\section{Landmark events}
\label{sec:eval-lmdk}

% \kat{After discussing with Dimitris, I thought you are keeping one chapter for the proposals of the thesis. In this case, it would be more clean to keep the theoretical contributions in one chapter and the evaluation in a separate chapter. }
% \mk{OK.}
In this section, we present the experiments that we performed, to test the methodology that we presented in Section~\ref{subsec:lmdk-sol}, on real and synthetic data sets. 

With the experiments on the real data sets (Section~\ref{subsec:lmdk-expt-bgt}), we show the performance in terms of data utility of our three {\thething}  privacy budget allocation schemes: Skip, Uniform and Adaptive.
We define data utility as the Mean Absolute Error introduced by the privacy mechanism.
We compare with the event and user differential privacy, and show that in the general case, {\thething} privacy allows for better data utility than user differential privacy.

With the experiments on the synthetic data sets (Section~\ref{subsec:lmdk-expt-cor}) we show the privacy loss \kat{in the previous set of experiments we were measuring the MAE, now we are measuring the privacy loss... Why is that? Isn't it two sides of the same coin? }by our framework when tuning the size and statistical characteristics of the input {\thething} set $L$ with special emphasis on how the privacy loss under temporal correlation is affected by the number and distribution of the {\thethings}.
\kat{mention briefly what you observe}



\subsection{Budget allocation schemes}
\label{subsec:lmdk-expt-bgt}

Figure~\ref{fig:real} exhibits the performance of the three mechanisms, Skip, Uniform, and Adaptive applied on the three data sets that we study.
% For the Geolife data set (Figure~\ref{fig:geolife}), Skip has the best performance (measured in Mean Absolute Error, in meters) because it invests the most budget overall at every regular event, by approximating the {\thething} data based on previous releases.
% Due to the data set's high density (every $1$--$5$ seconds or every $5$--$10$ meters per point) approximating constantly has a low impact on the data utility.
% On the contrary, the lower density of the T-drive data set (Figure~\ref{fig:t-drive}) has a negative impact on the performance of Skip.

For the Copenhagen data set (Figure~\ref{fig:copenhagen}), Adaptive has a constant\kat{it is not constant, for 0 it is much lower} overall performance and performs best for $0$\%, $60$\%, and $80$\% {\thethings} \kat{this is contradictory: you say that it is constant overall, and then that it is better for certain percentages. }.
We notice that for $0$\% {\thethings}, it achieves better utility than the event-level protection.\kat{what does this mean? how is it possible?}
The Skip model excels, compared to the others, at cases where it needs to approximate $20$\%--$40$\% or $100$\% of the times.\kat{it seems a little random.. do you have an explanation? (rather few times or all?)}

The combination of the small range of measurements in HUE ($[0.28$, $4.45]$ with an average of $0.88$kWh) and the large scale in the Laplace mechanism, results in a low mean absolute error for Skip (Figure~\ref{fig:hue}).
In general, a scheme that favors approximation over noise injection would achieve a better performance in this case.
\kat{why?explain}
However, the Adaptive model performs by far better than Uniform and strikes a nice balance\kat{???} between event- and user-level protection for all {\thething} percentages.

In the T-drive data set (Figure~\ref{fig:t-drive}), the Adaptive mechanism outperforms Uniform by $10$\%--$20$\% for all {\thething} percentages greater than $40$\% and Skip by more than $20$\%.
The lower density (average distance of $623$m) of the T-drive data set has a negative impact on the performance of Skip; republishing a previous perturbed value is now less accurate than perturbing the new location.


\begin{figure}[htp]
	\centering
	\subcaptionbox{Copenhagen\label{fig:copenhagen}}{%
		\includegraphics[width=.5\linewidth]{evaluation/copenhagen}%
	}%
	\hspace{\fill}
	\subcaptionbox{HUE\label{fig:hue}}{%
		\includegraphics[width=.5\linewidth]{evaluation/hue}%
	}%
	\subcaptionbox{T-drive\label{fig:t-drive}}{%
		\includegraphics[width=.5\linewidth]{evaluation/t-drive}%
	}%
	\caption{The mean absolute error (a)~as a percentage, (b)~in kWh, and (c)~in meters of the released data for different {\thething} percentages.}
	\label{fig:real}
\end{figure}

In general, we can claim that the Adaptive is the most reliable and best performing mechanism with minimal tuning\kat{what does minimal tuning mean?}, if we take into consideration the drawbacks of the Skip mechanism mentioned in Section~\ref{subsec:lmdk-mechs}. \kat{you can mention them also here briefly, and give the pointer for the section}
Moreover, designing a data-dependent sampling scheme \kat{what would be the main characteristic of the scheme? that it picks landmarks how?} would possibly\kat{possibly is not good enough, if you are sure remove it. Otherwise mention that more experiments need to be done?} result in better results for Adaptive.


\subsection{Temporal distance and correlation}
\label{subsec:lmdk-expt-cor}

As previously mentioned, temporal correlations are inherent in continuous publishing, and they are the cause of supplementary privacy leakage in the case of privacy preserving data publication.
In this section, we are interested in studying the effect that the distance of the {\thethings} from every event have on the leakage caused by temporal correlations. 

Figure~\ref{fig:avg-dist} shows a comparison of the average temporal distance of the events from the previous/next {\thething} or the start/end of the time series for various distributions in our synthetic data.
More specifically, we model the distance of an event as the count of the total number of events between itself and the nearest {\thething} or the series edge.

\begin{figure}[htp]
  \centering
  \includegraphics[width=.5\linewidth]{evaluation/avg-dist}%
  \caption{Average temporal distance of the events from the {\thethings} for different {\thethings} percentages within a time series in various {\thethings} distributions.}
  \label{fig:avg-dist}
\end{figure}

We observe that the uniform and bimodal distributions tend to limit the regular event--{\thething} distance.
This is due to the fact that the former scatters the {\thethings}, while the latter distributes them on both edges, leaving a shorter space uninterrupted by {\thethings}.
% and as a result they reduce the uninterrupted space by landmarks in the sequence.
On the contrary, distributing the {\thethings} at one part of the sequence, as in skewed or symmetric, creates a wider space without {\thethings}.
This study provides us with different distance settings that we are going to use in the subsequent temporal leakage study.

Figure~\ref{fig:dist-cor} illustrates a comparison among the aforementioned distributions regarding the overall privacy loss under (a)~weak, (b)~moderate, and (c)~strong temporal correlation degrees.
The line shows the overall privacy loss---for all cases of {\thethings} distribution---without temporal correlation.

\begin{figure}[htp]
  \centering
  \subcaptionbox{Weak correlation\label{fig:dist-cor-wk}}{%
    \includegraphics[width=.5\linewidth]{evaluation/dist-cor-wk}%
  }%
  \hspace{\fill}
  \subcaptionbox{Moderate correlation\label{fig:dist-cor-mod}}{%
    \includegraphics[width=.5\linewidth]{evaluation/dist-cor-mod}%
  }%
  \subcaptionbox{Strong correlation\label{fig:dist-cor-stg}}{%
    \includegraphics[width=.5\linewidth]{evaluation/dist-cor-stg}%
  }%
  \caption{Privacy loss \kat{what is the unit for privacy loss? I t should appear on the diagram} for different {\thethings} percentages and distributions under (a)~weak, (b)~moderate, and (c)~strong degrees of temporal correlation.
  The line shows the overall privacy loss without temporal correlation.}
  \label{fig:dist-cor}
\end{figure}

In combination with Figure~\ref{fig:avg-dist}, we conclude that a greater average event--{\thething} event \kat{it was even, I changed it to event but do not know what youo want ot say} distance in a distribution can result into greater overall privacy loss under moderate and strong temporal correlation.
This is due to the fact that the backward/forward privacy loss accumulates more over time in wider spaces without {\thethings} (see Section~\ref{sec:correlation}).
Furthermore, the behavior of the privacy loss is as expected regarding the temporal correlation degree: a stronger correlation degree generates higher privacy loss while widening the gap between the different distribution cases.
On the contrary, a weaker correlation degree makes it harder to differentiate among the {\thethings} distributions.
The privacy loss under a weak correlation degree converge \kat{with what?}.
