# Spatial regionalization as optimal data compression

Alec Kirkley  Department of Physics, University of Michigan, Ann Arbor,
Michigan 48109, USA  School of Data Science, City University of Hong Kong,
Hong Kong

###### Abstract

The process of aggregating areal units into contiguous clusters, known as
regionalization, is central to the analysis of spatial data. Regionalization
provides a means to reduce the effect of noise or outliers in sampled data,
identify socioeconomically homogeneous areas for policy development, and
simplify the visualization of data in maps among many other applications. Most
existing regionalization methods require a substantial amount of manual input,
such as the number of desired regions or a similarity measure among regional
populations, which may be desirable for some applications but does not allow
us to extract the natural regions defined solely by the data itself. Here we
view the problem of regionalization as one of data compression. We define the
optimal partition of spatial units with corresponding distributional data as
the one that minimizes the description length required to transmit the data,
and develop an efficient, parameter-free greedy optimization algorithm to
identify this partition. We demonstrate that our method is capable of
recovering planted spatial clusters in noisy synthetic distributional data,
and that it identifies meaningful ethnoracial boundaries in real demographic
data. Using our description length formulation, we find that the information
contained in spatial ethnoracial data in metropolitan areas across the U.S.
has become more difficult to compress over the period from 1980 to 2010, which
reflects the rising complexity of urban segregation patterns of these metros.
We identify the increasing overall diversity of these metros as a major
contributor to this lower data compressibility, while the spatial scale of
ethnoracial clustering does not appear to be a significant factor.

##  I Introduction

From the growth of economies fujita1999spatial  to the systemic segregation of
human populations brown2006spatial  to the environmental adaptation of
ecological species legendre1989spatial , many social and natural phenomena
manifest themselves in space with high levels of clustering among similar
agents or entities. Precisely defining the spatial boundaries of these
clusters and observing their evolution can shed light on the fundamental
processes driving the dynamics of these systems, aid in the reduction of noise
in spatially sampled data spielman2015reducing ; spielman2015studying , and
facilitate the identification of regions for spatially targeted policy
interventions rahman2004regionalization  among numerous other applications.
Regionalization methods—techniques to perform spatially constrained clustering
by aggregating spatial units—are typically the tools of choice for
partitioning spatial data into areas of interest for such analysis.
Consequently, regionalization methods have been adapted for applications
across fields as diverse as climatology fovell1993climate , urban sociology
garreton2016santiago , hydrology peterson2011hydro , geoecology
niesterowicz2016unsupervised , and political science george1997political .

Many approaches to regionalization typically require a significant amount of
input from the user to adjust various parameters prior to performing the
clustering. These tunable parameters can be used to constrain the size or
shape of clusters, or to avoid crossing administrative or geographical
boundaries duque2007supervised ; li2013efficient . User preferences are also
commonly incorporated into regionalization methods through the choice of a
similarity or distance function between adjacent regions
assunccao2006efficient ; wei2021efficient . Additionally, as is the case with
any clustering method, a key factor existing regionalization methods consider
is the choice of the number of regions, which is typically fixed by the user
duque2007supervised ; aydin2021quantitative  but is sometimes determined
endogeneously based on user-defined thresholds for covariates of interest or
other heuristics that depend or one’s choice of dissimilarity between spatial
units duque2012max ; wei2021efficient . An increased level of user control is
desirable for many applications of regionalization, as researchers can ensure
that the identified regions are suitable for the task at hand and do not
violate any necessary constraints. For example, clusters extracted from
regionalization methods may be used to define zones designated for different
aspects of urban development, and it may be preferred that these zones do not
cross significant geographical or infrastructural boundaries.

In other applications of regionalization, however, such as identifying
characteristic scales over which segregation or other socioeconomic phenomena
persist wright2014patterns ; olteanu2019segregation ; grainger2004role ;
kirkley2020information , one may be interested in imposing as few assumptions
as possible about how the data clusters into regions, and instead rely on the
data itself to naturally define these clusters. We adopt this approach for our
method, defining the best partition of a set of spatial units as the one that
allows us to optimally compress the information contained in a covariate of
interest. We employ the minimum description length concept from information
theory grunwald2007minimum ; cover2012elements  to quantify the extent to
which a given partition allows for compression of the data. By viewing the
problem of regionalization from this perspective, our approach does not
require the specification of any free parameters, such as an explicit
dissimilarity function between spatial units or a particular value for the
number of regions we want the algorithm to return. Our method also takes into
account the full distribution of the covariate of interest in each spatial
unit, rather than summarizing each local distribution with a single statistic
such as its mode. In this way, through regionalization we aim to compress all
of the information contained in the spatial covariate, and rely only on the
data itself to accomplish the task.

There are a few existing methods for clustering data without spatial
constraints that have a similar motivation as our own. The minimum description
length principle has been applied for clustering categorical data
li2004entropy , real-valued vector data georgieva2011cluster , and other sets
of objects KirkleyNewman2021  in aspatial contexts. In Rosvall07 , an
algorithm for community detection in (aspatial) network data is proposed that
identifies the partition minimizing the description length of an encoding of
the network. This method differs substantially from our own, however, in that
it takes only topological information into account, which is relatively
uninformative for planar networks of adjacent spatial regions (as is the case
in regionalization). By contrast, our method takes into account both
topological information (the adjacency between spatial units) as well as
spatial covariates. In chodrow2017structure , a regionalization algorithm is
proposed, which, in the same spirit as our own method, uses concepts from
information theory to define homogeneous aggregations of spatial units. This
method also employs a greedy optimization procedure as we do here, but differs
significantly from our method in that it requires the user to specify the
desired number of regions and chooses the class of Bregman divergences to
measure information loss. The method proposed here, on the other hand, allows
the user to automatically learn the optimal number of clusters from the data
if they choose and uses a simple combinatorial approach to define the
information content of a partition.

In this paper we first derive a regionalization objective function based
solely on fundamental combinatorial arguments and the minimum description
length principle. We then describe a greedy optimization procedure used to
obtain a partition that approximately minimizes this description length, which
involves iteratively merging the pair of spatial units that maximally reduces
the description length. We demonstrate our method on a series of experiments
using both real and synthetic spatial data. In the first experiment, we
illustrate how our method can effectively recover synthetically planted
clusters in spatial distributional data, even in the presence of substantial
noise. We move on to show that our method extracts meaningful regions and
their evolution in real ethnoracial data by analyzing the New Haven-Milford
metropolitan area of the U.S. as a case study, covering the decades between
1980 and 2010. Finally, in an experiment using a set of 110 large metropolitan
areas across the U.S., we demonstrate that our method reveals the increasing
complexity of urban segregation patterns over this same time period, and that
this trend can be well explained by the overall increase in ethnoracial
diversity of these metros rather than by the manifestation of clustering at
different spatial scales.

##  II Materials and Methods

###  II.1 Description length formulation

We can represent our spatial data to be regionalized as a network $G=(V,E)$
consisting of a set of spatial units (nodes) $V$ and a set of edges $E$ that
connect adjacent units. More precisely, the edge $(u,v)\in E$ if and only if
units $u\in V$ and $v\in V$ share a length of common border. We denote the
number of units in any subset $V^{\prime}\subseteq V$ of the network as
$n(V^{\prime})$. Over this set of $n(V)$ units, there are $b(V)\geq n(V)$
individuals residing (we adopt analogous notation for $b(V^{\prime})$), and
each of these individuals is classified under one of $R$ categories
$r=1,2,...,R$. For example, the spatial units $u$ that comprise the network
may be census tracts or block groups, and the categories could represent race,
income bracket, or occupation type. Our formulation in general will depend on
what spatial units are used, and so it is recommended to use regions with the
finest granularity possible while also avoiding excessive sampling noise. We
also denote with $b_{r}(V^{\prime})$ the number of individuals of type $r$ in
subset $V^{\prime}\subseteq V$, such that
$\sum_{r=1}^{R}b_{r}(V^{\prime})=b(V^{\prime})$.

Now, suppose we want to transmit to a receiver the entire dataset
$D=\\{b_{r}(u):r=1,..,R;~{}u\in V\\}$ consisting of the distribution of types
$r$ among individuals in all units $u$. (Since we generally do not know the
value $r$ for each individual due to confidentiality concerns, these unit-
level distributions are the highest granularity we consider.) We will transmit
this data in multiple parts, first partitioning the units $u$ into $K$
disjoint, spatially contiguous clusters
$\mathcal{P}=\\{V_{1},V_{2},...,V_{K}\\}$ that allow us to describe the data
to the receiver at a coarse spatial scale. We then transmit the small-scale
details within each of these clusters by describing how the cluster’s
population attributes are distributed among its individual constituent units.
Our goal will be to identify a partition $\mathcal{P}$ of the units such that
most of the information we need to transmit is contained in the first part, or
in other words, that the clusters describe most of the variation in the data
and are internally homogeneous.

We assume that the receiver knows there are $n(V)$ units in total that will be
assigned to $K$ clusters, and that there are $b(V)$ individuals with $R$
distinct categories that will be assigned to units $u\in V$. (Transmitting
these requires a negligible amount of information, so we can safely ignore
them in our description length anyway.) We first need to transmit the
populations $b(V_{k})$ for each of the clusters $V_{k}$, which consists of a
configuration of $K$ non-negative integer values that sum to $b(V)$. Prior to
transmission of the data $D$, we must develop a common codebook with the
receiver, from which we will transmit a binary string representing the
particular configuration of the populations $\\{b(V_{k})\\}$. Assuming $K\ll
b(V)$, there are approximately ${b(V)-1\choose K-1}$ possible configurations
of these values, and so we will possibly have to send a bitstring of length
$\lceil\log_{2}{b(V)-1\choose K-1}\rceil$ to the receiver to transmit the
cluster-level populations $\\{b(V_{k})\\}$. ($\lceil x\rceil$ denotes the
smallest integer not less than $x$, and we will omit this transformation in
future considerations as its contribution is negligible for $x\gg 1$. For the
sake of brevity we will also denote $\log_{2}(x)\equiv\log(x)$.) Thus, the
information content (or “description length”) of this step in the transmission
procedure is

$\displaystyle\mathcal{L}(\\{b(V_{k})\\})=\log{b(V)-1\choose K-1}.$ (1)

Following the same logic, we can construct the description lengths for the
rest of the steps required to transmit $D$ according to this scheme. After
sending the populations $\\{b(V_{k})\\}$, we must transmit the number of units
within each cluster, $\\{n(V_{k})\\}$, for which we will construct a different
codebook. This step will have a description length of the same form as Eq. 1,
thus

$\displaystyle\mathcal{L}(\\{n(V_{k})\\})=\log{n(V)-1\choose K-1}.$ (2)

Now, for each cluster $V_{k}$ we need to transmit the size distribution
$\\{b_{r}(V_{k})\\}$ of categories within the population $b(V_{k})$, which
will have the same form as Eq.s 1 and 2. The description length of this step
will be a sum over such description lengths, or

$\displaystyle\mathcal{L}(\\{b_{r}(V_{k})\\})=\sum_{k=1}^{K}\log{b(V_{k})-1\choose
R-1}.$ (3)

Similarly, we need to transmit the populations $b(u)$ of the units $u\in
V_{k}$, for each cluster $V_{k}$, which will give a total description length
contribution of

$\displaystyle\mathcal{L}(\\{b(u)\\})=\sum_{k=1}^{K}\log{b(V_{k})-1\choose
n(V_{k})-1}.$ (4)

The receiver now knows how many units $u$ are in each cluster $V_{k}$, how
many individuals are in each of these units, and how categories are
distributed across the entire population of $V_{k}$. The only information left
to transmit is how the categories in each cluster $V_{k}$ are distributed
among the populations in $V_{k}$’s constituent units $u$. The number of ways
these values can be distributed is equivalent to the number
$\Omega(\bm{a}_{k},\bm{c}_{k})$ of non-negative integer-valued matrices with
row sums $\bm{a}_{k}=\\{b(u)\\}_{u\in V_{k}}$ and column sums
$\bm{c}_{k}=\\{b_{r}(V_{k})\\}_{r=1}^{R}$. We can see this by noting that
there are $b(V_{k})$ total individuals in cluster $V_{k}$, and using the
identities

$\displaystyle b(V_{k})=\sum_{u\in V_{k}}b(u)$ (5)

and

$\displaystyle b(V_{k})=\sum_{r=1}^{R}b_{r}(V_{k}).$ (6)

The description length for this final step is thus given by

$\displaystyle\mathcal{L}_{final}=\sum_{k=1}^{K}\log\Omega(\bm{a}_{k},\bm{c}_{k}).$
(7)

Computing $\Omega(\bm{a}_{k},\bm{c}_{k})$ is in general challenging, but it
can be approximated in the regime $R,n(V_{k})\ll b(V_{k})$, which is typically
the regime we encounter in practice (see Ref. NCY20  for details on this
approximation).

Taken all together, the total description length of the data $D$ is given by
the sum of Eq.s 1, 2, 3, 4, and 7, thus

$\displaystyle\mathcal{L}(D,\mathcal{P})$ $\displaystyle=\log{b(V)-1\choose
K-1}+\log{n(V)-1\choose K-1}$
$\displaystyle~{}~{}+\sum_{k=1}^{K}\log{b(V_{k})-1\choose
R-1}+\sum_{k=1}^{K}\log{b(V_{k})-1\choose n(V_{k})-1}$
$\displaystyle~{}~{}+\sum_{k=1}^{K}\log\Omega(\bm{a}_{k},\bm{c}_{k}).$ (8)

We can see that the first three terms in Eq. II.1 penalize us for having a
greater number of clusters $K$, as they will tend to contribute greater
description lengths as $K$ increases. The fourth term will scale like
$n(V)\log(b/K)$ for equally sized clusters, and so in general will tend to
penalize us slightly for having fewer clusters. In the extreme case where
there is only one category $r^{\ast}$ that is represented in the population of
the units $u\in V_{k}$ (i.e. $\bm{c}_{k}[r]=0$ for $r\neq r^{\ast}$), then we
have $\Omega(\bm{a}_{k},\bm{c}_{k})=1$ and the contribution from the last term
in Eq. II.1 term vanishes. More generally, there are fewer ways the categories
can be distributed among the populations in $V_{k}$’s constituent tracts if
$\bm{c}_{k}$ is more concentrated on a single category, and so the last term
in Eq. II.1 will penalize us for having a high level of diversity within the
clusters. (Or, conversely, this term encourages partitions $\mathcal{P}$ that
have homogeneous clusters.)

The optimal partition $\mathcal{P}=\\{V_{1},...,V_{k}\\}$ that minimizes the
description length in Eq. II.1 will allow us to communicate most of the
information about the data $D$ through the cluster-level distributions alone,
but penalize us for constructing these clusters at too small a scale, since
this will not save us much effort above and beyond simply transmitting all the
unit-level data individually. The goal of our regionalization algorithm is to
identify this partition, and we describe an algorithm to accomplish this task
in the next section. A diagram of an optimal partition $\mathcal{P}$ of a
synthetic set of example units, along with the variables used in the
information transmission scheme, is shown in Fig. 1.

Figure 1: Diagram of variables in the information transmission process, for
the partition $\mathcal{P}$ of the (synthetic) unit-level distributions that
gives the minimum description length according to Eq. II.1. The optimal
partition of this space results in aggregated regions of spatial units that
capture most of the information content of the data.

###  II.2 Optimization and model selection

Minimization of the description length in Eq. II.1, like many other
regionalization objectives duque2007supervised , is a combinatorial
optimization problem that can be approached in a number of ways to obtain an
approximate solution. Here, we opt for a greedy solution that consists of
starting with each unit in its own cluster then iteratively merging the pair
of adjacent clusters whose aggregation results in the largest decrease in Eq.
II.1, until no merges produce a negative change in the description length.
This merging procedure also has the benefit of naturally ensuring that the
partition $\mathcal{P}$ produces only contiguous clusters of units.

For any pair of clusters $V_{k}$ and $V_{k^{\prime}}$, we can quickly compute
the change in Eq. II.1 that results from their aggregation into a single
cluster, $V_{k,k^{\prime}}$. Supposing there are $K$ clusters prior to the
proposed merge, the change in description length from merging clusters $V_{k}$
and $V_{k^{\prime}}$ is given by

$\displaystyle\Delta\mathcal{L}(k,k^{\prime})$
$\displaystyle=\log{b(V_{k,k^{\prime}})-1\choose
R-1}+\log{b(V_{k,k^{\prime}})-1\choose n(V_{k,k^{\prime}})-1}$
$\displaystyle+\log\Omega(\bm{a}_{k,k^{\prime}},\bm{c}_{k,k^{\prime}})-\log{b(V_{k})-1\choose
R-1}$ $\displaystyle-\log{b(V_{k})-1\choose
n(V_{k})-1}-\log\Omega(\bm{a}_{k},\bm{c}_{k})$ (9)
$\displaystyle-\log{b(V_{k^{\prime}})-1\choose
R-1}-\log{b(V_{k^{\prime}})-1\choose n(V_{k^{\prime}})-1}$
$\displaystyle-\log\Omega(\bm{a}_{k^{\prime}},\bm{c}_{k^{\prime}}).$

Here we have ignored the first two terms in Eq. II.1, as these terms change by
the same amount across all pairs $k,k^{\prime}$ and thus do not need to be
computed until the optimal pair $k,k^{\prime}$ is chosen. (Whether or not this
pair will be merged or the algorithm will terminate does depend on these first
two terms, which can be computed in constant time.) This expression can be
evaluated in $\mathrm{O}(n(V_{k})+n(V_{k^{\prime}}))$ time for each pair of
clusters $k,k^{\prime}$. Additionally, it only needs to be computed once for
each pair of clusters, and can be reused for future iterations of the
algorithm if the pair $k,k^{\prime}$ does not get merged (as long as each
newly formed cluster gets a unique label). Once no remaining pair of clusters
can be merged to reduce the description length
($\Delta\mathcal{L}(k,k^{\prime})>0$ for all adjacent cluster pairs
$V_{k},V_{k^{\prime}}$), the algorithm terminates.

The adjacency relations between clusters are updated as the algorithm
progress, by merging the neighbor sets of the two clusters that are merged.
This takes an additional $\mathrm{O}(d_{k}+d_{k^{\prime}})$ steps, where
$d_{k}$ is the number of adjacent clusters to cluster $k$, and is typically
much smaller than $\mathrm{O}(n(V_{k})+n(V_{k^{\prime}}))$ for large clusters,
since many clusters are only adjacent to a few others. We find in practice
that the algorithm scales well to large systems, running in much less than
order $\mathrm{O}(n(V)^{2})$ time.

The first few terms in Eq. II.1 penalize us for having a large number of
clusters, since we waste information describing all of the cluster-level
distributions in their entirety. Meanwhile, the last two terms penalize us for
having a small number of clusters, since we waste information describing the
small scale details of these clusters as they encompass to broad a variety of
units. The optimal balance, and thus the optimal value of $K$, lies somewhere
in between with an intermediate number of clusters, and the description length
in Eq. II.1 thus performs model selection for $K$ automatically. In our
example applications, we therefore choose to let the description length tell
us exactly how many clusters are in the data. However, in many applications it
may be preferable to have a fixed value of $K$ (duque2007supervised ;
KirkleyNewman2021 ), and this can easily be accommodated in our algorithm by
simply performing the greedy merge moves until there are the desired number of
clusters.

We can assess the quality of the information compression achieved through
partitioning the units into clusters by comparing the final description length
$\mathcal{L}(D,\mathcal{P})$ with the description length
$\mathcal{L}(D,\mathcal{P}_{0})$ for the trivial partition $\mathcal{P}_{0}$
in which each unit is in its own cluster (computed at the beginning of the
optimization algorithm). From this we can construct an inverse “compression
ratio” for the data $D$ as

$\displaystyle\eta(D)=\frac{\text{compressed size of $D$}}{\text{uncompressed
size of
$D$}}=\frac{\mathcal{L}(D,\mathcal{P})}{\mathcal{L}(D,\mathcal{P}_{0})}.$ (10)

$\eta(D)$ approaches its minimum value of $0$ when the data $D$ can be
compressed extremely efficiently through partitioning, and approaches its
maximum value of $1$ when there is no partition of the data that achieves any
level of information compression. Eq. 10 can thus be used as a measure of the
complexity of the spatial segregation of the data $D$, with more complex
spatial distributions of the covariate of interest resulting in higher inverse
compression ratios $\eta$. Intuitively, if the data $D$ is very easy to
compress (low $\eta$), then it is highly spatially segregated into homogeneous
clusters, and most of the information in $D$ is captured at large scales. On
the other hand, if the data is very hard to compress (high $\eta$), then much
of the information in the data is manifested at small spatial scales. This
could be due to the presence of small-scale segregated clusters or widespread
diversity, among numerous other factors that contribute to segregation
massey1988dimensions .

###  II.3 Ethnoracial data in U.S. metropolitan areas

In addition to synthetic data to examine the performance of our algorithm, we
test our method using ethnoracial data that take the form of distributions
within census tracts. Ethnoracial distributions for census tracts in U.S.
metro areas were obtained from the Longitudinal Tract Database
logan2014interpolating , which maps 2010 census tract boundaries to
ethnoracial distribution data for decades going back to 1970. (Data from 1970
are omitted from our analysis, as they do not include the designation of
Hispanic ethnicity.) The race/ethnicity categories considered are ‘Non-
Hispanic White’, ‘Non-Hispanic Black’, ‘Asian’, ‘Hispanic’, and ‘Other’, which
includes persons not categorized under the first four groups.

To process the census tract networks for each metropolitan area, we first map
each census tract to its corresponding core-based Metropolitan Statistical
Area (MSA) using the county designation of the tract. We then use TIGER
shapefile data united2019tiger  for the census tracts to determine the network
$G=(V,E)$ of adjacent tracts in each MSA. Finally, the longitudinal
ethnoracial distribution data is then mapped to the nodes in each network
using the census tract IDs. To reduce noise as much as possible in our
analysis, we kept only metros with at least 100 tracts that had complete
ethnoracial distribution estimates in all tracts for the four decades 1980,
1990, 2000, and 2010. After preprocessing, 110 metro networks remained for the
analysis in Sec. III.3, one of which was the New Haven-Milford metro used for
the case study in Sec. III.2.

We make the tract adjacency networks for each metro we used in our analysis
(with accompanying node metadata including ethnoracial distributions), as well
as code for executing our algorithm publicly available at
https://github.com/aleckirkley/MDL_regionalization.

##  III Results

###  III.1 Cluster recovery in synthetic data

As a first test of our method, we explore its capability of recovering
clusters in synthetic data. To do this, we create a synthetic model of spatial
distributional data that has four tunable parameters: the number of clusters
$K$, the number of covariate categories $R$, the level of statistical noise
between the cluster-level distributions $\epsilon_{between}$, and the level of
statistical noise within the clusters, $\epsilon_{within}$. The model requires
a spatial network $G=(V,E)$ representing the adjacencies among spatial units
(we use the census tract network for the New Haven-Milford metropolitan area,
see Sec. III.2 for details), although the specific choice of $G$ does not tend
to make a big qualitative difference in the results. It is also possible to
include variable unit populations $b(u)$ in this model, but for simplicity we
set $b(u)=10000$ for all $u\in V$ so that these values correspond roughly to
the values seen in the real U.S. census tract data in Sec. III.2 and Sec.
III.3.

To generate a realization of the model, we first randomly partition the units
into contiguous clusters by picking $K$ units (“seeds”) at random and
constructing the Voronoi tesselation of the network with respect to these
seeds. This Voronoi tesselation places each unit into the cluster
corresponding to the seed geographically nearest to the unit, and in doing so
guarantees that all clusters are contiguous. (This tesselation tends to
produce relatively compact regions, but there are other reasonable alternative
tesselations for generating the randomized contiguous partition.) We denote
this “planted” partition $\mathcal{P}_{planted}$, to distinguish it from the
partition $\mathcal{P}$ inferred using our minimum description length
algorithm.

Next, each cluster $V_{k}$ is assigned a vector $\bm{x}(V_{k})$ which tunes
the covariate distributions within the units that comprise $V_{k}$.
$\bm{x}(V_{k})$ is drawn from a Dirichlet distribution with length-$R$
concentration parameter $\bm{\alpha}=\epsilon_{between}^{-1}\bm{1}_{R}$. This
allows us to tune the level of differentiation between the cluster-level
distributions, as well as the localization of these distributions. For low
levels of between-cluster noise $\epsilon_{between}$, the distributions
$\bm{x}(V_{k})$ will all tend to distribute their probability relatively
equally around the $R$ categories, and there is little differentiation between
the clusters $V_{k}$. On the other hand, for high levels of between-cluster
noise $\epsilon_{between}$, there will be high between-cluster variance in the
distributions $\\{\bm{x}(V_{k})\\}$, which will each tend to localize around a
single category $r$. In general, the higher the between-cluster noise
$\epsilon_{between}$ is, the easier it should be to recover the planted
clusters in the synthetic data with our partitioning algorithm, since the
clusters are more easily distinguished.

To tune the level of noise within each cluster $V_{k}$, we generate the
distribution $\bm{x}(u)=\\{b_{r}(u)\\}_{r=1}^{R}/b(u)$ for each $u\in V_{k}$
using
$\bm{x}(u)=(1-\epsilon_{within})\bm{x}(V_{k})+\epsilon_{within}\bm{x}_{noise}$,
where $\bm{x}_{noise}$ is drawn from a Dirichlet distribution with
concentration parameters equal to $1$. If the level of within-cluster noise
$\epsilon_{within}$ is close to $0$, then each $\bm{x}(u)$ for $u\in V_{k}$
will be roughly the same as $\bm{x}(V_{k})$, and thus the units within the
cluster are highly homogeneous. On the other hand, if the level of within-
cluster noise $\epsilon_{within}$ is close to $1$, then the vectors
$\bm{x}(u)$ will have high variability within the cluster $V_{k}$ and contain
little information about each other. As opposed to the between-cluster noise,
higher values of the within-cluster noise $\epsilon_{within}$ correspond to it
being _harder_ to recover the planted clusters in the synthetic data, since
the clusters are not as homogeneous in composition.

To measure the performance of our algorithm for any particular draw from the
model, we compute the normalized mutual information vinh2010information
between our inferred minimim description length partition $\mathcal{P}$ and
the planted partition $\mathcal{P}_{planted}$. The mutual information tells us
how much information is shared between the two partitions, and its value is
then normalized to fall in $[0,1]$ so that $0$ corresponds to completely
uncorrelated partitions, and $1$ corresponds to identical partitions (up to an
arbitrary relabeling of the clusters). Letting $\mathcal{P}=\\{V_{k}\\}$ and
$\mathcal{P}_{planted}=\\{U_{k^{\prime}}\\}$, the mutual information
$MI(\mathcal{P},\mathcal{P}_{planted})$ is given by

$\displaystyle
MI(\mathcal{P},\mathcal{P}_{planted})=\sum_{k,k^{\prime}}\frac{|V_{k}\cap
U_{k^{\prime}}|}{n(V)}\log\frac{n(V)|V_{k}\cap
U_{k^{\prime}}|}{|V_{k}||U_{k^{\prime}}|}.$ (11)

The mutual information can be normalized to fall in $[0,1]$ by dividing by the
average of the entropies of the individual partitions $\mathcal{P}$ and
$\mathcal{P}_{planted}$, giving

$\displaystyle
NMI(\mathcal{P},\mathcal{P}_{planted})=2\frac{MI(\mathcal{P},\mathcal{P}_{planted})}{H(\mathcal{P})+H(\mathcal{P}_{planted})},$
(12)

with

$\displaystyle
H(\mathcal{P})=-\sum_{k}\frac{|V_{k}|}{n(V)}\log\frac{|V_{k}|}{n(V)}$ (13)

and

$\displaystyle
H(\mathcal{P}_{planted})=-\sum_{k^{\prime}}\frac{|U_{k^{\prime}}|}{n(V)}\log\frac{|U_{k^{\prime}}|}{n(V)}.$
(14)

The normalized mutual information is a standard and well-tested measure for
comparing partitions of networks DDDA05 ; lancichinetti2008benchmark , but it
has a critical shortcoming for our particular application in that it gives
very high baseline values to completely random contiguous partitions of
spatial networks. The reason for this is that Eq. 12 compares the partitions
$\mathcal{P}$ and $\mathcal{P}_{planted}$ relative to the ensemble of all
possible partitions of the network, contiguous or not, and the constraint of
contiguity induces a non-trivial level of correlation between the partitions.
To correct for this, we rescale the normalized mutual information to the
interval $[0,1]$ by subtracting off its minimum value over the range of
$\epsilon_{within}$ values, and dividing by one minus this minimum value. It
is then easy to see when we reach the NMI value at which the partitions are
minimally correlated, subject to the contiguity constraint. Our rescaling does
not map the highest value of the NMI over the $\epsilon_{within}$ range in
each panel to $1$, so that we have better differentiation of performance in
the low noise region. Indeed, we will see that the zero-noise values of the
rescaled NMI are slightly less than 1 in most cases, since some sampled model
realizations will by chance produce some adjacent clusters that are nearly
indistingushable.

Figure 2: Recovery of synthetic clusters generated by the model in Sec. III.1.
In each panel, the recovery performance of our algorithm, as measured by the
$NMI(\mathcal{P},\mathcal{P}_{planted})$ in Eq. 12 (rescaled to a zero
baseline value) is plotted against the level of within-cluster noise
$\epsilon_{within}$, for various values of between-cluster noise
$\epsilon_{between}$. The number of clusters $K$ and the number of variable
categories $R$ are varied across the panels.

In Fig. 2 we show the results of generating realizations of synthetic
contiguous partitions from our model, and running our regionalization
algorithm on each of these realizations to try to recover the planted
clusters. Each data point represents the average rescaled normalized mutual
information over $100$ of these cluster recovery experiments, with error bars
representing $2$ standard errors. We can see that as the level of within-
cluster noise $\epsilon_{within}$ increases, it becomes harder for us to
recover the planted partition (as expected), but that we still have recovery
better than the baseline value for levels of within-cluster noise up to around
$\epsilon_{within}\approx 0.4$ for $R=5$ and $\epsilon_{within}\approx 0.7$
for $R=10$. Recovery is better for $R=10$ than $R=5$, as it is less likely for
the modes of the distributions $\bm{x}(V_{k})$ to overlap if $R$ is larger. We
can also observe that the recovery task becomes easier as $\epsilon_{between}$
and $R$ increase, but how much easier it becomes depends on the number of
clusters $K$. It becomes more likely that the highly concentrated cluster
distributions $\bm{x}(V_{k})$ resulting from high $\epsilon_{between}$ have
overlapping modes as $K$ increases, and we thus see slightly worse recovery
performance for high values of $\epsilon_{between}$ at $K=20$ than at $K=5$,
since clusters will be harder to differentiate. In general, however, the
performance of our algorithm is not terribly sensitive to the number of
planted clusters, since we’ve varied $K$ by a factor of $4$ and only see a
modest reduction in recovery performance.

Overall, the results of Figure 2 indicate that our minimum description length
regionalization algorithm is able to successfully recover artificially planted
clusters, even in the presence of substantial noise. We now move on to examine
its performance on real ethnoracial distribution data.

###  III.2 Case study: Ethnoracial composition of the New Haven-Milford
metropolitan area

Figure 3: Ethnoracial distributions in census tracts (thin black borders)
within the New Haven-Milford, Connecticut metropolitan area, with inferred
cluster boundaries from the minimum description length regionalization
algorithm (thick black borders). Colored points are distributed at random
within each tract and each color covers an area proportional to the fraction
of the population within the tract that falls under the ethnoracial category
corresponding to that color.

To illustrate how the clusters obtained with our regionalization algorithm
capture meaningful patterns in real data, we look at a case study of the
ethnoracial evolution of the New Haven-Milford, Connecticut metropolitan area,
using the data described in Sec. II.3. This metro was chosen for the case
study analysis due to its relatively high diversity, distinct spatial
evolution for different ethnoracial groups, and low heterogeneity in census
tract density in comparison with other small metros, allowing for a clear
visual analysis of its temporal segregation patterns.

In Fig. 3, we show the evolution of the spatial distribution of ethnoracial
groups, along with the regional boundaries inferred from minimizing the
description length in Eq. II.1, for the census tracts in the New Haven-Milford
metro area between 1980 and 2010. Points are distributed randomly within each
tract in proportion to the fraction of the population in each ethnoracial
category. We can see that, in general, the clusters inferred through our
algorithm correspond to heterogeneities in the spatial densities of these
ethnoracial groups. The outlying tracts in the clusters, particularly in the
year 2000, do not have as high a proportion of minority ethnoracial groups as
the more densely packed areas of the clusters, but we can see these areas
begin to fill out with minority populations in 2010. (Their inclusion status
in the cluster is determined by their slightly higher relative concentrations
of the minority groups dominant in the core of their cluster, compared to
nearby areas.)

Two emerging Black/Hispanic clusters in the north and one in the south are the
predominant clusters captured by the algorithm, which assigns the rest of the
metro to a single more rural/suburban and predominantly White cluster. We see
that these clusters trend towards higher percentages of Hispanics relative to
Non-Hispanic Blacks, which is consistent with the high influx of Latinos to
the area between 1990 and 2000 NewHaven2003 . The spatial extent of these
Black/Hispanic clusters increases over time, reaching out into the less dense
region of the metro that was predominantly White in 1980, which is consistent
with ‘White flight’ during deindustrialization as well as the expanding
influence of Yale University in the south di2006there . In 2010, we see a
slightly different configuration of clusters, with the northern Black/Hispanic
clusters remaining largely in tact, but the southern-most cluster splitting
into a largely Black/Hispanic cluster and one relatively mixed cluster. In
2000, this mixed cluster was merged with a primarily Black cluster, but in
2010 we can see that the movement of Hispanic population into the previously
Black cluster provided a high enough level of Black/Hispanic mixing to create
a single dense southern-most cluster, and a separate cluster to the north with
smaller overall minority populations. In 2010 we also see the emergence of a
new largely Hispanic cluster to the west.

In addition to the contributions from the Black/Hispanic clusters, the
rural/suburban tracts diversified metro-wide due to an influx of Asian and
Hispanic populations to the area NewHaven2015 . For the most part these
outlying tracts do not have sufficient differentiation in their ethnoracial
distributions to necessitate separate clusters, and they are all grouped into
a similar majority-White cluster for all four decades. However, this
increasing overall diversity does result in greater difficulty compressing the
data, as there is a clear positive trend in the inverse compression ratio
$\eta$ (Eq. 10) over the four decades, with $\eta=\\{0.74,0.77,0.85,0.87\\}$
for $\\{1980,1990,2000,2010\\}$ respectively. We will show in the next section
that a similar trend is seen across all the metros in our dataset,
demonstrating the increasing complexity in urban segregation patterns country-
wide.

###  III.3 Compression of ethnoracial data across metros

Figure 4: Log-ratio of consecutive inverse compression ratios $\eta$ (Eq. 10)
versus the log-ratio of consecutive overall metro diversities $H_{metro}$ (Eq.
15), in U.S. metros over the decades spanning $t=1980$ to $t=2010$. Dotted
lines at $x=0$ and $y=0$ are displayed for reference, along with OLS
regression lines (solid black) and their coefficients of determination
$r^{2}$. The slopes of all regression lines were highly statistically
significant at the $0.01$ significance level. Without grouping the changes by
decade, we find $r^{2}=0.52$, and that the slope is again highly significant
at the $0.01$ significance level. Figure 5: Log-ratio of consecutive inverse
compression ratios $\eta$ (Eq. 10) versus the log-ratio of consecutive cluster
scales $\xi$ (Eq. 17), in U.S. metros over the decades spanning $t=1980$ to
$t=2010$. None of the regression line slopes were statistically significant,
even at the $0.10$ significance level. Without grouping the changes by decade,
we find $r^{2}=0.01$, and that the slope is again not significant at the
$0.10$ significance level.

Now that we have demonstrated that our regionalization method is capable of
identifying meaningful clusters in ethnoracial census data, we move on to a
large scale analysis of the metro area networks described in Sec. II.3.
Specifically, we look at the extent to which the data within each metro can be
compressed by our algorithm according to Eq. 10, which is an indicator of the
overall complexity of the segregation patterns in these areas, as discussed in
Sec. II.2.

From a purely visual analysis, one can easily argue that the segregation
patterns seen in the New Haven-Milford metro in Fig. 3 are becoming more
complex over time: describing to somebody the spatial distribution of
ethnoracial groups in this metro would require more effort in 2010 than in
1980. And although this concept is a bit difficult to express in precise
language due to the highly multifaceted nature of patterns in spatial data, we
can capture this intuition through the inverse compression ratio of Eq. 10,
which tells us how efficiently we can compress the data for its exact
description to a receiver. As discussed in Sec. III.2, the inverse compression
ratio $\eta(D)$ increases for the New Haven-Milford over the decades spanning
1980 to 2010.

Despite the difficulty we may have in succinctly discussing the overall
complexity of the observed segregation patterns, there are a few key features
that stand out in the plots of Fig. 3. The first is the increasing diversity
of a typical tract in the metro area, demonstrated by a greater and greater
fraction of area covered by colored points as time progresses. The second
feature that stands out is the changing spatial extent of the clustered areas,
seen through the gradual absorption of the primarily White outlying tracts in
1980 into the minority-dense clusters as these clusters expand. In this
section we explore the question of whether or not segregation patterns become
more complex (as quantified by Eq. 10) in metros other than New Haven-Milford,
and to what extent the patterns we observe across these metros are consistent
with the features of overall diversity and changing spatial scales of
clustering.

To measure the overall diversity of the data $D$ in each metro area, the first
feature of interest, we compute the entropy $H_{metro}(D)$ of the ethnoracial
distribution obtained by aggregating all tract-level distributions in the
metro, given by

$\displaystyle H_{metro}(D)=-\sum_{r=1}^{R}P_{r}\log P_{r},$ (15)

where

$\displaystyle P_{r}=\frac{\sum_{u\in V}b_{r}(u)}{b(V)}$ (16)

is the fraction of the overall population in the metro that are classified
under ethnoracial group $r$. Eq. 15 will take its minimum value of $0$ when
the population in $D$ is concentrated entirely into a single category $r$, and
its maximum value of $\log R$ when all categories have equal representation in
the metro.

To measure the second feature of interest, the spatial scale of clustering for
a metro area, we define the characteristic cluster scale $\xi(D)$ as

$\displaystyle\xi(D)=\sqrt{\frac{\sum_{k}A(V_{k})^{2}}{A(V)}},$ (17)

where $A(V^{\prime})$ is the area of tracts in the subset $V^{\prime}\subseteq
V$, and $\mathcal{P}=\\{V_{k}\\}_{k=1}^{K}$ is the minimum description length
partition of the metro. Eq. 17 will take its minimum value of
$\sqrt{A(V)/n(V)}$ when each cluster has a spatial extent of $A(V)/n(V)$—the
area of a single tract if the tracts were of equal size and each cluster only
consisted of a single tract. Conversely, Eq. 17 will take its maximum value of
$\sqrt{A(V)}$, the length scale of the entire metro, when the data $D$ is best
compressed with only a single cluster.

In Figures 4 and 5 we show how changes in the inverse compression ratio $\eta$
(Eq. 10) correspond to changes in $H_{metro}$ (Eq. 15) and $\xi$ (Eq 17),
across all 110 metros for each time period in our dataset (see Sec. II.3 for
details on the dataset). In order to identify statistical associations between
the variables, while removing temporal trends and accounting for potentially
nonlinear dependencies, ordinary least squares (OLS) regression analysis was
performed on the differences in the logarithm of each quantity over each of
the periods $1980-1990$, $1990-2000$, and $2000-2010$ (panels (a), (b), and
(c) respectively in each of the figures). All significance results reported in
the captions hold up under Bonferroni correction for multiple comparisons
miller1980simult .

We can see that the inverse compression ratio $\eta$ is in general increasing
over all time periods, as the majority of the points in both figures fall
above the line $y=0$ (the $y$-axes are the same in Figures 4 and 5). In
particular, the values of $\eta$ increased substantially between $t=1990$ and
$t=2000$, with all metros in our dataset having a positive change in this
quantity during this decade. This general pattern of increasing
compressibility, with the greatest change occurring during the $1990-2000$
period, is consistent with the case study analysis in Sec. III.2.

Looking at Fig. 4, we can observe a consistently increasing level of overall
diversity in the metro areas, as illustrated by the majority of points falling
above the line $x=0$ in the three plots. This observation is consistent with
findings that suburbs have generally become more racially diverse
orfield2013america , and that there are an increasing number of “no-majority”
communities in which no ethnoracial group makes up more than half of the
population farrell2018no . The Scranton Wilkes-Barre metro area represents a
clear outlier regarding changes in overall diversity, as its value of
$H_{metro}$ shot up in $2010$, with roughly a $116\%$ increase from relatively
low values in the first three decades. The coefficients of determination
$r^{2}$ for the regression analyses reveal that the temporal changes in
$H_{metro}$ are highly correlated with the changes in $\eta$ over the same
time periods, with the strongest correlation occurring between $2000$ and
$2010$. These $r^{2}$ values, along with the statistically significant
p-values of the corresponding regression line slopes (all of which had $p\ll
0.01$), suggest that the overall diversity of metros is an important factor
for determining the complexity in segregation patterns we see according to Eq.
10.

In Fig. 5, we observe no clear trend in the changes in the characteristic
cluster length scales $\xi$ across metros for each time period, with roughly
half of the metros in each time period having decreasing values $\xi$, and
half having increasing values of $\xi$. The metros that comprise these two
halves also differ across time periods: only $18$ of the $110$ metros studied
had monotonically increasing or decreasing values of $\xi$ across all time
periods (compared to $103$ of $110$ metros having a value of $H_{metro}$ that
increased throughout all decades). The $r^{2}$ values for the regression
analyses in Fig. 5 indicate that the temporal changes in $\xi$ are poorly
correlated with the changes in $\eta$ over the same time periods, with $r^{2}$
values in two of the decades even rounding to $0$ up to two decimal places.
The p-values corresponding to the slopes of the regression lines plotted do
not indicate any statistically significant linear relationship between the
plotted variables—$p=0.52$, $0.13$, and $0.89$ for panels (a), (b), and (c)
respectively. These results suggest that the increasing complexity of
segregation patterns we observe across metros is not substantially affected by
the characteristic spatial scale at which the tracts can be optimally
clustered in each metro (at least when considering census tracts as the
fundamental unit, which may blur out segregation patterns at even smaller
scales krupka2007big ).

Altogether, this analysis indicates that segregation patterns in large U.S.
metros are becoming more complex over time, as the ethnoracial data in these
areas is becoming more difficult to compress. The overall diversification of
these metros plays an important role in increasing the complexity of these
segregation patterns, while the manifestation of increasingly granular spatial
clustering among ethnoracial groups is likely not a major contributor.

##  IV Conclusion

Here we have presented a regionalization algorithm based on the minimum
description length principle for partitioning a set of spatial units with
distributional metadata into contiguous clusters. Our method requires no user
input, learning the natural clusters that result in a maximally compressed
representation of the data. We demonstrate that our approach can effectively
recover synthetically planted clusters in noisy spatial data, and that it
returns a useful partitioning of ethnoracial census data in U.S. metropolitan
areas. We uncover new insights about the ethnoracial segregation patterns in
these U.S. metros using our methodology, finding that these patterns have
become increasingly complex over time, in part due to the increasing
ethnoracial diversity of the metros over the time period studied.

There are a number of ways our method can be extended in future work. Our
current formulation requires the spatial data of interest to take the form of
a single discrete set of counts within each unit, but it may be possible to
adapt our description length calculation to multiple spatial covariates
simultaneously, or to ordinal or continuous distributions. This would allow us
to perform regionalization with respect to a variety of attributes of interest
with variable data types, for example race and income, all at once. It is also
possible to include additional penalties in the regionalization objective
function we use for clustering, to enforce constraints on the size, shape, or
populations of the clusters, which may make it more suitable for policy-driven
applications of regionalization. Extension of our transmission procedure to a
multi-step, hierarchical encoding scheme may also prove useful, as this would
allow for multiscalar regionalization.

Acknowledgments: The author thanks Mark Newman for useful discussions
regarding the use of mutual information for spatial clustering, and Phil
Chodrow for useful discussions about the data. This work was funded by the US
Department of Defense NDSEG fellowship program.

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