103 lines
8.7 KiB
TeX
103 lines
8.7 KiB
TeX
\section{Details}
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\label{sec:eval-dtl}
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In this section we list all the relevant details regarding the setting of the evaluation (Section~\ref{subsec:eval-setup}), and the real and synthetic data sets that we used(Section~\ref{subsec:eval-dat}), along with the corresponding configurations (Section~\ref{subsec:eval-conf}).
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\subsection{Setting}
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\label{subsec:eval-setup}
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We implemented our experiments\footnote{Code available at \url{https://git.delkappa.com/manos/the-last-thing}} in Python $3$.$9$.$7$ and executed them on a machine with an Intel i$7$-$6700$HQ at $3$.$5$GHz CPU and $16$GB RAM, running Manjaro Linux $21$.$1$.$5$.
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We repeated each experiment $100$ times and we report the mean over these iterations.
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\subsection{Data sets}
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\label{subsec:eval-dat}
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\subsubsection{Real}
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\paragraph{Copenhagen}~\cite{sapiezynski2019interaction}
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data set that was collected via the smartphone devices of $851$ university students over a period of $4$ week as part of the Copenhagen Networks Study.
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Each device was configured to be discoverable by and to discover nearby Bluetooth devices every $5$ minutes.
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Upon discovery each device registers, (i)~the timestamp in seconds, (ii)~the device's unique identifier, (iii)~the unique identifier of the device that it discovered ($- 1$ when no device was found or $- 2$ for any non-participating device), and (iv)~the Received Signal Strength Indicator (RSSI) in dBm.
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Half of the devices have registered data at at least $81\%$ of the possible timestamps.
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From this data set, we utilized the $1,000$ first contacts out of $12,167$ valid unique contacts of the device with identifier `$449$'.
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\paragraph{HUE}~\cite{makonin2018hue}
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contains the hourly energy consumption data of $22$ residential customers of BCHydro, a provincial power utility, in British Columbia.
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The measurements for each residence are saved individually and each measurement contains (i)~the date (YYYY-MM-DD), (ii)~the hour, and (iii)~the energy consumption in kWh.
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In our experiments, we used the first $1,000$ out of $29,231$ measurements of the residence with identifier `$1$', average energy consumption equal to $0.88$kWh, and value range $[0.28$, $4.45]$.
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\paragraph{T-drive}~\cite{yuan2010t}
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consists of $15$ million GPS data points of the trajectories of $10,357$ taxis in Beijing, spanning a period of $1$ week and a total distance of $9$ million kilometers.
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The taxis reported their location data on average every $177$ seconds and $623$ meters approximately.
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Each vehicle registers (i)~the taxi unique identifier, (ii)~the timestamp (YYYY-MM-DD HH:MM:SS), (iii)~longitude, and (iv)~latitude.
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These measurements are stored individually per vehicle.
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We sampled the first $1000$ data items of the taxi with identifier `$2$'.
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\subsubsection{Synthetic}
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We generated synthetic time series of length equal to $100$ timestamps, for which we varied the number and distribution of {\thethings}.
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We take into account only the temporal order of the points and the position of regular and {\thething} events within the series.
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\subsection{Configurations}
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\label{subsec:eval-conf}
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\subsubsection{{\Thething} percentage}
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For the Copenhagen data set, we achieve
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$0\%$ {\thethings} by considering an empty list of contact devices,
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$20\%$ by extending the list with $[3$, $6$, $11$, $12$, $25$, $29$, $36$, $39$, $41$, $46$, $47$, $50$, $52$, $56$, $57$, $61$, $63$, $78$, $80]$,
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$40\%$ with $[81$, $88$, $90$, $97$, $101$, $128$, $130$, $131$, $137$, $145$, $146$, $148$, $151$, $158$, $166$, $175$, $176]$,
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$60\%$ with $[181$, $182$, $192$, $195$, $196$, $201$, $203$, $207$, $221$, $230$, $235$, $237$, $239$, $241$, $254]$,
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$80\%$ with $[260$, $282$, $287$, $289$, $290$, $291$, $308$, $311$, $318$, $323$, $324$, $330$, $334$, $335$, $344$, $350$, $353$, $355$, $357$, $358$, $361$, $363]$, and
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$100\%$ by including all of the possible contacts.
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In HUE, we get $0$\%, $20$\% $40$\%, $60$\%, $80$\%, and $100$\% {\thethings} by setting the energy consumption threshold below $0.28$kWh, $1.12$kWh, $0.88$kWh, $0.68$kWh, $0.54$kWh, $4.45$kWh respectively.
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In T-drive, we achieved the desired {\thething} percentages by utilizing the method of Li et al.~\cite{li2008mining} for detecting stay points in trajectory data.
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In more detail, the algorithm checks for each data item if each subsequent item is within a given distance threshold $\Delta l$ and measures the time period $\Delta t$ between the present point and the last subsequent point.
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We achieve $0$\%, $20$\% $40$\%, $60$\%, $80$\%, and $100$\% {\thethings} by setting the ($\Delta l$ in meters, $\Delta t$ in minutes) pairs input to the stay point discovery method as [($0$, $1000$), ($2095$, $30$), ($2790$, $30$), ($3590$, $30$), ($4825$, $30$), ($10350$, $30$)].
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We generated synthetic data with \emph{skewed} (the {\thethings} are distributed towards the beginning/end of the series), \emph{symmetric} (in the middle), \emph{bimodal} (both end and beginning), and \emph{uniform} (all over the time series) {\thething} distributions.
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In order to get {\thethings} with the above distribution features, we generate probability distributions with appropriate characteristics and sample from them, without replacement, the desired number of points.
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%The generated distributions are representative of the cases that we wish to examine during the experiments.
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For example, for a left-skewed {\thething} distribution we would utilize a truncated distribution resulting from the restriction of the domain of a distribution to the beginning and end of the time series with its location shifted to the center of the right half of the series.
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For consistency, we calculate the scale parameter depending on the length of the series by setting it equal to the series' length over a constant.
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Notice that in our experiments, in the cases when we have $0\%$ and $100\%$ of the events being {\thethings}, we get the same behavior as in event- and user-level privacy respectively.
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This happens due the fact that at each timestamp we take into account only the data items at the current timestamp and ignore the rest of the time series (event-level) when there are no {\thethings}.
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Whereas, when each timestamp corresponds to a {\thething} we consider and protect all the events throughout the entire series (user-level).
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\subsubsection{Privacy parameters}
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To perturb the contact tracing data of the Copenhagen data set, we utilize the \emph{random response} technique~\cite{wang2017locally} to report with probability $p = \frac{e^\varepsilon}{e^\varepsilon + 1}$ whether the current contact is a {\thething} or not.
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We randomize the energy consumption in HUE with the Laplace mechanism (described in detail in Section~\ref{subsec:prv-mech}).
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We inject noise to the spatial values in T-drive that we sample from the Planar Laplace mechanism~\cite{andres2013geo}.
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We set the privacy budget $\varepsilon = 1$, and, for simplicity, we assume that for every query sensitivity it holds that $\Delta f = 1$.
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For the experiments performed on the synthetic data sets, the original values to be released do not influence the outcome of our conclusions, thus we ignore them.
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% Finally, notice that, depending on the results' variation, most diagrams are in logarithmic scale.
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\subsubsection{Temporal correlation}
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We model the temporal correlation in the synthetic data as a \emph{stochastic matrix} $P$, using a \emph{Markov Chain}~\cite{gagniuc2017markov}.
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$P$ is a $n \times n$ matrix, where the element $P_{ij}$
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%at the $i$th row of the $j$th column that
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represents the transition probability from a state $i$ to another state $j$.
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%, $\forall i, j \leq n$.
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It holds that the elements of every row $j$ of $P$ sum up to $1$.
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We follow the \emph{Laplacian smoothing} technique~\cite{sorkine2004laplacian} as utilized in~\cite{cao2018quantifying} to generate the matrix $P$ with a degree of temporal correlation $s > 0$ equal to
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% and generate a stochastic matrix $P$ with a degree of temporal correlation $s$ by calculating each element $P_{ij}$ as follows
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$$\frac{(I_{n})_{ij} + s}{\sum_{k = 1}^{n}((I_{n})_{jk} + s)}$$
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where $I_{n}$ is an \emph{identity matrix} of size $n$.
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%, i.e.,~an $n \times n$ matrix with $1$s on its main diagonal and $0$s elsewhere.
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% $s$ takes only positive values which are comparable only for stochastic matrices of the same size.
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The value of $s$ is comparable only for stochastic matrices of the same size and dictates the strength of the correlation; the lower its value,
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% the lower the degree of uniformity of each row, and therefore
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the stronger the correlation degree.
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%In general, larger transition matrices tend to be uniform, resulting in weaker correlation.
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In our experiments, for simplicity, we set $n = 2$ and we investigate the effect of \emph{weak} ($s = 1$), \emph{moderate} ($s = 0.1$), and \emph{strong} ($s = 0.01$) temporal correlation degree on the overall privacy loss.
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