# Regional Topics in British Grocery Retail Transactions

Mariflor<EMAIL_ADDRESS>[ Mirco
Musolesilabel=e2 [<EMAIL_ADDRESS>Jason O’Sullivanlabel=e3 [
<EMAIL_ADDRESS>Rosie Priorlabel=e4 [
<EMAIL_ADDRESS>Ioanna Manolopouloulabel=e5 [
<EMAIL_ADDRESS>Department of Computer Science, University
College London,  Dunnhumby,  Department of Statistical Sciences, University
College London,

###### Abstract

Understanding the customer behaviours behind transactional data has high
commercial value in the grocery retail industry. Customers generate millions
of transactions every day, choosing and buying products to satisfy specific
shopping needs. Product availability may vary geographically due to local
demand and local supply, thus driving the importance of analysing transactions
within their corresponding store and regional context. Topic models provide a
powerful tool in the analysis of transactional data, identifying topics that
display frequently-bought-together products and summarising transactions as
mixtures of topics. We use the Segmented Topic Model (STM) to capture customer
behaviours that are nested within stores. STM not only provides topics and
transaction summaries but also topical summaries at the store level that can
be used to identify regional topics. We summarised the posterior distribution
of STM by post-processing multiple posterior samples and selecting semantic
modes represented as recurrent topics. We use linear Gaussian process
regression to model topic prevalence across British territory while accounting
for spatial autocorrelation. We implement our methods on a dataset of
transactional data from a major UK grocery retailer and demonstrate that
shopping behaviours may vary regionally and nearby stores tend to exhibit
similar regional demand.

regional shopping behaviors,

market basket analysis,

topic models,

###### keywords:

, and

##  1 Introduction

In the grocery retail industry, millions of transactions are generated every
day by customers that choose and buy products to fulfil one or more needs.
Transactions typically contain few products out of thousands of available
items, reflecting the unseen customer motivations. For instance, customers go
to the grocery retailers to buy foods for breakfast, ingredients to cook a
roast dinner or popular products for a barbecue. Identifying customer
behaviours provides insights into high-resolution shopping patterns that may
help retailers to maximise efficiency while delivering value to all
stakeholders.

Customer motivations may be driven by geographical effects, i.e., showing
product combinations that are only relevant at specific stores. For example, a
store in Scotland may offer products from local brands and/or products that
are part of the local cuisine; these products might not have the same
popularity in other further constituent countries in the UK. In response,
retailers customise product assortments to include locally supplied products
and to fulfil local demand. Thus, we cannot generalise the prevalence of
customer behaviours over a large territory without accounting for spatial
variation.

In the UK, spatial analysis has been previously applied to grocery retail data
to study store catchment and store performance. For example, Sturley, Newing
and Heppenstall (2018) used an agent-based model to extract key consumer
behaviours about shopping frequency, shopping mission, store choice and
spending. Davies, Dolega and Arribas-Bel (2019) applied a spatial interaction
modelling (SIM) technique, to create catchment areas and investigate the
spatial variation on competition, sales area, trade intensity, among other
factors. With a SIM approach, Newing, Clarke and Clarke (2015) forecasted
store patronage and store revenues in two English regions. Waddington et al.
(2018) explored spatiotemporal fluctuations of store sales and catchment
areas. Berry et al. (2016) examined workplace geographies and census
statistics to investigate store trading characteristics in inner London.
However, none of the existing literature investigates spatial variations of
customer behaviours by modelling product combinations directly.

On the other hand, regional food consumption has been discussed in
anthropological and sociological works. For instance, Kuznesof, Tregear and
Moxey (1997) found that ‘regional foods’ are perceived as ‘regional products’
or ‘regional recipes’, which are associated with high‐value, speciality, or
hand‐crafted products and with dishes that require home preparation and
cooking. Groves (2005) defined ‘regional food’ as the food of a particular
area of the country, often representing a regional speciality. However, these
studies were not carried out using transactional data, but instead employed
market research methods such as focus groups and questionnaires.

This paper identifies regional topics and their geographical distributions
exploiting grocery transactions in a Bayesian modelling framework. Regional
topics represented as product combinations reflect customer behaviours, which
may help retailers launch marketing campaigns, customise store assortments and
layout, and may also support the prediction of thematic composition for new
stores. Besides, geographical resolution of shopping patterns may aid the
investigation of eating habits driven by social and cultural factors that
otherwise rely on expensive ad hoc studies. Thus, our overarching goal is to
identify combinations of products in high demand in specific areas and to
determine their spatial prevalence, characterising customer behaviours from
different regions and constituent countries of the UK. To this end, two main
ingredients are needed: a model for capturing customer behaviours through
transactional data, combined with a model that captures the spatial
distribution of these customer behaviours.

Customer behaviours can be modelled through topic modelling (TM) applied to
transactional data of product combinations. TM is a scalable statistical
framework that was originally introduced to analyse and summarise large
collections of text corpora. In retail analytics, TM allows describing
transactions and groups of transactions as probabilistic mixtures of topics,
which are distributions over a fixed product assortment. Different topics
exhibit different combinations of products with high probability, expressing
different customer behaviours. Topic models have been applied to model
customer behaviour over highly aggregated product assortments (Christidis,
Apostolou and Mentzas, 2010; Hruschka, 2014; Jacobs, Donkers and Fok, 2016;
Hruschka, 2016; Schröder, 2017), but were only recently applied to
transactional data at the full product resolution (Vega-Carrasco et al., 2020;
Hornsby et al., 2019).

We capture topics with geographical variability by accounting for the
dependency of transactions on store-specific product assortment, i.e.,
transactions can only contain products that are available at their associated
stores. Modelling this dependency implies a hierarchy in which stores are one
level above transactions. Without store hierarchy, regionally purchased
products would be drowned out by the sheer volume of nationally supplied
products, hampering the identification of regional topics. Thus, we apply the
segmented topic model (STM) (Du, Buntine and Jin, 2010), which enables the
identification of product combinations within the store context. STM provides
topic distributions, transaction-specific topical mixtures, and store-specific
topical mixtures. Topic distributions describe products that are frequently
bought together with high probabilities, reflecting different customer
behaviours. Topical mixtures summarise purchased products according to their
topical composition, i.e., a very popular topic in a store-specific topical
mixture would show a high probability.

We characterise the spatial distribution of regional topics using linear
Gaussian Process regression (LGPR) (Banerjee, Carlin and Gelfand, 2014;
Cressie and Wikle, 2015; Williams and Rasmussen, 2006) on store-specific topic
probabilities, modelling topical prevalence over the UK. LGPR accounts for
store meta-data and the geographical proximity between stores, both of which
STM does not account for. Specifically, we employ a spatial Gaussian process
within a linear model on regional covariates, where spatial dependence is
represented by the square exponential covariance function. LGPR allows us to
identify and characterise variation in the topic probabilities, which are
explained by spatial autocorrelation as well as regional covariates. We
demonstrate that the LGPR approach naturally achieves a better out-of-sample
predictive behaviour than a linear model by borrowing information from
neighbouring stores while affording an interpretable model with quantifiable
uncertainty.

Fully integrating STM and LGPR by assuming a Gaussian Process distribution
over store-specific topical mixtures, is feasible but computationally
prohibitive. The joint model would be similar to the correlated topic model
(Blei and Lafferty, 2006) with two layers of topical mixtures and a covariance
matrix defined over geographical distance. We do not pursue this approach as
the non-conjugacy of the Gaussian Process poses a challenge for posterior
inference. Instead, we feed an LGPR with a topical posterior summary obtained
from STM, taking advantage of the closed form Gibbs sampler of STM

Summarising the posterior distribution of STM is needed but it is not an easy
task. Topic models are often highly multi-modal, resulting in topics that may
not reappear among posterior samples (Chuang et al., 2015; Rosen-Zvi et al.,
2004). Here, we summarise posterior topic distributions by identifying
thematic modes following the clustering methodology in Vega-Carrasco et al.
(2020). This methodology fuses topic distributions from multiple posterior
samples to identify recurrent topics and their associated uncertainties.
Topics are grouped into clusters, which are represented by their average
distribution, named clustered topics, and by their cluster size, named
recurrence. Users evaluate subsets of clustered topics and select a posterior
topical summary depending on generalisation and quality metrics.

This paper is organised as follows: STM and LGPR are introduced in Sections 2
and 3. Regional topics in British grocery retail transactions are presented in
Section 4. Spatial analysis of regional topics is discussed in Section 5.
Finally, we conclude and summarise our findings in Section 6.

##  2 Topic modelling

Topic modelling was originally introduced to automatically organise,
understand, and summarise large collections of text corpora. Latent Dirichlet
Allocation (LDA) (Blei, Ng and Jordan, 2003; Blei, 2012) is one of the most
popular topic modelling techniques, which represents documents as mixtures of
topics, and topics as distributions over a fixed vocabulary. The segmented
topic model (STM) (Du, Buntine and Jin, 2010) extends LDA to include
hierarchical structure within documents, thereby STM represents documents as
collections of paragraphs (segments). Both documents and paragraphs are
represented as mixtures of topics, where a paragraph-specific topical mixture
derives from its document-specific topical mixture. LDA and STM interpret
documents as bags of words, disregarding word order.

STM has not been applied in retail analytics to the best of our knowledge, but
instead has been mainly used in text applications. For instance, STM has been
used to match experts with questions Riahi et al. (2012) and to analyse multi-
aspect sentiment in customer reviews Lu et al. (2011). We apply STM in the
context of grocery retail data, interpreting stores as documents, transactions
as segments and topics as distributions over a fixed assortment of products.
Transaction-specific topical mixtures derive from the corresponding store-
specific topical mixture. Thus, transactions and stores share the space of
latent topics. The bag of words assumption organically fits the grocery retail
domain since products are registered at stores without an inherent order.

In the standard LDA model, topics display products that are frequently
purchased together. If a product is frequently purchased in few stores (and
rarely purchased due to unavailability or low preference in the majority of
stores), then the product is unlikely to rank highly within a topic. Thus,
analysing retail data through LDA might overlook topics that reflect regional
or local customer behaviours. In contrast, STM can harness meta information of
store hierarchy over transactions. Thereby, product co-occurrence is relative
to store context and transactions taking place at the same store are expected
to exhibit more similar topical mixtures than transactions from other stores.

###  2.1 Segmented topic model

STM (Du, Buntine and Jin, 2010) consider the following hidden variables: topic
distributions, store-specific topical mixtures and transaction-specific
topical mixtures. In detail, $K$ topic distributions,
$[\phi_{1},....\phi_{K}]$, are sampled from a Dirichlet distribution governed
by hyperparameters $\boldsymbol{\beta}$; each $\phi$ is a $V$-dimensional
vector, and $V$ is the size of the product assortment. $D$ store-specific
topical mixtures, $\theta_{1},...,\theta_{D}$, are sampled from a Dirichlet
distribution governed by hyperparameters $\boldsymbol{\alpha}$; each $\theta$
is a $K-$dimensional vector. $P$ transaction-specific topical mixtures,
$\nu_{1,d},...,\nu_{P,d}$, are sampled from a Poisson-Dirichlet Process
distributed with discount parameter $a$, strength parameter $b$ and base
measure $\theta_{d}$; each $\nu$ is also a $K$-dimensional vector.

STM follows a generative process in which each transaction is created by
sampling products from topics, which are also sample from a transaction-
specific topical mixture. This generative process has two steps. First, a
topic assignment $z_{n,p,d}$ is sampled from a transaction-specific topical
mixture $\nu_{p,d}$. Second, a product $w_{n,p,d}$ is sampled from the
assigned topic distribution $\phi_{z_{n,p,d}}$, where $n$ is the $n^{th}$ item
in transaction $p$ in store $d$. Mathematically,

$\displaystyle\phi_{k}$
$\displaystyle\sim\textrm{Dirichlet}(\boldsymbol{\beta})$ (1)
$\displaystyle\theta_{d}$
$\displaystyle\sim\textrm{Dirichlet}(\boldsymbol{\alpha})$
$\displaystyle\nu_{p,d}$ $\displaystyle\sim\textrm{PDP}(a,b,\theta_{d})$
$\displaystyle z_{n,p,d}$ $\displaystyle\sim\textrm{Multinomial}(\nu_{p,d})$
$\displaystyle w_{n,p,d}$
$\displaystyle\sim\textrm{Multinomial}(\phi_{z_{n,p,d}}),$

The Poisson-Dirichlet process (PDP) (Buntine and Hutter, 2010; Ishwaran and
James, 2001; Pitman and Yor, 1997) is a generalisation of the Dirichlet
Process, also called the Pitman-Yor process. PDP is useful to handle conjugacy
between Dirichlet and Multinomial distributions.

####  2.1.1 Inference

PDP has a useful representation called the Chinese restaurant process (CRP)
(Aldous, 1985). CRP follows an intuitive analogy in which a Chinese restaurant
with infinite table capacity receive customers who choose to sit around an
occupied table or to open a new table; customers sitting around the same table
share the same dish. Interpreting the CRP in the retail context, customers are
products and dishes are customer behaviours; thus products that fulfil the
same customer need are grouped around the same topic. Note that customers and
dishes are linked through tables and a dish can be served by multiple tables.
Thus, the CRP introduces ‘table counts’, constrained latent variables
$\boldsymbol{t}$, that represent the number of tables serving the same dish.

Marginalising transaction-specific variables $\nu$ introduces the constrained
latent variables $\boldsymbol{t}$ and leaves the store-specific variables
$\theta$ in conjugate form. Integrating out topic distributions $\phi$ and
topical mixtures $\theta,\nu$, the joint conditional distribution of STM is:

$\displaystyle
p(\mathbf{z},\mathbf{w},\mathbf{t}\mid\boldsymbol{\alpha},\boldsymbol{\beta},a,b)=$
(2)
$\displaystyle\prod_{d}\frac{\textrm{Beta}_{K}(\boldsymbol{\alpha}+\sum_{p}\mathbf{t}_{p,d})}{\textrm{Beta}_{K}(\boldsymbol{\alpha})}\prod_{p,d}\frac{(b|a)_{\sum_{k}t_{p,d,k}}}{(b)_{N_{p,d}}}\prod_{p,d,k}S^{N_{k|p,d}}_{t_{p,d,k},a}\prod_{k}\frac{\textrm{Beta}_{V}(\boldsymbol{\beta}+\mathbf{N}_{k})}{\textrm{Beta}_{V}(\boldsymbol{\beta})}$

where $t_{p,d,k}$ is the table count for transaction $p$, store $d$ and topic
$k$. $\textrm{Beta}_{K}(\boldsymbol{\alpha})$ is the $K-$dimensional beta
function that normalises the Dirichlet distribution;
$\mathbf{t}_{p,d}=[t_{p,d,1},...,t_{p,d,K}]$ is a $K-$dimensional vector of
table count; $(x|y)_{N}$ denotes the Pochhammer symbol; $N_{p,d}$ size of
transaction $p$ in store $d$; $S^{N}_{M,a}$ is a generalised Stirling number;
$N_{k|p,d}$ number of topic assignments of topic $k$ in transaction $p$ in
store $d$. $\textrm{Beta}_{V}(\boldsymbol{\beta})$ is $V$ dimensional beta
function that normalises the Dirichlet distribution;
$\mathbf{N}_{k}=[N_{1|k},...,N_{v|k},...,N_{V|k}]$ is a $V-$dimensional vector
of term counts, which is the number of products of type $v$ assigned to topic
$k$. Detailed definitions of the Pochhammer symbol and generalised Stirling
number are explained in Du, Buntine and Jin (2010).

Due to the intractable computation of marginal probabilities, the posterior
distribution of latent variables cannot be computed directly. Thus, inference
of STM is solved by a Gibbs sampler. Du, Buntine and Jin (2010); Buntine and
Hutter (2010) proposed a Gibbs sampler algorithm, which samples topic
assignments and table counts. Later, (Du, Buntine and Jin, 2010; Buntine and
Hutter, 2010) proposed a more effective algorithm that jointly samples topic
assignments and ‘table indicators’ for each term. Table indicators are
constraint variables that reconstruct table counts through summation. We use
this block Gibbs sampler algorithm in our application of STM. See Appendix D
for more inference details.

The block Gibbs sampler algorithm does not explicitly sampled topics
$\boldsymbol{\phi}$, stores-specific topical mixtures $\boldsymbol{\theta}$ or
transaction-specific topical mixtures $\boldsymbol{\nu}$. Instead, hidden
variables are approximated using a posterior sample $s$ of topic assignments
and table counts. Then, hidden variables are approximated by their conditional
posterior means:

$\displaystyle\widehat{\theta}^{s}_{d,k}$
$\displaystyle=E(\theta^{s}_{d,k}\mid\textbf{t}^{s},\boldsymbol{\alpha})=\frac{\alpha_{k}+\sum_{p}t^{s}_{p,d,k}}{\alpha+\sum_{p,k}t^{s}_{p,d,k}},$
(3) $\displaystyle\widehat{\nu}^{s}_{p,d,k}$
$\displaystyle=E(\nu^{s}_{p,d,k}\mid\textbf{z}^{s},\textbf{t}^{s},a,b)=\frac{N^{s}_{p,d,k}-a\times
t^{s}_{p,d,k}}{b+N^{s}_{p,d}}+\theta_{d,k}\frac{\sum_{k}t^{s}_{p,d,k}\times
a+b}{b+N^{s}_{p,d}},$ (4) $\displaystyle\widehat{\phi}^{s}_{k,v}$
$\displaystyle=E(\phi^{s}_{k,v}\mid\textbf{z}^{s},\boldsymbol{\beta})=\frac{\beta_{v}+N^{s}_{k,v}}{\beta+N^{s}_{k}},$
(5)

where $\alpha=\sum_{k}^{K}\alpha_{k}$ and where $\beta=\sum_{v}^{V}\beta_{v}$.

###  2.2 Summarising topic distributions

Summarising the posterior distribution of a topic model is challenging because
the posterior distribution is often highly multi-modal; resulting in posterior
samples that capture different semantic modes. Thus, component-wise posterior
averaging may merge topic distributions that respond to different semantic
concepts. In addition, topics that capture different semantic modes may appear
and disappear across posterior samples of MCMC chains (Chuang et al., 2015;
Rosen-Zvi et al., 2004).

In response, we follow the methodology described in (Vega-Carrasco et al.,
2020) to construct a summary of topical modes using multiple posterior samples
from various MCMC chains. The methodology clusters topics using a bottom-up
hierarchical clustering method. At each step, the algorithm finds the pair of
clusters with the lowest cosine distance and merge clusters if their topic
distributions come from different samples. The algorithm keeps merging
clusters up to a cosine distance threshold. Each resulting cluster is
represented by the average topic distribution, named clustered topic, and by
the number of topics gathered in the same cluster, named cluster size. Cluster
size is a measure of recurrence and denotes (un)certainty, i.e., a topic that
has occurred in every posterior sample is highly recurrent, showing no
uncertainty.

###  2.3 Evaluation of clustered topics

Depending on the cosine distance threshold, the clustering algorithm produces
a set of clustered topics, which can be selected according to their
recurrence. Thus, multiple subsets of clustered topics can be formed by
varying cosine distance threshold and recurrence (setting a minimum cluster
size).

We evaluate each subset of clustered topics on 4 aspects: generalisation or
predictive power of a subset of topics, coherence of individual topics, the
distinctiveness of a topic with respect to the other topics in the same
posterior sample, and credibility of a topic with respect to the topics from
other posterior samples.

Topic coherence, distinctiveness and credibility are measured as described in
(Vega-Carrasco et al., 2020). Model generalisation, however, is measured by
the perplexity of unseen transactions given topics, store-specific topical
mixtures and PDP parameters:

$\textrm{Perplexity}=-\frac{\log
P(\mathbf{w}^{\prime}_{d}\mid\Phi,\theta_{d},a,b)}{N^{\prime}},$ (6)

where $\mathbf{w}^{\prime}_{d}$ is a set of products in a held-out transaction
at store $d$, $N^{\prime}$ is the number of products in
$\mathbf{w}^{\prime}_{d}$, $\Phi=[\phi_{1},\phi_{2},\ldots,\phi_{K}]$ the set
of inferred topics, $\theta_{d}$ is the store-specific topical mixtures
associated to store $d$, $a$ and $b$ are the PDP parameters.

We aim to select a subset of clustered topics that shows low perplexity,
gathering topics that are coherent, distinctive and credible (low
uncertainty).

##  3 Linear Gaussian process regression

According to Tobler’s first law of geography (Tobler, 1970): ‘everything is
related to everything else, but near things are more related than distant
things’. Thus, we expect that nearby stores show similar shopping patterns and
that some specific patterns may be limited to particular geographical areas.
STM does not take into account store location or proximity between stores.
Although a topic model that simultaneously accommodate store hierarchy over
transactions and store location would be mathematically possible, it would be
computationally prohibitive at the level of resolution of interest. Instead,
we use the summarised posterior distributions of topics obtained from STM and
take a spatial modelling approach to capture their geographical structure and
regional behaviour.

###  3.1 Model

A linear regression with a spatial process is defined as:

$\textbf{Y}=\textbf{X}\boldsymbol{\beta}+\boldsymbol{\eta}+\boldsymbol{\varepsilon},$
(7)

where Y is the dependent variable, X is the matrix of $p$ covariates
associated with locations $\textbf{s}_{1},...,\textbf{s}_{n}$,
$\boldsymbol{\beta}$ is a $p$-dimensional fixed effect, $\boldsymbol{\eta}$ is
a spatial process, which captures spatial residual, and
$\boldsymbol{\epsilon}$ is an independent process, which models pure error,
also known as the nugget effect.

The spatial process $\eta(\textbf{s}_{1}),...,\eta(\textbf{s}_{n})$ is
distributed as a zero-mean Gaussian process $GP(0,C_{\boldsymbol{\eta}})$ with
positive definitive covariance matrix $C_{\boldsymbol{\eta}}$. Residuals
$\varepsilon(\textbf{s}_{1}),...,\varepsilon(\textbf{s}_{n})$ are assumed
$iid$ with $\epsilon(\textbf{s}_{i})\sim N(0,\sigma^{2})$. Thus, observations
are distributed as:

$\textbf{Y}\sim N(\textbf{X}\boldsymbol{\beta},\Sigma),$ (8)

where $\Sigma=C_{\boldsymbol{\eta}}+\sigma^{2}I$.

Here, we use the positive definitive square exponential covariance function,

$C_{\boldsymbol{\eta}}(\textbf{s}_{i},\textbf{s}_{j}|\alpha,\rho)=\alpha^{2}\exp\Big{(}-\frac{\textrm{dist}(\textbf{s}_{i},\textbf{s}_{j})^{2}}{2\rho^{2}}\Big{)},$
(9)

where parameters $\alpha$ and $\rho$ control the amplitude and length-scale of
the spatial dependence, respectively.
$\textrm{dist}(\textbf{s}_{i},\textbf{s}_{j})$ is a measure of distance
between locations.

###  3.2 Methods

Linear Gaussian process regression specified in equation 7 is fitted using
Stan (Carpenter et al., 2017). Stan is a state-of-the-art platform for
statistical modelling and high-performance statistical computation. Stan
facilitates Bayesian inference by gradient-based sampling techniques such as
Hamiltonian Monte Carlo methods (Betancourt, 2017) and variational inference
(Blei, Kucukelbir and McAuliffe, 2017). In our study, the inference is
computed by the default Stan algorithm No-U-Turn Sampler (NUTS) (Hoffman and
Gelman, 2014). NUTS is an extension of the Hamiltonian Monte Carlo (HMC)
algorithm that effectively explores the parameter space by avoiding retaking
previously sampling paths in a U-turn style.

###  3.3 Predictions

Predicted topic probabilities
$\textbf{Y}^{\star}=[Y^{\star}(\textbf{s}_{1}),...,Y^{\star}(\textbf{s}_{n})]$
at new locations $\textbf{s}^{\star}_{1},...,\textbf{s}^{\star}_{n}$ are
distributed as:

$\textbf{Y}^{\star}|\textbf{Y},\boldsymbol{\beta},\Theta,\textbf{X}^{\star},\textbf{X}\sim
N(\textbf{X}^{\star}\boldsymbol{\beta}+\Sigma_{21}\Sigma_{11}^{-1}(\textbf{Y}-\textbf{X}\boldsymbol{\beta}),\Sigma_{22}-\Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12}),$
(10)

where $\textbf{X}^{\star}$ is the matrix of $p$ covariates at the new
locations. Here, $\Sigma_{11}$ is the covariance matrix of
$\textbf{s}_{1},...,\textbf{s}_{n}$ locations, $\Sigma_{12}=\Sigma_{21}$ the
covariance matrix between $\textbf{s}_{1},...,\textbf{s}_{n}$ and
$\textbf{s}^{\star}_{1},...,\textbf{s}^{\star}_{n}$, and $\Sigma_{22}$,
covariance matrix of $\textbf{s}^{\star}_{1},...,\textbf{s}^{\star}_{n}$.

Note that expected topic probabilities $E(\textbf{Y}^{\star})$ are computed by
two quantities. The first quantity is obtained by multiplying the covariate
matrix by the fixed effects as in multiple linear regression. The second
quantity pulls the expected value at a new store towards the values of the
nearby stores if spatial dependence is significant.

##  4 Identifying regional grocery topics

We analyse grocery transactions from a major retailer in the UK. Transactions
are sampled randomly, covering 100 nationwide superstores between September
2017 and August 2018. Transactions with less than 3 products are filtered out.
The training data set contains 36,000 transactions and a total of 392,840
products and the test data set contains 3,600 transactions and a total of
38,621 products. Transactions contain 10 products on average. The product
assortment contains 10,000 products, which are the most monthly frequent,
ensuring the selection of seasonal and non-seasonal products. We count unique
products in transactions, disregarding the quantities of repetitive products.
For instance, 5 loose bananas count as 1 product (loose banana). We do not use
an equivalent of stop words list (highly frequent terms), as we consider that
every product or combination of them tell different customer needs. We
disregard transactions with fewer than 3 products assuming that smaller
transactions do not have enough products to exhibit a regional topic. No
personal customer data were used for this research.

###  4.1 STM posterior summary

We explore STM with 100 topics to capture as many topics as possible without
making inference too computationally prohibited. As shown in (Vega-Carrasco et
al., 2020), a topic model with 100 topics identifies a variety of customer
behaviours in the domain of our application. Exploring STM with a smaller or
larger number of topic is out the scope of this paper.

We use symmetric priors with hyperparameters $\alpha_{k}=1000/K$ and
$\beta_{v}=0.01$, and PDP hyperparameters $b=3.0$ and $a=0.5$. We run four
MCMC chains for 100,000 iterations with a burn-in of 80,000 iterations,
samples were recorded every 5,000 iterations, obtaining 20 thinned posterior
samples (five samples for each chain). MCMC trace plots are presented in
Appendix A, where the convergence is satisfactory.

Posterior topic distributions are summarised by clustering a bag of 2,000
topics obtained from the aforementioned 20 posterior samples. As shown in
Appendix C, we observe that the subset formed with a minimum cluster size 10
(which represent 50% of the samples) and a cosine distance threshold $\geq
0.35$ show greater coherence, credibility and generalization, concurring with
Vega-Carrasco et al. (2020). Based on these results, we choose this subset
which contains 104 clustered topics.

Store-specific topical mixtures are then obtained by training STM with the
identified 104 clustered topic distributions. The inference process goes as
described in Section 2.1.1, but only Equations 3 and 4 are updated. A MCMC
chain runs with a burn-in period of 1,000 iterations, recording posterior
samples with a thin of 500 iterations. The MCMC trace plot in Appendix B shows
satisfactory convergence. We collect 30 posterior samples which are then
averaged to estimate store-specific topical mixtures for 500 stores across the
UK.

###  4.2 Interpreting topic distributions

We interpret six out of the 104 clustered topics as they capture a clear,
interpretable regional pattern. The remaining topics show ubiquitous
distributions over the UK. We interpret topics by analysing the product
descriptions of the 15 products with the largest probabilities. Topics are
manually named after the regional pattern or customer preference reflected on
the product descriptions. Note that the illustrated topics appeared
consistently across the 20 posterior samples (size = 20), indicating low
posterior uncertainty.

Product descriptions in Figures 1(a), 1(b) and 1(c) suggest foods supplied
locally and local brands associated to Scotland, Northern Ireland and Wales.
For instance, the Scottish topic includes ‘Scottish-branded skinless sausages’
and ‘Scottish-branded potato scones’, the Northern Irish topic shows the
‘North Ireland semi-skimmed milk’, ‘white potatoes packed in North Ireland’,
and the Welsh topic contains ‘Welsh jacket potatoes’ and ‘Welsh-branded
bread’. Hence, we name the Scottish topic, Northern Irish topic and Welsh
topic after the nationality that their product descriptions suggest.

Figures 1(d) and 1(e) show a variety of products such as types of milk, types
of bread, fruits and vegetables, etc. Close inspection of product descriptions
such as ‘oven bottom muffin’, ‘fruit teacake’, and ‘potato and meat pie’ in
Figure 1(d), and ‘pork pies’ and ‘scotch eggs’ in Figure 1(e) may reveal a
regional topic when regional expertise is available. Since these product
descriptions do not provide interpretations that can be directly associated
with specific regions, we momentarily name these topics ‘Mixed basket I’ and
‘Mixed basket II’. Figures 1(f) shows ‘organic’ quality foods, indicating a
specific customer preference, however, the topic does not suggest any specific
regional pattern.

Interpreting topic descriptions is not sufficient to identify geographically
driven shopping motivations, reinforcing the need for exploring store-specific
topical mixtures.

(a) Scottish

(b) Northern Irish

(c) Welsh

(d) Mixed basket I

(e) Mixed basket II

(f) Organic

Figure 1: Most probable products in grocery regional topics. Each topic is
interpreted using the 15 products with the largest probabilities.
Probabilities and products are sorted in descending order. General brand names
have been replaced by XXX. Local brands in North Ireland, Scotland, Wales and
North of England have been replaced by NI-XXX, SC-XXX, WE-XXX, NE-XXX. NPMI
and size are measures of topic coherence and recurrence.

(a) Scottish

(b) Northern Irish

(c) Welsh

(d) North and Centre

(e) South and Midlands

(f) Organic

Figure 2: Topic probabilities of clustered grocery topics in the UK. Purple
and yellow points reflect the largest and smallest topic probabilities,
respectively.

###  4.3 Mapping topical mixtures

Interpreting product descriptions may reveal the existence of regional topics,
i.e., the Northern Irish/ Scottish/ Welsh topic. Topic interpretations may
dismiss regional topics that exhibit products that are not directly linked
with specific areas. Thus, we visualise the store-specific topic probabilities
at the store’s location, aiming to find topics with a regional pattern. We
link store postcodes with location coordinates through querying stores’
postcodes in the lookup table from the Office for National Statistics (Office
for National Statistics, ). Figure 2 shows the topic probabilities of the six
clustered topics mapped across the UK.

Figures 2(a), 2(b), 2(c) clearly confirm that the Scottish, Northern Irish and
Welsh topics are more likely in their respective constituent countries. More
interestingly, Figure 2(c) shows the prevalence of the Welsh topic over
neighbouring regions. Figure 2(d) shows high topic probabilities concentrated
in the North West and surrounding regions, and Figure 2(e) shows high topic
probabilities in the central and southern English regions. We rename both
topics as North and Centre and South and Midlands due to their cross-regional
prevalence. Figure 2(f), which maps the Organic topic, shows significant
probabilities concentrated in London.

In comparison to the Scottish, Northern Irish and Welsh topics,
interpretations of the most probable topics in the North and Centre, South and
Midlands and Organic topics do not easily suggest a geographical pattern.
Mapping the store-specific topic probabilities aids the analysis and
identification of topics with spatial patterns.

###  4.4 STM vs LDA

STM shows two advantages over LDA. Firstly, STM provides topical summaries for
stores, by including the store hierarchy above transactions. Secondly, and
less obvious, STM discovers topics that are relevant within their store
context. In comparison, LDA finds products that are frequently bought together
across all transactions. Thus, a product combination that is only frequent in
few stores may not be shown among LDA topics. The ability to capture store-
specific topics is key to our subsequent spatial modelling analysis.

We compare the 104 STM clustered topics (HC-STM-100) against the posterior
summaries of the LDA model with 100 and 200 topics. The posterior summaries of
LDA were obtained using the same training data and following the clustering
methodology in Vega-Carrasco et al. (2020). The posterior summary of LDA with
100 topics (HC-LDA-100) gathers 96 clustered topics and the posterior summary
of LDA with 200 topics (HC-LDA-200) gathers 198 clustered topics.

(a) HC-STM-100 vs HC-LDA-100

(b) HC-STM-100 vs HC-LDA-200

Figure 3: Cosine similarity between clustered topics obtained from posterior
summaries of STM with 100 topics and LDA with 100 and 200 topics. Topics have
been aligned following a greedy algorithm that at each step searches and pairs
topics (that have not been paired) with the highest cosine similarity.

(a) HC-STM-100 vs HC-LDA-100

(b) HC-STM-100 vs HC-LDA-200

Figure 4: Distributions of the maximum cosine distance obtained from each
cosine similarity matrix in Figure 4. Figure 4(a) plots maximum cosine
distances between clustered STM topics (HC-STM-100) against the posterior
summary of LDA with 100 topics (HC-LDA-100) (left); and from HC-LDA-100 to HC-
STM-100 (right). Figure 4(b) plots maximum cosine distances between HC-STM-100
against the posterior summary of LDA with 200 topics (HC-LDA-200) (left); and
from HC-LDA-200 to HC-STM-100 (right).

Figure 3(a) shows the cosine similarity between (HC-STM-100) 104 clustered
topics and (HC-LDA-100) 96 clustered topics. Clustered topics are ordered to
visualise high similarities in the diagonal. As observed, the majority of
clustered topics are identified in both models, STM and LDA, with high cosine
similarity $>0.7$. Figure 4(a) shows that 70% of the (HC-STM-100) clustered
topics are found among HC-LDA-100 clustered topics, and 85% of the HC-LDA-100
clustered topics are found among the HC-STM-100 clustered topics with high
similarity. For instance, the Northern Irish topic is found in both models
with high cosine similarity (0.97). As depicted in Figure 5(a), Northern
Ireland related products rank in the top 15 products in both topics. The
Organic topic was also found among HC-LDA-100 clustered topics with high
cosine similarity (0.95).

(a) The Northern Irish topic in STM and LDA.

(b) The Welsh topic in STM and its most similar topic in LDA

Figure 5: Comparison of topics identified in STM and LDA posterior samples.
Highlighted products appear in both topics. While the Northern Irish topic is
clearly identified by both models, the Welsh topic is only found by the STM
model.

We also compare the 104 STM clustered topics against the 198 LDA clustered
topics obtained from summarising LDA posterior samples of 200 topics. This
comparison allows identifying the regional topics that were not caught in LDA
samples with 100 topics. For instance, the Scottish topic described in Figure
1(a), is not found in the HC-LDA-100 subset, but it is found in the HC-LDA-200
subset with a cosine similarity of 0.83. As observed in Figure 3(b), the
majority of the 104 clustered topics are found among the (HC-LDA-200) 198 LDA
clustered topics with high cosine similarity ($>0.7$). However, Figure 4(b)
shows that there are still some STM clustered topics that do not match with
any of the LDA clustered topic with high similarity. For example, the Welsh
topic described in Figure 1(c), is not found in either of the two subsets of
LDA clustered topics. The Welsh topic and the closest clustered topic in HC-
LDA-200 (with $0.67$ cosine similarity) are listed in Figure 5(b); as
observed, few products are shared by the topics but Welsh products are not
described in both topics. The North and Centre topic and the South and
Midlands topic were not found among the HC-LDA-200 clustered topics either.
Perhaps, these regional topics would appear among posterior samples of a
larger LDA model, i.e., LDA with 300 topics; however, increasing the model
complexity is not only more computationally expensive but also less efficient
since topics tend to show less distinctiveness (Vega-Carrasco et al., 2020).

In summary, three out of the six regional topics are identified by STM and the
other three regional topics are identified by both STM and LDA models. One of
these topics was captured by a larger LDA model. STM shows its strength over
LDA by identifying more regional topics.

##  5 Modeling regional prevelance

We aim to model topic probabilities across stores in the UK by constructing a
linear model with fixed effects associated with the constituent countries of
the UK (Wales, Scotland, Northern Ireland) and the nine English regions, and
imposing spatial dependency though a Gaussian process that captures residual
spatial association as defined in Equation 7. In this manner, we can quantify
the significance of a topic to a region or constituent country. This
administrative division was chosen assuming that each country and region would
broadly show differences in customer behaviour. Analysis over other
subdivisions is possible, but it is out of the scope of this paper.

The dependent variable $\textbf{Y}_{k}$ is the logit transformation of the
store-specific $k^{th}$ topic probabilities
$[\widehat{\theta}_{\textbf{s}_{1},k},\widehat{\theta}_{\textbf{s}_{2},k},...,\widehat{\theta}_{\textbf{s}_{n},k}]$,
given by:

$\textbf{Y}_{k}=\textrm{logit}([\widehat{\theta}_{\textbf{s}_{1},k},\widehat{\theta}_{\textbf{s}_{2},k},...,\widehat{\theta}_{\textbf{s}_{n},k}]),$
(11)

where each $\widehat{\theta}_{\textbf{s}_{i},k}$ is the average probability
over 30 posterior samples of the $k^{th}$ topic at store location
$\textbf{s}_{i}$ from Section 4.1. For simplicity, we assume independence
among topic probabilities and model each topic separately, i.e., for each
topic, a linear model is constructed. However, topic probabilities of a
topical mixture are not independent of each other since they need to sum to 1.

The logit transformation not only avoids predicting nonsensical values (i.e.,
topic probabilities $>1$ or $<0$), but also aids the visualisation of topic
probabilities that cannot be appreciated in the original scale. For instance,
Figure 6 (left panel) highlights stores in the South West that are not noticed
in Figure 2(c).

The covariate matrix X is defined by dummy variables responding to the
constituent countries: ‘North Ireland’, ‘Scotland’, ‘Wales’; and the English
regions: ‘North East’, ‘North West’, ‘Yorkshire and the Humber’, ‘East
Midlands’, ‘West Midlands’, ‘South West’, ‘South East’, and ‘East Anglia’,
where ‘London’ is the reference category.

The spatial distance $\textrm{dist}(\textbf{s}_{i},\textbf{s}_{j})$, which
define covariance between stores
$C_{\boldsymbol{\eta}}(\textbf{s}_{i},\textbf{s}_{j})$, is calculated by
firstly finding the latitude-longitude coordinates associated with the store’s
postcode, secondly computing the distance between pair of coordinates using
the Haversine formula (Robusto, 1957). The Haversine formula provides accurate
approximations of distance for locations over large areas. Postcode
coordinates are queried from the postcode lookup table from the Office for
National Statistics (Office for National Statistics, ). Spatial distance is
measured in kilometres.

We complement the Bayesian hierarchical model with weakly informative priors:
$\sigma^{2}\sim\textrm{half}N(0,1)$, $\beta\sim N(0,10)$; $\alpha\sim N(0,2)$,
and $\rho\sim IG(2,50)$.

Parameters of the linear Gaussian process regression are estimated with Stan,
using 2 MCMC chains which run for 2,000 iterations, 1,000 burn-in iterations,
and a thin of five iterations. Convergence of MCMC chains is satisfactory with
scale factor reduction $\widehat{R}=0.998$.

###  5.1 Prevalence of regional behaviours in the UK

Table 1 shows posterior summaries of the linear Gaussian process regression.
The intercept can be interpreted as how likely (in logit scale) a topic is at
a store in London and vice versa. Positive average coefficients indicate that
the topic is more likely than in London. Average coefficients that are
highlighted in red correspond to non-zero 95% credible intervals with $0>$
upper bound, and bold average coefficients correspond to non-zero 95% credible
intervals with $0<$ lower bound.

Unsurprisingly, the Scottish, Northern Irish and Welsh topics show positive
average coefficients with non-zero credibility intervals for the respective
constituent countries. This indicates that their topic probability largely
increases for stores in Scotland, North Ireland and Wales, respectively.

Table 1: Regression parameters for regional topics Red/bold mean estimates for coefficients with non-zero credibility intervals that decrease/increase the topic probability, respectively. | Northern | Scottish | Welsh | North | South | Organic
---|---|---|---|---|---|---
| Irish |  |  | and Centre | and Midlands |
Parameter | Avg. | SE | Avg. | SE | Avg. | SE | Avg. | SE | Avg. | SE | Avg. | SE
Intercept | -10.4 | 0.02 | -9.52 | 0.03 | -8.9 | 0.04 | -6.34 | 0.04 | -4.42 | 0.02 | -4.62 | 0.05
Northern Ireland | 8.67 | 0.03 | -0.72 | 0.04 | -1.44 | 0.07 | -4.11 | 0.05 | -5.77 | 0.03 | -1.25 | 0.06
Scotland | 0.19 | 0.02 | 6.84 | 0.04 | -1.12 | 0.05 | -1.93 | 0.04 | -1.82 | 0.03 | -1.34 | 0.06
Wales | -0.4 | 0.03 | -0.57 | 0.03 | 5.63 | 0.07 | 0.39 | 0.04 | -2.27 | 0.03 | -1.27 | 0.06
North West | 0.15 | 0.03 | 1.54 | 0.04 | 0.1 | 0.08 | 3.3 | 0.05 | -0.99 | 0.03 | -1.91 | 0.06
North East | -0.86 | 0.04 | 3.27 | 0.05 | -0.33 | 0.08 | 3.05 | 0.06 | -1.25 | 0.04 | -2.5 | 0.07
Yorkshire | 0.04 | 0.03 | 1.08 | 0.04 | -0.43 | 0.05 | 2.98 | 0.06 | -0.43 | 0.03 | -1.68 | 0.05
West Midlands | -0.15 | 0.02 | -0.24 | 0.03 | 1.89 | 0.07 | 1.95 | 0.05 | 0.26 | 0.03 | -1.01 | 0.05
East Midlands | -0.47 | 0.03 | 0.68 | 0.04 | 0.67 | 0.05 | 1.45 | 0.05 | 0.31 | 0.06 | -1.47 | 0.05
East Anglia | -0.27 | 0.02 | -0.28 | 0.03 | -0.38 | 0.05 | -0.31 | 0.04 | 0.99 | 0.02 | -1.03 | 0.05
South East | -0.21 | 0.2 | 0.56 | 0.03 | -0.25 | 0.04 | -1.07 | 0.04 | 0.66 | 0.02 | -0.51 | 0.05
South West | -0.26 | 0.2 | -0.1 | 0.03 | 1.26 | 0.05 | -0.64 | 0.04 | 0.71 | 0.03 | -0.02 | 0.05
Length-scale $\rho$ | 63.85 | 5.95 | 92.07 | 19.95 | 55.31 | 1.32 | 51.32 | 15.53 | 50.23 | 3.84 | 34.67 | 3.13
Amplitude $\alpha$ | 0.13 | 0.01 | 0.3 | 0.03 | 1.04 | 0.01 | 0.74 | 0.02 | 0.23 | 0.01 | 0.86 | 0.02
$\sigma$ | 0.78 | 0.01 | 1.38 | 0.01 | 1.43 | 0.01 | 1.37 | 0.01 | 1.15 | 0.01 | 1.58 | 0.01

Interestingly, Wales’s and Scotland’s neighbouring regions show positive
average coefficients with non-zero credibility intervals, i.e., North East and
North West to the Scottish topic and West Midlands and South West to the Welsh
topic. As shown in Figure 6 (central panel), probability estimates (in logit
scale) of the Welsh topic for stores in West Midlands and South West are
greater than the probability estimates of the Welsh topic at stores in further
regions. Moreover, the Gaussian process captures spatial residual
distinguishing the stores in the neighbouring regions that are close to Wales
from the stores (in the same regions) that are at further distances, as
demonstrated in Figure 6 (right panel).

Figure 6: Welsh topic: (left panel) observed topic probabilities in logit
scale; (central panel) probability estimates (in logit scale) using only fixed
effects; (right panel) spatial residuals captured by the Gaussian process.

The coefficients for the North and Centre topic clearly show that the topic is
more likely in the North West, North East, Yorkshire and West Midlands and is
less likely in Northern Ireland and Scotland. On the other hand, the
coefficients for the South and Midlands show that on average the topic is more
likely in the southern and central English regions; however, only the
coefficient of East England has a non-zero 95% credibility interval.

The Organic topic shows a different pattern, its average coefficients are
negative; this indicates that the probability of the Organic topic is on
average lower than the average topic probability in London. In other words,
the Organic topic is more likely in London than in any other region or
constituent country; however, the coefficients show 95% credible intervals
containing zero, suggesting that the regional effect may not be significant.

The covariance parameters length-scale $\rho$ and amplitude $\alpha$ model the
covariance between stores, which is stronger when the spatial distance is
smaller than $\rho$ and when $\alpha$ is significantly larger from zero. The
Welsh and the Organic topic show strong covariance as depicted in Figure 7. On
the other hand, the Northern Irish topic and the Scottish topic show small
values of $\alpha$ indicating weak covariance functions.

(a) Welsh topic

(b) Organic topic

Figure 7: Covariance function of the Welsh topic and Organic topic. Lines are
computed with posterior samples of $\alpha$ and $\rho$.

###  5.2 Linear Gaussian process regression vs Linear regression

Here, we compare mean squared error and the log of the probability density on
held-out data obtained from model topic prevalence using linear Gaussian
process regression (LGPR) and the linear regression (LR). We will show that
the former model retrieves more accurate estimates and better predictive
likelihood by modelling residual spatial effect.

Table 2 shows that LGPR improves the prediction of topic probabilities of the
Welsh, English-Northern and Centre, South and Midlands and Organic topics. The
difference between the mean squared error of these topics is statistically
significant at the 0.05 level, indicating that the Gaussian process provides
significant model improvement. Similarly, the log predictive likelihood of the
four aforementioned topics is significantly better at the 0.05 level. On the
contrary, the LGPR doesn’t show significantly improved predictions of the
Scottish and Northern Irish topics. The difference of their mean squared
errors is not statistically significant at the 0.05 level; however, the LGPR
shows significantly better predictive log-likelihood at the 0.05 level.

Table 2: Comparison of the linear Gaussian process regression (LGPR) vs linear regression (LR). lppd: log posterior predictive density on test data. p-values are computed for the pointwise difference of the two methods at each observation in the test set. | Northern | Scottish | Welsh | English-North | English-South | Organic
---|---|---|---|---|---|---
| Irish |  |  | and Centre | and Midlands |
LR: MSE (SE) | 0.64 (0.001) | 2.16 (0.004) | 3.18 (0.006) | 3.36 (0.007) | 1.65 (0.004) | 3.39 (0.007)
LGPR: MSE (SE) | 0.63 (0.001) | 2.15 (0.004) | 2.64 (0.005) | 3.24 (0.004) | 1.62 (0.003) | 3.18 (0.006)
p-value | 0.5877 | 0.1664 | 0.0000 | 0.0000 | 0.0000 | 0.0000
LR lppd (SE) | -298.3 (0.30) | -450.2 (0.25) | -499.1 (0.23) | -513.5 (0.31) | -418.8 (0.38) | -506 (0.26)
LGPR lppd (SE) | -296.5 (0.28) | -449.3 (0.26) | -476.9 (0.26) | -504.9 (0.43) | -412.6 (0.40) | -493.9 (0.27)
p-value | 0.0000 | 0.0169 | 0.0000 | 0.0000 | 0.0000 | 0.0000

Examining LGPR residuals in Figure 8, we still observe spatial patterns that
are not captured by the Gaussian process. For example, concentrations of
underestimated probabilities around North West in Figure 8(a), around the
centre of Scotland in Figure 8(b), around South West and East Anglia in Figure
8(c); and overestimated probabilities around South East in Figure 8(b).
Further work could explore the Gaussian process with non-stationary covariance
to capture local spatial patterns.

(a) Welsh topic

(b) North and Centre topic

(c) Organic topic

Figure 8: Residuals of modelling the Northern Irish/ North and Centre/ Organic
topic with LGPR.

##  6 Conclusions

In this paper, we showed that STM is powerful in the analysis of transaction
retail data, identifying topics that characterise various customer needs,
particularly, those that reflect regional demand. STM harnesses store
structure, describing transactions and stores as topical mixtures. More
importantly, STM can identify regional topics that otherwise would be overseen
by the widely used topic model, the LDA. Aggregating multiple samples of the
posterior distribution and selecting topic modes allow the identification of
certain and meaningful topics, achieving better data representations and
capturing posterior variability. Topic analysis, through LGPR, quantifies
regional effects and captures spatial dependence through the squared
exponential covariance function. Further work could explore the analysis of
spatial topics using other geographical hierarchies such as middle layer super
output areas (MSOAs) and non-stationary models such as the non-stationary
Gaussian process in Heinonen et al. (2016), which may capture local spatial
residuals.

##  Appendix A MCMC convergence of STM with 100 topics

We evaluate four Markov chains of STM with 100 topics. Markov chains are run
for 100,000 iterations with a burn-in period of 80,000 iterations. Log-
likelihood is measured at every 10 iterations. We calculate the potential
scale reduction factor using 8,000 samples.

Figure 9: Markov Chains of STM with 100. Potential scale reduction factor
$\hat{R}:1.07$.

##  Appendix B MCMC convergence of Clustered STM topics

We run STM with 104 clustered topics known a priori for 1,500 iterations and
burn-in period of 1,000 iterations.

Figure 10: Markov Chain of STM with 104 clustered topics known a priori.
Potential scale reduction factor $\hat{R}:0.998$.

##  Appendix C Clustering of STM topics.

(a) Generalisation

(b) Coherence

(c) Distinctiveness

(d) Credibility

Figure 11: Evaluation of subsets of clustered topics. Subsets are formed with
combinations of minimum cluster size and cosine distance thresholds.
Horizontal lines and dotted lines show the average measures (± one standard
error) of the STM posterior samples. Subsets of clusters formed with a minimum
cluster size of 10 show greater coherence and credibility, and the subsets
formed with a cosine distance threshold larger than 0.3 show better
generalisation (in comparison to the average generalisation of the STM
posterior samples). Subsets with a minimum cluster size of 10 show less
distinctive clustered topics, which might result from filtering out
distinctive but uncertain topics. A cosine distance threshold larger than 0.35
cosine distance does not significantly improve perplexity.

##  Appendix D Block Gibbs sampler

Chen, Du and Buntine (2011) proposes a block Gibbs sampling algorithm that
jointly samples topic assignments and table indicators, leading to a more
efficient sampling method. Table counts are not sampled, instead reconstructed
by summation of the table indicators.

$\displaystyle t_{k}=\sum^{N}_{n=1}u_{n}1_{z_{n}=k},$ (12)

Using the table indicator representation, the PDP posterior distribution is:

$\displaystyle p(z,t\mid a,b,\theta)=\prod_{k}\frac{n_{k}!}{t!(n-t)!}p(z,u\mid
a,b,\theta),$ (13)

responding to $\frac{n_{k}!}{t!(n-t)!}$ sitting arrangements.

The joint distribution of topic assignments and table indicators can be
obtained by using Equation 13 in Equation 2 resulting in:

$\displaystyle
p(\mathbf{z},\mathbf{w},\mathbf{t}\mid\boldsymbol{\alpha},\boldsymbol{\beta},a,b)=$
(14)
$\displaystyle\prod_{d}\frac{\textrm{Beta}_{K}(\boldsymbol{\alpha}+\sum_{p}\mathbf{t}_{p,d})}{\textrm{Beta}_{K}(\boldsymbol{\alpha})}\prod_{p,d}\frac{(b|a)_{\sum_{k}t_{p,d,k}}}{(b)_{N_{p,d}}}\prod_{p,d,k}S^{N_{k|p,d}}_{t_{p,d,k},a}\frac{t_{p,d,k}!(N_{p,d,k}-t_{p,d,k})!}{n_{p,d,k}!}\prod_{k}\frac{\textrm{Beta}_{V}(\boldsymbol{\beta}+\mathbf{N}_{k})}{\textrm{Beta}_{V}(\boldsymbol{\beta})}$

The block Gibbs sampling algorithm firstly samples a table indicator
$u_{z_{n}}=1$ or $u_{z_{n}}=0$ with probabilities:

$\displaystyle p(u_{z_{n}}=1\mid z_{n}=k)=\frac{t_{k}}{n_{k}}\quad
p(u_{z_{n}}=0|z_{n}=k)=1-\frac{t_{k}}{n_{k}},$ (15)

and discounts the current assignment $z_{n}$ from $N_{p,d,k}$ and reduces
$t_{p,d,k}$ by 1 if $u_{z_{n}}=1$.

Then, the full conditional distribution is computed taking into account two
scenarios: the probability of opening a new table (Equation 16) and the
probability of choosing an occupied table (Equation 17) if
$t_{p,d,k}^{\prime}>0$.

$\displaystyle
p(z_{n}=k,u_{n}=1\mid\mathbf{z}-\\{z_{n}\\},\mathbf{u}-\\{u_{n}\\},\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\beta},a,b)\propto$
(16)
$\displaystyle\frac{\alpha_{k}+t_{d,k}^{\prime}}{\alpha+t_{d}^{\prime}}\frac{b+at_{p,d}^{\prime}}{b+N_{p,d}^{\prime}}\frac{S^{N_{p,d,k}^{\prime}+1}_{t_{p,d,k}^{\prime}+1}}{S^{N_{p,d,k}^{\prime}}_{t_{p,d,k}^{\prime}}}\frac{t^{\prime}_{p,d,k}+1.0}{n^{\prime}_{p,d,k}+1.0}\frac{\beta_{v}+M_{k,w_{p,d,n}}^{\prime}}{\beta+M_{k}^{\prime}},$

$\displaystyle
p(z_{n}=k,u_{n}=0|\mathbf{z}-\\{z_{n}\\},\mathbf{u}-\\{u_{n}\\},\mathbf{w},\boldsymbol{\alpha},\boldsymbol{\beta},a,b)$
$\displaystyle\propto$ (17)
$\displaystyle\frac{S^{N_{p,d,k}^{\prime}+1}_{t_{p,d,k}^{\prime}}}{S^{N_{p,d,k}^{\prime}}_{t_{p,d,k}^{\prime}}}\frac{1}{b+N^{\prime}_{p,d}}\frac{n_{p,d,k}^{\prime}-t_{p,d,k}^{\prime}+1.0}{n_{p,d,k}^{\prime}+1.0}\frac{\beta_{v}+M_{k,w_{p,d,n}}^{\prime}}{\beta+M_{k}^{\prime}},$

where the dash indicates statistics after excluding the current assignment.

Finally, update the counts of $n_{p,d,k}$ and $t_{p,d,k}$ with the sampled
topic assignment $z_{n}$ and table indicator $u_{n}$.

##  Appendix E Hierarchical clustering

The hierarchical clustering algorithm takes a bag of topics, a list with
sample indexes, and a cosine distance threshold. The bag of topics gathers
topic distributions from various posterior samples from various MCMC. The list
of sample indices records a sample index for each topic, i.e., assuming that
the first 50 topics in the bag of topics come from posterior sample 1 and the
next 50 topics come from posterior sample 2, then the first 50 elements in the
list of samples indices are 1 and the next 50 elements are 2. The cosine
distance threshold indicates the limit up to which topics would be merged.

The algorithm will start by forming clusters with each of the topics in the
bag of topics. So, if there are $N$ topics, there are $N$ initial clusters.
Then, a list $L$ is created to record the cosine distance between two
clusters. This list contains the indexes of the two compared clusters and the
cosine distance between the clustered topics. A clustered topic is the average
topic distributions of the cluster members.

At each step, the algorithm finds the pair of clusters in $L$ with the minimum
cosine distance. Then, the algorithm evaluates if the members of both clusters
are from different posterior samples using the list of sample indices. If so,
a new cluster is created by merging the evaluated pair of clusters. Then, the
algorithm removes from the $L$ all comparisons that had any of the identified
clusters and adds comparisons from all the remaining clusters to the new
cluster. But, If the evaluation is false, the algorithm updates the cosine
distance between the pair of clusters with 1. Thereby, the algorithm would not
take the same pair of clusters in the next step.

The algorithm will keep merging clusters until the minimum cosine distance is
larger than the cosine distance threshold. The algorithm then retrieves all
the remaining clusters (clusters that are not eliminated because they do not
get merged).

## References

  * Aldous (1985) [author] Aldous, David JD. J. (1985). Exchangeability and related topics. In École d’Été de Probabilités de Saint-Flour XIII—1983 1–198. Springer.
  * Banerjee, Carlin and Gelfand (2014) [author] Banerjee, SudiptoS., Carlin, Bradley PB. P. and Gelfand, Alan EA. E. (2014). Hierarchical modeling and analysis for spatial data. CRC Press.
  * Berry et al. (2016) [author] Berry, TomT., Newing, AndyA., Davies, DeborahD. and Branch, KirstyK. (2016). Using workplace population statistics to understand retail store performance. The International Review of Retail, Distribution and Consumer Research 26 375–395.
  * Betancourt (2017) [author] Betancourt, MichaelM. (2017). A conceptual introduction to Hamiltonian Monte Carlo. arXiv preprint arXiv:1701.02434.
  * Blei (2012) [author] Blei, David MD. M. (2012). Probabilistic topic models. Communications of the ACM 55 77–84.
  * Blei, Kucukelbir and McAuliffe (2017) [author] Blei, David MD. M., Kucukelbir, AlpA. and McAuliffe, Jon DJ. D. (2017). Variational inference: A review for statisticians. Journal of the American Statistical Association 112 859–877.
  * Blei and Lafferty (2006) [author] Blei, DavidD. and Lafferty, JohnJ. (2006). Correlated topic models. NIPS’06 18 147\.
  * Blei, Ng and Jordan (2003) [author] Blei, David MD. M., Ng, Andrew YA. Y. and Jordan, Michael IM. I. (2003). Latent Dirichlet allocation. Journal of Machine Learning Research 3 993–1022.
  * Buntine and Hutter (2010) [author] Buntine, WrayW. and Hutter, MarcusM. (2010). A Bayesian view of the Poisson-Dirichlet process. arXiv preprint arXiv:1007.0296.
  * Carpenter et al. (2017) [author] Carpenter, BobB., Gelman, AndrewA., Hoffman, Matthew DM. D., Lee, DanielD., Goodrich, BenB., Betancourt, MichaelM., Brubaker, MarcusM., Guo, JiqiangJ., Li, PeterP. and Riddell, AllenA. (2017). Stan: A probabilistic programming language. Journal of Statistical Software 76\.
  * Chen, Du and Buntine (2011) [author] Chen, ChangyouC., Du, LanL. and Buntine, WrayW. (2011). Sampling table configurations for the hierarchical Poisson-Dirichlet process. In ECML-KDD 296–311. Springer.
  * Christidis, Apostolou and Mentzas (2010) [author] Christidis, KonstantinosK., Apostolou, DimitrisD. and Mentzas, GregorisG. (2010). Exploring customer preferences with probabilistic topics models. In European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases 12–24.
  * Chuang et al. (2015) [author] Chuang, JasonJ., Roberts, Margaret EM. E., Stewart, Brandon MB. M., Weiss, RebeccaR., Tingley, DustinD., Grimmer, JustinJ. and Heer, JeffreyJ. (2015). TopicCheck: Interactive alignment for assessing topic model stability. In NAACL HLT’15 175–184.
  * Cressie and Wikle (2015) [author] Cressie, NoelN. and Wikle, Christopher KC. K. (2015). Statistics for spatio-temporal data. John Wiley & Sons.
  * Davies, Dolega and Arribas-Bel (2019) [author] Davies, AlecA., Dolega, LesL. and Arribas-Bel, DanielD. (2019). Buy online collect in-store: exploring grocery click&collect using a national case study. International Journal of Retail & Distribution Management.
  * Du, Buntine and Jin (2010) [author] Du, LanL., Buntine, WrayW. and Jin, HuidongH. (2010). A segmented topic model based on the two-parameter Poisson-Dirichlet process. Machine learning 81 5-19.
  * (17) [author] Office for National Statistics http://geoportal1-ons.opendata.arcgis.com/datasets/bf7701adbfc74565a3de7feb414184e8.
  * Groves (2005) [author] Groves, AngelaA. (2005). The local and regional food opportunity. Institute of Grocery Distribution.
  * Heinonen et al. (2016) [author] Heinonen, MarkusM., Mannerström, HenrikH., Rousu, JuhoJ., Kaski, SamuelS. and Lähdesmäki, HarriH. (2016). Non-stationary gaussian process regression with hamiltonian monte carlo. In Artificial Intelligence and Statistics 732–740. PMLR.
  * Hoffman and Gelman (2014) [author] Hoffman, Matthew DM. D. and Gelman, AndrewA. (2014). The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research 15 1593–1623.
  * Hornsby et al. (2019) [author] Hornsby, Adam NA. N., Evans, ThomasT., Riefer, Peter SP. S., Prior, RosieR. and Love, Bradley CB. C. (2019). Conceptual organization is revealed by consumer activity patterns. Computational Brain & Behavior 1–12.
  * Hruschka (2014) [author] Hruschka, HaraldH. (2014). Linking multi-category purchases to latent activities of shoppers: analysing market baskets by topic models. Journal of Research and Management 36 267–273.
  * Hruschka (2016) [author] Hruschka, HaraldH. (2016). Hidden Variable Models for Market Basket Data. Statistical Performance and Managerial Implications Technical Report.
  * Ishwaran and James (2001) [author] Ishwaran, HemantH. and James, Lancelot FL. F. (2001). Gibbs sampling methods for stick-breaking priors. Journal of the American Statistical Association 96 161–173.
  * Jacobs, Donkers and Fok (2016) [author] Jacobs, Bruno JDB. J., Donkers, BasB. and Fok, DennisD. (2016). Model-based purchase predictions for large assortments. Marketing Science 35 389–404.
  * Kuznesof, Tregear and Moxey (1997) [author] Kuznesof, SharronS., Tregear, AngelaA. and Moxey, AndrewA. (1997). Regional foods: a consumer perspective. British Food Journal 99 199–206.
  * Lu et al. (2011) [author] Lu, BinB., Ott, MyleM., Cardie, ClaireC. and Tsou, Benjamin KB. K. (2011). Multi-aspect sentiment analysis with topic models. In ICDM’ 11 81–88.
  * Newing, Clarke and Clarke (2015) [author] Newing, AndyA., Clarke, Graham PG. P. and Clarke, MartinM. (2015). Developing and applying a disaggregated retail location model with extended retail demand estimations. Geographical Analysis 47 219–239.
  * Pitman and Yor (1997) [author] Pitman, JimJ. and Yor, MarcM. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. The Annals of Probability 855–900.
  * Riahi et al. (2012) [author] Riahi, FatemehF., Zolaktaf, ZainabZ., Shafiei, MahdiM. and Milios, EvangelosE. (2012). Finding expert users in community question answering. In WWW ’12 791–798.
  * Robusto (1957) [author] Robusto, C CarlC. C. (1957). The cosine-haversine formula. The American Mathematical Monthly 64 38–40.
  * Rosen-Zvi et al. (2004) [author] Rosen-Zvi, MichalM., Griffiths, ThomasT., Steyvers, MarkM. and Smyth, PadhraicP. (2004). The Author-Topic Model for Authors and Documents. In UAI’04 487–494. AUAI Press, Arlington, Virginia, USA.
  * Schröder (2017) [author] Schröder, NadineN. (2017). Using multidimensional item response theory models to explain multi-category purchases. Marketing ZFP 39 27–37.
  * Sturley, Newing and Heppenstall (2018) [author] Sturley, CharlotteC., Newing, AndyA. and Heppenstall, AlisonA. (2018). Evaluating the potential of agent-based modelling to capture consumer grocery retail store choice behaviours. The International Review of Retail, Distribution and Consumer Research 28 27–46.
  * Tobler (1970) [author] Tobler, Waldo RW. R. (1970). A computer movie simulating urban growth in the Detroit region. Economic Geography 46 234–240.
  * Vega-Carrasco et al. (2020) [author] Vega-Carrasco, MariflorM., O’sullivan, JasonJ., Prior, RosieR., Manolopoulou, IoannaI. and Musolesi, MircoM. (2020). Modelling Grocery Retail Topic Distributions: Evaluation, Interpretability and Stability.
  * Waddington et al. (2018) [author] Waddington, Thomas BPT. B., Clarke, Graham PG. P., Clarke, MartinM. and Newing, AndyA. (2018). Open all hours: spatiotemporal fluctuations in UK grocery store sales and catchment area demand. The International Review of Retail, Distribution and Consumer Research 28 1–26.
  * Williams and Rasmussen (2006) [author] Williams, Christopher KIC. K. and Rasmussen, Carl EdwardC. E. (2006). Gaussian Processes for Machine Learning. MIT Press Cambridge, MA.