# Connectivity of graphs that do not have the edge-Erdős-Pósa property

Henning Bruhn and Raphael Steck

###### Abstract

We show that we can assume graphs that do not have the edge-Erdős-Pósa
property to be connected. Then we strengthen this result to $2$-connectivity
under the additional assumptions of a minor-closed property and a generic
counterexample.

A class $\mathcal{F}$ has the _edge-Erdős-Pósa property_ if there exists a
function $f:\mathbb{N}\rightarrow\mathbb{R}$ such that for every graph $G$ and
every integer $k$, there are $k$ edge-disjoint graphs in $G$ each isomorphic
to some graph in $\mathcal{F}$ or there is an edge set $X\subseteq E(G)$ of
size at most $f(k)$ meeting all subgraphs in $G$ isomorphic to some graph in
$\mathcal{F}$. The edge set $X$ is called the _hitting set_. If we replace
vertices with edges in the above definition, that is, we look for a vertex
hitting set or vertex-disjoint graphs, then we obtain the _vertex-Erdős-Pósa
property_. The class $\mathcal{F}$ that is studied in this paper arises from
taking minors: For a fixed graph $H$, we define the set

$\mathcal{F}_{H}=\\{G\,|\,H\text{ is a minor of }G\\}.$

In other words, $\mathcal{F}_{H}$ is the set of _$H$ -expansions_. The vertex-
Erdős-Pósa property for $\mathcal{F}_{H}$ is well understood: Robertson and
Seymour [1] proved that the class $\mathcal{F}_{H}$ has the vertex-Erdős-Pósa
property if and only if $H$ is planar. This implies that the vertex-Erdős-Pósa
property is closed under taking minors, which in turn implies that

If for a graph $H$ the class $\mathcal{F}_{H}$ has the vertex-Erdős-Pósa
property, then so does the class $\mathcal{F}_{C}$ for every component $C$ of
$H$. (1)

For the edge-Erdős-Pósa property, it is not known whether it is minor-closed
or not, and it is not at all clear whether it should be. Thus, we tackle (1)
for the edge-Erdős-Pósa property. We show that

######  Theorem 1.

If for a graph $H$ the class $\mathcal{F}_{H}$ has the edge-Erdős-Pósa
property, then so does $\mathcal{F}_{C}$ for every component $C$ of $H$.

Interestingly, Robertson and Seymour proved their result about the vertex-
Erdős-Pósa property of planar graphs in two steps: First, they proved it for
every connected planar graph. In a second step, they lifted the connectivity
requirement. Thus Theorem 1 might also provide some help in verifying for
which graphs $H$ the class $\mathcal{F}_{H}$ has the edge-Erdős-Pósa property.
For example, it might help to prove that $\mathcal{F}_{H}$ does not have the
edge-Erdős-Pósa property if $H$ has large treewidth (for larger or arbitrary
maximum degree of $H$).

To this end, we also attempt a strengthening of Theorem 1 which allows us to
not only focus on connected, but $2$-connected graphs $H$. This, however, is
not achieved in full generality, and we prove it only by imposing some
additional assumptions (see Theorem 3).

For a graph $H$, a connected graph $G\subseteq H$ with at least one vertex
$v\in V(G)$ with $d_{H}(v)\geq 3$ and an integer $r\in\mathbb{N}$, we define
$G^{\times}$ to be the following graph: Starting with the empty graph, we add
a copy of every vertex $v\in V(G)$ with $d_{H}(v)\geq 3$ to $G^{\times}$. For
every non-trivial path $P$ of length $l$ in $G$ between two vertices such
vertices, we add $r$ internally disjoint paths of length $\max\\{l,2\\}$
between their corresponding copies in $G^{\times}$. Finally, for every path
$P$ of length $l$ in $G$ between a vertex $u$ with $d_{G}(u)=1$, $d_{H}(u)\leq
2$ and its closest vertex $v\in V(G)$ with $d_{H}(v)\geq 3$, we add $r$ paths
of length $l$ which are disjoint except for the copy of $v$. See Figure 1 for
an example.

$v$$w$$u$$v^{*}$$w^{*}$$G$$G^{\times}$ Figure 1: Construction of $G^{\times}$,
with $u\in V(H)\setminus V(G)$.

Let us check that for every edge set $X$ of size at most $r-1$, $G^{\times}-X$
contains a $G$-expansion. Every $v\in V(G)$ with $d_{H}(v)\geq 3$ can be
mapped to its copy $v^{\prime}\in V(G^{\times})$. Every $u$–$v$ path between
two such vertices can be mapped to one of its copies in $G^{\times}$ that is
disjoint from $X$. For every vertex $u\in V(G)$ with $d_{G}(u)=1$ and
$d_{H}(u)\leq 2$, there is vertex $v\in V(G)$ with $d_{H}(v)\geq 3$ that is
closest to $u$. Among all copies of the $u$–$v$ path, we pick a copy
$P^{\prime}$ that is disjoint from $X$ and map $P$ to $P^{\prime}$ such that
$v$ is mapped to $v^{\prime}$. The prerequisite of $G$ being connected and
containing a vertex $v$ with $d_{H}(v)\geq 3$ implies that every $v\in V(G)$
with $d_{G}(v)=d_{H}(v)=2$ lies on a path in $G$ between vertices of degree
other than $2$ in $H$, with at least one endvertex of the path having degree
at least $3$ in $H$. We conclude that the above mapping yields a $G$-expansion
in $G^{\times}-X$.

For $r\geq 3$, the number of vertices of degree at least $3$ in $G^{\times}$
is the same as the number of vertices $v$ with $d_{H}(v)\geq 3$ in $G$.

######  Remark 2.

Let $H$ be a connected graph for which $\mathcal{F}_{H}$ does not have the
edge-Erdős-Pósa property. Then $H$ contains vertices of degree at least $3$.

###### Proof.

Suppose $H$ would only contain vertices of degree $2$ or less. Since $H$ is
connected, $H$ must be a cycle, a path or an isolated vertex. However, for all
of those graphs, $\mathcal{F}_{H}$ is already known to have the edge-Erdős-
Pósa property. ∎

For two graphs $A$ and $B$, we define $\preceq$ by

$A\preceq B\Leftrightarrow A\text{ is a minor of }B.$

Similarly, we define $\not\preceq$ by

$A\not\preceq B\Leftrightarrow A\text{ is not a minor of }B.$

##  1 1-Connectivity

###### Proof of Theorem 1.

Let $A$ be some component of $H$ such that $\mathcal{F}_{A}$ does not have the
edge-Erdős-Pósa property. Thus, there exists an integer $k\in\mathbb{N}$ such
that for every $r\in\mathbb{N}$, there exists a graph $A^{*}_{r}$ such that
$A^{*}_{r}$ neither contains $k$ edge-disjoint expansions of $A$ nor an edge
set $X$ of size at most $r-1$ such that $A^{*}_{r}-X$ contains no expansion of
$A$. We separate the other components of $H$ into two disjoint sets
$\mathcal{B}$ and $\mathcal{C}$, which we define by

$\displaystyle\mathcal{B}$ $\displaystyle=\\{B\text{ component of
}H\,|\,A\not\preceq B\\}\text{ and}$ $\displaystyle\mathcal{C}$
$\displaystyle=\\{C\text{ component of }H\,|\,A\preceq C\\}\setminus\\{A\\}.$

To prove that $\mathcal{F}_{H}$ does not have the edge-Erdős-Pósa property,
let $r$ be given. We prove that there exists a graph $H^{*}$ that contains
neither $k$ edge-disjoint $H$-expansions nor an edge set $X$ of size at most
$r-1$ such that $H^{*}-X$ contains no $H$-expansion. We define $H^{*}$ to be
the disjoint union of

  * •

$A^{*}=A^{*}_{r}$,

  * •

for every $B\in\mathcal{B}$: $r$ distinct copies of $B$ and

  * •

for every $C\in\mathcal{C}$: one $C^{\times}$.

First we show that $H^{*}$ does not contain an edge-hitting set meeting all
$A$-expansions. Let $X\subseteq E(H^{*})$ be an edge set of size at most
$r-1$. We claim that $H^{*}-X$ still contains an $H$-expansion: Indeed, there
is an $A$-expansion in $A^{*}-X$ by choice of $A^{*}$. For every
$B\in\mathcal{B}$, at least one of the $r$ copies of $B$ in $H$ is disjoint
from $X$. Thus, there is a $B$-expansion in that copy. Finally, for every
$C\in\mathcal{C}$, there is a $C$-expansion in $C^{\times}-X$ by construction
of $C^{\times}$. Together, this yields an $H$-expansion in $H^{*}-X$.

We claim

Every $H$-expansion in $H^{*}$ contains an $A$-expansion in $A^{*}$. (2)

Note that (2) finishes the proof of the theorem: Indeed, by choice of $A^{*}$,
there can be no $k$ edge-disjoint $A$-expansions in $A^{*}$. To prove the
claim, consider an expansion $H^{\prime}$ in $H^{*}$, and suppose that (2) is
false for $H^{\prime}$.

Since every component of $H^{\prime}$ is connected, it must be contained in a
single component of $H^{*}$. Further note that every expansion of a
$C\in\mathcal{C}$ in some component of $H^{*}$ contains an expansion of $A$ by
definition of $\mathcal{C}$. Thus, by definition of $\mathcal{B}$, no
$A$-expansion (and thus no $C$-expansion for any $C\in\mathcal{C}$) can be
contained in some copy of some $B\in\mathcal{B}$. On top of that, if an
$A$-expansion (or a $C$-expansion for any $C\in\mathcal{C}$) is embedded in
$A^{*}$, this proves the above claim. Thus, suppose all $A$-expansion in
$H^{\prime}$ (and thus all $C$-expansions for every $C\in\mathcal{C}$) are
contained in $\bigcup\limits_{C\in\mathcal{C}}C^{\times}$. Every $H$-expansion
in $H^{*}$ contains at least one vertex of degree $\geq 3$ in $H^{*}$ for
every vertex of degree $\geq 3$ in $H$. However, by construction of
$C^{\times}$, $\bigcup\limits_{C\in\mathcal{C}}C^{\times}$ contains no more
vertices of degree $\geq 3$ than $\bigcup\limits_{C\in\mathcal{C}}C$. Since
$A$ contains vertices of degree $\geq 3$ by Remark 2 and the branch sets of
all vertices in $A\cup\left(\bigcup\limits_{C\in\mathcal{C}}C\right)$ must be
contained in $\bigcup\limits_{C\in\mathcal{C}}C^{\times}$, this is a
contradiction. Thus (2) holds, proving the theorem. ∎

##  2 2-Connectivity

If we want to prove that for some graph $H$, $\mathcal{F}_{H}$ does not have
the edge-Erdős-Pósa property, the above theorem implies that it suffices to
check the components of $H$ individually. Thus, without loss of generality, we
can assume that $H$ is $1$-connected. To improve this to $2$-connectivity of
$H$, we need two additional assumptions.

First, we define a graph property $\mathcal{P}$ to be _hereditary_ if
$\bar{\mathcal{P}}$ is closed under taking minors, that is, for every graph
$H$ without property $\mathcal{P}$, no minor of $H$ has property
$\mathcal{P}$. An example for a hereditary property would be large treewidth:
If a graph $H$ does not have treewidth at least $t$, then no minor of $H$ has
treewidth at least $t$.

Second, for a block $A$ of a graph $H$, let $S$ be the set of cutvertices of
$H$ that lie in $A$. We say that there is a _generic counterexample_ for $A$
if there exists some $k\in\mathbb{N}$ such that for every $r\in\mathbb{N}$,
there is a graph $A^{*}$ with the following properties: There are no $k$ edge-
disjoint $A$-expansions in $A^{*}$. Furthermore, for every $s\in S$, there is
a $s^{\prime}\in V(A^{*})$ such that for every edge set $X$ of size at most
$r-1$, there is an embedding of $A$ in $A^{*}$ such that for every vertex
$s\in S$, the branch set $B_{s}$ contains $s^{\prime}$. (Together, this
implies that $\mathcal{F}_{A}$ does not have the edge-Erdős-Pósa property.)
Every known construction that shows that some class $\mathcal{F}_{A}$ does not
have the edge-Erdős-Pósa property is a generic counterexample.

######  Theorem 3.

Let $\mathcal{P}$ be a hereditary graph property and let $H$ be a graph that
contains a block with property $\mathcal{P}$. Furthermore, for every block $A$
of $H$ with property $\mathcal{P}$, let there be a generic counterexample.

Then $\mathcal{F}_{H}$ does not have the edge-Erdős-Pósa property.

An example for an application of Theorem*3 could be the following. Suppose we
want to show that:

For every graph $H$ of treewidth at least $10^{100}$, $\mathcal{F}_{H}$ does
not have the edge-Erdős-Pósa property. (3)

Having large treewidth is a hereditary graph property. Assume we were able to
show that for every $2$-connected graph $H$ of treewidth at least $10^{100}$,
$\mathcal{F}_{H}$ does not have the edge-Erdős-Pósa property. Then we will
most likely do that by giving a generic counterexample. Thus we can apply
Theorem 3 to drop the connectivity requirement and we obtain (3). Now let us
prove Theorem 3.

###### Proof.

We can assume $H$ to be $1$-connected: Indeed, if $H$ contains a block $A$
with property $\mathcal{P}$, then there is a component $Q$ of $H$ that
contains $A$. Furthermore, if all block of $H$ with property $\mathcal{P}$
allow for a generic counterexample, then this includes the blocks of $Q$. If
we are able to prove that $\mathcal{F}_{Q}$ does not have the edge-Erdős-Pósa
property, then using Theorem 1, we conclude that $\mathcal{F}_{H}$ does not
have the edge-Erdős-Pósa property.

We consider the block tree $T$ of $H$. Let $T_{\mathcal{P}}$ be the minimal
subtree of $T$ that contains all blocks with property $\mathcal{P}$. We pick
$A$ to be a leaf of $T_{\mathcal{P}}$. Note that $A$ is a block of $H$ that
has the property $\mathcal{P}$. Observe that $A$ is not a trivial block, i.e.
$A$ is not a $K_{2}$ due to Remark 2.

We define

$\displaystyle\mathcal{C}$ $\displaystyle=\\{C\text{ block of }H\,|\,A\preceq
C\\}\setminus\\{A\\}.$

Since $\mathcal{P}$ is hereditary, it holds for every $C\in\mathcal{C}$ that
$C$ has the property $\mathcal{P}$, too. Let $T_{\mathcal{C}}$ be the minimal
subtree of $T$ that contains all blocks of $\mathcal{C}\cup\\{A\\}$. We
observe that $T_{\mathcal{C}}$ is a subgraph of $T_{\mathcal{P}}$. Thus,

$A$ is a leaf of $T_{\mathcal{C}}$. (4)

We define

$\displaystyle\mathcal{B}=$ $\displaystyle\\{B\text{ block of
}H\,|\,A\not\preceq B\\}\cap V(T_{\mathcal{C}})\text{ and}$
$\displaystyle\mathcal{D}=$ $\displaystyle\\{D\text{ component of
}\bigcup_{\begin{subarray}{c}B\text{ block of }H\\\ B\not\in
V(T_{\mathcal{C}})\end{subarray}}B\\}.$

Note that since $T$ is a tree, $T-T_{C}$ is a forest. For each component
$T^{\prime}$ in $T-T_{C}$, $V(T^{\prime})$ are the blocks and cutvertices of
one element of $\mathcal{D}$.

To show that $H$ does not have the edge-Erdős-Pósa property, let $r\geq 3$ be
some integer. We define our counterexample graph $H^{*}$ to be the union of

  * •

$A^{*}$,

  * •

for every $C\in\mathcal{C}$: one $C^{\times}$,

  * •

for every $D\in\mathcal{D}$: $r$ distinct copies of $D$,

  * •

for every non-trivial $B\in\mathcal{B}$: one $B^{\times}$ and

  * •

for every component $P$ of
$\bigcup\limits_{\begin{subarray}{c}B\in\mathcal{B}\\\
B=K_{2}\end{subarray}}B$: one $P^{\times}$.

We pick the above graphs to be disjoint except for those vertices which are
copies of the same vertex $v\in H$, which we identify with each other in
$H^{*}$, too. In $A^{*}$, we pick the vertex $s^{\prime}$ for every $s\in S$
and identify it with all copies of $s$. Note that for all blocks
$B\in\mathcal{B}$, there is a path $P$ in $H$ whose endvertices are in a block
in $\\{A\\}\cup\mathcal{C}$ and $P$ contains an edge of $B$. With $A$ and $C$
being non-trivial blocks for all $C\in\mathcal{C}$, their cutvertices have
degree at least $3$ in $H$. Thus the union of all trivial blocks in $B$ is a
collection of paths $P$ whose endvertices are in non-trivial blocks and have
degree at least $3$ in $H$. We denote the union of all $P^{\times}$ and all
$B^{\times}$ for every non-trivial $B\in\mathcal{B}$ by
$\mathcal{B}^{\times}$. Note that every vertex $v^{\prime}$ in
$\mathcal{B}^{\times}\cup\bigcup_{C\in\mathcal{C}}C^{\times}$ with
$d_{H^{*}}(v^{\prime})\geq 3$ is the copy of some vertex $v$ in
$\bigcup_{B\in\mathcal{B}}B\cup\bigcup_{C\in\mathcal{C}}C$ with $d_{H}(v)\geq
3$.

Now we show that $H^{*}$ does not contain an edge-hitting set meeting all
$A$-expansions. Let $X\subseteq E(H^{*})$ be an edge set of size at most
$r-1$. We claim that $H^{*}-X$ still contains an embedding of $H$: Indeed, we
can embed $A$ in $A^{*}-X$ such that for every $s\in S$, its branch set
$B_{s}$ contains $s^{\prime}$ by choice of $A^{*}$. For every
$D\in\mathcal{D}$, at least one of the $r$ copies of $D$ is disjoint from $X$.
Thus, we can embed $D$ in that copy. For every non-trivial
$B\in\mathcal{B}\cup\mathcal{C}$, we can embed $B$ in $B^{\times}-X$ by
construction of $B^{\times}$. We observed above that trivial blocks of $B$ are
contained in some paths between vertices of degree at least $3$, which can be
embedded in their copy in $H^{*}-X$. Together, this yields an embedding of $H$
in $H^{*}-X$ by construction of $H^{*}$.

It remains to show that there are no $k$ edge-disjoint embeddings of $H$ in
$H^{*}$. For this, we claim:

Every $H$-expansion in $H^{*}$ contains an $A$-expansion in $A^{*}$. (5)

Note that (5) proves the theorem: Indeed, by choice of $A^{*}$, there can be
no $k$ edge-disjoint embeddings of $A$ in $A^{*}$. Let us prove (5). Way say
that a block $B$ of $H$ is _embedded in a block $B^{*}$ of $H^{*}$_, when for
every $v\in V(B)$ with $d_{H}(v)\geq 3$, the branch set $B_{v}$ contains a
vertex $v^{*}\in V(B^{*})$ with $d_{B^{*}}\geq 3$. In this sense, every block
$B$ of $H$ is embedded in a single block $B^{*}$ of $H^{*}$. Note that every
embedding of a $C\in\mathcal{C}$ in some block of $H^{*}$ contains an
embedding of $A$ by definition of $\mathcal{C}$. Thus neither $A$ nor any
$C\in\mathcal{C}$ can be embedded in any copy of some $D\in\mathcal{D}$.
Additionally, if some $C\in\mathcal{C}$ is embedded in $A^{*}$, this includes
an embedding of $A$ in $A^{*}$, which was what we wanted. Thus, we may assume
that neither $A$ nor any $C\in\mathcal{C}$ is embedded in $A^{*}$.

We conclude that $A\cup\bigcup\limits_{C\in\mathcal{C}}C$ is embedded in
$\mathcal{B}^{\times}\cup\bigcup\limits_{C\in\mathcal{C}}C^{\times}$. Let
$B\in\mathcal{B}$. By definition of $\mathcal{B}$, $B$ is on the unique path
in $T$ that connects two blocks $C_{1},C_{2}\in\mathcal{C}\cup\\{A\\}$. Since
$A$ is a leaf of $T_{\mathcal{C}}$ by (4) and we assumed that neither $C_{1}$
nor $C_{2}$ are embedded in $A^{*}$, $B$ cannot be embedded in $A^{*}$. For
every $D\in\mathcal{D}$, there is a cutvertex separating $D$ from all
$C\in\\{A\\}\cup\mathcal{C}$. Therefore, $B$ cannot be embedded in $D$. We
conclude that $B$ is embedded in
$\mathcal{B}^{\times}\cup\bigcup\limits_{C\in\mathcal{C}}C^{\times}$.

To sum up,
$A\cup\bigcup\limits_{B\in\mathcal{B}}B\cup\bigcup\limits_{C\in\mathcal{C}}C$
is embedded in
$\mathcal{B}^{\times}\cup\bigcup\limits_{C\in\mathcal{C}}C^{\times}$. However,
the number of vertices $v$ in
$\bigcup\limits_{B\in\mathcal{B}}B\cup\bigcup\limits_{C\in\mathcal{C}}C$ with
$d_{H}(v)\geq 3$ is the same as the number of vertices $v^{\prime}$ in
$\mathcal{B}^{\times}\cup\bigcup\limits_{C\in\mathcal{C}}C^{\times}$ with
$d_{H^{*}}(v^{\prime})\geq 3$.

By by Remark 2, $A$ contains at least one vertex $v$ with $d_{A}(v)\geq 3$.
Since $A$ is $2$-connected, we conclude that it must contain at least two
vertices $v$ with $d_{A}(v)\geq 3$. Since $A$ is a leaf of $T_{C}$, it shares
exactly one vertex with
$\bigcup\limits_{B\in\mathcal{B}}B\cup\bigcup\limits_{C\in\mathcal{C}}C$.
Thus,
$A\setminus\left(\bigcup\limits_{B\in\mathcal{B}}B\cup\bigcup\limits_{C\in\mathcal{C}}C\right)$
contains at least one vertex $v$ with $d_{H}(v)\geq d_{A}(v)\geq 3$. But then
it is impossible to embed
$A\cup\bigcup\limits_{B\in\mathcal{B}}B\cup\bigcup\limits_{C\in\mathcal{C}}C$
in $\mathcal{B}^{\times}\cup\bigcup\limits_{C\in\mathcal{C}}C^{\times}$.

Thus, Claim (2) holds, proving the theorem. ∎

## References

  * [1] Neil Robertson and P.D Seymour, _Graph minors. V. Excluding a planar graph_ , Journal of Combinatorial Theory, Series B 41 (1986), no. 1, 92 – 114.
  * [2] Raphael Steck and Arthur Ulmer, _On the edge-Erdős-Pósa property of Ladders_ , 2020.


1329 Research application/design study L. Chen and Y. Ouyang are with School
of Information Science and Technology, ShanghaiTech University. E-mail:
<EMAIL_ADDRESS>They contributed equally to this work. S.
Hong is with School of Entrepreneurship and Management, ShanghaiTech
University. E-mail<EMAIL_ADDRESS>H. Zhang and Q. Li are with
School of Information Science and Technology, ShanghaiTech University. Quan Li
is the corresponding author. E-mail: zhanghp,liquan@@shanghaitech.edu.cn. Biv
et al.: Global Illumination for Fun and Profit

# RISeer: Inspecting the Status and Dynamics of Regional Industrial Structure
via Visual Analytics

Longfei Chen  Yang Ouyang  Haipeng Zhang  Suting Hong  and Quan Li

###### Abstract

Restructuring the regional industrial structure (RIS) has the potential to
halt economic recession and achieve revitalization. Understanding the current
status and dynamics of RIS will greatly assist in studying and evaluating the
current industrial structure. Previous studies have focused on qualitative and
quantitative research to rationalize RIS from a macroscopic perspective.
Although recent studies have traced information at the industrial enterprise
level to complement existing research from a micro perspective, the ambiguity
of the underlying variables contributing to the industrial sector and its
composition, the dynamic nature, and the large number of multivariant features
of RIS records have obscured a deep and fine-grained understanding of RIS. To
this end, we propose an interactive visualization system, RISeer, which is
based on interpretable machine learning models and enhanced visualizations
designed to identify the evolutionary patterns of the RIS and facilitate
inter-regional inspection and comparison. Two case studies confirm the
effectiveness of our approach, and feedback from experts indicates that RISeer
helps them to gain a fine-grained understanding of the dynamics and evolution
of the RIS.

###### keywords:

Spatiotemporal dynamics, multivariate time series, regional industrial
structure, visualization

K.6.1Management of Computing and Information SystemsProject and People
ManagementLife Cycle; K.7.mThe Computing ProfessionMiscellaneousEthics

Introduction

Regional industrial structure (RIS) refers to the proportional relationship
between industrial sectors with different development functions in a specific
region, and it is the result of the combination of the spatial layout of the
national economy in a specific region [2]. In other words, the reason why a
certain industrial structure emerges is determined both by the advantages of a
particular region and by the overall requirements of the spatial layout of the
national economy. Adjusting the RIS is fundamental to curbing the economic
downturn and achieving economic revitalization. Various empirical evidences
show that the economic development of regions around the world is always
linked to the optimization of industrial structure [34, 53, 69]. For example,
the relative decline of the United States and the rise of Japan in the late
$1970$s and early $1980$s, as well as the late recovery of the United States
and the decline of Japan in the $1990$s, can be attributed to a concerted
restructuring of regional industries.

Understanding the current status and dynamics of the RIS will greatly help to
examine and assess the corresponding rationality of the industrial structure
and the benefits to stakeholders. It is worth noting that for entrepreneurs,
identifying an appropriate timing to enter the market during the dynamic
change of the RIS is crucial. For investors, an objective evaluation of the
current RIS can provide insight into the allocation of rational upstream or
downstream industries. For regional planners and policy makers, an examination
of the dynamics of the RIS can help them make feasible assessments, develop
regional planning strategies, and attract more potential investment
opportunities. Given the characteristics of RIS, there is therefore an urgent
need for a methodology that can adequately reflect and examine the current
situation and dynamics of RIS. Previous studies on RIS have mainly focused on
qualitative and quantitative research to analyze and adjust the rationality
from a macro perspective. Generally speaking, most studies start from the
changes of industrial structure and national policies, and conduct statistical
analysis of historical data. Then, with the help of econometric models, they
estimate the evolutionary trends of RIS. Some studies use information on
regional industrial enterprises and production resources to study the dynamic
proportional relationships between industries [18, 20, 43, 48]. Clustering and
comparative analysis was also performed based on the industrial sectors
corresponding to enterprises registered in a specific region [1, 21, 42],
which performed well in understanding RIS dynamics.

This work focuses on tracing the current status and dynamics of the RIS
consisting of and reflecting the enterprise activities behind it [6, 12, 17,
38]. This can complement the analysis of RIS from a microscopic perspective.
However, it still poses several challenges. First, industrialization and
modernization have enabled countless businesses to be established, survive and
grow. During this process, profile information on newly established businesses
and all corresponding major industrial and commercial changes are recorded.
Traditional methods usually aggregate the proportions of the various
industries in a given region and observe trends over time. However, they
cannot assess whether and how the underlying variables affect these trends.
While some works consider correlations between different industrial sectors,
they still do not directly show the relationships between input
characteristics and output ratios of industries, as well as potential
problems, such as anomalies. Second, while most previous works are beneficial
for monitoring the evolutionary patterns of RIS, fine-grained analyses such as
industrial agglomeration, radiation and migration, which are often hidden
behind large-scale business activities, are challenging. An effective approach
should provide clear visual cues to capture potential RIS patterns, identify
anomalous changes, and make cross-sectional and regional comparisons. Third,
archival records of enterprises are inherently multi-attribute items with a
variety of data types, such as timestamp, text, numeric, and geographical
pairs, which together determine where a business belongs in a given industrial
structure. More and more enterprises are being registered and their
information is being added/updated over time. Given the long lead time of
industrial structure studies, capturing the impact of these attributes on the
dispersion and temporal changes of RIS requires careful visualization.

In this study, we propose a visual analytics approach to understanding the
current status and dynamics of RIS from the perspective of industrial
enterprise-level information. In particular, to address the first challenge,
we treat the dynamics of the enterprise establishments as a regression problem
and use multiple machine learning predictive models to explain potential
associations between input features and output RIS patterns as well as
possible anomalies. To address the second and third challenges, we employ an
enhanced timeline design with novel glyphs to encode the dynamics of RIS in
geographical space over time and reveal the impact of temporal changes in
multiple attributes on the region at different time periods. We propose
RISeer, an interactive visualization system that identifies the evolutionary
patterns of RIS and facilitates inter-regional comparisons based on visual
designs. We demonstrate the usefulness of RISeer using real-world datasets and
case studies. In addition, we collected feedback on our approach from a number
of domain experts. Our main contributions are: 1) We use information at the
industrial enterprise-level to characterize these issues in the context of
studying the current state and dynamics of the RIS through thorough discussion
of design requirements with domain experts; 2) We develop interactive
visualizations with new visual features to support the study of the current
state and dynamics of RIS. To the best of our knowledge, RISeer is the first
endeavor of this kind.

##  1 Related Work

###  1.1 Regional Industrial Structure Analysis

Previous studies on regional structural analysis have focused on the
rationalization and adjustment of the RIS. Notably, they would use econometric
models to estimate the evolutionary trend of RIS [39, 53, 65] or conduct
qualitative and quantitative analysis from a macro perspective [34, 62, 69].
For example, Lu et al. [44] used a panel data model to measure the
contribution of different industries to economic growth and concluded that
economic growth is highly correlated with the development of secondary and
tertiary industries. Zhu et al. [30] used a dynamic shift-share model to study
the different regional industry components with different characteristics.
These works combined with historical data and applied economic methods to
analyze and summarize some patterns of RIS. However, the analysis of some key
dimensions, such as the spatio-temporal dimension, is still missing.
Visualization techniques have also been introduced to help analyze industrial
structure. For instance, Zhang et al. [66] applied CiteSpace to study
industrial restructuring-related literature in the Chinese Social Sciences
Citation Index (CSSCI) database between $1998$ and $2017$ by mapping the
visual knowledge of high-impact authors, research institutions, and co-
occurring keywords. Jiang [70] revealed regional development patterns by
visualizing agglomeration and equilibrium evolution characteristics and
regional differences. Zhou et al. [68] visualized spatiotemporal and
multivariate GDP data to capture the dynamic characteristics of economic
developments. Later, they further utilized input-output table (IOT) data to
measure industry correlations and help users identify relevant temporal
changes in the underlying IOT data. Their goal is to combine the extraction of
economic clusters over a time period with the tracking of the evolution of
dynamic features across time. The characteristics of different clusters are
projected through MDS, and the proposed visualization methods similar to
Sankey diagram [56] allow analysts to visualize the dynamics of the clusters.
However, their work examines GDP data on a provincial basis, and their
visualization does not provide a good depiction of the internal structure of
spatial clusters, making it difficult for analysts to access the data in a
comprehensive way. In this work, we study the current state and dynamics of
RIS from the perspective of fine-grained business registration data through
elaborate visualization, which inherently influences and depicts the evolution
of industrial structure in a given region.

###  1.2 Time-Series Prediction and Evaluation

Forecasting the number of registered enterprises over a period of time belongs
to the category of time series forecasting. In the past decades, various time
series forecasting methods have been proposed, including classical statistical
methods and machine learning methods, aiming to predict future trends based on
empirical and data [64]. For example, different time series forecasting models
based on univariate time-series have been proposed, such as auto-regression-
based methods [10, 61] and exponential smoothing (ES) methods [31]. Later,
these traditional methods were extended to handle multivariate time series
[33] considering relevant properties. Vector Error Correction Model (VECM) was
proposed to predict the stock market [51], and Random Forest (RF) was applied
to predict H5N1 avian influenza outbreaks [35]. With the development of deep
learning and its variants, advanced models such as recurrent neural networks
(RNNs) [27] have been applied to solve multivariate time series forecasting.
In this work, we have implemented some of the most representative algorithms
of different classes in the scenario of industrial structure analysis. Also,
various methods have been proposed to evaluate and compare different time
series forecasting models [5, 11, 32, 37]. We used mean absolute percentage
error (MAPE) as a measure to evaluate our multivariate forecasting models
because it overcomes the scale dependence [7, 57]. We further provide an
explanation for the comparison and evaluation of RIS by revealing the
importance of different features in the forecasting process.

###  1.3 Visualization of Spatiotemporal Data

Visual analysis of information related to regional enterprises, such as
location, belongs to the visualization of spatiotemporal data with multiple
attributes. In particular, when analyzing the evolutionary dynamics of RIS,
well-designed visualizations are needed to integrate the spatiotemporal
aspects of the data. For example, Andrienko et al. [4] surveyed the literature
on existing experimental techniques related to spatiotemporal data and
corresponding analysis tasks. In general, spatial data distributions are
typically represented using heat maps [25, 47, 49], choropleth maps [60], and
glyph-based visualization techniques [9] to exploit, which are intuitive for
users to perceive and understand spatial data features. However, location
data, usually represented by pins on the map, will inevitably hide temporal
information. Efforts such as color-coding different times of the day [22] and
drawing connecting lines or curves through spatiotemporal cubes indicate
connectivity. However, they can introduce visual clutter and occlusion, making
the results quite challenging to perceive. The use of animations [40] also
limits overview capabilities and introduces overloaded thinking. In this work,
we divide the temporal evolution of the enterprises into multiple parts and
visualize the aggregated enterprises as elaborate glyphs based on their
aggregated geographical locations. We then use connected paths to model the
evolutionary dynamics of the RIS.

##  2 Observational Study

###  2.1 Background

In order to understand how (well) domain experts analyze in practice the
current state and dynamics of RIS, we worked with a team of experts from a
partner university and two governmental functions. They were two researchers
specializing in innovation-oriented entrepreneurship and fintech (E1, female,
age: $33$; E2, male, age: $34$), a business administrator (E3, male, age:
$38$), and a local industrial park planner (E4, female, age: $31$). Their
research and work have significant overlap in providing insight into local
economic markets through qualitative and quantitative analysis of information
on business and industry activities [16, 28, 29]. For example, E1 uses
theoretical models and empirical analysis to examine the different forces that
influence the development of enterprises and startups, and ultimately the RIS.
She also mentioned an effort proposed by the Harvard Growth
Lab111https://growthlab.cid.harvard.edu/videos/introducing-metroverse-growth-
labs-urban-economy-navigator called Metroverse, which aims to provide
stakeholders with unprecedented economic data on more than $1,000$ cities
around the world to help identify pathways to growth and diversification. She
likes the idea of using visualizations to bring research to the city level and
answer questions such as “what is the economic makeup of a city and where does
that city fit in the industrial space?” E2, on the other hand, focuses on
using advanced automated methods to predict future trends in business and the
accompanying industrial structure in terms of proportions in each industry and
cross-industry transformation. Based on their findings and recommendations, E3
and E4 will work together to assess the current state of RIS in conjunction
with a spreadsheet of relevant businesses and startups registered with local
government functions to infer key elements of RIS dynamics to gain insights
and develop strategies accordingly.

Nevertheless, in analyzing RIS dynamics, the experts had some concerns. First,
E3 and E4 commented, “we are not sensitive to updates in information about
registered companies.” Given the nature of the data, experts would like to
have some visual cues about potential patterns of interest in the historical
data and possible future trends for further decision-making needs. Second, in
exploring information related to enterprise registration, E1 and E2 attempted
to plot all enterprises on a geographical map based on latitude and longitude
coordinates, which created significant visual confusion. “In addition to the
scale, the points may have shapes that represent categories of enterprises and
colors that show another attribute, making them difficult to present, let
alone understand,” said E.2. In addition, these records exhibit spatiotemporal
variation, which “is not easy to address with a general summary based on
statistics.” Experts wondered if we could visualize these records in a more
abstract way, while maintaining details of dynamic and multivariate
attributes. Third, current practices focuses on RIS from a global and high
level perspective. For example, E3 and E4 often examine RIS dynamics from the
perspective of sectoral proportions and cross-sectoral shifts, which is
shallow. E4 shared one of his observations about the analysis of
agglomeration-performance relationship in certain regions: the initial goal of
an industrial park or region is to target a specific industry and integrate
upstream and downstream enterprises in the chain to form spatial aggregation
of property rights for all enterprises in the industrial chain. However, most
regional industrial parks encounter a similar problem: “Most industrial parks,
due to initial financial pressures, eventually drift into a spatial
aggregation of property in different and unrelated industries”. Thus, experts
were curious to perform a retrospective analysis of past and existing RIS in
an attempt to answer questions such as “does regional industrial agglomeration
really benefit enterprises?” Thus, a nontrivial visualization is needed to
clarify the “agglomeration-performance” relationship behind enterprise
activities.

Figure 1: RISeer pipeline. (A) RIS projection view obtains the overall
distribution of RIS patterns over the entire periods; (B) Registration
prediction view predicts the number of business registration and reveal
feature importance; (C) RIS evolution view explores regions’ characteristics
and evolution over time; (D) Spatiotemporal view projects the identified
regional clusters onto the map. (E) Box plots of growth rate of business
registration activity.

###  2.2 Experts’ Needs and Expectations

We interviewed E1 – E4 to identify their concerns and potential barriers to
using business and industry information to effectively analyze RIS dynamics.
At the end of the interviews, the need for a visualization system was a key
theme in the feedback gathered. We followed a user-centered design framework
using discussion, brainstorming, and design. After several iterations, we
gathered their input and condensed it into a set of following requirements.
The three Industrial Sector-Level requirements (R.1 – R.3) provide
stakeholders with a general sense of the dynamics of the industrial sectors
and the three Geospatial Region-Level requirements (R.4 – R.6) enable
stakeholders to understand the aggregation, migration, and evolution of
industrial structures over time within a given geospatial region.

R.1 Explain the temporal dynamics in the industrial and business sectors. The
traditional approach is to transfer the change in information from the
enterprise level to its corresponding sector level by calculating its
contribution to the sector. While understanding the change in ratios for each
sector provides an overview of the dynamics of all sectors over time, it only
considers absolute numbers and ignores factors that may influence change in
RIS dynamics. E1 is interested in explaining specific increases, outbreaks, or
decreases on the time axis. Therefore, equipping the system with factor
analysis allows to observe and speculate on the factors that influence the RIS
temporal dynamics.

R.2 Explore general patterns and potential outliers in the dynamics of sector
change. Experts need to quickly browse through large amounts of enterprise
data to identify areas of interest. For example, sectors with unprecedented
growth are more likely to attract the expert’s attention. Further
investigation is needed to explain general patterns and potential outliers
across historical periods. For example, E1 – 2 would like to find out if there
are standard characteristics and outliers between industrial sectors and the
possible reasons behind them.

R.3 Predict the future composition of the RIS and explore the influencing
factors behind it. Another conventional approach to RIS analysis is to
understand the composition of the industrial sector, for example, the ratio
and the appropriate number of establishments. Based on the history and current
status of RIS, experts want to know the future trends and influencing factors
of the quality of the composition of each industrial sector. Thus, they can
make speculations and assumptions about the future before developing the
corresponding strategies.

R.4 Summarize the dynamics and multivariance of enterprises. As mentioned
earlier, directly mapping millions of enterprises on a geographic map without
any abstraction or simplification is bound to create visual clutter,
compromise effective understanding, and affect the performance of the system.
In addition, as time goes on, more and more new businesses are being
established and added to the existing environment. Integrating these factors
requires careful design to summarize the dynamics and multiple attributes of
large-scale enterprises.

R.5 Reveal the regional “agglomeration-performance” relationship resided in
various groups of enterprises. According to E3 – 4, enterprise-level
performance may be strongly related to the geospatial locations of
agglomerations. Although theoretical studies have shown predominantly on
positive performance effects as an incentive for enterprise collocation “to
explain the emergence of agglomerations” [8, 50], experts have also focused on
adverse performance effects, “some enterprises may benefit from agglomeration,
while others may be harmed by aggregation and relocate elsewhere,” said E4.
Thus, in the face of these conflicting empirical findings, the net performance
effects of enterprises located in geographic agglomerations remain ambiguous
[38, 52, 59], calling for clarification of, e.g., “the agglomeration-
performance” relationship hidden in the large-scale activities of enterprises.

R.6 Compare and track performance at the region level. Having summarized the
dynamics and multiple attributes of enterprise at the regional level, experts
want to conduct in-depth analyses of RIS, such as determining regional
performance, comparing multiple regions at a specific point in time, and
tracking the evolution of a particular region over a long period of time. For
example, comparing two regions at a specific timestamp makes it possible to
identify differences in their positioning, while tracking their performance
over time can facilitate research into the reasons behind booms or busts.

##  3 Approach Overview

We propose RISeer, an interactive visual analytics system that helps to
understand the current status and dynamics of the RIS from the perspective of
industrial enterprise registration information. Figure 1 shows the
architecture of RISeer, which consists of a back-end engine in terms of a data
processing module and a forecasting module, and a front-end visualization. In
the back-end engine, the features hidden in the original raw data should first
be quantified and transformed into attributes acceptable to the time-series
multivariate inputs in the data processing module before building the
prediction model. In the prediction module, various machine learning models
are first trained based on the input data and features, and then these models
are employed to make registration predictions. In addition, we determine the
importance of the features for each model. We also divide the business
registration data into segments and generate clusters for each part. In the
front-end visualization, We provide several elaborate visual designs for
analysis.

##  4 Back-end Engine

###  4.1 Data Description and Processing

The collaboration with the domain experts provided us with a publicly
available dataset of business registration records collected from the National
Enterprise Credit Information Publicity System222http://www.gsxt.gov.cn,
covering the period from $1980$ to $2015$. Each record contains information
such as the enterprise’s primary identification code, name, address, and
registered capital. Based on suggestions from domain experts, we developed and
studied the following attributes in this work: 1) address: it records detailed
geographic information about a business; 2) start and end dates of operation:
it allows us to track the the operations of businesses that existed during a
specific period for further analysis; and 3) basic information, including
industry category, credit rating, registered capital, and enterprise status,
among others. This information can be used to describe a specific business.

To understand the evolutionary pattern of RIS, we transform the registration
records of existing enterprises into a time-series dataset. Then, we use a
multivariant time series prediction machine learning model to obtain the basic
features of the evolutionary patterns. Forming time series data from raw
business registration data is nontrivial due to the large volume and complex
attributes of raw business registration data. Moreover, the temporal
characteristics, i.e., the start and end dates of the business, are hidden in
each record independently of each other. Therefore, considering the
computational efficiency and speeding up the processing, we change the index
of the data from the main identity code of the enterprise to the date and
construct a hash table to reorganize the original information. Before modeling
the time series data, several non-numerical dimensions, such as industry
category and credit rating, must be quantified. Since these attributes have a
limited and relatively fixed range of values, it is acceptable to map them to
numbers in order to maintain the information encoded using one-hot encoding.

###  4.2 Time-series Forecasting Machine Learning Models

We define the input to a time series prediction machine learning model as a
sequence of historical data $X_{n}={x_{n-L},x_{n-(L-1)},...,x_{n-1},x_{n}}$,
where $L$ is the parameter that is adapted for different datasets and $x_{n}$
is a multidimensional feature vector with timestamp $n$ (i.e., the feature
vector consists of $7$ dimensions, including year, month, enterprise
classification code, registered capital, credit rating, enterprise property,
and enterprise state) denoted as $x_{n}^{f}\in R$. The goal of a multivariate
time series forecasting models is to predict specific values at some future
time stamp. In our case, we provide only a single-step forecast with an output
market of $y_{n}=x_{n+1}$. Considering that different countries divide the
industrial structure in different ways, but basically divided into three main
categories, we first divide the original dataset into three significant
industries based on industry category, namely, primary industry, secondary
industry, and tertiary industry. Then, we calculate the number of enterprises
existing in each month from $1980$ to $2015$. We use the data from $1980$ to
$1990$ as the first training set to predict the trend in $1991$, followed by
the data before $1991$ as the training set to predict the trend for $1992$,
and so on. The reason for performing stepwise forecasting is based on the
practical requirements of most real-world time series forecasting tasks. Thus,
the final prediction curve is spelled out for each time unit. We use the
classical MAPE metric to evaluate the performance of the forecasting model.
There are several representative time series forecasting models available for
evaluation, such as Arima [61], Vector Auto-Regression Model (VAR) [46],
Random Forest Model (RF) or Random Decision Forests [26] and Long Short-Term
Memory Recurrent Neural Networks (LSTM), covering linear, nonlinear regression
methods, and machine learning models [64]. In this work, we considered both
model performance and model interpretability, and finally select two machine
learning models for multivariate time series prediction, namely, RF and
XGBoost [13], as they have sufficiently high model accuracy and good
interpretability.

To explore the dynamic evolution of the time-series-based prediction model for
each business sector, we further provide model explanations to help users
understand the model performance and its differences in terms of feature
importance [64]. Notably, we chose to use SHapley Additive exPlanations (SHAP)
values to check the correlation between the input feature importance and the
output prediction results for business registrations [45]. We chose SHAP
because the contribution of features to the predicted value is easily to
observe. Moreover, it proves to be more consistent with human intuition [45]
than other interpretable machine learning techniques, such as LIME [55]. Also,
the value of SHAP can be directly used to add up to the actual predictive
value, explaining the importance of the feature, which is consistent with our
motivation for the subsequent visual design.

###  4.3 Spatiotemporal Clustering

Two challenges exist in tracking the evolution of RIS over long periods of
time and large geographic scales. First, correctly delineating long time
periods is critical to identifying potential patterns in the RIS evolution as
reflected by business registration activity. Simply retrieving records by year
may lead to duplicate analyses if no significant changes occur in the two
subsequent periods. Second, even if we manage to divide the evolution of the
RIS into different intervals, for each interval we are still confronted with a
large amount of data that shows different spatial locations by the records of
the enterprises belonging to that interval. It is important to abstract the
characteristics of the corresponding enterprises while maintaining dynamic and
multi-attribute details. In the following subsections, we perform spatio-
temporal clustering in terms of time series segmentation and geospatial
clustering.

Figure 2: Three sub-charts from the top to the bottom represent primary,
secondary and tertiary industry, respectively. Each sub-chart contains a
dashed line representing the ground-truth value, a solid line representing the
predicted value and stacked bars representing feature importances.

Partitioning of Time Series. To address the first challenge, we formulate the
partitioning of long periods of RIS evolution as a segmentation problem, which
partitions long time series into segments that can be formulated as follows:
given a time series $T$, generate the best representation such that the
combined errors (which can be obtained by calculating the sum of the maximum
error for all segments) is less than a user-specified threshold. Generating
segments for such time series data is the key to an efficient and effective
solution [36]. Notably, we take a top-down approach, considering every
possible partition of the time series and segmenting it at the optimal
location. The two subsegments are then tested to see if their approximation
error is below a user-specified threshold. If not, the algorithm recursively
splits the subsequence until the approximation error of all segments is below
the threshold. Thus, the original time series registration is divided into
several piecewise linear representations after time series segmentation.

Geospatial Clustering. To address the second challenge of the various
enterprise activities within each segment, we analyze the business data from a
cluster perspective rather than an individual perspective. The goal of
clustering is to make the similarity between objects within the same category
as large as possible and, conversely, the similarity between different
categories as small as possible. It is worth noting that we tried to cluster
individual enterprises belonging to different time periods using the division-
based KMeans algorithm and the density-based clustering algorithm DBSCAN,
respectively. However, they both exhibit specific advantages and
disadvantages. For example, although KMeans can achieve clustering results,
the determination of $K$ is quite challenging because the location of
enterprises is unpredictable. In addition, KMeans is developed based on the
centroid location of the aggregated clusters rather than the actual geographic
locations of enterprises. On the other hand, DBSCAN requires manual input of
two parameters, namely $[Eps]$ and $[minPts]$, so the accuracy of the
clustering results directly depends on the users’ parameter selection. Also,
the obtained clusters are spatially arbitrary shaped and are just sets of
geographical points rather than aggregated clusters. In addition, DBSCAN
suffers from indistinguishable noisy data. For these reasons, we propose a
hybrid algorithm based on DBSCAN and KMeans algorithms. First, the geographic
locations of enterprises are clustered into several clusters using the density
reachability of DBSCAN according to the specific settings of the two
parameters. The data in each cluster is taken as a new input. The centroid
positions are then obtained using KMeans to minimize the Sum of Squared Error
(SSE) between the data points in each cluster and the centroids of the
clusters they are in using the iterative aggregation of KMeans and $K$ is set
to $1$. In addition, we use KANN-DBSCAN [41], to automatically find a stable
interval of cluster number variation by generating candidate $[Eps]$ and
$[MinPts]$ using the distribution characteristics of the dataset. We take the
$[Eps]$ and $[MinPts]$ parameters corresponding to the minimum density
threshold of this interval as the optimal parameters. Specifically, first, the
corresponding $[Eps]$ candidate list can be obtained using KANN-DBSCAN. Then,
for the given $[Eps]$ candidate list, the number of neighboring objects can be
calculated sequentially and the expected values of these $[Eps]$ are used as
the $[MinPts]$ candidate list: $MinPts=\frac{1}{n}\sum_{i=1}^{n}P_{i},$
$P_{i}$ is the number of $[Eps]$ neighboring objects of the $i_{th}$ object,
and $n$ is the total number of objects of the data. Finally, the list of these
two parameters is given as input to DBSCAN, and then the number of clusters
generated with different parameter settings is obtained separately and
quickly. The result can be considered stable when the number of clusters is
the same three times in a row. After deciding the clusters, we need to further
define the centroids of each cluster. To summarize, we first apply KANN-DBSCAN
to cluster the geographic locations of enterprises into several clusters, and
then use KMeans to find the centroid location of each cluster.

##  5 Front-end Visualization

RISeer uses or enhances familiar visual metaphors so that domain experts can
focus on analysis. We strictly follow the mantra of “overview first, zoom and
filter, then details-on-demand” [58]. As shown in Figure 1, we develop five
main visualizations to help domain experts explore the evolutionary patterns
of RIS at the overview, evolution pathway, and regional cluster levels.
Specifically, we design the RIS Projection Overview to obtain the overall
distribution of RIS patterns over the entire periods (R.1). The Registration
Prediction View captures the correlation between the predicted registrations
and ground truth over time, and understands the latent features driving the
changes from the perspective of the time series forecasting models (R.2 –
R.3). The RIS Evolution View explores the regional features and their
evolutionary paths over time (R.4). The Spatiotemporal View visually projects
the identified regional clusters onto their corresponding locations to
facilitate understanding map-specific RIS patterns such as mitigation,
radiation, and aggregation (R.4 – R.5). The Regional Comparison View helps
domain experts to make detailed comparisons in terms of geospatial indicators
across multiple regions (R.6).

###  5.1 RIS Projection View

The RIS Projection View helps to identify potential anomalies and “clusters”
in RIS snapshots. Dimensionality reduction techniques such as t-SNE, PCA, and
MDS have been widely adopted to create low-dimensional representations that
preserve local similarity to express neighborhood structure [24, 54]. We
follow the conventional practice of projecting all RIS snapshots for the
entire period into two-dimensional space to see potential clusters and
outliers. After discussions with experts, we use the following metrics,
namely, enterprise classification code, registered capital, credit rating,
enterprise property, and enterprise state, to evaluate the RIS snapshots at
the overview level. After obtaining the above metrics, we can obtain the
corresponding feature vectors for each RIS snapshot. We use t-SNE as the
dimensional projection because “it reveals meaningful insights about the data
and shows superiority in generating two-dimensional projection” [40]. Similar
to [63], as shown in Figure 3, each RIS snapshot in two-dimensional space is
represented by a point, and the color indicates the snapshot. The first and
last snapshots are highlighted (red for the first snapshot and blue for the
last) and a black curve connects all snapshots in chronological order. The
user can hover over any particular RIS snapshot and a tooltip will display a
detailed timestamp. The user can lasso any entity on the projection space for
interaction and for further “link + view” analysis.

Figure 3: The RIS Projection View facilitates identifying the potential
anomalous and “clusters” of RIS snapshots. Figure 4: Design alternatives for
the feature importance. Figure 5: The RIS Evolution View helps domain experts
conduct metric comparisons at the level of regions. Figure 6: The
Spatiotemporal View: (a) Packed circles based on physical simulator; (b)
Regional evolution path; (c) Five-colored bubble sets indicate clusters at
different time periods; (d) The glyph represents an overview of regional
information. After clicking one glyph, more information will be shown in a
separate panel: (1) Geospatial distribution of six metrics in glyph; (2)
Registration information; (3) Enterprise livability; (4) Business category,
and (5) Heat map represents actual geographical coverage of the regional
cluster.

###  5.2 Registration Prediction View

The Registration Prediction View (Figure 2) helps experts understand the
detailed associations between RIS influencing factors and registration amounts
based on several representative time-series forecasting models. It consists of
three subplots, arranged from top to bottom, representing business activities
in the primary, secondary and tertiary sectors, respectively. In each subplot,
the dotted line represents the ground truth registration of enterprises. The
solid black line represents the predicted business activity compared to the
ground truth volume. Thus, users can observe the gap that represents the
difference between the actual and the predicted value at different timestamps.
In addition, the importance of the $7$ features is calculated, normalized, and
represented with different classification colors stacked on top of each other.
In each snapshot, the glyphs of feature importance are placed along the
Y-axis. Each bar in the stacked glyphs represents a feature. Bars stacked
above the predicted values represent features with negative effects, pushing
the predicted value down [64]. The features stacked below the predicted value
are positive, pushing the predicted value higher. The user can select other
models on the right side for comparison.

Design Alternatives. We consider several alternative designs as shown in
Figure 4. Figure 4(a) simply uses a multilinear graph to represent the
importance of features over time. When the number of features is large, visual
clutter can be severe. To alleviate this problem, we propose Figure 4(b),
which replaces the polylines with stacked bars. However, it cannot show the
continuous change of features because the order of different features is not
fixed at each timestamp. Figure 4(c) is an improvement on the second one. The
height of the bars encodes the absolute feature importance value, and the bars
corresponding to negative features are highlighted with strikes. In this
design, the order of features is fixed, allowing to track the changes in
feature importance over time. However, it mixes positive and negative features
and does not relate the predicted values to the importance of the features. We
finally chose Figure 4(d) because it better distinguishes between positive and
negative features and embeds the predicted values in the design.

###  5.3 RIS Evolution View

Although the above RIS Projection View conveys additional information about
clusters and potential outliers in all RIS snapshots, it is still unclear how
the RIS patterns evolve at the regional level. In addition, domain experts
often need to explore the properties of each regional cluster in order to
compare and evaluate them in one or more temporal snapshots, which can help
them to assess the performance of RIS. Therefore, based on discussions with
experts, we adopt and standardize the following indicators for a uniform
comparison: 1) number of primary industries; 2) number of secondary
industries; 3) number of tertiary industries; 4) aggregation index ($AI(x)$),
which is calculated by the Coefficient of Variation [19], to measure the
similarity of the distribution of indicators as follows:
$CV(x)=\frac{\sqrt{\frac{1}{N}\sum_{i=0}^{N}{(x_{i}-\mu)}^{2}}}{\mu},$ and
$AI(x)=\frac{1}{CV(x)}$; 5) average registered capital; 6) total registered
capital; 7) credit rating; 8) livability, i.e., the number of surviving
enterprises divided by the total number of enterprises; and 9) mortality,
complementary to livability.

Inspired by Lineup [23], we design a RIS Evolution View to help domain experts
to compare metrics at the regional level. Notably, as shown in Figure 5, we
present the metric values for each region cluster as a combined bar, where the
length of a single colored bar indicates the normalized metric value for the
corresponding region cluster. We rank the region clusters generated in all
consecutive snapshots. The ranking is based on the value of a specific metric.
When a region cluster is clicked, all “identical” region clusters across time
snapshots are highlighted. We determine whether a regional cluster is
identical between two subsequent snapshots based on the amount of overlaps,
and we only depict the evolutionary path between two subsequent “identical”
region clusters with the largest number of overlaps. In the two subsequent
snapshots, we place an axis with two different scales along the left and right
sides of the axis. The left axis represents the scale of the number of
enterprises that transition from one regional cluster to another in the next
period. The right axis represents the scale of the distance between the two
centroids of the regional clusters in the two subsequent snapshots. For
example, as shown in in Figure 5(1), these two figures indicate that $155,950$
enterprises in one regional cluster between $1992$ and $2006$ stay in the
other regional cluster between $2006$ and $2012$, while the distance between
the centroids of the two regional clusters is $1.87km$ each. Intuitively, a
small value indicates that the main distribution of enterprises in the early
snapshot is largely consistent with the distribution of enterprises, including
the existing enterprises in the early snapshot and newly established ones in
the later snapshot.

Figure 7: The RIS projection view projects Shenzhen’s enterprises’
registration data into several clusters.

###  5.4 Spatiotemporal View

We apply the classical map-centric visual exploration approach to represent
clusters of enterprise geographic locations. As shown in Figure 6, we design
radar-like glyphs at the top of the map and place them in the corresponding
geographic regions of the regional clusters generated by the geospatial
clustering described above. The indicators represented by the axes in the
radar-like glyphs are the same as those in the RIS evolution view, namely,
aggregation index, average registered capital, total registered capital,
credit rating, livability, and mortality. We did not choose a bar chart
because we need to have the radius of the glyphs represent the number of
enterprises in the cluster. In addition, in the Regional Comparative View (see
Section 6.5), we need to compare the same indicators between two or three
clusters, and radar-like glyphs are easier to perceive than bar charts for
comparison.

Note that we place all the regional clusters generated by DBSCAN and KMeans on
the map at different time intervals, inevitably causing visual clutter and
overlap problems. Therefore, we use a force-directed layout to pack those
overlapping clusters [15]. As shown in Figure 6(a), a physics-based simulator
will be used to find the optimal circle positions by 1) optimizing the
distance between the circle and the force center, 2) attracting each other
slightly, and 3) avoiding overlap. We warp the packed circles with an
additional black ring (Figure 6(a)), indicating that these groups of regions
have some kind of unavoidable overlap. To emphasize the representation of
region groups from a particular time snapshot and inspired by [3], we borrow
the Bubble Sets technique [14] to provide continuous boundary contours that
allow us to examine scenarios with semantically important spatial organization
and important set membership relationships for enterprises belonging to the
snapshot. Notably, as shown in Figure 6(c), the five colored bubble sets
represent the geospatial locations in the five snapshots. We also provide
regional evolution paths in the form of black curves that connect clusters of
regions in different snapshots in chronological order (Figure 6(b)).

To elaborate more detailed information about region-specific clusters, we
provide a separate panel (Figure 6(1 – 5)) showing the geospatial metrics
(Figure 6(1)), registration (Figure 6(2)), livability (Figure 6(3)), and
business category (Figure 6(4)). The actual geographical coverage of the
regional clusters is displayed as a heat map (Figure 6(5)), where domain
experts can observe the most dense points and the scale of distribution.

###  5.5 Regional Comparative View

While it is crucial to get a general impression and assess the overall
distribution of indicators for regional clusters on a map, experts also want
to understand how they differ from each other and whether one specific
regional cluster is better than another. That is, multiple regional clusters
should be placed in the context of each other in order to thoroughly compare
relevant information and differences across regions simultaneously. When users
find several ideal candidate regional clusters through the Spatiotemporal
View, they can add them in for detailed comparison. Inspired by Skylens [67],
we design a Regional Comparative View that allows users to take a closer look
at the differences between several regional clusters (Figure 1(E)).
Specifically, we invite a radar plot-based glyph to perceive the factor values
of different regional clusters. For our specific scenario, we enhance the
traditional radar plot in several ways. First, the axes show the same
indicators as in the spatiotemporal view. However, since most of these
indicators are numerical values rather than distributions, it is not feasible
to represent the detailed distribution of each indicator at the regional
cluster level. Inspired by the RIS agglomeration phenomenon, we decide to
calculate the indicator distribution based on geographic agglomeration. In
other words, each glyph is represented as a series of rings, representing
geographic distances. Specifically, after analyzing the location distribution,
we arrange the distances from the cluster centroid into segments, i.e., $0$ –
$1.5$km, $1.5$ – $2$km, $2$ – $4$km, $4$ – $6$km, and $6$ – $10$km. Thus, each
indicator can be restructured according to distance segments in an attempt to
answer questions such as “is there an industrial agglomeration in such
regional clusters?” To mitigate the distortion caused by the visual overlap of
different metrics, the user can highlight a particular metric by hovering over
it. The differences between two regional clusters are described by radar
glyphs illustrating the indicator distributions in the two corresponding
regional clusters for comparison. In addition to comparing selected regional
clusters, the growth rates for each industry along the evolving RIS paths
provide insight into regional development, especially if the region is
expanding its geographic size. As shown in Figure 1(E), we plot the growth
rate of business registration across all snapshots using box plots, with each
representing one region of a particular RIS path. Multiple box plots can be
aligned to facilitate comparative analysis of growth rates across regions in a
single snapshot.

###  5.6 Interaction Among the Views

Rich interactions are integrated to facilitate exploring and assessing RIS
patterns. Users can select regional clusters of interest in either RIS
Evolution View or Spatiotemporal View, and RISeer will highlight the
corresponding information in other views. The time range in Registration
Prediction View can be adjusted interactively to highlight specific periods of
RIS evolution. The coordinated interaction between comparable regional
clusters facilitates the inspection of a variety of information at different
granularities and other contexts. Users can also hover the mouse over the path
of a bubble set and the covered regional clusters will be highlighted for
visualization.

##  6 Evaluation

###  6.1 Case I: Shenzhen Leading China’s Economic Miracle

Below are a series of activities that occurred in E1 – 2 while exploring
Shenzhen business registration records over the past $30$ years using RISeer.
In particular, Case 1 uncovers RIS evolution patterns and key observations of
Shenzhen that lead China’s economic miracle.

Figure 8: Identifying two country-level districts were merged into one
district by observing two RIS evolving and merging paths. Figure 9: The
financial sector in the selected regional cluster (i.e., Futian District,
Shenzhen) is gradually occupying a dominant position over time.

After experts loaded a total of $1.1$ million in business registration
activity in Shenzhen over the past $30$ years into RISeer, the RIS Projection
View breaks down Shenzhen business registration data by month. It describes
the corresponding business registration data for each month based on a feature
vector consisting of enterprise classification code, enterprise state,
enterprise property, credit rating and registered capital. In other words, a
feature vector is generated for the business registration data in Shenzhen in
the monthly snapshot. After projecting the high-dimensional feature vectors
onto a two-dimensional plane, experts observed several different clusters
(Figure 7(1)). E2 then lassoed this view and determined that each cluster
corresponded to a specific continuous period. The experts were quite
interested in those projected observations with large distance spans, so E2
directly performed a larger lasso operation on this view (Figure 7(2)). As
shown in Figure 7, after the lasso operation, the registration prediction view
predicts the registration data of enterprises for the corresponding time
horizon, including ground truth, model predictions, and feature importance. As
shown in Figure 7(3), E1 observed that the number of enterprises in Shenzhen
grows very rapidly from $1993$ to $1998$. E1 also found that at the beginning
of this rapid rise, the difference between the model prediction and the ground
truth is large, but this difference slowly decreases over time. Therefore, E1
believes that some different events occurred during this period and that this
pattern of change is new to the model and will take some time to learn. E2
further noticed that the feature importance captured by the model during this
period shows a sharp increase in the contribution of features of enterprise
attributes in the primary sector, while the changes in the secondary and
tertiary sectors are the result of a combination of features. E2 then moved to
the RIS Evolution View, focusing on the paths corresponding to that period. As
shown in Figure 8, after clicking on multiple evolutionary paths, the expert
found a tendency for two of them to merge during this period (Figure 8(1)).
The expert then clicked on each cluster glyph to view the geographic coverage
through the heat map and found that the starting areas of the two paths were
in Henggang District and Longgang District (Figure 8(2)). The expert added
that before $1993$, Henggang and Longgang were two township-level districts in
Shenzhen, respectively, and after $1993$, they were merged into Longgang
District due to administrative planning (Figure 8(3)). Moreover, E1 mentioned
that the rapid growth of newly registered enterprises in this area has led to
the relocation of cluster centroids, “the rural shareholding cooperative
system has been implemented in this region.” That is, the nature of the
primary sector has changed. This also explains the predictive model believes
that the feature of enterprise property was the main contributing factor in
this period. Experts also mentioned that the “‘Three-Plus-One’ trading-mix
strategy (i.e., customized production and compensatory trade in materials,
designs, or samples) stimulated the development of the local economy, leading
to a rapid growth in the number of regional businesses.“

In addition to observing the above phenomena of RIS path evolution and
merging, experts would like to explore how the RIS evolves along the RIS
paths. Notably, as shown in Figure 9(a – c), E1 relocated a RIS path, selected
several regional clusters in chronological order along the path, and added
them to the Cluster Comparison View for further comparison. E1 witnessed that
over time, the registered capital, the livability, and the credit rating of
regional enterprises are increasing, which indicates a positive momentum of
industrial development in the region. By looking closely at the information of
each cluster, experts found that the financial industry in the region (i.e.,
Futian District, Shenzhen) is becoming dominant over time. This development
trend also reflects the spirit of the “Shenzhen Special Economic Zone
Development Plan”, promulgated in $1982$, which clearly defines Futian as a
commercial, financial and administrative center.

Figure 10: Two paths: (a) is based on the primary industry development and (b)
is based on the secondary/tertiary industries.

###  6.2 Case II: Development Comparison Between Wuhou District and Qionglai
District in Chengdu

This case documents our experts’ use of RISeer to study business registration
activities in Chengdu. E1 and E2 first turned their attention to the RIS
Evolution View and interactively examined the evolutionary patterns of the
different paths. They found that one of the paths (path1 (Figure 10(a))) is
mainly based on the development of the primary sector. In contrast, the other
path (path2 (Figure 10(b))) is mainly based on the development of the
secondary and tertiary sectors. Based on the geographical location information
related to the regional cluster provided by the Spatiotemporal View, the
experts witnessed that the enterprises covered by path1 are mainly located in
Qionglai (a country-level city affiliated with Chengdu). As shown in Figure
10, in the RIS Evolution View, the experts further observed that the primary
industry of path1 has been proliferating since $2004$. Based on the above
information, experts investigated and studied the relevant periods and
locations and found that in order to balance urban and rural development,
Chengdu established a model of arable land protection fund system in $2008$,
and Qionglai was the key area of arable land protection. Especially, “after
the implementation of the farmland protection fund system, the arable land
area in Qionglai has increased year by year”, and corresponding agricultural
output value has also been increasing. These policies have greatly stimulated
the development of the primary industry. Meanwhile, through the heat map
distribution of the clusters corresponding to path2 (Figure 11), the experts
could see that the enterprises participating in path2 were mainly in the urban
area of Chengdu. They also found that the number of enterprises in path2
increased significantly after entering period2. In particular, the number of
enterprises that existed in $2004$ – $2012$ was twice as many as in $1995$ –
$2003$. The secondary industry grew rapidly, “the deepening industrial reforms
in Chengdu are due to the impact of large-scale development policy in the
western region.”

Figure 11: Comparing the starting/ending regional clusters for two paths.

The experts also found different growth rates for the two paths from the Path
Comparison View. It can be seen that the growth rate of the primary sector for
path1 is generally higher than that of path2. On the contrary, the growth
rates of the secondary and tertiary sectors of path2 are significantly higher
than those of path1 after $2004$ (Figure 10). In fact, the regions
corresponding to path2 have entered a rapid development phase after $2004$. By
comparing the starting aggregation information of the two paths (Figure 11),
the experts found that the starting aggregation of path2 is more uniform than
that of path1 in terms of the distribution of various indicators. The
aggregation index of path2 is also relatively higher than that of path1. This
conclusion can also be verified by the heat map corresponding to cluster,
where the experts determined that the geographical distribution of the
starting cluster of path1 is more discrete. In contrast, the geographic
distribution of the starting cluster of path2 is more dense. Likewise, experts
can find that the average registered capital and total registered capital of
the end cluster of path2 are significantly higher than those of the end
clusters of path1. This is due to the fact that path2 develops mainly
secondary and tertiary industries, while path1 develops primary industries.

##  7 Discussion and Limitation

We conducted one-hour semi-structured interviews with E1 – E4 to verify the
efficacy of our methodology to study RIS dynamics.

System Performance. All experts appreciated the ability of our system to
support exploration of RIS evolution patterns in large volumes of business
registration data. “RISeer is very cool because it provides a highly
interactive exploration to detect regional clusters and time periods,” said
E1. RISeer can examine a large number of enterprises activities at different
levels of granularity, which can “substantially reduce the amount of work
consumed by traditional data exploration.” E2 liked the design of
Spatiotemporal View because he had tried to present the location of all
enterprises on a map, but there was a serious visual clutter. RISeer embeds
spatiotemporal factors into the map, which “resolves my concerns when doing
customized exploration and decision making.” E3 commented that “it inspired me
when it provided some visual cues to remind me of some old messages and
strategies.” E4 said that RISeer has the potential to be deployed to real-
world scenarios and help RIS researchers with retrospective analysis. E1 and
E2 liked the idea of using predictions to examine the underlying data. E3
echoed his opinions, saying “the projection overview is very intuitive because
it fits my intuition about the evolving RIS in Shenzhen.” We conducted a user-
centric design process that involved experts in the initial design phase so
they soon became familiar with the visual encoding.

Contributions Over Previous Work. E3 indicated that RISeer can effectively
“extract the main features and factors that drive regional economic and
industrial restructuring”. Experts can identify the patterns of regional
development from the static historical data. If the industrial structure has
changed significantly in a certain period, RISeer can give direct hints to
capture this signal in time. In addition, traditional RIS analysis aggregates
the companies involved on a map based on their geographic locations and
divides them in fixed-size blocks. Nevertheless, such fixed divisions can
undermine the integrity of company aggregation. RISeer can effectively display
the phenomenon of industrial aggregation, radiation and migration, allowing
users to select specific areas and track RIS changes within that area.

Generalizability and Scalability. We discussed with the experts which parts of
RISeer could be used directly for other scenarios and which parts needed
customization. They said that the system is already very versatile and can be
applied to any region if the data is pre-processed beforehand. Also, other
economic data, such as GDP, can be easily integrated into RISeer. This work
has the potential to be applied to a variety of usage scenarios that leverage
“alternative data” to e.g., understand economic status, guide investment
strategy and find a market edge. Regarding the scalability, RISeer handles
millions of data records for a single city by abstracting to regional
clusters. Therefore, scalability issues for front-end visualization are
minimal. The bottleneck of algorithmic complexity lies in clustering. When the
data is huge and complex, the density of locations extracted each time may not
be consistent, which may not produce satisfactory results.

Limitation. First, there is a lag in updating enterprise information due to
occasional lax enforcement and incomplete filling. Our data may inevitably
contain such outdated information, which may bias the objective evaluation and
analysis of the current state of the RIS. Second, we divided all enterprises
into three significant industries without further subdivision. This is due to
the wide variety of subsectors within the three significant industries and the
different criteria in each country. However, this also prevents our system
from performing a more careful analysis of the substructures within the
sectors. Third, although we use multivariate time-series forecasting models to
predict the activities of enterprises, registration information may have
different criteria at an early stage, resulting in some missing data
attributes and inaccurate results. And we classified the target period based
on the changes in the overall number of business registration activities,
which may ignore other features. We could try the same time series
segmentation algorithm in a higher dimensional space, or support users to
segment the data themselves. Fourth, among the nine metrics abstracted from
regions, the livability and extinction rates of enterprises are time-varying
and should not be put together with the other indicators. Moreover, different
color encoding in different views may cause visual confusion.

##  8 Conclusion and Future Work

We propose a visual analytics approach to exploring and analyzing the
evolutionary dynamics of RIS revealed by large-scale regional business
registration activities. We first summarize the design requirements through
in-depth discussions with domain experts. The proposed RISeer facilitates the
examination and exploration of key time periods, dynamic correlation between
predictions and ground truth by feature importance, and regional clusters and
their spatiotemporal properties. Two case studies and expert interviews
demonstrate the effectiveness of our approach. Currently, RISeer is more like
a confirmatory analysis tool for observing and analyzing the changes in the
existing RIS. We plan to extend our approach to a broader scope, capturing
dynamic RIS evolution patterns using time series forecasting models better
suited to change and city-level comparisons.

###### Acknowledgements.

We are grateful for the valuable feedback and comments provided by the
anonymous reviewers. This work is partially supported by the Key Projects of
Shanghai Soft Science Research Program from the Science and Technology
Commission of Shanghai Municipality (No. 22692112900, 21692108700).

## References

  * [1] R. Abramov.  Regional economic policy based on industrial sector clustering in the context of sustainable.  Research journal of pharmaceutical, biological and chemical sciences, 7(2):2100–2106, 2016.
  * [2] M. Aleksander.  Great soviet encyclopedia, 1979.
  * [3] B. Alsallakh, L. Micallef, W. Aigner, H. Hauser, S. Miksch, and P. Rodgers.  The state-of-the-art of set visualization.  In Computer Graphics Forum, vol. 35, pp. 234–260. Wiley Online Library, 2016.
  * [4] N. Andrienko, G. Andrienko, and P. Gatalsky.  Exploratory spatio-temporal visualization: an analytical review.  Journal of Visual Languages & Computing, 14(6):503–541, 2003.
  * [5] J. S. Armstrong and F. Collopy.  Error measures for generalizing about forecasting methods: Empirical comparisons.  International journal of forecasting, 8(1):69–80, 1992.
  * [6] J. Atack.  Firm size and industrial structure in the united states during the nineteenth century.  The Journal of Economic History, 46(2):463–475, 1986.
  * [7] D. K. Barrow, S. F. Crone, and N. Kourentzes.  An evaluation of neural network ensembles and model selection for time series prediction.  In The 2010 international joint conference on neural networks (IJCNN), pp. 1–8. IEEE, 2010.
  * [8] G. G. Bell.  Clusters, networks, and firm innovativeness.  Strategic management journal, 26(3):287–295, 2005.
  * [9] R. Borgo, J. Kehrer, D. H. S. Chung, E. Maguire, R. S. Laramee, H. Hauser, M. Ward, and M. Chen.  Glyph-based Visualization: Foundations, Design Guidelines, Techniques and Applications.  In M. Sbert and L. Szirmay-Kalos, eds., Eurographics 2013 - State of the Art Reports. The Eurographics Association, 2013. doi: 10 . 2312/conf/EG2013/stars/039-063
  * [10] G. E. Box, G. M. Jenkins, G. C. Reinsel, and G. M. Ljung.  Time series analysis: forecasting and control.  John Wiley & Sons, 2015.
  * [11] F. Camastra and M. Filippone.  A comparative evaluation of nonlinear dynamics methods for time series prediction.  Neural Computing and Applications, 18(8):1021–1029, 2009.
  * [12] J. Chen.  Development of chinese small and medium-sized enterprises.  Journal of small business and enterprise development, 2006.
  * [13] T. Chen and C. Guestrin.  Xgboost: A scalable tree boosting system.  In Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining, pp. 785–794, 2016.
  * [14] C. Collins, G. Penn, and S. Carpendale.  Bubble sets: Revealing set relations with isocontours over existing visualizations.  IEEE Transactions on Visualization and Computer Graphics, 15(6):1009–1016, 2009.
  * [15] C. R. Collins and K. Stephenson.  A circle packing algorithm.  Computational Geometry, 25(3):233–256, 2003.
  * [16] S. Delanoë.  From intention to start-up: the effect of professional support.  Journal of Small Business and Enterprise Development, 2013.
  * [17] J. Drucker and E. Feser.  Regional industrial structure and agglomeration economies: An analysis of productivity in three manufacturing industries.  Regional Science and Urban Economics, 42(1-2):1–14, 2012.
  * [18] J.-l. Du, Y. Liu, and W.-x. Diao.  Assessing regional differences in green innovation efficiency of industrial enterprises in china.  International Journal of Environmental Research and Public Health, 16(6):940, 2019.
  * [19] B. Everitt.  The Cambridge Dictionary of Statistics.  Cambridge University Press, 1998.
  * [20] A. Evseev, N. Morozova, and I. Vasileva.  Digital activity of industrial enterprises in the context of ensuring sustainable economic development of territories.  In Proceedings of the 2nd International Scientific Conference on Innovations in Digital Economy: SPBPU IDE-2020, pp. 1–5, 2020.
  * [21] E. Feser, H. Renski, and H. Goldstein.  Clusters and economic development outcomes: An analysis of the link between clustering and industry growth.  Economic Development Quarterly, 22(4):324–344, 2008.
  * [22] J. Fuchs, F. Fischer, F. Mansmann, E. Bertini, and P. Isenberg.  Evaluation of alternative glyph designs for time series data in a small multiple setting.  In Proceedings of the SIGCHI conference on human factors in computing systems, pp. 3237–3246, 2013.
  * [23] S. Gratzl, A. Lex, N. Gehlenborg, H. Pfister, and M. Streit.  Lineup: Visual analysis of multi-attribute rankings.  IEEE transactions on visualization and computer graphics, 19(12):2277–2286, 2013.
  * [24] F. Heimerl and M. Gleicher.  Interactive Analysis of Word Vector Embeddings.  Computer Graphics Forum, 2018. doi: 10 . 1111/cgf . 13417
  * [25] B. N. Hilton, T. A. Horan, R. Burkhard, and B. Schooley.  Saferoadmaps: Communication of location and density of traffic fatalities through spatial visualization and heat map analysis.  Information Visualization, 10(1):82–96, 2011.
  * [26] T. K. Ho.  Random decision forests.  In Proceedings of 3rd international conference on document analysis and recognition, vol. 1, pp. 278–282. IEEE, 1995.
  * [27] S. Hochreiter and J. Schmidhuber.  Long short-term memory.  Neural computation, 9(8):1735–1780, 1997.
  * [28] S. Hong, R. M. Hunt, and K. Serfes.  Dynamic pricing of credit cards and the effects of regulation.  2018\.
  * [29] S. Hong, K. Serfes, and V. Thiele.  Competition in the venture capital market and the success of startup companies: Theory and evidence.  Journal of Economics & Management Strategy, 29(4):741–791, 2020\.
  * [30] Z. Huaxiong and Y. Fang.  Research on regional economic and industrial structure based on dynamic shift-share analysis: an empirical analysis of six provinces in central china.  In 2011 International Conference on Business Computing and Global Informatization, pp. 62–66. IEEE, 2011.
  * [31] R. Hyndman, A. B. Koehler, J. K. Ord, and R. D. Snyder.  Forecasting with exponential smoothing: the state space approach.  Springer Science & Business Media, 2008.
  * [32] R. J. Hyndman and A. B. Koehler.  Another look at measures of forecast accuracy.  International journal of forecasting, 22(4):679–688, 2006.
  * [33] S. Johansen et al.  Likelihood-based inference in cointegrated vector autoregressive models.  Oxford University Press on Demand, 1995.
  * [34] C. Kai-gen, Z. Cong-sheng, and C. Jia-bing.  A study on the standard of the rationality of regional industrial structure based on environments and resources [j].  Human Geography, 6, 2003.
  * [35] M. J. Kane, N. Price, M. Scotch, and P. Rabinowitz.  Comparison of arima and random forest time series models for prediction of avian influenza h5n1 outbreaks.  BMC bioinformatics, 15(1):1–9, 2014.
  * [36] E. Keogh, S. Chu, D. Hart, and M. Pazzani.  Segmenting time series: A survey and novel approach.  In Data mining in time series databases, pp. 1–21. World Scientific, 2004.
  * [37] U. Khair, H. Fahmi, S. Al Hakim, and R. Rahim.  Forecasting error calculation with mean absolute deviation and mean absolute percentage error.  In Journal of Physics: Conference Series, vol. 930, p. 012002. IOP Publishing, 2017.
  * [38] J. Knoben, A. Arikan, F. van Oort, and O. Raspe.  Agglomeration and firm performance: One firm’s medicine is another firm’s poison.  Environment and Planning A, 48(1):132–153, 2016.
  * [39] C. Li, X. Gao, J. Wu, and K. Wu.  Demand prediction and regulation zoning of urban-industrial land: Evidence from beijing-tianjin-hebei urban agglomeration, china.  Environmental Monitoring and Assessment, 191(7):1–14, 2019.
  * [40] Q. Li, K. S. Njotoprawiro, H. Haleem, Q. Chen, C. Yi, and X. Ma.  Embeddingvis: A visual analytics approach to comparative network embedding inspection.  In 2018 IEEE Conference on Visual Analytics Science and Technology (VAST), pp. 48–59. IEEE, 2018.
  * [41] W. Li, S. Yan, Y. Jiang, S. Zhang, and C. Wang.  Research on method of self-adaptive determination of dbscan algorithm parameters.  Comput. Eng. Appl, 55(5):1–7, 2019.
  * [42] G. Liu, J. Yang, Y. Hao, and Y. Zhang.  Big data-informed energy efficiency assessment of china industry sectors based on k-means clustering.  Journal of cleaner production, 183:304–314, 2018.
  * [43] E. Lopatin.  Methodological approaches to research resource saving industrial enterprises.  670216917, 2019.
  * [44] F. Lu and Z. Cao.  Empirical analysis on the contribution of the industrial structure to china’s economic growth.  In 2008 International Symposium on Intelligent Information Technology Application Workshops, pp. 886–889. IEEE, 2008.
  * [45] S. M. Lundberg and S.-I. Lee.  A unified approach to interpreting model predictions.  In Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, p. 4768–4777. Curran Associates Inc., Red Hook, NY, USA, 2017.
  * [46] H. Lütkepohl.  Vector autoregressive models.  In Handbook of research methods and applications in empirical macroeconomics. Edward Elgar Publishing, 2013.
  * [47] R. Maciejewski, S. Rudolph, R. Hafen, A. Abusalah, M. Yakout, M. Ouzzani, W. S. Cleveland, S. J. Grannis, and D. S. Ebert.  A visual analytics approach to understanding spatiotemporal hotspots.  IEEE Transactions on Visualization and Computer Graphics, 16(2):205–220, 2009.
  * [48] K. Materna, J. Hasman, and D. Hána.  Acquisition of industrial enterprises and its relations with regional identity: The case of the beer industry in central europe.  Norsk Geografisk Tidsskrift-Norwegian Journal of Geography, 73(4):197–214, 2019.
  * [49] A. Mehler, Y. Bao, X. Li, Y. Wang, and S. Skiena.  Spatial analysis of news sources.  IEEE transactions on visualization and computer graphics, 12(5):765–772, 2006.
  * [50] F. X. Molina-Morales and M. T. Martínez-Fernaández.  The impact of industrial district affiliation on firm value creation.  European Planning Studies, 11(2):155–170, 2003.
  * [51] T. K. Mukherjee and A. Naka.  Dynamic relations between macroeconomic variables and the japanese stock market: an application of a vector error correction model.  Journal of financial Research, 18(2):223–237, 1995.
  * [52] J. Myles Shaver and F. Flyer.  Agglomeration economies, firm heterogeneity, and foreign direct investment in the united states.  Strategic management journal, 21(12):1175–1193, 2000.
  * [53] M. Peneder.  Industrial structure and aggregate growth.  Structural change and economic dynamics, 14(4):427–448, 2003.
  * [54] P. E. Rauber, S. G. Fadel, A. X. Falcao, and A. C. Telea.  Visualizing the hidden activity of artificial neural networks.  IEEE Transactions on Visualization and Computer Graphics, 23(1):101–110, Jan. 2017. doi: 10 . 1109/TVCG . 2016 . 2598838
  * [55] M. T. Ribeiro, S. Singh, and C. Guestrin.  ” why should i trust you?” explaining the predictions of any classifier.  In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, pp. 1135–1144, 2016.
  * [56] P. Riehmann, M. Hanfler, and B. Froehlich.  Interactive sankey diagrams.  In IEEE Symposium on Information Visualization, 2005. INFOVIS 2005., pp. 233–240. IEEE, 2005.
  * [57] R. Sharda and R. B. Patil.  Connectionist approach to time series prediction: an empirical test.  Journal of Intelligent Manufacturing, 3(5):317–323, 1992.
  * [58] B. Shneiderman.  The eyes have it: A task by data type taxonomy for information visualizations.  In The craft of information visualization, pp. 364–371. Elsevier, 2003.
  * [59] J. B. Sørensen and O. Sorenson.  From conception to birth: Opportunity perception and resource mobilization in entrepreneurship.  In Geography and Strategy. Emerald Group Publishing Limited, 2003\.
  * [60] M. Stein, H. Janetzko, A. Lamprecht, D. Seebacher, T. Schreck, D. Keim, and M. Grossniklaus.  From game events to team tactics: Visual analysis of dangerous situations in multi-match data.  In 2016 1st International Conference on Technology and Innovation in Sports, Health and Wellbeing (TISHW), pp. 1–9. IEEE, 2016.
  * [61] E. Stellwagen, L. Tashman, et al.  Arima: The models of box and jenkins.  Foresight: The International Journal of Applied Forecasting, (30):28–33, 2013.
  * [62] P. M. Townroe.  Rationality in industrial location decisions.  Urban Studies, 28(3):383–392, 1991.
  * [63] S. van den Elzen, D. Holten, J. Blaas, and J. J. van Wijk.  Reducing snapshots to points: A visual analytics approach to dynamic network exploration.  IEEE transactions on visualization and computer graphics, 22(1):1–10, 2015.
  * [64] K. Xu, J. Yuan, Y. Wang, C. Silva, and E. Bertini.  Mtseer: Interactive visual exploration of models on multivariate time-series forecast.  In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems, CHI ’21. Association for Computing Machinery, New York, NY, USA, 2021. doi: 10 . 1145/3411764 . 3445083
  * [65] Y. Yu, Y.-r. Deng, and F.-f. Chen.  Impact of population aging and industrial structure on co2 emissions and emissions trend prediction in china.  Atmospheric Pollution Research, 9(3):446–454, 2018.
  * [66] P. Zhang, R. Wang, and J. Wang.  Research evolution and frontier analysis on the industrial structure adjustment – based on the visualization analysis using citespace.  Industrial Economy Review, 5(2):16–28, 2018.
  * [67] X. Zhao, Y. Wu, W. Cui, X. Du, Y. Chen, Y. Wang, D. L. Lee, and H. Qu.  Skylens: Visual analysis of skyline on multi-dimensional data.  IEEE transactions on visualization and computer graphics, 24(1):246–255, 2017.
  * [68] Z. Zhou, H. Li, F. Liu, Y. Liu, C. Huang, Y. Tao, H. Lin, and W. Su.  Visual analytics of economic features for multivariate spatio-temporal gdp data.  Journal of Visualization, 21(2):337–350, 2018.
  * [69] B. Zhu, M. Zhang, Y. Zhou, P. Wang, J. Sheng, K. He, Y.-M. Wei, and R. Xie.  Exploring the effect of industrial structure adjustment on interprovincial green development efficiency in china: A novel integrated approach.  Energy Policy, 134:110946, 2019.
  * [70] J. Zilong.  Analysis of agglomeration and equilibrium characteristics of regional economic in china with spatial visualization in science computing technique.  In 2014 Sixth International Conference on Measuring Technology and Mechatronics Automation, pp. 91–94. IEEE, 2014.