A Multiple Subspaces-Based Model: Interpreting Urban Functional Regions with Big Geospatial Data

Abstract

Analyzing the urban spatial structure of a city is a core topic within urban geographical information science that has the ability to assist urban planning, site selection, location recommendation, etc. Among previous studies, comprehending the functionality of places is a central topic and corresponds to understanding how people use places. With the help of big geospatial data which contain affluent information about human mobility and activity, we propose a novel multiple subspaces-based model to interpret the urban functional regions. This model is based on the assumption that the temporal activity patterns of places lie in a high-dimensional space and can be represented by a union of low-dimensional subspaces. These subspaces are obtained through finding sparse representations using the data science method known as sparse subspace clustering (SSC). The paper details how to use this method in the context of detecting functional regions. With these subspaces, we can detect the functionality of urban regions in a designated study area and further explore the characteristics of functional regions. We conducted experiments using real data in Shanghai. The experimental results and outperformance of our proposed model against the single subspace-based method prove the efficacy and feasibility of our model.

Full text

International Journal of
Geo-Information
Article
A Multiple Subspaces-Based Model: Interpreting Urban
Functional Regions with Big Geospatial Data
Jiawei Zhu 1, Chao Tao 1, Xin Lin 2, Jian Peng 1, Haozhe Huang 1, Li Chen 1and Qiongjie Wang 3,*


Citation: Zhu, J.; Tao, C.; Lin, X.;
Peng, J.; Huang, H.; Chen, L.; Wang,
Q. A Multiple Subspaces-Based
Model: Interpreting Urban Functional
Regions with Big Geospatial Data.
ISPRS Int. J. Geo-Inf. 2021,10, 66.
https://doi.org/10.3390/ijgi10020066
Academic Editor: Wolfgang Kainz
Received: 30 December 2020
Accepted: 2 February 2021
Published: 6 February 2021
Publisher’s Note: MDPI stays neu-
tral with regard to jurisdictional clai-
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nal affiliations.
Copyright: © 2021 by the authors. Li-
censee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and con-
ditions of the Creative Commons At-
tribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1School of Geosciences and Info-Physics, Central South University, South Lushan Road,
Changsha 410083, China; jw_zhu@csu.edu.cn (J.Z.); kingtaochao@csu.edu.cn (C.T.);
PengJ2017@csu.edu.cn (J.P.); hz_huang@csu.edu.cn (H.H.); vchenli@csu.edu.cn (L.C.)
2Central and Southern China Municipal Engineering Design & Research Institute Co., Ltd,
Wuhan 430010, China; linxin@citic.com
3China Center for Information Industry Development, Beijing 100083, China
*Correspondence: wangqiongjie@ccidthinktank.com; Tel.: +86-010-68209225
Abstract:
Analyzing the urban spatial structure of a city is a core topic within urban geographical
information science that has the ability to assist urban planning, site selection, location recommen-
dation, etc. Among previous studies, comprehending the functionality of places is a central topic
and corresponds to understanding how people use places. With the help of big geospatial data
which contain affluent information about human mobility and activity, we propose a novel multi-
ple subspaces-based model to interpret the urban functional regions. This model is based on the
assumption that the temporal activity patterns of places lie in a high-dimensional space and can
be represented by a union of low-dimensional subspaces. These subspaces are obtained through
finding sparse representations using the data science method known as sparse subspace cluster-
ing (SSC). The paper details how to use this method in the context of detecting functional regions.
With these subspaces, we can detect the functionality of urban regions in a designated study area
and further explore the characteristics of functional regions. We conducted experiments using real
data in Shanghai. The experimental results and outperformance of our proposed model against the
single subspace-based method prove the efficacy and feasibility of our model.
Keywords:
urban functional regions; multiple subspaces; big geospatial data; urban spatial structure;
sparse subspace clustering
1. Introduction
Analyzing the urban structure of a city is a primary research topic in the urban
geographical information sciences. Therein, understanding different functions and their
corresponding spatial distribution, i.e., functional regions, has drawn a lot of interest [
1
–
3
].
Functional regions describe urban space usage like when and why people visit a place and
provide insights for a group of people. First, it helps decision-makers and researchers to
calibrate the urban planning, assess the urban spatial structure, and study social and spatial
disparities [
4
–
6
]. Second, it provides strangers like tourists with a quick understanding of
the city and benefits social recommendations [
7
]. Moreover, it helps local people to expand
their knowledge of urban operating regularity and regions with similar functions in the
city [
8
,
9
]. However, the functionality of a region may change and develop through human
mobility and activity [
10
], which can result in a functional difference from its original or
intended use. Therefore, it is necessary to update the knowledge about the urban functional
regions in time to capture the urban growth and thus facilitate better urban planning, site
selection, location recommendation, etc.
Related works date back to the 20th century when Goddard [
11
] conducted a case
study on functional regions in central London, using taxi flows collected in a traffic survey.
However, these traditional methods were based on social surveys [
12
,
13
], the access to data
ISPRS Int. J. Geo-Inf. 2021,10, 66. https://doi.org/10.3390/ijgi10020066 https://www.mdpi.com/journal/ijgi

ISPRS Int. J. Geo-Inf. 2021,10, 66 2 of 16
was costly both in time and labor, and the objectivity of the data as well as the accuracy of
study could easily be affected by subjective factors. Fortunately, with the development of
information and telecommunication techniques, it is convenient to fetch big geospatial data
such as GPS (Global Position System) data, GSM (Global System for Mobile) data, Smart
Card Data, Social Media data, etc. This kind of data contains affluent, fine-grained, and
large-scale information on human mobility and activity in both time and space dimensions.
With the influx of available information as such, the study of urban spatial structure
enters a new paradigm that allows an increasing number of studies to combine data mining
techniques with big geospatial data to reveal urban functional regions through human daily
mobility and activity. Qi et al. [
14
] extracted temporal variation of get-on/off amount from
taxi traces data and applied agglomerative clustering to characterize the social function
of city regions. Pan et al. [
15
] used an iterative DBSCAN to recognize the social function
of a region from the patterns of pick-up/set-down dynamics of taxi trajectories within
the area. Yuan et al. [
7
] discovered the function of regions using both POIs data and taxi
trajectory datasets by a topic-based inference model Latent Dirichlet Allocation (LDA).
Liu et al. [
16
] classified Shanghai into six traffic “source-sink” areas that strongly link
to six different functionalities by investigating the temporal patterns of taxi trajectory
data in a seven-day timespan. Pei et al. [
17
] validated that land use information can
be derived from mobile phone data in a case study of Singapore by implementing the
Fuzzy C-means clustering on vectors of which consist of normalized hourly call volume
and the total call volume of aggregated mobile phone data. Zhi et al. [
18
] used social
media check-in data to infer urban internal functional regions by the coupling of Low Rank
Approximation (LRA) and the K-means algorithm. Gao et al. [
19
] applied a popularity-
based LDA model on POIs and incorporated user check-ins to derive latent thematic topics,
functional regions are thus extracted by using clustering methods to group places that are
semantically similar.
Wang et al. [4]
incorporated taxi GPS data with POIs to identify urban
functional regions via the non-negative Matrix Factorization (NMF) and spatial semantic
mining. Chen et al. [
20
] proposed a dynamic time warping (DTW) distance-based k-
medoids method to delineate functional regions based on building-level social media data.
Yu et al. [21]
identified functional regions through the clustering method according to the
travel characteristics derived from the taxi data and utilized POI data to label functionality.
In summary, in terms of data source, researchers have conducted experiments mainly
with taxi data, social media check-in data, mobile phone record data, as well as combi-
nations of data. However, with mobility data like taxi data solely, the characteristics of
functional regions cannot be fully captured because of missing the information of what
people visit a place for. Due to this reason, after detection, some studies introduced
POI data or check-in data to provide travel demands information and label functional
regions, the former is implicit and needed inference while the latter can offer explicit
information. Therefore, in this study, we use taxi data and check-in data as data sources.
Moreover, instead of separately using the mobility and activity information, we incorporate
and transform these two into a form that can simultaneously reveal the dynamic of when
and why people visit places to facilitate the identification of functional regions.
As for the techniques used to derive functional regions, usually, after preprocessing,
the statistics of the data are stored in matrix form, where each vector represents characteris-
tics of one place in terms of human temporal activity. Finding the functional regions, in
reality, is equivalent to finding the underlying patterns in the vector space. Some studies
directly applied clustering techniques. However, the intrinsic dimension of data is often
much smaller. Patterns can be cluttered by redundant information. Therefore, dimen-
sionality reduction is necessary [
22
], some studies introduced methods such as Singular
Value Decomposition (SVD), Low Rank Approximation (LRA), and Non-negative Matrix
Factorization (NMF) to extract valid information. These methods basically assume that
data points distribute in a single lower-dimensional subspace, which is a subset of its
original space. This idea explores the relationships between the data points and the bases
of subspace. Then, the relationships can be represented by new lower-dimensional vectors.

ISPRS Int. J. Geo-Inf. 2021,10, 66 3 of 16
In regard to detecting functional regions, the bases represent patterns of human temporal
activity and each low-dimensional vector denotes how the patterns are combined into a real
place. Therefore, they use shared bases of the low-dimensional space to model observations
and further explore functional regions by identifying clustered places with similar patterns.
Unfortunately, these methods have a disadvantage. As the functional regions can be differ-
ent in nature and have intrinsic complexities, diverse bases with distinct dimensionality
are needed to depict those regions more concisely and accurately. This challenges methods
based on the single space assumption that share the same set of bases.
Therefore, a more general model should consider data lies in a union of low-
dimensional subspaces rather than one single low-dimensional subspace [
23
]. In order to
improve our model, we assert the assumption that all observations lie in a vector subspace
that can be represented by a union of multiple subspaces, such that they can have diverse
sets of bases, and each subspace represents one type of functional region. To discover the
subspaces and subordinate points, we introduce a clustering method called Sparse Sub-
space Clustering (SSC) [
24
,
25
]. SSC acknowledges at the same time that the data belongs to
a mixture of clusters and each having a different low dimensional structure. It imposes
a sparse restriction on the self-expressiveness property of data [
26
]. That is, data point
can only be represented by a linear combination of other points in the same subspace.
By this means, data points that share similar patterns are designated to the same subspace.
Consequently, the functional regions detected using this method characterize themselves
and help us to exhibit the functional structure of the study area. Our contributions are to
express how to use SSC to explore functional regions and to propose the metrics termed
uniqueness degree and richness degree in order to evaluate the conditions of detected
urban functional regions. The metrics are derived from the geometric properties of the
subspaces’ representations.
The rest of the paper is organized as follows. Section 2introduces our proposed model,
gives an overview, and explains some preliminaries. Section 3elaborates on the experiment
settings. Section 4presents the results of our experiment as well as a comparison with a
method based on the single space assumption. Finally, the conclusions of the study are in
Section 5.
2. Methods and Definitions
2.1. Frequency Matrix
Providing the coverage of the study area, taxi GPS points dataset, and check-in dataset,
we can form a frequency matrix
C∈RM×N
. Specifically, first, we partition the study area
into
N
geographical units and divide the timespan of one day into
T
time slots. Then, we
remove abnormal GPS points caused by data recording or transfer error and identify taxi
trips with passengers to further extract drop-off events. At the same time, we filter the
social media check-in dataset to obtain valid check-in events. Next, we match the check-in
events with drop-off events based on both coordinates and timestamps. When finished, the
demand-tags of check-in events are used to label the purposes of matched drop-off events,
and
D
is used to denote the number of categories of demands. All the labeled records are
then mapped onto the geographical units and time slots to which they belong. After that,
for each geographical unit, we can form a column vector
gn∈RM×1
, where
M=T×D
.
Each entry in
gn
denotes the frequency of people visiting this geographical unit in one
time slot for a certain demand, such as going home, working, etc. That is, each column
vector
gn
presents the dynamic activity patterns of one geographical unit. In the end,
C
which represents the dynamic activity patterns of the study area is constructed by stacking
[g1,g2, . . . , gn, . . . , gN].
2.2. Sparse Subspace Clustering
Given the frequency matrix
C
, Sparse subspace clustering (SSC) is then utilized to
reveal underlying relationships between geographical units and thus to detect functional
regions. The key idea of the SSC algorithm in this scenario is to find the sparsest combi-

ISPRS Int. J. Geo-Inf. 2021,10, 66 4 of 16
nation of geographical units
gj
in the matrix
C
to constitute a geographical unit
gi
which
belongs to a subspace
Sk
, i.e.,
gi=∑j6=iZij gj
, where
Z
is a coefficient matrix. Finding the
sparsest combination restricts to use the geographical units in the same subspace to con-
stitute each other only, which means
Zij =
0 if
gj/∈Sk
, and this characteristic is called
self-expression [26]
. As finding the sparsest solution is usually NP-hard,
Z
is optimized by
solving a relaxed convex l1optimization problem to guarantee sparsity.
minimize kZkl1, s.t. CZ =C,Zii =0 (1)
Notably, the restrain Zii =0 is to avoid trivial solution that each geographical unit is
only represented by itself.
Once finished the optimization, we can obtain the matrix
Z
and construct a simi-
larity matrix
W=|Z|+|Z|>
, where non-zero values denote the relationships between
geographical units. Then, we apply spectral clustering techniques to
W
to acquire the
subspaces segmentation. During this process, the number of subspaces is determined from
the normalized Laplacian matrix of
W
with the eigengap indication [
27
]. The Laplacian
matrix is defined as
L=I−D−1/2W D−1/2
, where
I
is an identity matrix and
D=∑iWij
is a degree matrix. Shen and Cheng [
27
] pointed out that when the eigenvalues
λ
of the
normalized Laplacian matrix are in ascending order, and the length of the i-th eigengap
which defined as
λi+1−λi
is the largest, then i is viewed as the appropriate candidate for
the number of subspaces.
Finally,
C
can be divided into K sub-matrices
[S1,S2, . . . , Sk, . . . , SK]
, which standing
for K subspaces according to the clustering results, while each geographical unit
gn
belongs
to only one subspace.
C=[g1,g2, . . . , gn, . . . , gN]=[S1,S2, . . . , Sk, . . . , SK]·Γ(2)
where
Γ
is a permutation operation that specifies the segmentation of geographical units.
Based on above conditions and sparse restrictions, a subspace is formed from geographical
units that share the most similar patterns, thus a subspace virtually represents a functional
region.
2.3. Eigenplace and Significant Eigenplace
Eigenplaces,
Ek=[e1,e2, . . . , er, . . . , eR]k
, represent bases of
Sk
which are derived
from the Singular Value Decomposition (SVD), and the count
R
of bases can be different
with respect to subspaces. Eigenplaces reveal which kinds of human temporal activity
patterns exist in the functional region
Sk
. With proper coefficient vectors, any place can be
represented by a set of eigenplaces. Subsequently, a significant eigenplace is defined as one
of the top
r
eigenplaces
[e1,e2, . . . , er]k
, which determines the nature of a functional region
Sk
. Notably, the functions of a region are dynamically changing with human mobility
and activity. The nature of a functional region refers to the most prominent dynamic
characteristic that people would visit the place for.
In summary, the relationships between the subspace
Sk
(i.e., functional regions), the
bases Ek, and the geographical units gnare expressed as follows:
Sk=EkV>
k=[e1,e2, . . . , er, . . . , eR]k·hV>
1,V>
2, . . . , V>
r, . . . , V>
Ri>
k(3)
gn∈Sk,gn=Ek·Vr=v1re1+v2re2+. . . +vRr eR(4)
where each row of
Vk
denotes the relationship between one unit and its related set of bases.
2.4. Definitions
Some notations and definitions used throughout are elaborated as follows.
1.
Affinity between subspaces. The affinity between subspaces is computed from the
principal angles between subspaces [
25
], which depicts the similarity between two

ISPRS Int. J. Geo-Inf. 2021,10, 66 5 of 16
subspaces. It is defined as
aff(Sk, Sl)=qcos2θ(1)
kl +. . . +cos2θ(dk∩dl)
kl
, where
Sk
and
Sl
are two subspaces of dimensions
dk
and
dl
,
θ(i)
kl
is the principle angle between
subspaces,
cos θ(i)
kl
is the
i
-th largest singular values of
U(k)>U(l)
, while
U(k)
and
U(l)
are orthobases of
Sk
and
Sl
, respectively. Note that
aff(Sk, Sl)
is high if two compared
subspaces are similar.
2.
Area Proportion (AP). AP denotes the ratio of the area that one functional region
covers in the study area. It is used to complement the assessment of the urban
spatial structure.
3.
Uniqueness Degree of a Functional Region (UDR). The UDR describes the special-
ization level of functionality in one functional region and shows how one type of
functional region is distinct from the others. If each region has a high UDR, there will
be a clear functional division, with less vagueness and overlap between functional
regions. The higher the UDR is, the more accurately the detection result will describe
the urban structure. As significant eigenplaces characterize a functional region, the
UDR is related to significant eigenplaces. It is designed to be inversely proportional
to the affinity between subspaces restricted by significant eigenplaces and is defined
as
UDR(Si)=√K−1/q∑K−1
1a f f 2(Si,S−i)
, where
K
is the number of subspaces
(i.e., regions) and S−idenotes all subspaces except Si.
4.
Richness Degree of a Functional Region (RDR). The RDR originates from the recon-
struction error of using significant eigenplaces to approximate original functional
regions. It is defined as
RDR(Si)=
C(Si)−˜
C(Si)
F/kC(Si)kF
, where
C(Si)
is the
matrix constituted of vectors belonging to subspace
Si
, and
˜
C(Si)
is the reconstructed
matrix created by significant eigenplaces of
Si
. If the reconstruction error is large,
more eigenplaces are needed to depict the functional region besides just the dom-
inant ones. Thus, the
RDR(Si)
implies the pluralistic development and function
diversity of a region. When considering all regions, the
RD A
is the summation of
each functional region weighted by corresponding AP. Therefore, the calculation is
RD A =∑K
i=1AP(Si)×RDR(Si)
. The
RD A
examines whether the overall develop-
ment of the urban spatial structure in the study is balanced.
2.5. Workflow
Figure 1presents a detailed workflow of our model which is comprised of three major
parts: In the first part, geographical units, drop-off events, and check-in events are obtained
after preprocessing the coverage of the study area, taxi GPS points dataset, and check-in
dataset, respectively. Then, the frequency matrix which indicates the dynamic activity
pattern of geographical units is constructed through mapping and matching between
drop-off events and check-in events. Later, in the detection part, the frequency matrix
is fed into the SSC to detect multiple subspaces, i.e., functional regions. Then Singular
Valued Decomposition (SVD) is performed to find the eigenplaces of each functional region,
while significant eigenplaces used to label the functionality of detected regions are found
by extracting dominant eigenplaces. Finally, based on the results of detection, we can
discover the nature of each region and evaluate the condition of urban functional regions
by computing the proposed Uniqueness Degree of a Functional Regio (UDR) and Richness
Degree of a Functional Region (RDR). Besides, we compare area proportion among all
regions to complement the assessment of urbanization in the city.

ISPRS Int. J. Geo-Inf. 2021,10, 66 6 of 16
Figure 1. Workflow of the proposed model in this study.
3. Experiment Settings
3.1. Study Area
Shanghai is one of the four municipalities of China. It also plays a role as the com-
mercial and financial hub and has a rapid urbanization growth [
28
]. The study area we
chose covers a large portion of the urban area of Shanghai includes the center area and
major transportation hubs, which is shown in Figure 2. Notably, the center area is the
core of Shanghai’s development with a dense distribution of residential, business, and
commercial facilities.
It is inevitable to partition the study area into geographical units to discover functional
regions. However, the choice of research units still lacks standards [
29
]. A majority of
studies divide their study area into regular grids [
14
,
18
,
30
–
32
], and the grid size varies
from 250
×
250 to 1000
×
1000 m
2
. In this study, we divided the study area into regular
grids with a moderate size of 500
×
500 m
2
. After removing the water zones, the study
area is composed of 3166 geographical units.

ISPRS Int. J. Geo-Inf. 2021,10, 66 7 of 16
Figure 2. Study area (Shanghai, China).
3.2. Datasets
We used a GPS points dataset generated by 6600 taxicabs in Shanghai from 3 different
periods in 2009: 1 June to 5 June, 8 June to 12 June, and 15 June to 19 June. The number
of GPS points reaches approximately 714 million at the average sampling rate of 10 s per
point within a position accuracy of about
±
10 m. In addition, the social media check-in
dataset we use has about 15 million social check-in records.
3.3. Data Processing
After removing abnormal GPS points, the trajectories with passengers were extracted.
In total, 7.85 million trips were extracted from the taxi GPS points. Based on the trip records,
drop-off events are extracted. All the drop-off events are then mapped into geographical
units according to their drop-off location. To acquire the dynamic activity pattern of one
geographical unit, we select 24 h as the time span where every hour is a single time slot.
We derive the categories of demands-tags for each check-in event using the same methods
as Zhi et al. [
18
]. The demands are then divided into six types: Home (H), Transportation
(Tr), Work (W), Dining (D), Entertainment (E), and Others (O; refers to going to parks,
museums, libraries, and other places). Thus, each grid has a 144-dimensional vector, where
every 24 h entry describes the dynamic visiting frequency of a certain demand. In total, the
final frequency matrix Cwas 144 by 3166.
4. Results
4.1. Results of Detection
As shown in Figure 1, the detection procedure has two steps: using SSC to obtain the
functional regions and using SVD to extract eigenplaces. The main results in this section
are the spatial distribution of functional regions, the set of significant eigenplaces, and the
dynamic characteristics of functional regions.

ISPRS Int. J. Geo-Inf. 2021,10, 66 8 of 16
4.1.1. Functional Region
As computed using SSC, we obtain the relationship between geographical units and
generate a similarity matrix. Next, we apply the spectral clustering technique to the
similarity matrix in order to acquire clusters, that is, the functional regions in this scenario.
By calculating the eigengap mentioned in Section 2.2, the number of functional regions is
determined as five. The similarity matrix is visualized in Figure 3, it is permutated to allow
geographical units that share similar representations to be indexed sequentially. As the
similarity between geographical units indicates that they belong to the same functional
region, the number of subspaces can also be inferred as five based on the number of
darker blocks.
Figure 3.
Visualization of the similarity matrix. Non-zero entries in the matrix are painted black,
while zero entries are not colored. The numbers aside are the indexes of permutated geographical
units. Dashed lines are added to highlight the five darker blocks.
Then, we allocated the clustering results to the regular grids, the spatial distribution
of the derived functional regions is displayed in Figure 4. As shown, the central zone
of the study area is mainly covered by Region 5, while the outer zones contain different
functional regions.

ISPRS Int. J. Geo-Inf. 2021,10, 66 9 of 16
Figure 4.
Geographical distribution of functional regions (five detected functional regions are
rendered in different colors, and the numbered red dots are points of interest (POI) in Shanghai).
4.1.2. Significant Eigenplaces and Characteristics of Each Region
Through SVD, we obtain eigenplaces of each region from the corresponding sub-
matrix. As significant eigenplaces reflect the dominant functions of the region, we find the
significant ones by evaluating the distribution of eigenvalues. The first five eigenvalues
account for more than 90% of the total data in Regions 1, 2, 3, and 4, but less than 90% in
Region 5, as shown in Figure 5. Therefore, we choose the top 5 eigenplaces as the significant
eigenplaces in each region except for Region 5. As for Region 5, we consider the influence
of the top 10 eigenplaces.
Figure 5.
Distribution of eigenvalues in corresponding covariance matrix (the vertical axis indicates
normalized eigenvalue and the horizontal axis denotes indexes of descending eigenvalues).

ISPRS Int. J. Geo-Inf. 2021,10, 66 10 of 16
Based on the significant eigenplaces, we obtain the patterns of activity and infer
the functionality of regions from their dominant activities in corresponding significant
eigenplaces. As shown in Figure 6, Home activity is the most active among all significant
eigenplaces in Region 1. Dining activity takes second place, followed by Entertainment
activity. Therefore, we associate Region 1 with the Residential Area, which is developed
with facilities like restaurants and entertainment. Evaluated in the same way, Region 2 is
the Transportation Hub due to the high Transportation activity; Region 3 is the Work Space
as the major activity is Work; and Region 4 is the Other zones for parks, museums, gas
stations, and so on. As for Region 5, we weighed the influence of the top 10 eigenplaces
and determined it to be the Business District because Dining and Entertainment hold a
large proportion of high activity. Our detected functionality is in line with the functionality
of POIs in Figure 4, which verifies our methods.
Figure 6.
The significant eigenplaces of each region (each bar represents the dynamic activity level
for a certain demand in a day; H, Tr, W, D, E, and O are the abbreviations for activities relating to
Home, Transportation, Work, Dining, Entertainment, and Others, respectively).
Combined with the analysis result in Figure 4, we can see the spatial distribution
of each functional region. The Business District occupies the center of the study area
and sprawls outward, while the outer zones contain other types of functional regions.
Furthermore, the separation of Work Space and Residential Area is also implied by
Figure 4.
4.2. Udr and Rdr Assessments
We use the proposed UDR and RDR, as well as AP, to explore the condition of the
functional regions and urbanization in our study area. The results are shown in Table 1.
AP is derived by computing the ratio of the grid number of each detected region
to the total number of grids. As shown in Table 1, the Residential Area is slightly larger
than the Work Space. The Business District has the highest proportion of land occupancy,
which implies the prosperity in the hospitality industry of this section of Shanghai, and
the great demand of urban residents in Dining and Entertainment functions. Notably, the

ISPRS Int. J. Geo-Inf. 2021,10, 66 11 of 16
Transportation Hub AP takes up 0.14 of the total area, which means there is a high traveling
demand in the area.
In regard to UDR, the Work Space reveals a high uniqueness, which implies a high
specialization of workplaces. The lowest UDR belongs to the Residential Area. It can be
explained by the Dining and Entertainment demands of residents leaving the residential
area to be mixed with other facilities like restaurants and entertainment. This makes it
similar to the Business District and therefore less specialized. Still, the overall high UDR
shows that the functional regions of the study area are significantly different.
The highest RDR belongs to the Other Zones, meaning it contains the most compli-
cated activity pattern. This is a logical conclusion considering it supplies diverse demands.
The low RDR of the Residential Area and Business District can be explained by their
relatively simple functionalities which mainly concentrate on Home, Dining, and Enter-
tainment activity. As for the entire area, the value of RDA is 0.538, which is a middle level
compared to functional regions, indicating a comprehensive and balanced development of
functional regions.
Table 1. Assessment results of functional regions.
Region Residential Area Transportation Hub Work Space Other Zones Business District
AP 0.195 0.140 0.192 0.086 0.387
UDR 0.898 1.080 1.151 1.047 0.946
RDR 0.510 0.571 0.554 0.607 0.518
In summary, urban spatial structure, specifically the function structure, of the study
area is predominant in Dining and Entertainment. Home and Work are important com-
ponents, Transportation is a necessity for the sound growth of this area, and Other Zones
adorns the structure.
4.3. Comparison with the Single Subspace-Based Method
In this section, we conduct a comparative experiment to exhibit the differences be-
tween our model and single subspace-based methods. The method we used for compari-
son detects functional regions via a low-rank approximation (LRA) model based on SVD
(Zhi et al., 2016). The computations of the LRA method are introduced as
minimize kC−ˆ
CkF, s.t. rank (ˆ
C)≤k(5)
ˆ
C=ˆ
Uˆ
Sˆ
V>=ˆ
Uˆ
S1
2ˆ
Vˆ
S1
2>(6)
where
ˆ
C
is the best rank-
k
approximation of a
M×N
matrix
C
, and SVD is applied to
decompose
ˆ
C
, the resulted
ˆ
U
is a
M×M
unitary matrix,
ˆ
V
is a
N×N
unitary matrix,
ˆ
S
is a diagonal matrix constraining the singular values
σi
of
ˆ
C
. Rows in
ˆ
Uˆ
S1
2
represent the
characteristics distribution of temporal activity pattern in one eigenplace, while rows in
ˆ
Vˆ
S1
2
denote the characteristics distribution of eigenplaces. Then, the similarity matrix can
be constructed as
W=

ˆ
Vˆ
S1
2

·

ˆ
Vˆ
S1
2


>
, the number of detected regions is determined as six
using the aforemenyioned eigengap method. The distribution and significant eigenplaces
of the functional regions are shown in Figures 7and 8, respectively.

ISPRS Int. J. Geo-Inf. 2021,10, 66 12 of 16
Figure 7.
Geographical distribution of functional regions detected by the low-rank approximation
(LRA) Method.
Figure 8. Significant eigenplaces of each region (LRA Method).

ISPRS Int. J. Geo-Inf. 2021,10, 66 13 of 16
Comparing Figure 7with Figure 4, the distribution of functional regions detected by
the LRA method is similar to that discovered by our model, with the exception that the
LRA method separates our Region 5 (Business District) into two regions: Region 5 and
Region 6. The significant eigenplaces in Figure 8also show that Region 5 and Region 6
detected by the LRA method have overlapped functionality as Dining and Entertainment
are both active in each region. The values of the UDR in Table 2are generally lower than
those in Table 1, which implies that the functions of each region are more overlapped and
less distinguished.
Table 2. Uniqueness Degree of a Functional Region (UDR) Assessment of the LRA method.
Region 1 2 3 4 5 6
UDR 0.838 0.918 0.888 0.962 0.836 0.878
To further verify whether Region 5 and Region 6 should be integral regions, we
use t-SNE to visualize clustering results [
33
]. t-SNE is a technique commonly used to
project high-dimensional data to two-dimensional for visualization. It utilizes joint prob-
abilities to model similarities between data points. To ensure the similarities between
data points remain unchanged after dimensioning reduction, t-SNE uses gradient descent
to minimize the Kullback–Leibler divergence between the joint probabilities of the low-
dimensional embedding and the high-dimensional data. With the frequency matrix as
input, t-SNE generates two-dimensional scatter plots, where points represent geographical
units. Then, points are rendered with different colors according to cluster labels generated
by our model and the LRA method, as shown in Figure 9a,b respectively. Figure 9a shows
that the five clusters are well separated, only the blue cluster (Region 3) and the orange
cluster (Region 1) are slightly overlapped with the yellow cluster (Region 5). Meanwhile,
there shows no clear gap within the yellow cluster, which suggests there is no need for
division. In contrast, Figure 9b is more cluttered. A considerable proportion of green points
(Region 6) are mixed with yellow ones (Region 5). Additionally, it not only presents a larger
mingle among blue (Region 3), orange (Region 1), and yellow (Region 5) clusters, but also
a messier distribution of the green cluster (Region 6) and yellow cluster (Region 5) in the
area possessed by the pink cluster (Region 2).
Moreover, the #5 eigenplace of Region 5 detected by LRA is dominated by the work
activity, while other eigenplaces of the same region are dominated by entertainment and
dining. The #3 eigenplace in Region 4 is also inconsistent with others. These inconsistencies
within the same region reveal a fuzzy function detection of LRA. In conclusion, our model
outperforms LRA in a clearer functional division.
Figure 9.
Two-dimensional visualization of the regions using the t-SNE algorithm. Subfigure (
a
) is
colored according to our results; subfigure (b) is colored according to results of LRA.

ISPRS Int. J. Geo-Inf. 2021,10, 66 14 of 16
5. Conclusions
In this study, we proposed a model based on the idea that urban spatial structure
could be inferred from human temporal activity with the aid of big geospatial data.
Specifically, the human temporal activity within a geographical unit tells when and why
people visit it, which helps in interpreting the social functionality of the geographical
unit. We assumed that features of all geographical units in terms of human temporal
activity lie in a high-dimensional space which can be represented by a union of multiple
low-dimensional subspaces. Moreover, subspaces are found under the constraint that a
geographical unit can be only represented by the combination of other geographical units
within the same subspace. By this means, geographical units with the same temporal
activity patterns are in the same subspaces. Therefore, recovering subspaces leads to the
detection of functional regions.
Our model is divided into three parts: data processing, detection, and assessment.
First, we reproduce the human temporal activity of an area with geographical units, drop-
off events, and check-in events obtained from big geospatial data, and store it in a matrix
form. In the detection procedure, aiming to identify functional regions, we apply Sparse
Subspace Clustering to the matrix to discover relationships between geographical units
and obtain subspaces (functional regions) using spectral clustering. We adopt a multiple
subspaces-based technique under the consideration that different functional regions can
be different in nature and have intrinsic complexities. Therefore, diverse bases with
distinct dimensionality are needed to depict those regions more concisely and accurately.
Therefore, our model is more flexible compared with other single space-based models
which adopt the assumption that different functional regions share the same set of bases.
After detection, we extract significant eigenplaces which illustrate major activities and their
patterns within a region to analyze the characteristics of each detected region and label
the functionality of it. Furthermore, we propose calculating UDR and RDR by measuring
the affinity of subspaces and reconstruction error to assess the spatial structure of the
study area.
This study was performed in an area in Shanghai, China. Experimental results and the
comparison with the LRA method prove the outperformance of our model when focusing
on distinct functional regions. However, additional works are needed. First, spatial
aggregation operations are required in this study, but the scale and shape of geographical
units may exert impacts on research results, which is the so-called Modifiable areal unit
problem [
34
]. Though we chose a common scheme under the consideration that larger units
may have mixed functions, while the smaller unit may both be computational unfriendly
and trivial in generating numerous adjacent parcels that can be merged, quantitative
research of MAUP is needed for further exploration. Second, the number of clusters is
a parameter needs to be predefined; here, we choose the eigengap as a heuristic and the
result is in accordance with the number revealed by the similarity matrix in this study.
However, the heuristic may fail in more complicated conditions. As for the main data
source of human mobility, we chose the GPS trajectory of taxicabs while daily travel
transportation also includes buses, subways, etc. We will improve on our work in a future
study by incorporating other sources of big geospatial data and extend our research to
more cities.
Author Contributions:
Conceptualization, Jiawei Zhu; Data curation, Jian Peng, Haozhe Huang
and Li Chen; Methodology, Jiawei Zhu and Chao Tao; Software, Jiawei Zhu and Xin Lin; Supervi-
sion, Qiongjie Wang; Visualization, Jiawei Zhu; Writing—original draft, Jiawei Zhu and Xin Lin.
All authors have read and agreed to the published version of the manuscript.
Funding:
This research was funded by National Natural Science Foundation of China (grant num-
bers 41571397, 41871364); and the Fundamental Research Funds (No. 2019zzts881) for the Central
Universities of Central South University.
Conflicts of Interest: The authors declare no conflict of interest.

ISPRS Int. J. Geo-Inf. 2021,10, 66 15 of 16
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