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wk7/week7.tex
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\section{Results} \label{sec:results} \section{Results} \label{sec:results}
\subsection{The advantage of height}
\begin{table}[ht]
\centering
\label{tab:chi-height}
\begin{tabular}{|l|r|r|r|r|}
\hline
& \textbf{M: 168 - 188} & \textbf{M: 189 - 210} & \textbf{F: 155 - 171} & \textbf{F: 172 - 189} \\ \hline
\textbf{1 - 99} & 67 / 73 & 32 / 26 & 38 / 42 & 60 / 55 \\
\textbf{100 - 199} & 69 / 72 & 30 / 26 & 31 / 27 & 32 / 36 \\
\textbf{200 - 299} & 75 / 68 & 17 / 25 & 18 / 17 & 22 / 23 \\
\textbf{300 - 399} & 61 / 60 & 21 / 23 & 11 /12 & 17 / 16 \\
\textbf{400 - 499} & 59 / 60 & 22 / 22 & 7 / 6
& 7 / 8 \\
\hline
\end{tabular}
\caption{Observed / Expected values used for the $\chi^2$-test. The groups are divided by their rank (vertical) and, per gender, their height (horizontal).}
\end{table}
$$
\chi^2 \approx 7.697606186049128
$$
$$
df = (5-1)(4-1) = 12
$$
$$
\chi^2(7.69\dots,12) \approx 0.8082925814979871
$$
\textbf {t-test men:} T score: 1.711723, P score: 0.043815
\textbf {t-test women:} T score: 1.860241, P score: 0.032030
\subsection{The advantage of left-handedness}
\begin{table}[ht]
\centering
\label{tab:chi-hand}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
& \textbf{1 - 99} & \textbf{100 - 199} & \textbf{200 - 299} & \textbf{300 - 399} & \textbf{400 - 499} \\
\hline
\textbf{L} & 22 / 21 & 23 / 18 & 17 / 15 & 6 / 12 & 8 / 10 \\
\textbf{R} & 174 / 177 & 139 / 144 & 117 / 119 &
105 / 98 & 88 / 86 \\
\hline
\end{tabular}
\caption{Observed / Expected values used for the $\chi^2$-test. The groups are divided by which hand they use (vertical) and their rank (horizontal).}
\end{table}
$$
\chi^2 \approx 6.467312944404331
$$
$$
df = (2-1)(5-1) = 4
$$
$$
\chi^2(6.46\dots,4) \approx 0.1668616190847413
$$
\textbf {t-test:} T score: 0.451694, P score: 0.325815
\section{Discussion} \label{sec:discussion} \section{Discussion} \label{sec:discussion}
\end{document} \end{document}

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\usepackage[utf8]{inputenc} %support umlauts in the input \usepackage[utf8]{inputenc} %support umlauts in the input
% Easier compilation % Easier compilation
\usepackage{bookmark} \usepackage{bookmark}
\usepackage{natbib}
\usepackage{graphicx}
\begin{document} \begin{document}
\title{Week 8 - Quantitative data analysis} \title{Week 8 - Quantitative data analysis}
@ -25,8 +27,161 @@
\section{Method} \label{sec:method} \section{Method} \label{sec:method}
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. 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
aformentioned, 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} \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} \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} \end{document}

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\usepackage[utf8]{inputenc} %support umlauts in the input \usepackage[utf8]{inputenc} %support umlauts in the input
% Easier compilation % Easier compilation
\usepackage{bookmark} \usepackage{bookmark}
\usepackage{graphicx}
\begin{document} \begin{document}
\title{Week 9 - Correlation and Regression} \title{Week 9 - Correlation and Regression}
@ -22,6 +23,12 @@
\maketitle \maketitle
\section{Introduction} \label{sec:introduction} \section{Introduction} \label{sec:introduction}
We present a report on the relationship between the heights and weights of the
top tennis players as catalogued in provided data. We use statistical analysis
techniques to numerically describe the characteristics of the data, to see how
trends are exhibited within the data set. We conclude the report with a brief
discussion of the implications of the analysis and provide insights on
potential correlations that may exist.
\section{Method} \label{sec:method} \section{Method} \label{sec:method}
Provided with a set of 132 unique records of the top 200 male tennis players, Provided with a set of 132 unique records of the top 200 male tennis players,
@ -34,21 +41,33 @@
samples and samples of ranking ranges within the top 200. To this end, we made samples and samples of ranking ranges within the top 200. To this end, we made
use of Microsoft Excel tools and functions of the Python library SciPy. use of Microsoft Excel tools and functions of the Python library SciPy.
We specifically have made use of these separate statistical analysis tools in the
interest of sanity checking our findings. To do this, we simply replicated the
correlation tests within other software environments.
\section{Results} \label{sec:results} \section{Results} \label{sec:results}
We performed seperate statistical analyses on 10 different samples of the We performed separate statistical analyses on 10 different samples of the
population, as well as the population itself. This included 5 separate subsets population, as well as the population itself. This included 11 separate
of the rankings (top 20 and 50, middle 20, bottom 20 and 50) and 5 seperate subsets of the rankings:
randomly chosen samples of 20 players. \begin{itemize}
\\ \\ \item The top 20 entries
\Cref{tab:excel-results} shows the the results for the conducted tests. \item The middle 20 entries
\item The bottom 20 entries
\item The top 50 entries
\item The bottom 50 entries
\item 5 randomly chosen sets of 20 entries
\end{itemize}
\vspace{1em}
Table \ref{tab:excel_results} shows the the results for the conducted tests.
\begin{table}[ht] \begin{table}[ht]
\centering \centering
\label{tab:excel_results}
\begin{tabular}{|l|r|r|} \begin{tabular}{|l|r|r|}
\hline \hline
\textbf{Test Set} & \textbf{Pearson's Coefficient} & \textbf{Spearman's Coefficient} \\ \textbf{Test Set} & \textbf{Pearson's Coefficient} & \textbf{Spearman's Coefficient} \\
\hline \hline
\textbf{Population} & 0.77953 & 0.73925 \\ \textbf{Full Population} & 0.77953 & 0.73925 \\
\textbf{Top 20} & 0.80743 & 0.80345 \\ \textbf{Top 20} & 0.80743 & 0.80345 \\
\textbf{Middle 20} & 0.54134 & 0.36565 \\ \textbf{Middle 20} & 0.54134 & 0.36565 \\
\textbf{Bottom 20} & 0.84046 & 0.88172 \\ \textbf{Bottom 20} & 0.84046 & 0.88172 \\
@ -61,11 +80,20 @@
\textbf{Random Set \#5} & 0.86203 & 0.77832 \textbf{Random Set \#5} & 0.86203 & 0.77832
\\ \hline \\ \hline
\end{tabular} \end{tabular}
\caption{TODO: Insert better caption for this table. All data is rounded to 5 decimal \caption{Table showing the correlation coefficients between height and
weight using different test sets. All data is rounded to 5 decimal
places} places}
\label{tab:excel-results}
\end{table} \end{table}
\begin{figure}[ht]
\centering
\label{fig:scipy}
\includegraphics[width=0.6\textwidth]{pearson.png}
\includegraphics[width=0.6\textwidth]{spearman.png}
\caption{The Pearsion (top) and Spearman (bottom) correlations coefficients
of the data set as computed by the Pandas Python library}
\end{figure}
\section{Discussion} \label{sec:discussion} \section{Discussion} \label{sec:discussion}
The results generally indicate that there is a fairly strong positive The results generally indicate that there is a fairly strong positive
correlation between the weight and weight of an individual tennis player, correlation between the weight and weight of an individual tennis player,

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{
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"execution_count": 1,
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"outputs": [
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"text": [
"Using matplotlib backend: MacOSX\n",
"Populating the interactive namespace from numpy and matplotlib\n"
]
}
],
"source": [
"%pylab\n",
"%matplotlib inline\n",
"import pandas as pd\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
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"\n",
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" return ['background-color: %s' % color for color in c]\n",
"\n",
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" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row0_col2 {\n",
" background-color: #fee91d;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row0_col3 {\n",
" background-color: #f4f242;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row1_col0 {\n",
" background-color: #ffd20c;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row1_col1 {\n",
" background-color: #fc7f00;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row1_col2 {\n",
" background-color: #e4ff7a;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row1_col3 {\n",
" background-color: #eafa63;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row2_col0 {\n",
" background-color: #fee91d;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row2_col1 {\n",
" background-color: #e4ff7a;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row2_col2 {\n",
" background-color: #fc7f00;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row2_col3 {\n",
" background-color: #ff9d00;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row3_col0 {\n",
" background-color: #f4f242;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row3_col1 {\n",
" background-color: #eafa63;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row3_col2 {\n",
" background-color: #ff9d00;\n",
" } #T_727bef98_4f3e_11e8_a315_787b8ab7acb2row3_col3 {\n",
" background-color: #fc7f00;\n",
" }</style> \n",
"<table id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2\" > \n",
"<thead> <tr> \n",
" <th class=\"blank level0\" ></th> \n",
" <th class=\"col_heading level0 col0\" >DOB</th> \n",
" <th class=\"col_heading level0 col1\" >RANK</th> \n",
" <th class=\"col_heading level0 col2\" >HEIGHT</th> \n",
" <th class=\"col_heading level0 col3\" >Weight</th> \n",
" </tr></thead> \n",
"<tbody> <tr> \n",
" <th id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2level0_row0\" class=\"row_heading level0 row0\" >DOB</th> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row0_col0\" class=\"data row0 col0\" >1</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row0_col1\" class=\"data row0 col1\" >0.280386</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row0_col2\" class=\"data row0 col2\" >0.122412</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row0_col3\" class=\"data row0 col3\" >0.00769861</td> \n",
" </tr> <tr> \n",
" <th id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2level0_row1\" class=\"row_heading level0 row1\" >RANK</th> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row1_col0\" class=\"data row1 col0\" >0.280386</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row1_col1\" class=\"data row1 col1\" >1</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row1_col2\" class=\"data row1 col2\" >-0.160006</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row1_col3\" class=\"data row1 col3\" >-0.0908714</td> \n",
" </tr> <tr> \n",
" <th id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2level0_row2\" class=\"row_heading level0 row2\" >HEIGHT</th> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row2_col0\" class=\"data row2 col0\" >0.122412</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row2_col1\" class=\"data row2 col1\" >-0.160006</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row2_col2\" class=\"data row2 col2\" >1</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row2_col3\" class=\"data row2 col3\" >0.739246</td> \n",
" </tr> <tr> \n",
" <th id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2level0_row3\" class=\"row_heading level0 row3\" >Weight</th> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row3_col0\" class=\"data row3 col0\" >0.00769861</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row3_col1\" class=\"data row3 col1\" >-0.0908714</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row3_col2\" class=\"data row3 col2\" >0.739246</td> \n",
" <td id=\"T_727bef98_4f3e_11e8_a315_787b8ab7acb2row3_col3\" class=\"data row3 col3\" >1</td> \n",
" </tr></tbody> \n",
"</table> "
],
"text/plain": [
"<pandas.io.formats.style.Styler at 0x111a3b198>"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"spearman = data.corr(method=\"spearman\")\n",
"spearman.style.apply(background_gradient,\n",
" cmap='Wistia',\n",
" m=spearman.min().min(),\n",
" M=spearman.max().max()\n",
")"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.4"
}
},
"nbformat": 4,
"nbformat_minor": 2
}