# Urban mapping in Dar es Salaam using AJIVE

Rachel J. Carrington  Department of Mathematics and Statistics, Lancaster
University  Ian L. Dryden  School of Mathematical Sciences, University of
Nottingham  Department of Statistics, University of South Carolina  Madeleine
Ellis  N/Lab, University of Nottingham  James O. Goulding  N/Lab, University
of Nottingham  Simon P. Preston  School of Mathematical Sciences, University
of Nottingham  David J. Sirl  School of Mathematical Sciences, University of
Nottingham

###### Abstract

Mapping deprivation in urban areas is important, for example for identifying
areas of greatest need and planning interventions. Traditional ways of
obtaining deprivation estimates are based on either census or household survey
data, which in many areas is unavailable or difficult to collect. However,
there has been a huge rise in the amount of new, non-traditional forms of
data, such as satellite imagery and cell-phone call-record data, which may
contain information useful for identifying deprivation. We use Angle-Based
Joint and Individual Variation Explained (AJIVE) to jointly model satellite
imagery data, cell-phone data, and survey data for the city of Dar es Salaam,
Tanzania. We first identify interpretable low-dimensional structure from the
imagery and cell-phone data, and find that we can use these to identify
deprivation. We then consider what is gained from further incorporating the
more traditional and costly survey data. We also introduce a scalar measure of
deprivation as a response variable to be predicted, and consider various
approaches to multiview regression, including using AJIVE scores as
predictors.

Keywords: Dimension reduction; Multiview data; Urban mapping

##  1 Introduction

Mapping levels of deprivation in urban areas is important, for example for
identifying areas with greatest need, in order to plan intervention work and
decide where to allocate resources (Smith et al.,, 2013; Blumenstock et al.,,
2015; Steele et al.,, 2017). However, estimates are traditionally based on
census data, which for many countries is unavailable or outdated, or household
surveys, which are expensive and time-consuming to collect (Smith-Clarke et
al.,, 2014). In addition, for cities that are growing rapidly in population,
the urban landscape can change rapidly, so results drawn from these sorts of
data may be quickly out of date. There is a need therefore to exploit sources
of data that are readily available and current.

The goal of this paper is to map poverty in the city of Dar es Salaam in
Tanzania by exploiting non-traditional types of data including satellite
imagery and phone call detail record (CDR) data. Using Angle-Based Joint and
Individual Variance Explained (Feng et al.,, 2018), we detect sources of joint
and individual variation within the data, which allows us to understand what
information is contained in both data sets, and what is unique to each. Later,
we incorporate data from a survey conducted in Dar es Salaam to investigate
what additional information, if any, we can gain from this.

Previous studies have shown the potential for using these sorts of data to map
poverty in different locations. For example, Xie et al., (2016), Engstrom et
al., (2017), and Babenko et al., (2017) use satellite imagery to map poverty.
Several studies have used CDR data to derive socioeconomic information: for
example, Blumenstock et al., (2015) uses the call histories of individuals to
predict their socioeconomic status in Rwanda; Smith-Clarke et al., (2014) uses
features derived from CDR data to map deprivation in different regions of Cote
d’Ivoire.

Steele et al., (2017) combines remote sensing and CDR data to map poverty in
Bangladesh, by using features derived from both these types of data to fit a
model to predict poverty. Our approach differs from theirs in the way we use
the image data (using a convolutional neural network to generate feature
vectors, rather than using hand-crafted features such as roads and
vegetation). We also aim to get more insight into the data by finding both
joint and individual sources of variation, which their approach does not.

The data that we consider are: (i) cell-phone call detail record (CDR) data;
(ii) high-resolution satellite imagery (image data) covering most of the city;
and (iii) survey data (see Section 2.3 for details). We focus mainly on the
first two, which are easily available and entail little time or expense to
collect. However, later on we also incorporate the survey data, to investigate
what information contained within the survey data is common to or distinct
from the other two data types, and reflect on how much benefit comes from
collecting and incorporating it. Our aim is to understand the relationships
amongst these different data types, and to exploit them to predict deprivation
for different administrative divisions of Dar es Salaam.

The data are high-dimensional and of different types, and so to identify
useful low-dimensional structure amongst the data — and ultimately to map
deprivation — we use the approach of Angle-Based Joint and Individual
Variation Explained (AJIVE) (Feng et al.,, 2018). The goal of AJIVE is to
identify a small number of components that explain a large proportion of
variation within the data, and for these components to be interpretable as
reflecting individual variation (unique to each data type) and joint variation
(belonging to both data types). The idea is that there is value in
incorporating multiple data types into a joint “multiview” analysis, rather
than performing separate individual analyses, both in identifying a stronger
signal of poverty and in understanding the information contributed by the
different data types.

In Section 2, we explain some further context about Dar es Salaam and describe
the available data, plus other derived measures of poverty in Dar es Salaam
that we later use for comparison. In Section 3, we summarise the AJIVE
algorithm and explain how we estimate the number of components. Section 4
contains results of applying AJIVE and its extensions to the Dar es Salaam
data. We investigate using AJIVE with CDR and image data, and also with the
survey data in addition to these. We give an interpretation of what each of
the joint and individual components shows, and what this tells us about the
data sources, comparing AJIVE to results from applying simpler Principal
Component Analysis (PCA) to the two data sets independently, emphasising the
additional insights that AJIVE provides. We also compare the resulting
deprivation estimates to estimates computed using a completely different data
set and method (Seymour et al.,, 2022), and investigate how well we can
predict deprivation from the AJIVE output using regression modelling.

##  2 Background & Data

Dar es Salaam is located on the eastern coast of Tanzania. According to World
Bank data, the population of Dar es Salaam in 2020 was 6.7 million, having
almost tripled from just under 2.3 million in 2000. It is the largest city in
Tanzania and the fastest growing city in Africa, with some projections
suggesting it could reach a population of over 60 million by 2050 (Locke and
Henley,, 2016). Figure 1 (a) and (b) show the locations of Tanzania and Dar es
Salaam.

The city of Dar es Salaam is divided into 452 administrative regions called
_subwards_. In this paper, we treat the subward as the observational unit; in
each data set each subward is represented by its own feature vector. Figure 1
(c) displays a map of the city, with subward boundaries shown. The image data
only covers 383 subwards, so we include only these subwards in the study;
these are shown in blue in Figure 1 (c). Throughout, we use
$\boldsymbol{X}_{1}$ and $\boldsymbol{X}_{2}$ to refer to the CDR and image
data respectively. Each of these matrices has $n=383$ rows representing the
subwards; the number of columns, representing feature dimensions, is
$p_{1}=20$ (CDR) and $p_{2}=1536$ (image).

The survey data is denoted by $\boldsymbol{X}_{3}$. There are two subwards
missing from this data set, so we leave these out when we incorporate the
survey data into the analysis, giving $n=381$. The survey data has column
dimension $p_{3}=31$.

(a)(b)(c)

Figure 1: (a). Location of Tanzania within Africa. (b). Location of Dar es
Salaam on the east coast of Tanzania. (c). A map of Dar es Salaam showing
boundaries of the subwards (administrative divisions). The blue subwards are
those which are included in the analysis; grey subwards were excluded due to
lack of image data.

###  2.1 CDR data

Figure 2: Locations of cell towers and tower regions in Dar es Salaam. Towers
are shown as red dots with region boundaries marked in red. Subwards and
subward boundaries are shown in blue. As in Figure 1, subwards which were
excluded from the analysis due to lack of available image data are shaded in
grey.

The phone call detail record (CDR) data were collected from $n_{T}=593$ base
transceiver station (BTS) towers located over Dar es Salaam (shown in Figure
2), during 122 days in 2014111For more details, see Engelmann et al., (2018)..
For each tower, information was collected about all call and SMS interactions
recorded, including the time of the interaction, and the user IDs of both
sender and receiver. This information was used to calculate a set of features
for each user, such as the number of calls and SMS sent and received, the
number of BTS towers visited, and the mean time between interactions; Table 2
in the supplementary material gives the full list of features. There are also
5 features calculated at tower level (such as the total number of interactions
recorded by each tower). To convert user-level features to tower features,
each user ID was assigned a “home” tower based on where the majority of their
night-time interactions (between 10pm and 6am) took place; the tower features
were calculated as the mean values across individuals who were assigned to
that tower.

To calculate feature vectors for each subward, it was first necessary to
determine the areas of overlap between the subwards and the regions served by
each tower. The tower regions were calculated by constructing a Voronoi
diagram, in which a polygon is placed around each tower in such a way that any
point contained within it is closer to that tower than to any other. We refer
to these polygons as the tower regions. The subward feature vectors were then
calculated as weighted averages over the tower feature vectors, with weights
proportional to the areas of overlap between the tower regions and subwards.

To describe this mathematically, we label the subwards $S_{1},\ldots,S_{n}$,
and the tower regions $T_{1},\ldots,T_{n_{T}}$. Let $\boldsymbol{\Gamma}$
denote an $n\times n_{T}$ matrix with entries $\gamma_{ij}$, where

$\gamma_{ij}=area\left(S_{i}\cap
T_{j}\right),~{}~{}~{}i=1,\ldots,n;j=1,\ldots,n_{T}.$

Then the feature vector corresponding to subward $S_{i}$ is given by

$\boldsymbol{x}_{1,i}=\frac{\boldsymbol{\gamma}_{i}^{T}\boldsymbol{X}_{T}}{\boldsymbol{\gamma}_{i}^{T}\boldsymbol{1}},$

where $\boldsymbol{\gamma}_{1},\ldots,\boldsymbol{\gamma}_{n}$ are the row
vectors of $\boldsymbol{\Gamma}$, and $\boldsymbol{1}$ is an $n_{T}\times 1$
vector with all elements equal to 1. After calculating the matrix of subward
CDR features $\boldsymbol{X}_{1}$, each column is $98\%$ winsorized (i.e. the
largest and smallest 1% are replaced with the values of the 99th and 1st
percentiles) to remove the effect of large outliers, and then standardized so
that each feature has mean 0 and variance 1 across subwards.

###  2.2 Image data

The image data consists of high-resolution, georeferenced satellite imagery of
Dar es Salaam, which is made available by MAXAR (at
https://www.maxar.com/open-data/covid19). We divided the images into subwards
to create an image for each subward, discarding any sections that were covered
by cloud. As a result, we have full or partial images for 383 subwards; the
remaining 69 had to be excluded from the analysis, due to either being outside
the area covered by the satellite imagery, or to the corresponding section of
the image being entirely obscured by cloud. Figure 3 shows an example of the
image data for a particular subward.

Figure 3: Image for the subward of Idrisa, Dar es Salaam.

The subward images are high-dimensional and are of different sizes and shapes.
To apply AJIVE or PCA, we need to construct a matrix representation of the
data, where a row vector corresponds to each subward. To do this, we follow
Carmichael et al., (2021) by randomly sampling a set of uniformly-sized
patches from each subward image, and applying a pre-trained convolutional
neural network (CNN) to map each patch to a lower-dimensional vector. We then
take the mean feature vector across the patches for each subward. Figure 4
illustrates this process.

For each subward, we took 10 patches of dimension $200\times 200$ pixels,
sampling the top-left pixel uniformly from all possible locations (i.e. the
image with the 199 right-most and bottom-most pixels excluded). Patches could
overlap. Any patches containing more than $1\%$ black background pixels were
replaced, to ensure that the image backgrounds were not a prominent feature in
the patches.

For the CNN, we used InceptionResNet-V2 Szegedy et al., (2016) trained on
ImageNet; this is one of the best-performing models on ImageNet. The schema is
given in Figure 15 of Szegedy et al., (2016). A CNN takes as its input a
$d_{1}\times d_{2}\times d_{3}$ dimensional array of pixel values (where
$d_{1}\times d_{2}$ are the dimensions of the image, and $d_{3}$ is the number
of colour channels, in our case 3), where each element has an integer value
between 0 and 255, and outputs a $v_{1}\times v_{2}\times v_{3}$ dimensional
feature array, where $v_{1}v_{2}v_{3}<<d_{1}d_{2}d_{3}$. If we let
$\mathcal{I}$ denote the space of possible images, the CNN can be defined as a
mapping

$f:\mathcal{I}\rightarrow\mathbb{R}^{v_{1}\times v_{2}\times v_{3}}.$

The values of $v_{1}$, $v_{2}$, and $v_{3}$ are determined by the model
architecture. This array is then usually fed into a final classification
layer, which converts it into a vector; however, as the classification task
the model was trained on is not directly relevant to our situation, we did not
use this. Given the input patch dimensions and choice of model architecture,
the output we obtain for each patch is a $4\times 4\times 1536$ dimensional
array. To reduce the dimension of this further, we used maxpooling, taking the
maximum value in each $4\times 4\times 1$ dimensional sub-array to produce a
1536-dimensional vector. We then took the mean across patches to generate a
feature vector for each subward. As with the CDR features, the image features
are then $98\%$ winsorized and standardized to have mean 0 and variance 1.

Figure 4: Illustration of how image features are generated for a subward.

###  2.3 Survey data

The survey data were collected in two phases in May and August 2019. In each
subward, data were collected from approximately 8 respondents who were asked
to answer questions on that subward. Most subwards had 8 respondents, although
some had more or fewer; all had between 2 and 17.

The questions cover various aspects such as poverty and unemployment,
education, and medical facilities. The data were originally collected (Ellis,,
2021) to create a set of fine-grained, socio-demographic data about Dar es
Salaam, which could be used to inform research on topics such as deprivation
and forced labour risk, and to provide a ground truth against which models
could be compared. Not all of the questions and responses could be used in our
analysis, as some questions had categorical or write-in answers: we only
included questions which had numerical or ordinal responses. Tables 3 and 4 in
the supplementary material give the full list of survey questions in the data
set used for this paper, and possible responses. For each subward we take the
median responses for the respondents. The variables are then centred and
scaled so that each response variable has mean 0 and variance 1.

Figure 5: Correlation plots of CDR variables (left) and survey variables
(right). The features are ordered using complete-linkage hierarchical
clustering, in order to highlight relationships between features. Figure 6:
Correlations between image features. The features are ordered using complete-
linkage hierarchical clustering, in order to highlight relationships between
features.

Figures 5 and 6 show pairwise correlations between each of the CDR, image, and
survey features. In each case there are several clusters of correlated
variables, which suggests that dimension reduction should be useful.

###  2.4 Deprivation estimates based on comparative judgments

Our aim is to explore how we can use the CDR and image data to map
deprivation. To provide a comparison, we use deprivation estimates for Dar es
Salaam calculated from comparative judgment data, using the Bayesian Spatial
Bradley-Terry model (Seymour et al.,, 2022). These are available in the R
package BSBT and consist of a score between -1.2 and 2.2 for each subward in
Dar es Salaam, where higher scores indicate less deprived subwards. Figure 7
shows a plot of these deprivation scores for the subwards that are included in
our analysis.

Figure 7: Deprivation scores by subward, estimated from comparative judgment
data using the Bayesian Spatial Bradley-Terry (BSBT) model (see (Seymour et
al.,, 2022)). Subwards with higher scores (shown in blue on the figure) are
those which are considered to be less deprived. Although these data cover all
of Dar es Salaam, the figure only shows scores for the subwards included in
our analysis.

##  3 Methodology

In this section we give an overview of the Angle-Based Joint and Individual
Variation Explained (AJIVE) algorithm (Feng et al.,, 2018), which we use to
obtain the main part of our results. We also use Principal Component Analysis
(PCA) for comparison; as AJIVE can be viewed as an extension of PCA to
multiple data sets, we outline this first in Section 3.1, followed by AJIVE in
Section 3.2. We then, in Section 3.3, discuss how to choose the ranks for low-
rank approximation to our data, on which both PCA and AJIVE depend.

###  3.1 Principal Components Analysis (PCA)

For a centred $n\times p$ data matrix, $\boldsymbol{X}$, PCA identifies the
subspace of a chosen dimension, $r$, in which the data have greatest variance.
It can be computed in terms of the Singular Value Decomposition (SVD):

$\boldsymbol{X}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^{\top},$

where, with $m=\min\\{n,p\\}$, $\boldsymbol{\Sigma}$ is an $m\times m$
diagonal matrix containing the $m$ singular values of $\boldsymbol{X}$ in
decreasing order; and $\boldsymbol{U}$ and $\boldsymbol{V}$ are respectively
$n\times m$ and $p\times m$ matrices with columns corresponding respectively
to the left and right singular vectors of $\boldsymbol{X}$, such that
$\boldsymbol{U}^{\top}\boldsymbol{U}=\boldsymbol{V}^{\top}\boldsymbol{V}=\boldsymbol{I}_{m}$
(the $m\times m$ identity matrix).

Reducing the dimension of the data to the chosen dimension $r$ $(<m)$ by PCA
entails approximating $\boldsymbol{X}$ using the rank-$r$ truncated SVD, that
is

$\boldsymbol{X}\approx\boldsymbol{U}_{r}\boldsymbol{\Sigma}_{r}\boldsymbol{V}_{r}^{\top}$
(1)

where $\boldsymbol{U}_{r}$ and $\boldsymbol{V}_{r}$ are matrices containing
the first $r$ columns of $\boldsymbol{U}$ and $\boldsymbol{V}$ respectively,
and $\boldsymbol{\Sigma}_{r}$ is the $r\times r$ diagonal matrix containing
the first $r$ singular values of $\boldsymbol{X}$. The PC scores are the rows
of the $n\times r$ matrix $\boldsymbol{V}_{r}$.

###  3.2 AJIVE

Angle-Based Joint and Individual Variation Explained (AJIVE) (Feng et al.,,
2018) is a dimension reduction algorithm which can be applied to two or more
data sets, where the data correspond to the same group of individuals, but are
of different types. It is applicable when we believe there is some joint
structure common to both data sets, but also some information which is unique
to each: in contrast to methods such as Principal Component Analysis, where
each data matrix is decomposed separately, or Canonical Correlation Analysis,
which finds only joint components, AJIVE allows us to analyse both joint and
individual variation within the data. It was developed as a more efficient
version of Joint and Individual Variation Explained (JIVE) (Lock et al.,,
2013).

AJIVE (Feng et al.,, 2018) aims to decompose each data matrix
$\boldsymbol{X}_{i}$ as

$\boldsymbol{X}_{i}=\boldsymbol{J}_{i}+\boldsymbol{A}_{i}+\boldsymbol{E}_{i}$

where
$\boldsymbol{J}=\left(\boldsymbol{J}_{1}~{}\cdots~{}\boldsymbol{J}_{k}\right)$
is the joint variation matrix (of rank $r_{J}$),
$\boldsymbol{A}_{1},\ldots,\boldsymbol{A}_{k}$ are the individual variation
matrices (each of which have rank $r_{i}$), and
$\boldsymbol{E}_{1},\ldots,\boldsymbol{E}_{k}$ represent noise.

We can decompose $\boldsymbol{J}$ as

$\boldsymbol{J}=\boldsymbol{U}_{J}\boldsymbol{\Sigma}_{J}\boldsymbol{V}_{J}^{\top},$
(2)

which we obtain by taking the (exact) rank-$r_{J}$ SVD of $\boldsymbol{J}$.
$\boldsymbol{U}_{J}$ is the $n\times r_{J}$ matrix of joint scores (in our
case, each row corresponds to a subward), and $\boldsymbol{V}_{J}$ is the
$p\times r_{J}$ matrix of joint loadings (each row corresponds to a CDR or
image feature). $\boldsymbol{\Sigma}_{J}$ contains the singular values of
$\boldsymbol{J}$ and controls the relative weights of each component. Figure 8
illustrates this decomposition.

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2Subward
3F1F2F3F4$\boldsymbol{U}_{J}$$\boldsymbol{\Sigma}_{J}$$\boldsymbol{V}_{J}^{\top}$$\boldsymbol{J}$$\boldsymbol{=}$JC1JC2JC1JC2JC1JC2JC
= Joint componentF = Feature Figure 8: Decomposition
$\boldsymbol{J}=\boldsymbol{U}_{J}\boldsymbol{\Sigma}_{J}\boldsymbol{V}_{J}^{\top}$
in the case where $r_{J}=2$. Each of the joint components has an
$n$-dimensional score vector (column of $\boldsymbol{U}_{J}$) and a
$p$-dimensional loading vector (row of $\boldsymbol{V}_{J}^{\top}$) associated
with it; in this example, $n=3$ and $p=4$. Each subward has an
$r_{J}$-dimensional score vector (row of $\boldsymbol{U}_{J}$) associated with
it, and each feature has a $r_{J}$-dimensional loading vector (column of
$\boldsymbol{V}_{J}^{\top}$). Given that the singular values in
$\boldsymbol{\Sigma}_{J}$ are distinct and ordered from largest to smallest,
the decomposition is identifiable up to multiplication of components by -1: we
can multiply any column of $\boldsymbol{U}_{J}$ and the corresponding column
of $\boldsymbol{V}_{J}$ (row of $\boldsymbol{V}_{J}^{\top}$) by -1 without
changing the value of
$\boldsymbol{U}_{J}\boldsymbol{\Sigma}_{J}\boldsymbol{V}_{J}^{\top}$.

Similarly, for the $i$th individual component ($i=1,\ldots,k$) we decompose
$\boldsymbol{A}_{i}$ as

$\boldsymbol{A}_{i}=\boldsymbol{U}_{i}\boldsymbol{\Sigma}_{i}\boldsymbol{V}_{i}^{\top},$

where $\boldsymbol{U}_{i}$ is the $n\times r_{i}$ score matrix,
$\boldsymbol{V}_{i}$ the $p_{i}\times r_{i}$ loading matrix, and
$\boldsymbol{\Sigma}_{i}$ contains the $r_{i}$ non-zero singular values of
$\boldsymbol{A}_{i}$.

The AJIVE algorithm is given in detail in (Feng et al.,, 2018); we give an
outline here. The algorithm has three main stages:

_Stage 1_. We choose initial signal ranks $\tilde{r}_{1},\ldots,\tilde{r}_{k}$
(see Section 3.3) and approximate each $\boldsymbol{X}_{i}$ by its
rank-$\tilde{r}_{i}$ truncated SVD:

$\boldsymbol{X}_{i}\approx\tilde{\boldsymbol{X}}_{i}=\tilde{\boldsymbol{U}}_{i}\tilde{\boldsymbol{\Sigma}}_{i}\tilde{\boldsymbol{V}}_{i}^{\top},$

_Stage 2_. To estimate the joint scores $\boldsymbol{U}_{J}$, we combine the
score matrices into one matrix $\boldsymbol{M}$, from which we will extract
the joint signal:

$\boldsymbol{M}=\left(\tilde{\boldsymbol{U}}_{1}~{}~{}\ldots~{}~{}\tilde{\boldsymbol{U}}_{k}\right),$

and calculate its SVD:

$\boldsymbol{M}=\boldsymbol{U}_{M}\boldsymbol{\Sigma}_{M}\boldsymbol{V}_{M}^{\top}.$

We then choose the joint rank $r_{J}$, based on the singular values of
$\boldsymbol{M}$, (again, see Section 3.3), and set the estimate of the joint
scores to be $\tilde{\boldsymbol{U}}_{J}=\tilde{\boldsymbol{U}}_{M}$, where
$\tilde{\boldsymbol{U}}_{M}$ is a matrix containing the first $r_{J}$ columns
of $\boldsymbol{U}_{M}$.

To estimate the joint matrix $\boldsymbol{J}$, we project
$\boldsymbol{X}=\left(\boldsymbol{X}_{1},\boldsymbol{X}_{2}\right)$ onto
$\tilde{\boldsymbol{U}}_{J}$:

$\hat{\boldsymbol{J}}=\tilde{\boldsymbol{U}}_{J}\tilde{\boldsymbol{U}}_{J}^{\top}\boldsymbol{X},$

and decompose $\hat{\boldsymbol{J}}$ as
$\hat{\boldsymbol{J}}=\hat{\boldsymbol{U}}_{J}\hat{\boldsymbol{\Sigma}}_{J}\hat{\boldsymbol{V}}_{J}^{\top}$,
by taking its rank-$r_{J}$ SVD (which gives an exact decomposition since
$\hat{\boldsymbol{J}}$ has rank $r_{J}$):

$\hat{\boldsymbol{J}}=\hat{\boldsymbol{U}}_{J}\hat{\boldsymbol{\Sigma}}_{J}\hat{\boldsymbol{V}}_{J}^{\top}.$

(Note that although generally
$\tilde{\boldsymbol{U}}_{J}\neq\hat{\boldsymbol{U}}_{J}$ (unless $r_{J}=1$),
we have
$\tilde{\boldsymbol{U}}_{J}\tilde{\boldsymbol{U}}_{J}^{\top}=\hat{\boldsymbol{U}}_{J}\hat{\boldsymbol{U}}_{J}^{\top}$:
they are both estimates of the score space of $\hat{\boldsymbol{J}}$.)

_Stage 3_. To estimate the individual matrices $\boldsymbol{A}_{i}$, we note
that for each $i$, we require the joint scores $\boldsymbol{U}_{i}$ to be
orthogonal to $\hat{\boldsymbol{U}}_{J}$. Hence, we project each
$\boldsymbol{X}_{i}$ onto the orthogonal complement of $\hat{\boldsymbol{J}}$:

$\tilde{\boldsymbol{X}}_{i}=\left(\boldsymbol{I}_{n}-\tilde{\boldsymbol{U}}_{J}\tilde{\boldsymbol{U}}_{J}^{\top}\right)\boldsymbol{X}_{i}$

where $\boldsymbol{I}_{n}$ is the $n\times n$ identity matrix. We then take a
final rank-$r_{i}$ SVD of this matrix $\tilde{\boldsymbol{X}}_{i}$:

$\hat{\boldsymbol{A}}_{i}=\tilde{\boldsymbol{X}}_{i}\approx\hat{\boldsymbol{U}}_{i}\hat{\boldsymbol{\Sigma}}_{i}\hat{\boldsymbol{V}}_{i}^{\top}.$

###  3.3 Rank estimation

To implement AJIVE we must provide estimates of the initial ranks
$\tilde{r}_{1},\ldots,\tilde{r}_{k}$, joint rank $r_{J}$, and individual ranks
$r_{1},\ldots,r_{k}$. We describe here how we do this. For simplicity, we will
refer to the case where we have the CDR and image data, so $k=2$: we later
incorporate the survey data as well, but the methods are the same.

_Initial ranks_. In selecting the initial ranks, we aim to distinguish signal
from noise in each data set. We do this by inspecting scree plots of
$\boldsymbol{X}_{1}$ and $\boldsymbol{X}_{2}$ (Figure 9 (a)–(b)). On both
plots there is a sizeable jump after the third singular value, so we take
$\tilde{r}_{1}=\tilde{r}_{2}=3$.

Scree plot: CDR(a)Scree plot: Image(b)Joint rank estimation(c)

Figure 9: Estimation of initial and joint ranks. (a), (b): Scree plots showing
the singular values of the CDR feature matrix $\boldsymbol{X}_{1}$, and the
image feature matrix $\boldsymbol{X}_{2}$. (c): Estimation of $r_{J}$, given
initial ranks of 3 for both $\boldsymbol{X}_{1}$ and $\boldsymbol{X}_{2}$. The
red points are the singular values of $\boldsymbol{M}$. The black points
correspond to the largest singular value of $\tilde{\boldsymbol{M}}$ for each
of 1000 random samples. The black line shows the $95\%$ threshold, above which
singular values are assumed to correspond to joint components.

_Joint rank_. After combining the initial score matrices into one matrix
$\boldsymbol{M}$ (see previous section), we want to find out which components
of $\boldsymbol{M}$ have sufficiently large singular values to be regarded as
part of the joint signal. To do this we use a random direction bound, as in
Feng et al., (2018). To recap, we have

$\boldsymbol{M}=\left(\tilde{\boldsymbol{U}}_{1}~{}~{}\tilde{\boldsymbol{U}}_{2}\right),$

where $\boldsymbol{M}$ is an $n\times\left(\tilde{r}_{1}+\tilde{r}_{2}\right)$
dimensional matrix.

We create 1000 random matrices of the same dimensions as $\boldsymbol{M}$,
with elements sampled independently from $U(0,1)$, and calculate the largest
singular value in each case. We take the 95th percentile of these values as a
lower bound for the singular values of $\boldsymbol{M}$ that correspond to
joint signal. Figure 9 (c) illustrates this process for the CDR and image
data. The red points show the ordered singular values of $\boldsymbol{M}$
(excluding those that are equal to 0), whilst the black points correspond to
the maximum singular values of each randomly generated matrix, as described
above. The horizontal black line corresponds to the threshold for the joint
signal. There are two red points above the black line, so we take $r_{J}=2$.

_Individual ranks_. After calculating the joint components, we subtract from
each $\boldsymbol{X}_{i}$ the corresponding part of the joint signal matrix,
and calculate its SVD:

$\tilde{\boldsymbol{X}}_{i}=\boldsymbol{X}_{i}-\tilde{\boldsymbol{U}}_{J}\tilde{\boldsymbol{U}}_{J}^{\top}\boldsymbol{X}_{i}=\tilde{\boldsymbol{U}}_{i}\tilde{\boldsymbol{\Sigma}}_{i}\tilde{\boldsymbol{V}}_{i}^{T}.$

The initial rank $\tilde{r}_{i}$ was selected by inspecting a scree plot, but
we could equivalently have selected a threshold

$\nu_{i}=\frac{\lambda^{i}_{\tilde{r}_{i}}+\lambda^{i}_{\tilde{r}_{i}+1}}{2},$

where $\lambda_{\tilde{r}_{i}}^{i}$ and $\lambda_{\tilde{r}_{i}+1}^{i}$ are
the $\tilde{r}_{i}$th and $\tilde{r}_{i}+1$th largest singular values of
$\boldsymbol{X}_{i}$, such that we only keep components with singular values
greater than or equal to $\nu_{i}$. For the individual components, we use the
same threshold, so we keep the individual components which have corresponding
singular values greater than or equal to $\nu_{i}$ and discard those which are
smaller. For the CDR and image data, this gives $r_{1}=r_{2}=1$.

##  4 Results

We implement the AJIVE algorithm in R. We first inspect scree plots (Figure 9)
for $\boldsymbol{X}_{1}$ and $\boldsymbol{X}_{2}$ to choose the initial ranks
$\tilde{r}_{1}$ and $\tilde{r}_{2}$: we set $\tilde{r}_{1}=\tilde{r}_{2}=3$.
To select the joint rank $r_{J}$, we use a random direction bound (see Section
3.3), with 1000 random samples, which leads to a value of $r_{J}=2$. The
individual ranks for both the CDR and image data ($r_{1}$ and $r_{2}$) are
estimated to be 1. Throughout, we refer to the joint components as JC1 and
JC2, and the individual components as IC1CDR and IC1Image.

(a)(b)(c)Joint component 1Patches from most positive subwardsPatches from most
negative subwardsCDR feature loadings Figure 10: (a) Subward scores for JC1.
Red corresponds to negative scores and blue to positive scores. Grey subwards
are those for which we do not have image data. (b) CDR feature loadings,
sorted from most positive to most negative. (c) Patches from the two most
positive and two most negative subwards in JC1, with arrows showing to which
subwards they correspond. (a)(b)(c)Joint component 2Patches from most positive
subwardsPatches from most negative subwardsCDR feature loadings Figure 11: (a)
Subward scores for JC2. Red corresponds to negative scores and blue to
positive scores. Grey subwards are those for which we do not have image data.
(b) CDR feature loadings, sorted from most positive to most negative. (c)
Patches from the two most positive and two most negative subwards in JC2 with
arrows showing to which subwards they correspond.

We look first at each of the components individually (Figures 10 to 13). For
each component, the figure shows the subward scores (columns of
$\boldsymbol{U}_{J}$ or $\boldsymbol{U}_{i}$), patches from the subwards at
the extreme positive and negative ends of each component, and, where
applicable, the CDR feature loadings (rows of $\boldsymbol{U}_{J}$ or
$\boldsymbol{U}_{i}$). (Although the relative values of scores and loadings
are of interest, it is arbitrary which end is negative and which is positive.)

###  4.1 Joint components

Figure 10 shows the results for the first joint component (JC1). The positive
end of the component shows green, rural areas; at the negative end, the
patches show built-up areas with small, residential buildings. We can also see
from Figure 10(a) that most subwards with negative scores are small and
located close to the city centre. The most positive CDR features are the
number of calls initiated by users, the average distance between users and the
people they call, and the ratio of calls to SMS. This suggests that residents
in these areas are more likely to initiate calls and to make calls rather than
sending SMS, compared to those at the negative end of the component. It also
seems reasonable that people living in rural areas, with low population
density, would tend to communicate with people further away, as there are
fewer people living close by. The most negative CDR features are the number of
day, evening, and night interactions, and the number of active users; these
are likely to be correlated with population.

For the second joint component (JC2) (Figure 11), the subwards with the most
positive scores again appear to be more rural areas, but the negative end
appears to show mostly commercial and industrial areas. The most negative
subwards are located close to the city centre and on the coast, whilst the
subwards in the south-west and south-east of the city tend to be those with
the most positive scores. The most positive CDR features are the percentage of
contacts that account for $80\%$ of a user’s call interactions, entropy of
contacts — which are both measures of the diversity of contacts with whom a
user interacts — and the number of active users. Looking at the CDR features
with the most negative loadings, we infer that at this end of the component,
users move around more (to different towers), make more calls, and use their
phones on more days.

(a)(b)(c)CDR individual componentPatches from most positive subwardsPatches
from most negative subwardsCDR feature loadings Figure 12: (a) Subward scores
for IC1CDR. Red corresponds to negative scores and blue to positive scores.
Grey subwards are those for which we do not have image data. (b) CDR feature
loadings, sorted from most positive to most negative. (c) Patches from the two
most positive and two most negative subwards in IC1CDR, with arrows showing to
which subwards they correspond.

###  4.2 Individual components

For the CDR individual component (IC1CDR) (Figure 12), the most positive CDR
features are those related to entropy of locations and contacts, and the total
number of SMS sent. At the negative end are the ratio of calls to texts, and
the standard deviation of interevent times. This end of the component also
seems to contain built-up areas with taller buildings, although it is less
clear what the patches from the positive end show. (This is perhaps not
surprising, as this component does not directly use information from the
patches.)

The most positive subward seems to have a particularly extreme value: we can
see that it is much brighter than all the other subwards in Figure 12 (a).
This subward contains the University of Dar es Salaam, as well as a large
shopping centre (part of the roof of this is shown in one of the patches in
Figure 12 (c)); it seems reasonable to assume that these would have an impact
on patterns of phone usage, as people’s activities in this subward will likely
be different to other areas of the city.

(a)(b)Image individual componentPatches from most positive subwardsPatches
from most negative subwards Figure 13: (a) Subward scores for IC1Image. Red
corresponds to negative scores and blue to positive scores. Grey subwards are
those for which we do not have image data. (b) Patches from the two most
positive and two most negative subwards in IC1Image, with arrows showing to
which subwards they correspond.

For the image individual component (IC1Image) (Figure 13), it looks like the
positive end corresponds to industrial and commercial areas, whereas the
negative end is mostly rural. There are no CDR feature loadings to display for
this component. As for IC1CDR, there appears to be one extreme subward, this
time at the negative end of the component: one of the patches from this
component (Figure 13 (b)) looks unusual, so this may be responsible. However,
when we re-implement AJIVE with this subward removed, there is not a
noticeable difference in the results for the other subwards, so it does not
seem to be having too much influence on the results. (This is also the case
for the outlier subward in IC1CDR.

| Positive | Negative
---|---|---
Joint component 1 | Rural | High-density, slum housing
| More initiation of calls | Total CDR interactions
Joint component 2 | Rural or semi-rural | Industrial/commercial
| Diversity of contacts |
CDR individual component | Low-density housing | Industrial/commercial
| High entropy CDR data | High calls to text ratio
Image individual component | Industrial/commercial | Rural
Table 1: Qualitative interpretation of the AJIVE joint and individual
components shown in Figures 10, 11, 12, and 13, summarising the discussion in
Sections 4.1 and 4.2

IC1CDR and IC1Image have virtually no correlation ($\rho=0.03$), so we can
surmise that all joint variation is adequately captured by JC1 and JC2:
correlation between the individual components would suggest that the joint
rank was too small. We can therefore assume that the information captured in
IC1CDR and IC1Image is unique to the CDR and image data sets respectively.

Table 1 briefly summarises our observations in this section.

Figure 14: Boxplots of CDR features, with blue triangles showing the values
for the subward of Chuo Kikuu.

###  4.3 Comparison with deprivation estimates

One of the main aims of this paper is to investigate whether AJIVE can be used
to predict deprivation. We are interested in whether our methods and data
sources can be used to do this in situations where alternative data sources
are unavailable; however, to assess how well it works, we here use deprivation
estimates created from a completely distinct data set (details were given in
Section 2.4).

Figure 15 shows correlations between the deprivation estimates and AJIVE and
PC scores. There is a fairly strong positive correlation between deprivation
and JC2 ($\rho=0.67$). Hence, this component seems to be picking up joint
variation in the data which is relevant for determining deprivation. In
particular, this indicates that deprivation information is present in both
$\boldsymbol{X}_{1}$ and $\boldsymbol{X}_{2}$. For each of the other AJIVE
components there is no or very little correlation with deprivation.

###  4.4 Comparison with Principal Components Analysis

One question is whether using AJIVE gives a significant advantage over using
Principal Components Analysis (PCA). PCA can be used either on
$\boldsymbol{X}_{1}$ and $\boldsymbol{X}_{2}$ separately, or on the
concatenated data matrix
$\boldsymbol{X}=\left(\boldsymbol{X}_{1},\boldsymbol{X}_{2}\right)$. We
consider both of these options here, and compare their output to that of
AJIVE.

As a reminder, from Section 3.1, using rank-$r$ PCA we represent an $n\times
p$ data matrix $\boldsymbol{X}$ as

$\boldsymbol{X}=\boldsymbol{U}_{r}\boldsymbol{\Sigma}_{r}\boldsymbol{V}_{r}^{\top},$

where $\boldsymbol{U}_{r}$ and $\boldsymbol{V}_{r}$ are $n\times r$ and
$p\times r$ matrices containing the first $r$ left and right singular vectors
of $\boldsymbol{X}$ respectively, and $\boldsymbol{\Sigma}_{r}$ is an $r\times
r$ diagonal matrix containing the first $r$ singular values of
$\boldsymbol{X}$ along its diagonal. As with AJIVE, we can interpret the
columns of $\boldsymbol{U}_{r}$ and $\boldsymbol{V}_{r}$ as score and loading
vectors respectively. The singular values in $\boldsymbol{\Sigma}_{r}$ control
the relative importance of each component.

To compare AJIVE with separate PC analyses of $\boldsymbol{X}_{1}$ and
$\boldsymbol{X}_{2}$, we computed the first three principal components
(referred to as PC1, PC2, and PC3) for each data set. (Each data set has three
AJIVE components related to it — two joint and one individual — so this seems
a fair comparison.) Figure 15 shows correlations between scores for the AJIVE
and PC components. (Note that JC1 and JC2 are forced to be orthogonal to each
other, as well as to the individual components, and the PC components for each
data matrix are also forced to be orthogonal.)

Figure 15: Correlations between AJIVE and PC scores, when both are applied to
the CDR and image data. (Since the signs of the score vectors, and hence the
directions of correlation between them, are arbitrary, the absolute values of
correlations are shown.) Deprivation scores estimated using the Bayesian
Spatial Bradley-Terry model (see Section 2.4) are also included for
comparison.

We see that JC1 is strongly correlated with PC2CDR and is also correlated with
both PC1Image and PC2Image, whilst JC2 is correlated with PC1CDR and PC3Image.
IC1CDR is strongly correlated with PC3CDR, and IC1Image is correlated with
PC1Image and PC2Image. IC1CDR appears to be uncorrelated with the image PC
scores, and the same applies to IC1Image and the CDR PC scores. AJIVE appears
to give similar overall results to doing separate PCAs of the CDR and image
data, but the division of components is different. Hence, AJIVE identifies
which parts of the components are common to both data sets and which are
unique to each data set.

When doing PCA on the concatenated data matrix
$\boldsymbol{X}=\left(\boldsymbol{X}_{1}\hskip
2.84544pt\boldsymbol{X}_{2}\right)$, we find that the first few PCs of
$\boldsymbol{X}$ are almost identical to the corresponding PCs of
$\boldsymbol{X}_{2}$. This is not surprising as $\boldsymbol{X}_{2}$ has a
much higher dimensionality than $\boldsymbol{X}_{1}$ ($p_{1}=20,p_{2}=1536$).
However, this shows that applying PCA to the concatenated data matrix
$\boldsymbol{X}$ is not useful for our data, as we essentially lose the
information from $\boldsymbol{X}_{1}$. This would also apply to other
situations where the number of dimensions differs greatly between data
matrices.

####  4.4.1 Proportion of variance explained

AJIVEPCA

Figure 16: Proportion of variance explained by AJIVE (left) and PCA (right)
for the CDR and image data.

Figure 16 shows the proportion of variance explained by each of the components
for the CDR and image data. For the CDR data, the three AJIVE components
together account for around $40\%$ of the variance in the data, with JC1
contributing the most. For the image data, the total variance explained is
much smaller, likely due to $\boldsymbol{X}_{2}$ being much more high-
dimensional ($p=1536$ compared to $p=20$ for the CDR data). The individual
component contributes more than either JC1 or JC2. In both cases the
proportion of variance explained by PCA is slightly greater than that
explained by AJIVE. (This is what would be expected, as PCA maximizes the
proportion of variance explained.)

###  4.5 Incorporating the survey data

So far, we have looked at applying AJIVE and PCA to the CDR and image data
sets. We now repeat the foregoing analysis but with the addition of the survey
data introduced earlier in Section 2.3. Recall that the survey data are slower
and more costly to collect than the CDR and image data, so it is natural to
ask: what extra information does the survey data provide, and does it warrant
the cost of collecting the data in the first place?

(a)(b)

Figure 17: Rank estimation for survey data. (a). Scree plot. (b). Estimation
of joint rank using the random samples method, with initial ranks of (3, 3,
2).

The scree plot (Figure 17) suggests an initial rank of 2 for the survey data.
Applying the random direction bound to estimate the joint rank (as in Section
3.3, but using all three data matrices) returns a joint rank $r_{J}=2$. We
also obtain individual rank estimates of $r_{1}=r_{2}=r_{3}=1$.

(a)(b)(c)Survey individual componentPatches from most positive subwardsPatches
from most negative subwardsSurvey feature loadings Figure 18: (a) Subward
scores for IC1Survey, the survey individual component. Red corresponds to
negative scores and blue to positive scores. Grey subwards are those for which
we do not have image data. (b) Survey feature loadings, sorted from most
positive to most negative. (c) Patches from the two most positive and two most
negative subwards in IC1Survey, with arrows showing to which subwards they
correspond.

Figure 18 shows a plot of the survey individual component (IC1Survey) scores
for each subward, and 5 patches from two most positive and most negative
subwards. Corresponding plots for the other components (JC1, JC2, IC1CDR, and
IC1Image) are shown in the supplementary material (Section C): they are very
similar to those we obtained using only the CDR and image data (Figures 10 to
13). Figure 19 shows correlations between AJIVE and PC components for the
three data matrices. Comparing with Figure 15, the relations between JC1, JC2,
IC1CDR and IC1Image, and the CDR and image PCs are virtually unchanged, so
incorporating the survey data does not have much effect on the decomposition.
However, looking at the survey individual component allows us to see what
information may be present in the survey data that we cannot get from the
other data sets. From Figure 18 (b), it appears that IC1Survey highlights a
risk of forced labour and exploitation: the variables with the most negative
loadings are those relating to forced work and arranged marriage, whilst as
the positive end we have higher rates of children and teenagers in education,
and more use of technology, which may be associated with lower risks of forced
labour.

Figure 19: Correlations between joint, individual and PCA components for the
CDR, image and survey data. Deprivation scores estimated using the Bayesian
Spatial Bradley-Terry model (see Section 2.4) are also included for
comparison.

###  4.6 Regression modelling to predict deprivation

We have investigated correlation between the dimension-reduced representations
of the data (AJIVE and PC scores) and deprivation scores, but the approaches
to dimension reduction were “unsupervised” in the sense that they did not
involve using the deprivation data. In this section, we consider the
“supervised” approach of using deprivation as a response variable in a
regression model geared towards prediction of deprivation from the CDR, image
and survey data. We particularly investigate whether it aids prediction to use
AJIVE and/or PCA as a step of dimension reduction to construct “features” for
the regression model.

A recently introduced approach (Ding et al.,, 2022) for incorporating multiple
data “views” into a regression model involves minimising the objective

$\frac{1}{2}\left\|\boldsymbol{y}-\sum_{m=1}^{M}\boldsymbol{Z}_{m}\boldsymbol{\beta}_{m}\right\|^{2}+\sum_{m=1}^{M}\lambda_{m}\|\boldsymbol{\beta}_{m}\|_{1}+\frac{\rho}{2}\sum_{m<m^{\prime}}\left\|\boldsymbol{Z}_{m}\boldsymbol{\beta}_{m}-\boldsymbol{Z}_{m^{\prime}}\boldsymbol{\beta}_{m^{\prime}}\right\|^{2},$
(3)

with respect to the regression parameters
$\boldsymbol{\beta}_{1},\ldots,\boldsymbol{\beta}_{M}$, in which
$\boldsymbol{\beta}_{m}$ is $p_{m}\times 1$ vector of parameters corresponding
to the $m$th view, $\boldsymbol{Z}_{m}$, and $\mathbf{y}$ is a response
variable to be predicted. For the special case with parameters $\rho=0$ and
$\lambda_{1}=\cdots=\lambda_{m}$, this is the objective for LASSO regression
of $\mathbf{y}$ on the concatenated data
$\boldsymbol{Z}=(\boldsymbol{Z}_{1},\ldots,\boldsymbol{Z}_{M})$. But when
$\rho>0$, the final term encourages “cooperation” between the predictions from
the different individual views, which in some circumstances leads to better
predictive performance for out-of-sample data, i.e. new observations not used
in fitting the model (Ding et al.,, 2022).

Figure 20: Values of the MSE plotted against $\rho$, where we apply the
regression model in Equation 3 with cross-validation. Different lines
correspond to different combinations of variables. For the case where we
include all AJIVE and PC scores, increasing $\rho$ also increases the MSE, so
the model performs worse. For other scenarios, the value of $\rho$ has little
or no effect on performance.

For a given choice of $\rho,\lambda_{1},\ldots,\lambda_{M}$, the fitted
parameters
$\hat{\boldsymbol{\beta}}=(\hat{\boldsymbol{\beta}}_{1}^{\top},\ldots,\hat{\boldsymbol{\beta}}_{M}^{\top})^{\top}$
are determined by minimising (3), and prediction of the response variable for
a new observation with predictor vector
$\mathbf{z}=(\mathbf{z}_{1}^{\top},\ldots,\mathbf{z}_{M}^{\top})^{\top}$ is
$\hat{\boldsymbol{y}}=\mathbf{z}^{\top}\hat{\boldsymbol{\beta}}$. One way to
select the hyper-parameters $\rho,\lambda_{1},\ldots,\lambda_{M}$ is by cross-
validation, i.e., repeatedly partitioning the observations into training and
“held-out” test sets, finding $\hat{\boldsymbol{\beta}}$ based on the training
data for various different values of the hyper-parameters, then selecting the
hyper-parameter values that minimise mean-squared error in predictions for the
held-out test data.

To compare the performance of different $\boldsymbol{Z}$’s, we select
$\lambda_{1},\ldots,\lambda_{M}$ using cross-validation with 20 folds, and
calculate the average Mean Squared Error (MSE) across folds: we choose the
value of $\boldsymbol{\lambda}$ which minimizes this. A lower MSE means the
model is more accurate at predicting deprivation $\boldsymbol{y}$. In various
numerical investigations exploring values $\rho\geq 0$ with different choices
of predictor data $\boldsymbol{Z}$, we found no circumstances where $\rho>0$
performed better than $\rho=0$ — see for example Figure 20 — so we set
$\rho=0$ in all following experiments.

Figure 21: Results of regression using different choices of $\boldsymbol{Z}$.
In each case we implemented the model 5 times with data split randomly into 20
folds in each case.

Figure 21 shows the results using $\rho=0$ and different choices for
$\boldsymbol{Z}$. Except for PCA on the CDR data (where we have a maximum of
20 components to work with), all AJIVE and PCA regressions are done with 30
total components, which are divided equally between data sets and, for AJIVE,
between joint and individual components.

We first consider using each data set individually, setting $\boldsymbol{Z}$
to be either the entire data set or the matrix of principal components. The
CDR data gives the lowest MSE, whilst the survey data does by far the worst.
In each case using PCA gives similar results to using the entire data set.

We then combine data sets: we consider using the CDR and image data, as we did
for our main analysis, and then adding in the survey data. Here, we find that
PCA regression does slightly better than using the entire data. The difference
between PCA and AJIVE is less clear: when we have just the CDR and image data,
AJIVE seems to do slightly better than PCA, but this is not the case once we
also add in the survey data, although the differences in MSE are very small,
so it is difficult to draw firm conclusions. However, as mentioned above AJIVE
with CDR, image and survey data gives the additional useful information about
forced labour risk and so this is our preferred approach.

Figure 22: Average MSE (across 10 runs) plotted against number of components
for several different combinations of variables. The horizontal black line
corresponds to the MSE for the null model. Where relevant, components are
divided equally (as far as possible) between joint and individual components,
and between CDR and image components.

Figure 22 shows a plot of the MSE versus the number of components, when we use
several combinations of components for the CDR and image data. The black
horizontal line corresponds to the null model. In each case the MSE decreases
until we have around 20 components, then levels off. There does not appear to
be much difference in performance between the three choices of
$\boldsymbol{Z}$ we use here.

##  5 Discussion

We have seen that AJIVE allows us to identify common features across the CDR
and image data sets, as well as features which are unique to each. The joint
components we have found are also useful for predicting deprivation.

Incorporating the survey data, in addition to CDR and image data, does not
much affect the joint components, or contribute to our ability to predict
deprivation. This is perhaps surprising, but it shows that much of the
information contained within the survey data can also be obtained from the
other two data types. However, it may also be the case that the methods we
have used are less well suited to the survey data, as it consists mostly of
categorical and ordinal variables, whereas PCA and AJIVE are designed for
continuous data. However, we do find that the survey data provides information
about forced labour risk, which we could not detect from the other data sets.

Future research could include conducting similar analyses in different
regions, to see whether the results are similar to those we have observed. In
addition, since the data are cheap to obtain and the analysis is
straightforward to implement, it should be possible to generate updated
analyses using data from different time periods (e.g. years), which could also
make way for further exploration of how the situation changes over time.

## Acknowledgements

This work was supported by the Engineering and Physical Sciences Research
Council [grant reference EP/T003928/1].

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## Supplementary material

##  Appendix A List of CDR features

Table 2 lists CDR features with definitions.

Variable | Level | Description
---|---|---
ratio_call_text | User | Number of calls made, divided by number of SMS sent
initiated_calls | User | Percent of calls which were outgoing
percent_initiated | User | Percent of calls and SMS which were outgoing
percent_pareto_calls | User | Percentage of contacts that account for $80\%$ of
|  | call interactions
entropy_contacts | User | Entropy of call contacts
total_sms | User | Total SMS sent or received
norm_entropy | User | Entropy of visited BTS (base transceiver stations)
unique_bts | User | Number of unique BTS at which calls/SMS sent or received
frequent_bts | User | Number of BTS that account for $80\%$ of locations where
|  | the user sent or received calls or SMS
interevent_time_sd* | User | Standard deviation of time between events (call or SMS)
active_days | User | Number of days on which the user sent or received a call
|  | or SMS
interevent_time_call_mean | User | Mean time between call initiations
mean_call_interactions | User | Mean number of call interactions with each contact
response_delay_mean | User | Mean response delay in seconds, for texts
response_delay_sd | User | Standard deviation of response delay in seconds, for texts
active_users | Tower | Number of users for whom this tower is their home tower
day_interactions_overall | Tower | Total interactions between 8am and 7pm
evening_interactions_overall | Tower | Total interactions between 7pm and 12pm
night_interactions_overall | Tower | Total interactions between 12pm and 8am
distance_overall | Tower | Mean distance to call recipient
Table 2: List of CDR features used in the analysis. The first 15 features are
calculated for individual users, and averaged to get a feature vector for each
tower. The last 5 features are calculated directly for each tower. Entropy is
calculated using Shannon entropy (Shannon,, 1948):
$H=-\sum_{l=1}^{N}P(c_{l})\log P(c_{l})$, where e.g. for entropy of contacts,
$c_{1},\ldots,c_{N}$ are the contacts of each user ($N$ is user dependent),
and $P(c_{l})$ is the proportion of a user’s interactions which take place
with contact $c_{l}$ (for $l=1,\ldots,N$). The features are calculated for
each subward (see below), each one then corresponding to a column of
$\boldsymbol{X}_{1}$.

##  Appendix B List of survey features

Tables 3 and 4 list the survey features with definitions.

How strongly do you agree with the following statement? (1-5)
---
overcrowding | Overcrowding is a problem in this subward.
litter | The level of litter in this subward is a problem.
day.safety | I would feel safe in this subward during the day.
night.safety | I would feel safe in this subward during the night.
unemployment | The level of unemployment in this subward is a problem.
unemployment.compare | The level of unemployment in this subward is high compared
| to the rest of Dar.
poverty | Poverty is a problem in this subward.
medical.care | There is a good availability to medical care in this subward.
theft.violence | Theft or violence is a problem in this subward.
well.paid | People are paid well in this subward compared to the rest of
| Dar es Salaam.
Percentage scale (0-20%, 20-40%, 40-60%, 60-80%, 80-100%)
move.work | In your opinion what percentage of people move to this subward
| for work?
teens.school | In your opinion what percentage of teenagers in this subward
| aged 13-18 are in school?
children.school | In your opinion what percentage of children in this subward
| aged 12 and under are in school?
mobile.money | What percentage of people in this subward use mobile money?
own.device | What percentage of people in this subward own a mobile device?
Table 3: List of questions in the survey (part 1). The first group of
questions are of the form “How strongly do you agree with this statement?”
with answers given on a Likert scale: possible responses are “Strongly
Disagree”, “Disagree”, “Neither Agree nor Disagree”, “Agree”, “Strongly
Agree.” The second group of questions ask for percentages of the population
where the respondent selects a range from one of 0-20%, 20-40%, 40-60%,
60-80%, 80-100%.

Variable name Question Possible responses street.lighting Is there street
lighting in this subward? No / Yes, but it’s limited and/or broken / Yes
formal.employment What is the most common type of employment in this subward?
Informal / Formal residence.length On average how long do people remain living
in this subward? Under a year / 1-5 years / 5-10 years / 10+ years
reason.travel.to What is the most common reason people travel to this subward?
Social / Mixture or Other / Work reason.travel.from What is the most common
reason people travel out from this subward? Social / Mixture or Other / Work
time.out.weekends How much of the day do residents spend outside Less than
half of the day / of this subward during the weekends? Half of the day / Most
of the day time.out.week How much of the day do residents spend outside Less
than half of the day / of this subward during week days? Half of the day /
Most of the day start.work In your opinion what age do people generally 0-11
years / 12-14 years / start paid work in this subward? 15-17 years / 18-20
years / 21+ years medical.facilities What medical facilities are available in
this Hospital / Small medical facility subward? Select all that apply. but not
hospital / Doctors are working without a building, no small medical facility
or hospital / None age.marriage What age do people tend to get married in this
subward? min.age.marriage What is the youngest age people get married in this
subward? religion What religion are most people in this subward? Christian /
Mixed or Other / Muslim education.level What level of education do most people
reach Primary / O-Level / A-Level / in this subward? University id.docs Do
most people in this subward have identification documentations (such as
passports, driving license)? Yes / No

Table 4: Survey questions (continued): questions that do not fit into either
of the first two groups (Table 3).

##  Appendix C AJIVE with survey data

Figure 23 displays the subward scores for JC1, JC2, IC1CDR, and IC1Image, when
the survey data is included in AJIVE. Figure 24 shows plots of the survey
feature loadings for the two joint components. Figure 25 shows a plot of the
CDR and survey loadings for JC1 and JC2.

Figure 23: Plots showing the subward scores for components JC1, JC2, IC1CDR
and IC1Image, when we run AJIVE also including the survey data.

JC1JC2

Figure 24: Survey feature loadings for the joint components. Figure 25:
Feature loadings for the joint components: survey features are in black, CDR
features in red.