Our problem setting consists of three entities: (i) data generators (users), (ii) data publishers (trusted non-adversarial entities), and (iii) data consumers (possibly adversarial entities).
Users generate sensitive data, which are processed in a secure and private way by a trusted curator and are later published in order to be consumed by potentially adversarial data analysts.
%The data unit produced by the users is an \emph{event}, i.e., a piece of timestamped user-related information.\kat{should we say geo-stamped?}.
Data are produced as a series of events, which we call time series.
An \emph{event} is defined as a triple of an identifying attribute of an individual and the possibly sensitive data at a timestamp.
%This workflow is repeated in a continuous manner, producing series of events, which we call time series.
%, producing, processing, publishing, and consuming events in a private manner.
%\kat{keep only the terms with a small description.}
\item\textbf{Data generators} (users) entity $E_g$ interacts with a crowdsensing application and produces continuously privacy-sensitive data items in an arbitrary frequency during the application's usage period $T =(t)_{t \in\mathbb{N}}$.
Thus, at each timestamp $t$, $E_g$ generates a data set $D_t \in\mathcal{D}$ where each of its members contributes a single data item.
\item\textbf{Data publishers} (trusted non-adversarial) entity $E_p$ receives the data sent by $E_g$ in the form of a series of events in $T$.
Following the \emph{global} processing and publishing scheme, $E_p$ collects at $t$ a data set $D_t$ and privacy-protects it by applying the respective privacy mechanism $\mathcal{M}_t$.
$\mathcal{M}_t$ uses independent randomness such that it satisfies $\varepsilon_t$-differential privacy.
\item\textbf{Data consumers} (possibly adversarial) entity $E_c$ receives the result $\mathbf{o}_t$ of the privacy-preserving processing of $D_t$ by $E_p$.
According to Theorem~\ref{theor:compo-seq-ind}, the overall privacy guarantee of the outputs of $\mathcal{M}$ is equal to the sum of all the privacy budgets of the respective privacy mechanisms that compose $\mathcal{M}$, i.e.,~$\sum_{t \in T}\varepsilon_t$.
\end{enumerate}
We assume that all the interactions between $E_g$ and $E_p$ are secure and private, and thus $E_p$ is considered trusted and non-adversarial by $E_g$.
Notice that, in a real life scenario, $E_g$ and $E_c$ might overlap with each other, i.e.,~data producers might be data consumers as well.
The identification of {\thething} events can be performed manually or automatically~\cite{zhou2004discovering, hariharan2004project}, and is an orthogonal problem to this current work.
In this work, we consider the {\thething} timestamps non-sensitive and provided by the user as input along with the privacy budget $\varepsilon$.
For example, the time series ($p_1$, \dots, $p_8$) with {\thethings} set the \{$p_1$, $p_3$, $p_5$\} is {\thething} neighboring to the time series of Figure~\ref{fig:scenario}.
We proceed to propose \emph{{\thething} privacy}, a configurable variation of differential privacy for time series (Definition~\ref{def:thething-prv}).
Let $\mathcal{M}$ be a privacy mechanism with range $\mathcal{O}$ that takes as input a time series.
We say that $\mathcal{M}$ satisfies {\thething}$\varepsilon$-differential privacy (or, simply, {\thething} privacy) if for all sets of possible outputs $O \subseteq\mathcal{O}$, and for every pair of {\thething}-neighboring time series $S_T$, $S_T'$,
Theorem~\ref{theor:thething-prv} proposes how to achieve the desired privacy for the {\thethings} (i.e.,~a total budget lower than $\varepsilon$), and in the same time provide better quality overall.
Let $\mathcal{M}$ be a mechanism with input a time series $S_T$, where $T$ is the set of the involved timestamps, and $L \subseteq T$ be the set of {\thething} timestamps.
$\mathcal{M}$ is decomposed to $\varepsilon$-differential private sub-mechanisms $\mathcal{M}_t$, for every $t \in T$, that apply independent randomness to the data item at $t$.
Then, given a privacy budget $\varepsilon$, $\mathcal{M}$ satisfies {\thething} privacy if for every $t$ it holds that
$$\sum_{i\in L \cup\{t\}}\varepsilon_i \leq\varepsilon$$
All mechanisms use independent randomness, and therefore for a time series $S_T ={D_1, \dots, D_T}$ and outputs $(\pmb{o}_1, \dots, \pmb{o}_T)\in O \subseteq\mathcal{O}$ it holds that
Since $S_T$ and $S'_T$ are {\thething}-neighboring, there exists $i \in T$ such that $D_i = D'_i$ for a set of {\thethings} with timestamps $L$.
Thus, we get
$$\frac{Pr[\mathcal{M}(S_T)=(\pmb{o}_1, \dots, \pmb{o}_T)]}{Pr[\mathcal{M}(S'_T)=(\pmb{o}_1, \dots, \pmb{o}_T)]}=\prod_{i \in L \cup\{t\}}\frac{Pr[\mathcal{M}_i(D_i)=\pmb{o}_i]}{Pr[\mathcal{M}_i(D'_i)=\pmb{o}_i]}$$
$D_i$ and $D'_i$ are neighboring for $i \in L \cup\{t\}$.
$\mathcal{M}_i$ is differential private and from Definition~\ref{def:dp} we get that $\frac{Pr[\mathcal{M}_i(D_i)=\pmb{o}_i]}{Pr[\mathcal{M}_i(D'_i)=\pmb{o}_i]}\leq e^{\varepsilon_i}$.
Hence, we can write
$$\frac{Pr[\mathcal{M}(S_T)=(\pmb{o}_1, \dots, \pmb{o}_T)]}{Pr[\mathcal{M}(S'_T)=(\pmb{o}_1, \dots, \pmb{o}_T)]}\leq\prod_{i \in L \cup\{t\}} e^{\varepsilon_i}= e^{\sum_{i \in L \cup\{t\}}\varepsilon_i}$$
For any $O \in\mathcal{O}$ we get $\frac{Pr[\mathcal{M}(S_T)\in O}{Pr[\mathcal{M}(S'_T)\in O]}\leq e^{\sum_{i \in L \cup\{t\}}\varepsilon_i}$.
If the formula of Theorem~\ref{theor:thething-prv} holds, then we get $\frac{Pr[\mathcal{M}(S_T)\in O}{Pr[\mathcal{M}(S'_T)\in O]}\leq e^\varepsilon$.
Due to Definition~\ref{def:thething-prv} this concludes our proof.
