\section{Selection of events}
\label{sec:lmdk-sel-eval}

In this section we present the experiments that we performed, to test the methodology that we presented in Section~\ref{subsec:lmdk-sel-sol}, on real and synthetic data sets.
With the experiments on the synthetic data sets (Section~\ref{subsec:sel-utl}) we show the normalized Euclidean and Wasserstein distances of the time series histogram for various distributions and {\thething} percentages.
This allows us to justify our design decisions for our concept that we showcased in Section~\ref{subsec:lmdk-sel-sol}.
With the experiments on the real data sets (Section~\ref{subsec:sel-prv}), we show the performance in terms of utility of our three {\thething} mechanisms in combination with the privacy preserving {\thething} selection component.


\subsection{{\Thething} selection utility metrics}
\label{subsec:sel-utl}

Figure~\ref{fig:sel-dist} demonstrates the normalized distance that we obtain when we utilize either (a)~the Euclidean or (b)~the Wasserstein distance metric to obtain a set of {\thethings} including regular events.

\begin{figure}[htp]
  \centering
  \subcaptionbox{Euclidean\label{fig:sel-dist-norm}}{%
    \includegraphics[width=.5\linewidth]{evaluation/sel-dist-norm}%
  }%
  \subcaptionbox{Wasserstein\label{fig:sel-dist-emd}}{%
    \includegraphics[width=.5\linewidth]{evaluation/sel-dist-emd}%
  }%
  \caption{The normalized (a)~Euclidean, and (b)~Wasserstein distance of the generated {\thething} sets for different {\thething} percentages.}
  \label{fig:sel-dist}
\end{figure}

Comparing the results of the Euclidean distance in Figure~\ref{fig:sel-dist-norm} with those of the Wasserstein in Figure~\ref{fig:sel-dist-emd} we conclude that the Euclidean distance provides more consistent results for all possible distributions.
% (0 + (0.25 + 0.25 + 0.3 + 0.3)/4 + (0.45 + 0.45 + 0.45 + 0.5)/4 + (0.5 + 0.5 + 0.7 + 0.7)/4 + (0.6 + 0.6 + 1 + 1)/4 + (0.3 + 0.3 + 0.3 + 0.3)/4)/6
% (0 + (0.15 + 0.15 + 0.15 + 0.15)/4 + (0.2 + 0.2 + 0.3 + 0.4)/4 + (0.3 + 0.3 + 0.6 + 0.6)/4 + (0.3 + 0.3 + 1 + 1)/4 + (0.05 + 0.05 + 0.05 + 0.05)/4)
The maximum difference is approximately $0.4$ for the former and $0.7$ for the latter between the bimodal and skewed {\thething} distribution.
While both methods share the same mean normalized distance of $0.4$, the Euclidean distance demonstrates a more consistent performance among all possible {\thething} distributions.
Therefore, we choose to utilize the Euclidean distance metric for the implementation of the privacy-preserving {\thething} selection in Section~\ref{subsec:lmdk-sel-sol}.


\subsection{Budget allocation and {\thething} selection}
\label{subsec:sel-prv}

Figure~\ref{fig:real-sel} exhibits the performance of Skip, Uniform, and Adaptive (see Section~\ref{subsec:lmdk-mechs}) in combination with the {\thething} selection component.

\begin{figure}[htp]
  \centering
  \subcaptionbox{Copenhagen\label{fig:copenhagen-sel}}{%
    \includegraphics[width=.5\linewidth]{evaluation/copenhagen-sel}%
  }%
  \hspace{\fill}
  \subcaptionbox{HUE\label{fig:hue-sel}}{%
    \includegraphics[width=.5\linewidth]{evaluation/hue-sel}%
  }%
  \subcaptionbox{T-drive\label{fig:t-drive-sel}}{%
    \includegraphics[width=.5\linewidth]{evaluation/t-drive-sel}%
  }%
  \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-sel}
\end{figure}

In comparison with the utility performance without the {\thething} selection component (Figure~\ref{fig:real}), we notice a slight deterioration for all three models.
This is natural since we allocated part of the available privacy budget to the privacy-preserving {\thething} selection component which in turn increased the number of {\thethings}.
Therefore, there is less privacy budget available for data publishing throughout the time series for $0$\% and $100$\% {\thethings}.
Skip performs best in our experiments with HUE, due to the low range in the energy consumption and the high scale of the Laplace noise which it avoids due to its tendency to approximate.
However, for the Copenhagen data set and T-drive it attains greater mean absolute error than the user-level protection scheme. 
Overall, Adaptive has a consistent performance in terms of utility for all of the data sets that we experimented with.
