial and temporal auto-correlation are incorporated in the modeling framework.
For example, spatio-temporal Kriging approaches have been used in a number of applications in
climate and environmental modeling [Cressie and Wikle 2015], where time is treated as another
dimension while learning covariance structures in space and time. A related area of research is
spatial item recommendation [Wang et al. 2017], where the preference over spatial items (e.g.,
restaurants or tourist attractions) have to be predicted in a time-varying manner, using social
network information and the history of preferences of every user.
4.3

Frequent Pattern Mining

Frequent pattern mining is the process of discovering patterns in a data set that occur frequently
over multiple instances in a data set, e.g., frequently bought groups of items in market-basket
transactions. Given the rich variety of data types and instances in ST applications, there are several
categories of frequent pattern mining problems that can be formulated in the presence of spatial
and temporal components, as described in the following.
4.3.1 Co-occurrence Patterns in ST Points. Given a collection of points such as ST events of
varying types, a common pattern of interest is co-occurrence patterns, which are basically subsets
of ST event types that occur in close spatial and temporal proximity of each other. For example,
given a data set of crime and other events in a city, we may be interested in finding ST event
types that occur together (e.g., bar closing and drunk driving). Co-occurrence patterns has been
studied in the context of spatial datasets as ‘co-location’ patterns for more than a decade [Huang
et al. 2004; Wang et al. 2013; Xiao et al. 2008]. In the spatial statistics literature, co-locations have
been studied using measures such as the Ripley’s cross-K function [Dixon 2002], which capture
statistical patterns of attraction or repulsion among pairs of spatial point types.
One of the first approaches for discovering co-occurrence patterns include an Apriori-principle
based approach developed by [Pillai et al. 2012]. This approach relies on an interestingness measured
termed as spatio-temporal co-occurrence co-efficient to capture co-occurrence in spatial and temporal
dimensions. A filter and refine approach was later proposed in [Pillai et al. 2013] to tackle the
problem of discovering co-occurrence patterns in large event databases. Constructing rules from cooccurrence patterns have also been studied in [Pillai et al. 2014]. Spatio-temporal extensions of the
cross-K function have been proposed in [Lynch and Moorcroft 2008], where the ST neighborhood
of ST points is used for identifying attraction or repulsion patterns among ST point types in both
space and time.
4.3.2 Sequential Patterns in ST Points. Sequential patterns have been studied in the context of
ST event data, where the occurrence of ST events of a particular type can trigger a sequence of ST
events of other types. For example, a car accident on a freeway could trigger a traffic jam in its ST
neighborhood. Sequential patterns have been originally defined in the context of market-basket
transactions where sequences of transactions from every customer are available and the goal is
to discover ordered list of items appearing with high frequency [Agrawal and Srikant 1995]. An
approach for discovering sequential patterns of ST event types was presented in [Huang et al. 2008].
This approach was able to discover ordered lists of event types such as f 1 → f 2 → . . . → fn , where
events belonging to type f 1 trigger events of type f 2 , that further triggers events of type f 3 and a
series of events resulting in events of type fk . They developed an Slicing-STS-Miner approach to
efficiently discover statistically significant sequential patterns of events. A partially-ordered subsets
of event types, referred to as cascading spatial temporal patterns [Mohan et al. 2010, 2012], have also
been studied to capture event sequences whose instance are located together and occur in successive
stages. Some of the key challenges in mining ST sequential patterns include defining interesting
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measures that capture meaningful non-spurious patterns and developing efficient approaches to
discover interesting patterns from an exponentially large space of candidate patterns.

4.3.3 Sequential Patterns in Trajectories. Sequential patterns in trajectories are sequences of
spatial locations or regions that are visited by multiple moving objects in the same order, and are
thus covered by multiple trajectory instances. For example, given the trajectories of tourists visiting
a city, we can find sequential patterns that represent the typical behavior of tourists as they visit an
attraction and move to another. Although trajectories can be treated as multi-dimensional sequences
where the spatial identifiers (e.g., latitude and longitude) of visited locations are used as dimensions,
traditional sequential pattern mining approaches are unsuitable for finding sequential patterns
of trajectories because of their inability to handle the spatial nature of information contained in
trajectories. For example, the locations visited by moving objects may not exactly match but be in
close spatial vicinity to be considered as part of the same pattern. Moreover, trajectories belonging
to a common sequential pattern are expected to follow similar spatial path but not exactly the
same path. To handle the spatial aspect of trajectories in the process of mining sequential patterns,
[Tsoukatos and Gunopulos 2001] used ordered sequences of spatial elements such as rectangular
regions to define sequential patterns. A trajectory is then considered to be part of a sequential
pattern if every line segment joining consecutive locations of the trajectory are enclosed within the
rectangular regions of the pattern. A similar approach for sequential pattern mining was developed
in [Cao et al. 2005] where pattern elements were defined as spatial regions around frequent line
segments of trajectories. Spatio-temporal extensions of association rules, termed as spatio-temporal
association rules (STARs) [Verhein and Chawla 2006], have also been explored to find frequent
patterns of geographic regions visited by moving objects [Verhein and Chawla 2008]. Approaches
for mining periodic patterns in trajectories, where moving objects periodically visit sequences of
locations in time, have been developed in [Li and Han 2014]. Approaches to handle uncertainty
have also been studied to handle the incompleteness of the data and artificial noise introduced in
privacy-sensitive applications [Chen et al. 2016; Li et al. 2013].
Another aspect of trajectories that is important to
consider while mining sequential patterns in some
applications is the presence of timing constraints
among visited locations. For example, we may be
interested in finding sequential patterns where the
time duration between consecutive locations are
similar across the trajectory instances, in addition
to the similarity of paths followed by the moving
objects [Giannotti et al. 2007]. This approach was
used to discover sequential patterns in GPS traces
of trucks in Athens, Greece. One resultant pattern
∆t 1

∆t 2

A −−−→ B′ −−−→ B′′ is shown in Figure 7 that captures Fig. 7. Example of a sequential patterns found in
how the trucks visit region A first, then visit region GPS traces of trucks in Athens, Greece. (Figure
B′, and then visit region B′′, with fixed time gaps taken from [Giannotti et al. 2007])
between the locations. Another way of introducing
timing constraints is to consider sequential patterns of trajectories that are in close temporal
proximity of each other, i.e., the moving objects visit a sequence of locations at similar points in
time. Such patterns are referred to as ‘flock patterns’ [Vieira et al. 2009] or ‘convoys’ [Jeung et al.
2008] that are used for finding groups of objects moving together.
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4.3.4 Motif Patterns in Time-series. In the context of ST raster data, time-series motifs represent repeated temporal measurements observed across multiple spatial locations. For example, in
vegetation time series data of agricultural farms, the harvesting cycles of crops can be discovered
as time-series motifs that recur at multiple farm locations belonging to the same crop type. Timeseries motifs have been extensively studied in applications such as ecology, medicine, finance, and
production industry [Mueen 2014; Torkamani and Lohweg 2017]. While the naïve approach for
discovering motifs is quadratic in the length of the time series, approximate approaches [Chiu et al.
2003; Meng et al. 2008; Minnen et al. 2007] have also been designed. Recently, the development
of efficient approaches for constructing ‘matrix profile’ [Yeh et al. 2016], a vector comprised of
minimum non-trivial distance for each subsequence in time series A with subsequences in time
seies B, have enabled an efficient approach for motif discovery [Zhu et al. 2016]. At the heart of this
approach for computing the matrix profile is a fast Fourier transform based algorithm for efficiently
computing z-normalized Euclidean distance between two time series subsequences. Using this
approach, subsequence similarity in any given time series can be efficiently computed and the
subsequences that are highly similar correspond to motifs.
While traditional formulations for motif discovery assume independence among the time-series,
in ST applications, there is a strong spatial auto-correlation among the time-series at nearby
locations, limiting the usefulness of existing formulations. This is because any time-series motif
observed over a set of locations may also show up in the neighborhood of those locations with minor
variations, resulting in a number of redundant patterns that look almost similar. One approach
to address this challenge is to enforce motif discovery algorithms to identify recurring temporal
patterns that appear synchronously in spatially coherent regions, in contrast to isolated points in
space. This would result in the discovery of physically relevant processes and events that span
both space and time, e.g., droughts or floods affecting an ST region in climate data sets. We can
call such ST motifs as ‘spatially coherent time-series motifs.’ It may also be useful to explore a
slightly modified version of this problem where locations belonging to the same region do not
synchronously show a common pattern but instead there is a gradual evolution of temporal activity
across them. Such ‘evolving ST motifs’ can help us detect frequent ST phenomena that gradually
moves across locations in a region.
An active research direction is to discover structural patterns in ST data that elucidate complex
spatial and temporal dynamics. For example, [Zhou and Matteson 2015] developed approaches
to discover ST dynamics in ambulance demand data for ambulance fleet management. A major
challenge in this direction is to handle the heterogeneity in the presentation of different patterns
in space and time. Sparsity of the underlying data also poses a challenge to capture patterns of
interest. While early efforts in using tensor-based factorization [Takahashi et al. 2017], multi-task
learning [Zhao et al. 2015], and spatio-temporal kernel density estimation [Zhou and Matteson
2015] approaches have shown promise, their effectiveness in handling heterogeneity and sparsity
is yet to be investigated.
4.3.5 Network Patterns in ST Rasters. Given a collection of ST rasters, we are interested in
discovering patterns of structure in network-based representations of ST data (described in Section
3.3.5), which occur frequently in multiple ST rasters. A commonly studied type of network structure
is the arrangement of communities in the network, where a community represents a group of
entities (nodes or links) that interact among themselves more than they do with the rest of the
entities [Leskovec et al. 2010]. Another type of network structure is network motifs [Milo et al.
2002], which are the basic building blocks of the connectivity patterns that frequently occur across
multiple network instances. The use of network motifs has been explored in several domains, e.g.,
bioinformatics [Shen-Orr et al. 2002] and social networks [Milo et al. 2002].
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In ST applications, network motif discovery has been used to identify the inherent ST structures
of brain fMRI data [Sporns and Kötter 2004]. Network communities in ST rasters can reveal
interesting high-level patterns. For example, in climate data, a community can represent a set of
distant locations that are experiencing similar climatic conditions and show consistent temporal
activity. While the discovery of network patterns can prove to be a valuable tool in the analysis of
ST raster data sets, care must be taken while using them as the spatial auto-correlation in the data
can result in a number of spurious edges that can lead to misleading results. Hence, during the
evaluation of patterns in network-based representations of ST rasters, it is important to filter out
patterns arising from spatially neighboring locations for the discovery of long-range dependencies
between distant locations.
4.4

Anomaly Detection

Anomalies are traditionally defined as instances that are remarkably different from the majority
of instances in the data set. In ST applications, detecting anomalies (also referred to as outliers)
can help us identify interesting but rare phenomena, e.g., an anomalous trajectory taken by a taxi
vehicle or the changing frequencies of floods and droughts in certain areas due to climate change
or human actions that change the ecology of a region. Given a method for computing pair-wise
similarities among ST instances (discussed in Section 3.4), anomalies can be identified as instances
that are highly dissimilar from the rest of the instances. Detecting anomalies can also be seen as
a by-product of clustering, since instances that do not easily become part of a cluster or form an
isolated cluster that is quite far from the other clusters can be termed as anomalies. There are two
areas of work that are relevant in the context of detecting anomalies in ST data. First, a number
of techniques have been developed for identifying anomalies in time series data, which are also
called as discords. This includes approaches for univariate time series [Bishop 1994; Keogh et al.
2005, 2002; Wei et al. 2005] as well as multivariate time series [Baragona and Battaglia 2007; Chen
et al. 2008; Galeano et al. 2006; Takeishi and Yairi 2014] A survey of anomaly detection approaches
for temporal data is presented in [Gupta et al. 2014]. Second, approaches for detecting anomalies
in spatial data, also termed as spatial outliers, have also been developed [Shekhar and Chawla
2003]. Spatial outliers are defined as points whose non-spatial attributes are different from those
that are in its close spatial neighborhood . Techniques for detecting spatial outliers vary in their
choice of method for constructing local neighborhood and assigning anomaly scores, e.g., using
spatial distance measures [Knorr and Ng 1997; Knorr et al. 2000; Lu et al. 2003a], graphical-distance
measures [Kou et al. 2007; Shekhar et al. 2001], and visualization tools such as the Moran’s I scatter
plot, pocket plots, and the Variogram Cloud plot [Anselin 1994; Haslett et al. 1991]. A review of
techniques for spatial outlier detection is presented in [Aggarwal 2017]. However, the joint presence
of spatial and temporal aspects introduces novel ways of describing anomalies in ST data, resulting
in unique anomaly detection problems, as described below.
4.4.1 ST Point Anomalies. The concept of spatial outliers can be easily extended to spatiotemporal domains, where the neighborhood of an ST point is defined with respect to both space
and time. Thus, a spatio-temporal outlier is basically an ST point that breaks the natural ST autocorrelation structure of the normal points. One approach to find ST outliers is to first cluster the
normal points using approaches such as ST-DBSCAN, and then report points that did not conform
well to the discovered clusters [Kut and Birant 2006]. Another category of approaches presented in
[Cheng and Li 2004, 2006] is to aggregate ST clusters at a coarser scale so that the effect of outliers
on the clustering is reduced. ST outliers are then detected by comparing the original clustering
with the coarsened clustering. Note that most ST outlier detection algorithms assume homogeneity
in neighborhood properties across space and time, which can be violated in the presence of ST
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heterogeneity. This can be handled by methods that model the variance of normal instances in the
local neighborhood of every point, along with its expected value [Sun and Chawla 2004].
4.4.2 Trajectory Anomalies. There are several ways we can identify a trajectory to be anomalous.
A common approach is to compute pair-wise similarities among trajectories (discussed in Section
3.4.2) and identify trajectories that are spatially distant from the others. For example, [Lee et al.
2008] explored varying notions of distances between trajectory sub-sequences (such as perpendicular distance, parallel distance, and angle distance) and identified a trajectory to be anomalous
if it had very few neighbors within a certain threshold. Distance based methods for detecting
anomalous trajectories have also been explored in [Bu et al. 2009], where trajectories that appear
in distant spatial regions than the rest of the trajectories are considered anomalous. Another class
of approaches is to aggregate the spatial shapes of trajectories using a coarse spatial grid, and
then compute useful characteristics of a trajectory in every spatial grid cell. For example, we can
compute the average direction of trajectories in every spatial grid and identify a trajectory as
anomalous if it deviates from the expected behavior in a number of grid cells that it covers, as
proposed in [Ge et al. 2010]. A graph-based approach for detecting outliers in traffic data streams
was developed in [Liu et al. 2011], where nodes are regions and edge weights represent the traffic
flow between regions. Traffic outliers were discovered as edge anomalies in this graph, which were
subsequently analyzed for causal interactions using causal outlier trees. Supervised methods for
detecting anomalous trajectory shapes (or motifs) have also been explored in [Li et al. 2006, 2007b].
While the techniques discussed above for identifying trajectory anomalies focused on the spatial
pattern (e.g., shape, distance, or angle) of trajectories, we can also consider a trajectory to be
anomalous if it deviates from its local neighbors as it switches from one cohort to other, e.g., a
zebra moving from one group to another. To identify such anomalies, the approach presented in
[Li et al. 2009] computed the spatial neighbors of every moving object at any given time stamp
(using trajectory information in the immediate history) and designed an anomaly score based
on deviations from the trajectories of neighbors. Such an approach can enable the discovery of
trajectories that are contextually anomalous in their local neighborhoods, and can thus be robust
to aggregate changes such as population shifts. It can also be useful for detecting anomalies in
an on-line fashion, as trajectories that begin to drift apart from their local neighbors will start to
accumulate large anomaly scores in a short number of time stamps.
4.4.3 Group Anomalies in ST Rasters. While techniques described in Section 4.4.1 are well-suited
for discovering ST point anomalies, it is often the case that anomalies appear in ST raster data
as spatially contiguous groups of locations (regions) that show anomalous values consistently
for a short duration of time stamps. Some examples of such group anomalies in ST raster data
include rare events such as cyclones, floods, and droughts that result in abnormally high or low
precipitation in a given region for a certain duration of time, or an abnormal number of tweets or
emergency calls from a spatial region in a small time window.
Most approaches for detecting group anomalies in ST raster data decompose the anomaly
detection problem by first treating the spatial and temporal properties of the outliers independently,
which are then merged together in a post-processing step. For example, the approach presented
in [Wu et al. 2010] made use of the spatial scan statistic to find the top-k contiguous groups of
locations that showed anomalous activity at every time stamp. These groups of locations were
then stitched together in time to find ST groups of anomalous values. A similar line of work has
been explored in [Lu et al. 2007], where the anomalous spatial regions were discovered using an
image segmentation algorithm and a regression technique was applied to track the movement
of the centers of the anomalous regions across consecutive time stamps. Another approach for
detecting anomalies in ST raster data was explored in [Faghmous et al. 2013a] for discovering ocean
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eddies in climate data. Ocean eddies are revolving masses of water in the ocean that appear as local
extremas (depressions or elevations) in a snapshot of sea surface height data on any given time
stamp, and these extremas keep moving over time. The approach presented in [Faghmous et al.
2013a] identified eddies as local extremas in space using a bottom-up thresholding scheme, which
could then be stitched in time using multiple hypothesis object tracking procedures [Faghmous
et al. 2013b]. An orthogonal approach to detecting anomalies in ST rasters is to find anomalies in
time series data that can then be merged across space [Faghmous et al. 2012].
Approaches for detecting anomalies in ST raster that jointly use information about the spatial
and temporal aspects of the data have also been developed. For example, in the application of
detecting abnormal activities in crowded scenes from surveillance camera videos, anomaly detection
approaches using models of normal activity have been developed in [Kratz and Nishino 2009; Li
et al. 2014a]. In this problem, a major difficulty is to come up with a notion of normal motion
activity that can be differentiated from anomalous activities, which can be quite complex in crowded
atmospheres. To address this challenge, [Li et al. 2014a] used a mixture of dynamic textures models
to capture the spatial and temporal salience of normal activities, borrowing from the vast literature
on activity recognition in the area of computer vision. Spatial and temporal anomaly maps are then
constructed at multiple spatial scales, which are then integrated together using a conditional random
field framework. On the other hand, the approach presented in [Kratz and Nishino 2009] model
the steady-state motion behavior of normal space-time volumes that can capture the variations in
the ST data and can compactly represent the overall video volume. These models are then used to
detect unusual motion activities as statistical deviations from the normal model.
A special type of group anomaly in ST rasters is bursts of activity in the time series of locations,
detected for short time periods. For example, Twitter queries involving specific terms such as
‘earthquake’ can be discovered as spatio-temporal burst events related to natural disasters. One of
the early works on detecting ST burst events includes the work by [Lappas et al. 2012], where both
the spatial and temporal nature of burst patterns were jointly taken into consideration. A system
for mining ST burst events, termed as the Spatio-TEmporal Miner (STEM), has been developed in
[Lappas et al. 2013].
4.5

Change Detection

The problem of change detection involves identifying the time point when the behavior of a system
undergoes a significant deviation from its past behavior. Detecting changes in ST data is useful
in several applications, e.g., discovering transitions from an El Nino phase to a La Nina phase in
climate science [Miralles et al. 2014] or understanding the switching of the state of the brain from
planning to cognition in brain imaging data [Tang et al. 2012].
Change detection has been extensively studied in the context of time-series data, where the
objective is to determine time intervals (segments) that exhibit homogeneous properties [Huang
et al. 2013; Keogh et al. 2004; Liu et al. 2008; Mithal et al. 2012]. Different types of homogeneity
in segments such as mean [Bernaola-Galván et al. 2001; Horváth 2001], variance [Andreou and
Ghysels 2002; Inclan and Tiao 1994; Whitcher et al. 2000], and distribution statistics [Carpena and
Bernaola-Galván 1999; Grosse et al. 2002] have been used for identifying time series segments and
their changes. Apart from time series segmentation, other approaches has also been explored for
identifying changes in time-series. For example, variational approaches for switching state space
models have been used to discover abnormal breathing episodes during sleep (also known as sleep
apnea) based on chest volume measurements [Ghahramani and Hinton 2000]. Another category of
approaches is change detection methods for periodic or semi-periodic data. [Boriah et al. 2010]
defined an iterative bottom-up approach for change detection in periodic time series for identifying
forest fires, where adjoining annual segments are merged if they are highly similar, after accounting
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for the noise and seasonal variability in the data. Recently, Chandola et al. [Chandola and Vatsavai
2011] defined different types of changes in ‘periodic’ time series (which exhibits annual cycles)
and proposed a Gaussian Process based approach for discovering such changes. Other techniques
for time series change detection have been explored in remote sensing applications [Karpatne
et al. 2016a], e.g., using anomaly detection approaches [Lunetta et al. 2006; Mithal et al. 2011b],
forecasting-based approaches [Chandola and Vatsavai 2011; Liang et al. 2014], and sub-sequence
pattern matching approaches [Salmon et al. 2011; Zhu et al. 2012].
In ST applications involving raster data, it is important to consider the spatial context of timeseries at every location to identify changes in both space and time. A recent work by [Chen et al.
2013] defined a new type of change called contextual change, where a time series is considered to
be changing if it deviates from other time series in its local context. The context of a time series
can be defined in a number of ways, e.g., by considering the group of time series that are similar to
the given time series for a period of time, or the time series observed at locations in close spatial
vicinity. Another generalization of change detection problems in ST applications is to determine
the spatial extent and temporal window in an ST raster where a change has likely manifested. For
example, the set of geographic locations and time intervals where loss in vegetation occurred due
to deforestation in a forest area is of interest to ecologists. Recent work has explored some efficient
approaches to capture specific type of events in global vegetation index data. An efficient approach
that enumerates and prunes candidate spatio-temporal windows has been proposed by [Zhou et al.
2013b]. This approach is referred to as a space-time window enumeration and pruning (SWEP)
approach and was shown to be efficient for discovering the space and time subspaces where there
is a consistent decrease in vegetation. [Zhou et al. 2011] defined change detection in the context of
a path (i.e., a highway or a longitude), where their goal is to determine sub-paths where abrupt
changes are seen. They proposed a a sub-path enumeration and pruning (SEP) approach and have
shown its promise in discovering abrupt changes in vegetation across longitudes in Africa.
Approaches that are relevant to spatio-temporal change detection problem have also been
studied in the domain of image analysis where the objective is to determine objects and segments
of videos where they are present [Grundmann et al. 2010; Lhermitte et al. 2008; Moscheni et al.
1998]. Suitability of these approaches to spatio-temporal data such as remote sensing and magnetic
resonance imaging is yet to be explored. A taxonomy of possible generalization of ‘change’ in
spatio-temporal data is provided by [Zhou et al. 2014], where they discuss applications that require
the spatial extent of change to be defined as a point, line-segment, polygon or a network, and the
temporal extent of changed to be defined as a time point or a time interval.
4.6

Relationship Mining

In the domain of time series data mining, relationships among pairs of time series can be discovered
using any of the similarity measures defined over time series instances as described in Section 3.4.3.
However, in ST applications where the time series are also associated with a spatial information
about the location of their observation, relationships in ST data can be defined in numerous ways.
One type of relationship is defined as a similarity/dissimilarity between two distant groups of
contiguous locations. Such long-range relationships are of interest in a number of applications such
as climate science and brain imaging. For example, while positive correlations have generally been
studied in brain imaging [Bassett and Bullmore 2006], negative correlations (also referred to as
teleconnections) among distant locations have been studied in climate datasets [Yang and DelSole
2012]. In such problems, the goal is to determine both the sets of contiguous locations, not just a
pair of locations, that participate in a relationship. Hence, it is important to discover the pairs of
regions as well as their relationships simultaneously. This is because if the regions are discovered
independently of the relationships between them, e.g., using a clustering algorithm, we may miss a
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Fig. 8. Example of a set of nodes and edges that are simultaneously discovered using a tensor decomposition
of spatio-temporal data proposed in [Davidson et al. 2013]. (Figure taken from [Davidson et al. 2013])

number of relationships whose one or both regions fall at the boundary of the clusters. [Davidson
et al. 2013] designed a tensor-based approach to discover regions and relationships among them
simultaneously, with applications in brain fMRI Data. Figure 8 shows the network discovered from
the resting-state fMRI data of a healthy subject, which appears to be highly similar to a commonly
found and widely studied network known as the default-mode network in resting subjects’ fMRI
scans [Raichle and Snyder 2007].
An important consideration when mining relationships in ST data is that the strength of relationships among pairs of regions may vary with time. For example, [Handwerker et al. 2012]
demonstrated that the correlation between time series from the Posterior Cingulate region and
other locations in the brain change with time. Due to this, it becomes necessary to determine the
pair of interacting regions as well as the time window within which they interact. [Atluri et al.
2014] designed a pattern mining based approach to study dynamic relationships among time series
from different regions (with homogeneous time series). This approach was used in the context of
brain fMRI data to discover sets of intermittently synergistic brain regions. An example of such a
set is shown in Figure 9. Another approach was explored in [Kawale et al. 2011] using graph-based
methods for climate science applications, where a new graph was constructed for each time interval
and the relationships within a time interval are discovered from the corresponding graph.
Another important consideration when discovering relationships in ST data is that the relationships among time-series could exist with time lags. This can be because one region has an influence
on the other and the lag is due to the time it takes to conduct the influence through the system.
Such relationships are referred to as lagged relationships and they have been studied in several
applications such as climate science [Chen et al. 2011; Lu et al. 2016]. A more complex version of
lagged relationships is dynamic lagged relationships, where the lagged relationships lasts for a small
interval rather than the entire time. In this case the goal is to determine the two sets of locations
as well as the intervals in each of them that exhibit the desired similarity. A central challenge in
discovering lagged relationships is the increase in the number of degrees of freedom available
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Atluri et al.

Fig. 9. Example of a time series pattern that shows three different brain regions exhibiting high similarity in
different time intervals. This patterns is discovered using the pattern mining approach proposed in [Atluri
et al. 2014]. (Figure taken from [Atluri et al. 2016])

for finding relationships, which can result in spurious detections unless statistical corrections for
multiple hypothesis testing have been carefully performed [Bland and Altman 1995].
While lagged relationships suggest some type of causal association between the two regions,
causal relationships that have been defined in time series data are known to capture truly causal
associations [Granger 1969]. Clive Granger who received Nobel prize for this work defined causality
based on the notion that a cause (x) can predict the effect (y) significantly better than an autoregressive model of the effect (y) itself. This work can be generalized for multivariate time series
data where the effect of all variables on a target variable can be studied. Such an approach does not
take into account the causal relationships that could exist between the causal factors themselves.
This observation motivated the need to develop Granger causal maps for multivariate time series
where the causal relationships between all variables are assessed [Eichler 2013]. These are further
extended to capture sparsity [Arnold et al. 2007] and group structure [Lozano et al. 2009a] in the
underlying model. The framework of Pearl causality has also been explored for identifying causal
interactions in climate science [Ebert-Uphoff and Deng 2012, 2017; Hannart et al. 2016], where
probabilistic graphical models were constructed using causal edges among the nodes.
5

CONCLUSIONS AND FUTURE WORK

In this article, we attempted to provide a broad overview of the field of spatio-temporal data mining
that covers research spanning more than several decades. However, given the vast literature on
this topic and space limitations, we have only been able to cover a small fraction of work in this
fast growing area of research. Still our hope is that this article provides the right foundation for
consolidating the rich and diverse literature on STDM research, that has been explored in different
application contexts for different types of ST data, under a common over-arching structure. The
organization of STDM research presented in this article can be used as a stepping stone for future
reviews on this topic that provide detailed empirical comparisons of different STDM methods,
e.g., in terms of accuracy, efficiency, and scalability. While such evaluations have been performed
in limited contexts for specific topics, e.g., for trajectory pattern mining [Giannotti et al. 2007;
Mamoulis et al. 2004] and time-series data mining [Ding et al. 2008; Esling and Agon 2012], an
extensive evaluation covering the full breadth of STDM problems and methods is missing.
We also hope that this article helps the reader in identifying the relevant set of questions that
can be posed for a given ST data set, and the right choice of methods for pursuing a research
question in STDM. This can be particularly helpful in translating STDM research ideas developed
in one community to another, so that new problems in novel application domains can be solved.
For example, in flow dynamics, data pertaining to the breakup of liquid droplets is generated using
simulations where droplets are present in some spatial locations in arbitrary shapes. The goal in
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these studies is to study droplet dynamics, i.e., changes in shape and position of the droplets with
time under various flow and ambient conditions. Such patterns can be discovered using approaches
for finding moving clusters of migratory animals, discussed in Section 4.1.2 in the context of
clustering trajectories.
The variety of data types, problems, and methods, and the range of emerging application areas
where ST data is being increasingly collected and new scientific questions explored makes spatiotemporal data mining a quintessential melting pot for new research in data mining. One of the
major emerging themes in STDM research is of studying novel representations of ST raster data
sets. Most recent work in this direction involves defining novel types of edges or relationships
between spatial entities and developing effective approaches for discovering them [Agrawal et al.
2017]. Much of the work in this area focuses on capturing ‘static’ edges, while there is increasing
recognition that ‘dynamic’ edges are better suited for representing ST raster data. Approaches for
effectively capturing such edges in spatial-temporal varying graphs have been explored in [Chen
et al. 2010] for applications in climate science.
Another emerging direction where there is tremendous interest is to integratively mine multimodal spatio-temporal datasets. Multi-modal ST datasets are available in domains such as neuroimaging and climate science. In neuroimaging, fMRI and ECG capture the same underlying brain
activity using very different technologies that offer different spatial and temporal resolution. On
the other hand, in climate science, different variables such as temperature, pressure, humidity, and
precipitation are available at the same spatial and temporal resolution. In the former case, the
objective of interest is to construct a best possible image of brain activity based on the images
from different modalities. Recent work in the form of BrainZoom [Fu et al. 2017] is one of the first
efforts in this direction where they developed an optimization based approach to construct the
best possible brain activity image based on fMRI and MEG images. In the latter case, the objective
of interest is to learn a model of the climate system by taking into account the different climate
variables and their inter-relationships [Tye et al. 2016].
Yet another direction that is most relevant in particular to ST data involves tackling the problem
of determining the level of granularity or resolution at which a phenomenon of interest needs to
be searched for, be it clusters, patterns or anomalies. Most of the existing work in ST data mining
overlooks this granularity problem. A closely related contribution is made in the context of network
science where it was found that existing community detection approaches might miss important
substructures in the network as they over-partition or under-partition the network [Fortunato and
Barthélemy 2007]. They also noted that the definition of modularity that these techniques rely upon
can have an inherent resolution limit and argued for the need of developing approaches without such
limits [Fortunato and Barthélemy 2007]. To address this problem, [Delvenne et al. 2010] proposed
an approach to discover clusters that are relevant at different granularities. Similarly, in the context
of ST data, there is a need for understanding the inherent limitations of existing approaches in
tackling the granularity problem and for developing new approaches that can overcome such
limitations.
Since many problems in physical sciences involve the study of processes that are spatio-temporal
in nature, e.g., the dynamics of turbulent flow or the evolution of climate states and weather
patterns, there is a growing interest to incorporate the scientific basis of such processes in the
STDM framework. While traditional STDM methods typically rely only on the information contained in the data, the complex nature of problems and the paucity of observations in scientific
applications requires a principled way of integrating data science methods with the wealth of
scientific knowledge, often encoded as physics (or theory)-based models. This is the paradigm of
theory-guided data science [Karpatne et al. 2017] that is gaining attention in several disciplines
for accelerating scientific discovery from data, and is particularly relevant for STDM applications.
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Atluri et al.

Overall, considering the emerging problems and promising research directions, we anticipate the
relatively young field of STDM to grow significantly in the next decade.
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