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\begin{document}
	\title{Week 8 - Quantitative data analysis}
	\author{
		Jai Bheeman \and Kelvin Davis \and Jip J. Dekker \and Nelson Frew \and Tony
		Silvestere
	}
	\maketitle

	\section{Introduction} \label{sec:introduction}
	
	The purpose of this report is to re-analyse the data presented in the paper by
	\cite{dong2018methods}, which investigates the effect that protests (as an
	example of disruptive social behaviours in general) have on consumer
	behaviours. \cite{dong2018methods} hypothesise that protests decrease
	consumer behaviour in the surrounding area of the event, and suggest that
	consumer spending could be used as an additional non-traditional economic
	indicator and as a gauge of consumer sentiment. Consumer spending was analysed
	using credit card transaction data from a metropolitan area within a country
	that is part of The Organisation for Economic Co-operation and Development
	(OECD). Although \cite{dong2018methods} investigate temporal and spatial
	effects on consumer spending, for the purposes of this analysis, only the
	spatial effect of variables (with relation to the geographical distance from
	the event) is considered.
	
	\section{Method} \label{sec:method}

	The dataset consists of variables measured as a	function of the distance 
	from the event (in km), including: the number of customers, the median 
	spending amount, the number of transactions, and the total sales amount.
	The re-analysis is conducted on the data provided in the
	paper\cite{dong2018methods}, using Python in conjunction with packages such as
	pandas, matplotlib, numpy and seaborn, to process and visualise the data. As
	aforementioned, only spatial data and the variables mentioned above are
	considered, for the reference days and the change occuring Day 62 (day of
	first socially disruptive event). The distribution of the difference between
	the reference period and Day 62 is visualised by plotting a histogram for each
	variable. Since the decrease of each the variables from the reference period
	to Day 62 is provided, the mean and the median of these distributions can be
	used to perform a one-sample (as we have are given the difference) hypothesis
	test to assess whether the protests on Day 62 had a discernable effect.

	Assuming the mean of each variable over the reference period is the midpoint
	between their respective maximum and minimum values, we can reconstruct
	approximate actual values for Day 62 (given the decrease in value on Day 62
	from the reference period). By comparing these value to the range over the
	reference period, another assessment can be made to determine whether the data
	presents a discernible effect on consumer spending as a result of social
	discuption, scaling with distance.

	Although time series data was not explicitely provided, by extrapolating
	information from a graph in \cite{dong2018methods} we can quantify the decrease
	in number of customers and median spending on Day 62 using information about the
	reference days (from 43 to 61). After collecting the values for each of the
	reference days (43-61), the mean and standard deviation of this sample can be
	calculated. Assuming a normal distribution of the data, we can calculate a
	z-score for each observation on Day 62, and use this to assess the original
	hypothesis.

	By performing each of the above test, a re-analysis will be conducted on
	\cite{dong2018methods}'s paper hypothesising that consumer spending decreases
	as a result of social events such as protests. In the Results section, we will
	perform the statistical analyses described above. The results of these tests
	will then be explored in the Discussion section, along with assumptions and
	limitations of the tests and what can be conclused from them.

	\section{Results} \label{sec:results}

	For each of the variables in the given data (number of customers, median
	spending amount, number of transactions, and sales totals) we construct a
	histogram of the decrease of each (on Day 62). We then compute the mean and
	median of the data so we can proceed to perform a one-sample hypothesis test.

	\begin{figure}[ht]
		\centering
		\label{fig:distr}
		\includegraphics[width=\textwidth]{distr.png}
		\caption{Distribution of each of the variables recorded in the data, as a function of the distance from an event}
	\end{figure}

	Using a mean/median of the reference period, obtained by taking the midpoint of the minimum and maximum values over for each distance measure, a value can be reconstructed for the measurement on Day 62 (for each location) using:

	\begin{equation}
		\textrm{value} = \frac{\textrm{min} + \text{max}}{2} - \textrm{decrease.}
		\tag{1}
	\end{equation}
\\
	We can then plot the maximum and minimum values for the reference period, as well as the reconstructed Day 62 variables to observe the behaviour of consumer spending after the event.

	\begin{figure}[ht]
		\centering
		\label{fig:effect}
		\includegraphics[width=\textwidth]{effect.png}
		\caption{The reconstructed values for Day 62 of each variable plotted against their respective minimums and maximums over the reference period}
	\end{figure}

	Using the data recorded, for each of the three distance recorded, the mean and standard deviation of the reference period can be calculated. The z-score for each observed value on Day 62 can be computed using:

	\begin{equation}
	\textrm{Z} = \frac{\textrm{X} - \mu}{\sigma},
	\tag{2}
	\end{equation}
\\
	where X is the observed value, $\mu$ and $\sigma$ are the mean and standard deviation (respectively) of the reference period.

	\begin{table}[ht]
		\centering
		\label{my-label}
		\begin{tabular}{|l|l|r|r|}
		\hline
		\textbf{Variable}        & \textbf{Distance} & \textbf{X} & \textbf{Z} \\
		\hline
		\textbf{Customers}       & \textless 2km     & -0.600     &  6.87798   \\
		\textbf{Customers}       & 2km - 4km         & -0.200     & -3.33253   \\
		\textbf{Customers}       & \textgreater 4km  & -0.100     & -3.70740   \\
		\textbf{Median Spending} & \textless 2km     & -0.200     & -3.05849   \\
		\textbf{Median Spending} & 2km - 4km         & -0.100     & -1.46508   \\
		\textbf{Median Spending} & \textgreater 4km  & -0.035     & -1.99199   \\
		\hline
		\end{tabular}
		\caption{The $Z$ score computed using equation 2 and the temporal data}
	\end{table}

	\section{Discussion} \label{sec:discussion}

	As shown in each of the subplots of Figure 1, the mean and median values of
	the decrease in each of the distributions are greater than zero (note: higher
	values of the decrease variable indicate a larger decrease/negative change).
	These mean and median values can be used to perform a one-sample hypothesis
	tests, which finds that since each of the mean/median values is greater than
	zero, we can infer that the event had a net decreasing affect on the number of
	customers, median spending amount, number of transactions, and total sales
	amount.

	In Figure \ref{fig:effect} values were approximated for each variable on Day
	62, using Equation 1, and plotted against the minimum and maximum values of
	the respective variables. This allows us to visually assess whether the
	reconstructed value for Day 62 lies outside the range of recorded values for
	the reference period, and presents uncharacteristic behaviour. A decrease is
	evident in each of the variables after the event has occurred (on Day 62)
	within a distance of approximately 2 km, and appears to stabilise thereafter.
	This provides support to \cite{dong2018methods}'s hypothesis that consumer
	spending is affected by socially disruptive events, and also provides evidence
	to the notion of spatial scaling of this effect (based on the event location).
	It is important to note that the approximation used in this technique is
	subject to a level of error due to the ideal calculation of the mean/median of
	the reference data as the midpoint between the minimum and maximum values
	provided.

	Extrapolating data from a graph in \cite{dong2018methods} provided time series
	data (divided into three radius') to analyse. This data was collected by
	visually estimating the values from the graph which will inherently introduce
	a source of error. However, by computing the z-score as described in Equation
	2, the table provided in Figure 3 was constructed. Each of the z-score values
	in the table are negative, indicating a decrease in both the number of
	customers and median spending on Day 62. The much larger magnitude of z-scores
	for the <2km distance ring for both variables is in agreement with earlier
	discussion, strengthening the hypothesis of the spatial correlation of
	consumer spending.

	Each of the above tests have agreed on the spatial and temporal correlation of
	consumer spending and socially disruptive events. With the limited data
	available, we can therefore concur with the hypothesis of Dong et al. that
	consumer spending decreases in the area around disruptive social behaviour,
	after finding the temporal correlation on Day 62, as well as the spatially
	decreasing effect further from the event.

	\bibliographystyle{humannat}
	\bibliography{references}

\end{document}