\section{Privacy}
\label{sec:privacy}

When personal data are publicly released, either as microdata or statistical data, individuals' privacy can be compromised, i.e,~an adversary becomes certain about an individual's personal information with a probability higher than a desired threshold.
In the literature, this compromise is know as \emph{information disclosure} and is usually categorized as~\cite{li2007t, wang2010privacy, narayanan2008robust}:

\begin{itemize}
  \item \emph{Presence disclosure}---the participation (or absence) of an individual in a data set is revealed.
  \item \emph{Identity disclosure}---an individual is linked to a particular record.
  \item \emph{Attribute disclosure}---new information (attribute value) about an individual is revealed.
\end{itemize}

In the literature, identity disclosure is also referred to as \emph{record linkage}, and presence disclosure as \emph{table linkage}.
Notice that identity disclosure can result in attribute disclosure, and vice versa.

To better illustrate these definitions, we provide some examples based on Table~\ref{tab:snapshot}.
Presence disclosure appears when by looking at the (privacy-protected) counts of Table~\ref{tab:snapshot-statistical}, we can guess if Quackmore has participated in Table~\ref{tab:snapshot-micro}.
Identity disclosure appears when we can guess that the sixth record of (a privacy-protected version of) the microdata of Table~\ref{tab:snapshot-micro} belongs to Quackmore.
Attribute disclosure appears when it is revealed from (a privacy-protected version of) the microdata of Table~\ref{tab:snapshot-micro} that Quackmore is $62$ years old.


\subsection{Levels}
\label{subsec:prv-levels}

The information disclosure that a data release may entail is linked to the protection level that indicates \emph{what} a privacy-preserving algorithm is trying to achieve.
More specifically, in continuous data publishing we consider the privacy protection level with respect to not only the users but also to the \emph{events} occurring in the data.
An event is a pair of an identifying attribute of an individual and the sensitive data (including contextual information) and we can see it as a correspondence to a record in a database, where each individual may participate once.
Data publishers typically release events in the form of sequences of data items, usually indexed in time order (time series) and geotagged, e.g.,~(`Dewey', `at home at Montmartre at $t_1$'), \dots, (`Quackmore', `dining at Opera at $t_1$').
We use the term `users' to refer to the \emph{individuals}, also known as \emph{participants}, who are the source of the processed and published data.
Therefore, they should not be confused with the consumers of the released data sets.
Users are subject to privacy attacks, and thus are the main point of interest of privacy protection mechanisms.
In more detail, the privacy protection levels are:

\begin{enumerate}[(a)]
  \item \emph{Event}~\cite{dwork2010differential, dwork2010pan}---limits the privacy protection to \emph{any single event} in a time series, providing maximum data utility.
  \item \emph{$w$-event}~\cite{kellaris2014differentially}---provides privacy protection to \emph{any sequence of $w$ events} in a time series.
  \item \emph{User}~\cite{dwork2010differential, dwork2010pan}---protects \emph{all the events} in a time series, providing maximum privacy protection.
\end{enumerate}

Figure~\ref{fig:prv-levels} demonstrates the application of the possible protection levels on the statistical data of Example~\ref{ex:continuous}.
For instance, in event-level (Figure~\ref{fig:level-event}) it is hard to determine whether Quackmore was dining at Opera at $t_1$.
Moreover, in user-level (Figure~\ref{fig:level-user}) it is hard to determine whether Quackmore was ever included in the released series of events at all.
Finally, in $2$-event-level (Figure~\ref{fig:level-w-event}) it is hard to determine whether Quackmore was ever included in the released series of events between the timestamps $t_1$ and $t_2$, $t_2$ and $t_3$, etc. (i.e.,~for a window $w = 2$).

\begin{figure}[htp]
  \centering
  \hspace{\fill}\subcaptionbox{Event-level\label{fig:level-event}}{%
    \includegraphics[width=.32\linewidth]{level-event}%
  }\hspace{\fill}
  \subcaptionbox{User-level\label{fig:level-user}}{%
    \includegraphics[width=.32\linewidth]{level-user}%
  }\hspace{\fill}
  \subcaptionbox{$2$-event-level\label{fig:level-w-event}}{%
    \includegraphics[width=.32\linewidth]{level-w-event}%
  }\hspace{\fill}
  \caption{Protecting the data of Table~\ref{tab:continuous-statistical} on (a)~event-, (b)~user-, and (c)~$2$-event-level. A suitable distortion method can be applied accordingly.}
  \label{fig:prv-levels}
\end{figure}

Contrary to event-level, that provides privacy guarantees for a single event, user- and $w$-event-level offer stronger privacy protection by protecting a series of events.
Event- and $w$-event-level handle better scenarios of infinite data observation, whereas user-level is more appropriate when the span of data observation is finite.
$w$-event- is narrower than user-level protection due to its sliding window processing methodology.
In the extreme cases where $w$ is equal to either $1$ or to the size of the entire length of the time series, $w$-event- matches event- or user-level protection, respectively.
Although the described levels have been coined in the context of \emph{differential privacy}~\cite{dwork2006calibrating}, a seminal privacy method that we will discuss in more detail in Section~\ref{subsec:prv-statistical}, it is possible to apply their definitions to other privacy protection techniques as well.


\subsection{Attacks}
\label{subsec:prv-attacks}

Information disclosure is typically achieved by combining supplementary (background) knowledge with the released data or by setting unrealistic assumptions while designing the privacy-preserving algorithms.
In its general form, this is known as \emph{adversarial} or \emph{linkage} attack.
Even though many works directly refer to the general category of linkage attacks, we distinguish also the following sub-categories, addressed in the literature:

\begin{itemize}
  \item \emph{Sensitive attribute domain} knowledge.
  Here we can identify \emph{homogeneity and skewness} attacks~\cite{machanavajjhala2006diversity,li2007t}, when statistics of the sensitive attribute values are available, and \emph{similarity attack}, when semantics of the sensitive attribute values are available.
  \item \emph{Complementary release} attacks~\cite{sweeney2002k} with regard to previous releases of different versions of the same and/or related data sets.
  In this category, we also identify the \emph{unsorted matching} attack~\cite{sweeney2002k}, which is achieved when two privacy-protected versions of an original data set are published in the same tuple ordering.
  Other instances include: (i)~the \emph{join} attack~\cite{wang2006anonymizing}, when tuples can be identified by joining (on the (quasi-)identifiers) several releases, (ii)~the \emph{tuple correspondence} attack~\cite{fung2008anonymity}, when in case of incremental data certain tuples correspond to certain tuples in other releases, in an injective way, (iii)~the \emph{tuple equivalence} attack~\cite{he2011preventing}, when tuples among different releases are found to be equivalent with respect to the sensitive attribute, and (iv)~the \emph{unknown releases} attack~\cite{shmueli2015privacy}, when the privacy preservation is performed without knowing the previously privacy-protected data sets.
  \item \emph{Data dependence}
  \begin{itemize}
    \item within one data set.
    Data tuples and data values within a data set may be correlated, or linked in such a way that information about one person can be inferred even if the person is absent from the database.
    Consequently, in this category we put assumptions made on the data generation model based on randomness, like the random world model, the independent and identically distributed data (i.i.d.) model, or the independent-tuples model, which may be unrealistic for many real-world scenarios.
    This attack is also known as the \emph{deFinetti's attack}~\cite{kifer2009attacks}.
    \item among one data set and previous data releases, and/or other external sources~\cite{kifer2011no, chen2014correlated, liu2016dependence, zhao2017dependent}.
    The strength of the dependence between a pair of variables can be quantified with the utilization of \emph{correlations}~\cite{stigler1989francis}.
    Correlation implies dependence but not vice versa, however, the two terms are often used as synonyms.
    The correlation among nearby observations, i.e.,~the elements in a series of data points, are referenced as \emph{autocorrelation} or \emph{serial correlation}~\cite{park2018fundamentals}.
    Depending on the evaluation technique, e.g.,~\emph{Pearson's correlation coefficient}~\cite{stigler1989francis}, a correlation can be characterized as \emph{negative}, \emph{zero}, or \emph{positive}.
    A negative value shows that the behavior of one variable is the \emph{opposite} of that of the other, e.g.,~when the one increases the other decreases.
    Zero means that the variables are not linked and are \emph{independent} of each other.
    A positive correlation indicates that the variables behave in a \emph{similar} manner, e.g.,~when the one decreases the other decreases as well.

    The most prominent types of correlations might be:
    \begin{itemize}
      \item \emph{Temporal}~\cite{wei2006time}---appearing in observations (i.e.,~values) of the same object over time.
      \item \emph{Spatial}~\cite{legendre1993spatial, anselin1995local}---denoted by the degree of similarity of nearby data points in space, and indicating if and how phenomena relate to the (broader) area where they take place.
      \item \emph{Spatiotemporal}---a combination of the previous categories, appearing when processing time series or sequences of human activities with geolocation characteristics, e.g.,~\cite{ghinita2009preventing}.
    \end{itemize}
    Contrary to one-dimensional correlations, spatial correlation is multi-dimensional and multi-directional, and can be measured by indicators (e.g.,~\emph{Moran's I}~\cite{moran1950notes}) that reflect the \emph{spatial association} of the concerned data.
    Spatial autocorrelation has its foundations in the \emph{First Law of Geography} stating that ``everything is related to everything else, but near things are more related than distant things''~\cite{tobler1970computer}.
    A positive spatial autocorrelation indicates that similar data are \emph{clustered}, a negative that data are dispersed and are close to dissimilar ones, and when close to zero, that data are \emph{randomly arranged} in space.
  \end{itemize}

  A common practice for extracting data dependencies from continuous data, is by expressing the data as a \emph{stochastic} or \emph{random process}.
  A random process is a collection of \emph{random variables} or \emph{bivariate data}, indexed by some set, e.g.,~a series of timestamps, a Cartesian plane $\mathbb{R}^2$, an $n$-dimensional Euclidean space, etc.~\cite{skorokhod2005basic}.
  The values a random variable can take are outcomes of an unpredictable process, while bivariate data are pairs of data values with a possible association between them.
  Expressing data as stochastic processes allows their modeling depending on their properties, and thereafter the discovery of relevant data dependencies.
  Some common stochastic processes modeling techniques include:

  \begin{itemize}
    \item \emph{Conditional probabilities}~\cite{allan2013probability}---probabilities of events in the presence of other events.
    \item \emph{Conditional Random Fields} (CRFs)~\cite{lafferty2001conditional}---undirected graphs encoding conditional probability distributions.
    \item \emph{Markov processes}~\cite{rogers2000diffusions}---stochastic processes for which the conditional probability of their future states depends only on the present state and it is independent of its previous states (\emph{Markov assumption}).
    \begin{itemize}
      \item \emph{Markov chains}~\cite{gagniuc2017markov}---sequences of possible events whose probability depends on the state attained in the previous event.
      \item \emph{Hidden Markov Models} (HMMs)~\cite{baum1966statistical}---statistical Markov models of Markov processes with unobserved states.
    \end{itemize}
  \end{itemize}

\end{itemize}

The first sub-category of attacks has been mainly addressed in works on snapshot microdata publishing, and is still present in continuous publishing; however, algorithms for continuous publishing  typically accept the proposed solutions for the snapshot publishing scheme (see discussion over $k$-anonymity and $l$-diversity in Section~\ref{subsec:prv-seminal}).
This kind of attacks is tightly coupled with publishing the (privacy-protected) sensitive attribute value.
An example is the lack of diversity in the sensitive attribute domain, e.g.,~if all users in the data set of Table~\ref{tab:snapshot-micro} shared the same \emph{running} Status  (the sensitive attribute).
The second and third subcategory are attacks emerging (mostly) in continuous publishing scenarios.
Consider again the data set in Table~\ref{tab:snapshot-micro}.
The complementary release attack means that an adversary can learn more things about the individuals (e.g.,~that there are high chances that Donald was at work) if he/she combines the information of two privacy-protected versions of this data set.
By the data dependence attack, the status of Donald could be more certainly inferred, by taking into account the status of Dewey at the same moment and the dependencies between Donald's and Dewey's status, e.g.,~when Dewey is at home, then most probably Donald is at work.
In order to better protect the privacy of Donald in case of attacks, the data should be privacy-protected in a more adequate way (than without the attacks).


\subsection{Operations}
\label{subsec:prv-operations}

Protecting private information, which is known by many names (obfuscation, cloaking, anonymization, etc.), is achieved by using a specific basic privacy protection operation.
Depending on the intervention that we choose to perform on the original data, we identify the following operations:

\begin{itemize}
  \item \emph{Aggregation}---group together multiple rows of a data set to form a single value.
  \item \emph{Generalization}---replace an attribute value with a parent value in the attribute taxonomy.
  Notice that a step of generalization, may be followed by a step of \emph{specialization}, to improve the quality of the resulting data set.
  \item \emph{Suppression}---delete completely certain sensitive values or entire records.
  \item \emph{Perturbation}---disturb the initial attribute value in a deterministic or probabilistic way.
  The probabilistic data distortion is referred to as \emph{randomization}.
\end{itemize}

For example, consider the table schema \emph{User(Name, Age, Location, Status)}.
If we want to protect the \emph{Age} of the user by aggregation, we may  replace it by  the average age in her Location; by generalization, we may replace the Age by age intervals; by suppression we may delete the entire table column corresponding to \emph{Age}; by perturbation, we may augment each age by a predefined percentage of the age; by randomization we may randomly replace each age by a value taken from the probability density function of the attribute.

It is worth mentioning that there is a series of algorithms (e.g.,~\cite{benaloh2009patient, kamara2010cryptographic, cao2014privacy}) based on the \emph{cryptography} operation.
However, the majority of these methods, among other assumptions that they make, have minimum or even no trust to the entities that handle the personal information.
Furthermore, the amount and the way of data processing of these techniques usually burden the overall procedure, deteriorate the utility of the resulting data sets, and restrict their applicability.
Our focus is limited to techniques that achieve a satisfying balance between both participants' privacy and data utility.
For these reasons, there will be no further discussion around this family of techniques in this article.


\subsection{Seminal works}
\label{subsec:prv-seminal}

For completeness, in this section we present the seminal works for privacy-preserving data publishing, which, even though originally designed for the snapshot publishing scenario, have paved the way, since many of the works in privacy-preserving continuous publishing are based on or extend them.


\subsubsection{Microdata}
\label{subsec:prv-micro}

Sweeney coined \emph{$k$-anonymity}~\cite{sweeney2002k}, one of the first established works on data privacy.
A released data set features $k$-anonymity protection when the sequence of values for a set of identifying attributes, called the \emph{quasi-identifiers}, is the same for at least $k$ records in the data set.
Computing the quasi-identifiers in a set of attributes is still a hard problem on its own~\cite{motwani2007efficient}.
$k$-anonymity is syntactic, it constitutes an individual indistinguishable from at least $k-1$ other individuals in the same data set.
In a follow-up work~\cite{sweeney2002achieving}, the author describes a way to achieve $k$-anonymity for a data set by the suppression or generalization of certain values of the quasi-identifiers.
Machanavajjhala et al.~\cite{machanavajjhala2006diversity} pointed out that $k$-anonymity is vulnerable to homogeneity and background knowledge attacks.
Thereby, they proposed \emph{$l$-diversity}, which demands that the values of the sensitive attributes are `well-represented' by $l$ sensitive values in each group.
Principally, a data set can be $l$-diverse by featuring at least $l$ distinct values for the sensitive field in each group (\emph{distinct} $l$-diversity).
Other instantiations demand that the entropy of the whole data set is greater than or equal to $\log(l)$ (\emph{entropy} $l$-diversity) or that the number of appearances of the most common sensitive value is less than the sum of the counts of the rest of the values multiplied by a user defined constant $c$ (\emph{recursive (c, l)}-diversity).
Later on, Li et al.~\cite{li2007t} indicated that $l$-diversity can be void by skewness and similarity attacks due to sensitive attributes with a small value range.
In such cases, \emph{$\theta$-closeness} guarantees that the distribution of a sensitive attribute in a group and the distribution of the same attribute in the whole data set is `similar'.
This similarity is bounded by a threshold $\theta$.
A data set features  $\theta$-closeness when all of its groups feature $\theta$-closeness.

The main drawback of $k$-anonymity (and its derivatives) is that it is not tolerant to external attacks of re-identification on the released data set.
The problems identified in~\cite{sweeney2002k} appear when attempting to apply $k$-anonymity on continuous data publishing (as we will also see next in Section~\ref{sec:micro}).
These attacks include multiple $k$-anonymous data set releases with the same record order, subsequent releases of a data set without taking into account previous $k$-anonymous releases, and tuple updates.
Proposed solutions include rearranging the attributes, setting the whole attribute set of previously released data sets as quasi-identifiers or releasing data based on previous $k$-anonymous releases.


\subsubsection{Statistical data}
\label{subsec:prv-statistical}

While methods based on $k$-anonymity have been mainly employed for releasing microdata, \emph{differential privacy}~\cite{dwork2006calibrating} has been proposed for releasing high utility aggregates over microdata while providing semantic privacy guarantees.
Differential privacy is algorithmic, it ensures that any adversary observing a privacy-protected output, no matter his/her computational power or auxiliary information, cannot conclude with absolute certainty if an individual is included in the input data set.
Moreover, it quantifies and bounds the impact that the addition/removal of the data of an individual to/from an input data set has on the derived privacy-protected aggregates.

\begin{definition}
  [Neighboring data sets]
  \label{def:nb-d-s}
  Two data sets are neighboring (or adjacent) when they differ by at most one tuple, i.e.,~one can be obtained by adding/removing the data of an individual to/from the other.
\end{definition}

More precisely, differential privacy quantifies the impact of the addition/removal of a single tuple in $D$ on the output $\pmb{o}$ of $\mathcal{M}$.
The distribution of all $\pmb{o}$, in some range $\mathcal{O}$, is not affected \emph{substantially}, i.e.,~it changes only slightly due to the modification of any one tuple in all possible $D \in \mathcal{D}$.
Thus, differential privacy is algorithmic, it ensures that any adversary observing any $\pmb{o}$ cannot conclude with absolute certainty whether or not any individual is included in any $D$.
Its performance is irrelevant to the computational power and auxiliary information available to an adversary observing the outputs of $\mathcal{M}$.

\begin{definition}
  [Differential privacy]
  \label{def:dp}
  A privacy mechanism $\mathcal{M}$, with domain $\mathcal{D}$ and range $\mathcal{O}$, satisfies $\varepsilon$-differential privacy, for a given privacy budget $\varepsilon$, if for every pair of neighboring data sets $D, D' \in \mathcal{D}$ and all sets $O \subseteq \mathcal{O}$:
  $$\Pr[\mathcal{M}(D) \in O] \leq e^\varepsilon \Pr[\mathcal{M}(D') \in O]$$
\end{definition}

\noindent $\Pr[\cdot]$ denotes the probability of $\mathcal{M}$ generating $\pmb{o}$ as output, from a set of $O \subseteq \mathcal{O}$, when given any version of $D$ as input.
The privacy budget $\varepsilon$ is a positive real number that represents the user-defined privacy goal~\cite{mcsherry2009privacy}.
As the definition implies, $\mathcal{M}$ achieves stronger privacy protection for lower values of $\varepsilon$ since the probabilities of $D$ and $D'$ being true worlds are similar, but the utility of $\pmb{o}$ is reduced since more randomness is introduced by $\mathcal{M}$.
The privacy budget $\varepsilon$ is usually set to $0.01$, $0.1$, or, in some cases, $\ln2$ or $\ln3$~\cite{lee2011much}.

\begin{definition}
  [Query function sensitivity]
  \label{def:qry-sens}
  The sensitivity of a query function $f$ for all neighboring data sets $D, D' \in \mathcal{D}$ is:
  $$\Delta f = \max_{D, D' \in \mathcal{D}} \lVert {f(D) - f(D')} \rVert_{1}$$
\end{definition}

The pertinence of differential privacy methods is inseparable from the query's function sensitivity.
The presence/absence of a single record can only change the result slightly, and therefore differential privacy methods are best for low sensitivity queries such as counts.
However, sum and max queries can be problematic since a single (very different) value could change the output noticeably, making it necessary to add a lot of noise to the query's answer.
Furthermore, asking a series of queries may allow the disambiguation between possible data sets, making it necessary to add even more noise to the outputs.
For this reason, after a series of queries exhausts the available privacy budget the data set has to be discarded.
Keeping the original guarantee across multiple queries that require different/new answers requires the injection of noise proportional to the number of the executed queries, and thus destroying the utility of the output.

\paragraph{Privacy mechanisms}
\label{subsec:prv-mech}
A typical example of differential privacy mechanism is the \emph{Laplace mechanism}~\cite{dwork2014algorithmic}.
It draws randomly a value from the probability distribution of $\textrm{Laplace}(\mu, b)$, where $\mu$ stands for the location parameter and $b > 0$ the scale parameter (Figure~\ref{fig:laplace}).
Here, $\mu$ is equal to the original output value of a query function, and $b$ is the sensitivity of the query function divided by $\varepsilon$.
The Laplace mechanism works for any function with range the set of real numbers.
A specialization of this mechanism for location data is the \emph{Planar Laplace mechanism}~\cite{andres2013geo}, which is based on a multivariate Laplace distribution.

\begin{figure}[htp]
  \centering
  \includegraphics[width=.7\linewidth]{laplace}
  \caption{A Laplace distribution for location $\mu = 2$ and scale $b = 1$.}
  \label{fig:laplace}
\end{figure}

For query functions that do not return a real number, e.g.,~`What is the most visited country this year?' or in cases where perturbing the value of the output will completely destroy its utility, e.g.,~`What is the optimal price for this auction?', most works in the literature use the \emph{Exponential mechanism}~\cite{mcsherry2007mechanism}.
This mechanism utilizes a utility function $u$ that maps (input data set $D$, output value $r$) pairs to utility scores, and selects an output value $r$ from the input pairs, with probability proportional to $\exp(\frac{\varepsilon u(D, r)}{2\Delta u})$,
where $\Delta u$ is the sensitivity of the utility function.

Another technique for differential privacy mechanisms is the \emph{randomized response}~\cite{warner1965randomized}.
It is a privacy-preserving survey method that introduces probabilistic noise to the statistics of a research by randomly instructing respondents to answer truthfully or `Yes' to a sensitive, binary question.
The technique achieves this randomization by including a random event, e.g.,~the flip of a fair coin.
The respondents reveal to the interviewers only their answer to the question, and keep as a secret the result of the random event (i.e.,~if the coin was tails or heads).
Thereafter, the interviewers can calculate the probability distribution of the random event, e.g.,~$\frac{1}{2}$ heads and $\frac{1}{2}$ tails, and thus they can roughly eliminate the false responses and estimate the final result of the research.

A special category of differential privacy-preserving algorithms is that of \emph{pan-private} algorithms~\cite{dwork2010pan}.
Pan-private algorithms hold their privacy guarantees even when snapshots of their internal state (memory) are accessed during their execution by an external entity, e.g.,~subpena, security breach, etc.
There are two intrusion types that a data publisher has to deal with when designing a pan-private mechanism: \emph{single unannounced}, and \emph{continual announced} intrusion.
In the first, the data publisher assumes that the mechanism's state is observed by the external entity one unique time, without the data publisher ever being notified about it.
In the latter, the external entity gains access to the mechanism's state multiple times, and the publisher is notified after each time.
The simplest approach to deal with both cases is to make sure that the data in the memory of the mechanism have constantly the same distribution, i.e.,~they are differentially private.
Notice that this must hold throughout the mechanism's lifetime, even before/\allowbreak after it processes any sensitive data item(s).

The notion of differential privacy has highly influenced the research community, resulting in many follow-up publications (\cite{mcsherry2007mechanism, kifer2011no, zhang2017privbayes} to mention a few).
We distinguish here \emph{Pufferfish}~\cite{kifer2014pufferfish} and \emph{geo-indistinguishability}~\cite{andres2013geo,chatzikokolakis2015geo}.
\emph{Pufferfish} is a framework that allows experts in an application domain, without necessarily having any particular expertise in privacy, to develop privacy definitions for their data sharing needs.
To define a privacy mechanism using \emph{Pufferfish}, one has to define a set of potential secrets $\mathcal{X}$, a set of distinct pairs $\mathcal{X}_{pairs}$, and auxiliary information about data evolution scenarios $\mathcal{B}$.
$\mathcal{X}$ serves as an explicit specification of what we would like to protect, e.g.,~`the record of an individual $x$ is (not) in the data'.
$\mathcal{X}_{pairs}$ is a subset of $\mathcal{X} \times \mathcal{X}$ that instructs how to protect the potential secrets $\mathcal{X}$, e.g.,~(`$x$ is in the table', `$x$ is not in the table').
Finally, $\mathcal{B}$ is a set of conservative assumptions about how the data evolved (or were generated) that reflects the adversary's belief about the data, e.g.,~probability distributions, variable correlations, etc.
When there is independence between all the records in the original data set, then $\varepsilon$-differential privacy and the privacy definition of $\varepsilon$-\emph{Pufferfish}$(\mathcal{X}, \mathcal{X}_{pairs}, \mathcal{B})$ are equivalent.
\emph{Geo-indistinguishability} is an adaptation of differential privacy for location data in snapshot publishing.
It is based on $l$-privacy, which offers to individuals within an area with radius $r$, a privacy level of $l$.
More specifically, $l$ is equal to $\varepsilon r$ if any two locations within distance $r$ provide data with similar distributions.
This similarity depends on $r$ because the closer two locations are, the more likely they are to share the same features.
Intuitively, the definition implies that if an adversary learns the published location for an individual, the adversary cannot infer the individual's true location, out of all the points in a radius $r$, with a certainty higher than a factor depending on $l$.
The technique adds random noise drawn from a multivariate Laplace distribution to individuals' locations, while taking into account spatial boundaries and features.

\paragraph{Composition}
\label{subsec:compo}

Mechanisms that satisfy differential privacy are \emph{composable}, i.e.,~the combination of their results satisfy differential privacy as well.
In this section, we provide an overview of the most prominent composition theorems that instruct data publishers \emph{how} to estimate the overall privacy protection when utilizing a series of differential privacy mechanisms.

\begin{theorem}
  [Composition]
  \label{theor:compo}
  Any combination of a set of independent differential privacy mechanisms satisfying a corresponding set of privacy guarantees shall satisfy differential privacy as well, i.e.,~provide a differentially private output.
\end{theorem}

Generally, when we apply a series of independent (i.e.,~in the way that they inject noise) differential privacy mechanisms on independent data, we can calculate the privacy level of the resulting output according to the \emph{sequential} composition property~\cite{mcsherry2009privacy, soria2016big}.

\begin{theorem}
  [Sequential composition on independent data]
  \label{theor:compo-seq-ind}
  The privacy guarantee of $m \in \mathbb{Z}^+$ independent privacy mechanisms, satisfying $\varepsilon_1$-, $\varepsilon_2$-, \dots, $\varepsilon_m$-differential privacy respectively, when applied over the same data set equals to $\sum_{i = 1}^m \varepsilon_i$.
\end{theorem}

Notice that the sequential composition corresponds to the worst case scenario where each time we use a mechanism we have to invest some (or all) of the available privacy budget.
In the special case that we query disjoint data sets, we can take advantage of the \emph{parallel} composition property~\cite{mcsherry2009privacy, soria2016big}, and thus spare some of the available privacy budget.

\begin{theorem}
  [Parallel composition on independent data]
  \label{theor:compo-par-ind}
  When $m \in \mathbb{Z}^+$ independent privacy mechanisms, satisfying $\varepsilon_1$-, $\varepsilon_2$-,\dots, $\varepsilon_m$-differential privacy respectively, are applied over disjoint independent subsets of a data set, they provide a privacy guarantee equal to $\max_{i \in [1, m]} \varepsilon_i$.
\end{theorem}

When the users consider recent data releases more privacy sensitive than distant ones, we estimate the overall privacy loss in a time fading manner according to a temporal discounting function, e.g.,~exponential, hyperbolic,~\cite{farokhi2020temporally}.

\begin{theorem}
  [Sequential composition with temporal discounting]
  \label{theor:compo-seq-disc}
  A set of $m \in \mathbb{Z}^+$ independent privacy mechanisms, satisfying $\varepsilon_1$-, $\varepsilon_2$-,\dots, $\varepsilon_m$-differential privacy respectively, satisfy $\sum_{i = 1}^m g(i) \varepsilon_i$ differential privacy for a discount function $g$.
\end{theorem}

% The presence of temporal correlations might result into additional privacy loss consisting of \emph{backward privacy loss} $\alpha^B$ and \emph{forward privacy loss} $\alpha^F$~\cite{cao2017quantifying}.
Cao et al.~\cite{cao2017quantifying} propose a method for computing the temporal privacy loss (TPL) of a differential privacy mechanism in the presence of temporal correlations and background knowledge.
The goal of their technique is to guarantee privacy protection and to bound the privacy loss at every timestamp under the assumption of independent data releases.
It calculates the temporal privacy loss as the sum of the backward and forward privacy loss minus the default privacy loss $\varepsilon$ of the mechanism (because it is counted twice in the aforementioned entities).
This calculation is done for each individual that is included in the original data set and the overall temporal privacy loss is equal to the maximum calculated value at every timestamp.
The backward/forward privacy loss at any timestamp depends on the backward/forward privacy loss at the previous/next timestamp, the backward/forward temporal correlations, and $\varepsilon$.

\begin{definition}
  [Temporal privacy loss (TPL)]
  \label{def:tpl}
  The potential privacy loss of a privacy mechanism at a timestamp $t \leq T$ due to a series of outputs $\pmb{o}_1$, \dots, $\pmb{o}_T$ and temporal correlations in its input $D_t$ with respect to any adversary, targeting an individual with potential data items $x_t$ (or $x'_t$) and having knowledge $\mathbb{D}_t$ equal to $D_t - \{x_t\}$ (or $D'_t - \{x'_t\}$), is defined as:

  \begin{equation}
    \label{eq:tpl}
    \alpha_t = \sup_{x_t, x'_t, \pmb{o}_1, \dots, \pmb{o}_T} \ln \frac{\Pr[\pmb{o}_1, \dots, \pmb{o}_T | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_1, \dots, \pmb{o}_T | x'_t, \mathbb{D}_t]}
  \end{equation}
\end{definition}
%
By analyzing Equation~\ref{eq:tpl} we get the following:

\begin{align}
  \label{eq:tpl-1}
  (\ref{eq:tpl}) = & \underbrace{\sup_{x_t, x'_t, \pmb{o}_1, \dots, \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_1, \dots, \pmb{o}_t| x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_1, \dots, \pmb{o}_t | x'_t, \mathbb{D}_t]}}_{\text{Backward privacy loss}\ (\alpha^B_t)} \nonumber \\
  & + \underbrace{\sup_{x_t, x'_t, \pmb{o}_t, \dots, \pmb{o}_T} \ln \frac{\Pr[\pmb{o}_t, \dots, \pmb{o}_T| x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_t, \dots, \pmb{o}_T | x'_t, \mathbb{D}_t]}}_{\text{Forward privacy loss}\ (\alpha^F_t)} \nonumber \\
  & - \underbrace{\sup_{x_t, x'_t, \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_t | x_t, \mathbb{D}_t] }{\Pr[\pmb{o}_t | x'_t, \mathbb{D}_t]}}_{\text{Present privacy loss}\ (\varepsilon_t)}
\end{align}

\begin{definition}
  [Backward privacy loss (BPL)]
  \label{def:bpl}
  The potential privacy loss of a privacy mechanism at a timestamp $t \leq T$ due to outputs $\pmb{o}_1$, \dots, $\pmb{o}_t$ and temporal correlations in its input $D_t$ with respect to any adversary, targeting an individual with potential data items $x_t$ (or $x'_t$) and having knowledge $\mathbb{D}_t$ equal to $D_t - \{x_t\}$ (or $D'_t - \{x'_t\}$), is called backward privacy loss and is defined as:

  \begin{equation}
    \label{eq:bpl-1}
    \alpha^B_t = \sup_{x_t, x'_t, \pmb{o}_1, \dots, \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_1, \dots, \pmb{o}_t | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_1, \dots, \pmb{o}_t | x'_t, \mathbb{D}_t]}
  \end{equation}

\end{definition}
%
From differential privacy we have the assumption that $\pmb{o}_1$, \dots, $\pmb{o}_t$ are independent events.
Therefore, according to the Bayesian theorem, we can write Equation~\ref{eq:bpl-1} as:

\begin{align}
  \label{eq:bpl-2}
  (\ref{eq:bpl-1}) = & \sup_{x_t, x'_t, \pmb{o}_1, \dots, \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1}| x_t, \mathbb{D}_t] \Pr[\pmb{o}_t | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x'_t, \mathbb{D}_t] \Pr[\pmb{o}_t | x'_t, \mathbb{D}_t]} \nonumber \\
  = & \sup_{x_t, x_t', \pmb{o}_1, \dots, \pmb{o}_{t - 1}} \ln \frac{\Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x_t', \mathbb{D}_t]} \nonumber \\
  & + \sup_{x_t, x_t', \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_t | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_t | x'_t, \mathbb{D}_t]}
\end{align}
%
Applying the law of total probability to the first term of Equation~\ref{eq:bpl-2} for all the possible data $x_{t - 1}$ (or  $x'_{t - 1}$) and $\mathbb{D}_{t - 1}$ we get the following:

\begin{align}
  \label{eq:bpl-3}
  (\ref{eq:bpl-2}) = &
  \adjustbox{max width=0.9\linewidth}{
    $\sup\limits_{x_t, x'_t, \pmb{o}_1, \dots, \pmb{o}_{t - 1}} \ln \frac{\sum\limits_{x_{t - 1}} \Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x_t, \mathbb{D}_t, x_{t - 1}, \mathbb{D}_{t - 1}] \Pr[x_{t - 1}, \mathbb{D}_{t - 1} | x_t, \mathbb{D}_t]}{\sum\limits_{x'_{t - 1}} \Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x'_t, \mathbb{D}_t, x'_{t - 1}, \mathbb{D}_{t - 1}] \Pr[x'_{t - 1}, \mathbb{D}_{t - 1} | x'_t, \mathbb{D}_t]}$
  } \nonumber \\
  & \adjustbox{max width=0.3\linewidth}{
    $+ \sup\limits_{x_t, x_t', \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_t | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_t | x'_t, \mathbb{D}_t]}$
  }
\end{align}
%
Since $\mathbb{D}_t$ is equal to $D_t - \{x_t\}$ (or $D'_t - \{x'_t\}$), and thus is constant and independent of every possible $x_t$ (or $x'_t$), $\forall t \leq T$, Equation~\ref{eq:bpl-3} can be written as:

\begin{align}
  \label{eq:bpl-4}
  (\ref{eq:bpl-3}) = &
  \adjustbox{max width=0.9\linewidth}{
    $\sup\limits_{x_t, x'_t, \pmb{o}_1, \dots, \pmb{o}_{t - 1}} \ln \frac{\sum\limits_{x_{t - 1}} \Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x_t, \mathbb{D}_t, x_{t - 1}, \mathbb{D}_{t - 1}] \Pr[x_{t - 1} | x_t, \mathbb{D}_t] \Pr[\mathbb{D}_{t - 1} | x_t, \mathbb{D}_t]}{\sum\limits_{x'_{t - 1}} \Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x'_t, \mathbb{D}_t, x'_{t - 1}, \mathbb{D}_{t - 1}] \Pr[x'_{t - 1} | x'_t, \mathbb{D}_t] \Pr[\mathbb{D}_{t - 1} | x'_t, \mathbb{D}_t]}$
  } \nonumber \\
  & \adjustbox{max width=0.275\linewidth}{
    $+ \sup\limits_{x_t, x_t', \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_t | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_t | x'_t, \mathbb{D}_t]}$
  } \nonumber \\
  = & \adjustbox{max width=0.825\linewidth}{
    $\sup\limits_{x_t, x'_t, \pmb{o}_1, \dots, \pmb{o}_{t - 1}} \ln \frac{\sum\limits_{x_{t - 1}} \Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x_t, \mathbb{D}_t, x_{t - 1}, \mathbb{D}_{t - 1}] \Pr[x_{t - 1} | x_t] \Pr[\mathbb{D}_{t - 1} | \mathbb{D}_t]}{\sum\limits_{x'_{t - 1}} \Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x'_t, \mathbb{D}_t, x'_{t - 1}, \mathbb{D}_{t - 1}] \Pr[x'_{t - 1} | x'_t] \Pr[\mathbb{D}_{t - 1} | \mathbb{D}_t]}$
  } \nonumber \\
  & \adjustbox{max width=0.275\linewidth}{
    $+ \sup\limits_{x_t, x_t', \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_t | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_t | x'_t, \mathbb{D}_t]}$
  } \nonumber \\
  = & \adjustbox{max width=0.7\linewidth}{
    $\sup\limits_{x_t, x'_t, \pmb{o}_1, \dots, \pmb{o}_{t - 1}} \ln \frac{\sum\limits_{x_{t - 1}} \Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x_t, \mathbb{D}_t, x_{t - 1}, \mathbb{D}_{t - 1}] \Pr[x_{t - 1} | x_t]}{\sum\limits_{x'_{t - 1}} \Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x'_t, \mathbb{D}_t, x'_{t - 1}, \mathbb{D}_{t - 1}] \Pr[x'_{t - 1} | x'_t]}$
  } \nonumber \\
  & \adjustbox{max width=0.275\linewidth}{
    $+ \sup\limits_{x_t, x_t', \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_t | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_t | x'_t, \mathbb{D}_t]}$
  }
\end{align}
%
The outputs $\pmb{o}_1$, \dots, $\pmb{o}_t$ and $x_t$ (or $x'_t$) are conditionally independent in the presence of
$x_{t - 1}$ (or $x'_{t - 1}$), and thus Equation~\ref{eq:bpl-4} can be written as:

\begin{align}
  \label{eq:bpl-5}
  (\ref{eq:bpl-4}) = &
  \adjustbox{max width=0.9\linewidth}{
    $\sup\limits_{x_t, x'_t, \pmb{o}_1, \dots, \pmb{o}_{t - 1}} \ln \cfrac{\sum\limits_{x_{t - 1}} \Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x_{t - 1}, \mathbb{D}_{t - 1}] \Pr[x_{t - 1} | x_t]}{\sum\limits_{x'_{t - 1}} \underbrace{\Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | x'_{t - 1}, \mathbb{D}_{t - 1}]}_{\alpha^B_{t - 1}} \underbrace{\Pr[x'_{t - 1} | x'_t]}_{P^B_{t - 1}}}$
  } \nonumber \\
  & \adjustbox{max width=0.4\linewidth}{
    $+ \underbrace{\sup\limits_{x_t, x'_t, \pmb{o}_t} \ln \cfrac{\Pr[\pmb{o}_t | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_t | x'_t, \mathbb{D}_t]}}_{\varepsilon_t}$
  }
\end{align}

\begin{definition}
  [Forward privacy loss (FPL)]
  \label{def:fpl}
  The potential privacy loss of a privacy mechanism at a timestamp $t \leq T$ due to outputs $\pmb{o}_t$,\dots,$\pmb{o}_T$ and temporal correlations in its input $D_t$ with respect to any adversary, targeting an individual with potential data item $x_t$ (or $x'_t$) and having knowledge $\mathbb{D}_t$ equal to $D_t - \{x_t\}$ (or $D'_t - \{x'_t\}$), is called forward privacy loss and is defined as:

  \begin{equation}
    \label{eq:fpl-1}
    \alpha^F_t = \sup_{x_t, x'_t, \pmb{o}_t, \dots, \pmb{o}_T} \ln \frac{\Pr[\pmb{o}_t, \dots, \pmb{o}_T | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_t, \dots, \pmb{o}_T | x'_t, \mathbb{D}_t]}
  \end{equation}
\end{definition}
%
Similar to the way that we concluded to Equation~\ref{eq:bpl-5} from Equation~\ref{eq:bpl-1} we can write Equation~\ref{eq:fpl-1} as follows:

\begin{align}
  \label{eq:fpl-2}
  (\ref{eq:fpl-1}) = &
  \adjustbox{max width=0.9\linewidth}{
    $\sup\limits_{x_t, x'_t, \pmb{o}_{t + 1}, \dots, \pmb{o}_T} \ln \cfrac{\sum\limits_{x_{t + 1}} \Pr[\pmb{o}_{t + 1}, \dots, \pmb{o}_T | x_{t + 1}, \mathbb{D}_{t + 1}] \Pr[x_{t + 1} | x_t]}{\sum\limits_{x'_{t + 1}} \underbrace{\Pr[\pmb{o}_{t + 1}, \dots, \pmb{o}_T | x'_{t + 1}, \mathbb{D}_{t + 1}]}_{\alpha^F_{t + 1}} \underbrace{\Pr[x'_{t + 1} | x'_t]}_{P^F_{t + 1}}}$
  } \nonumber \\
  & \adjustbox{max width=0.4\linewidth}{
    $+ \underbrace{\sup\limits_{x_t, x'_t, \pmb{o}_t} \ln \cfrac{\Pr[\pmb{o}_t | x_t, \mathbb{D}_t]}{\Pr[\pmb{o}_t | x'_t, \mathbb{D}_t]}}_{\varepsilon_t}$
  }
\end{align}

Equations~\ref{eq:tpl-1},~\ref{eq:bpl-5}, and~\ref{eq:fpl-2} apply to the global publishing schema.
In the local schema, $D$ (or $D'$) is a single data item and is the same with $x$ (or $x'$), i.e.,~the possible data item of an individual user.
Therefore, we calculate the extra privacy loss under temporal correlations, due to an adversary that targets a user at a timestamp $t$, based on the assumption that their possible data are $D_t$ or $D'_t$.
More specifically, the calculation of TPL (Equation~\ref{eq:tpl-1}) becomes:
\begin{align}
  \label{eq:tpl-local}
  & \underbrace{\sup_{D_t, D'_t, \pmb{o}_1, \dots, \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_1, \dots, \pmb{o}_t| D_t]}{\Pr[\pmb{o}_1, \dots, \pmb{o}_t | D'_t]}}_{\text{Backward privacy loss}\ (\alpha^B_t)} \nonumber \\
  & + \underbrace{\sup_{D_t, D'_t, \pmb{o}_t, \dots, \pmb{o}_T} \ln \frac{\Pr[\pmb{o}_t, \dots, \pmb{o}_T| D_t]}{\Pr[\pmb{o}_t, \dots, \pmb{o}_T | D'_t]}}_{\text{Forward privacy loss}\ (\alpha^F_t)} \nonumber \\
  & - \underbrace{\sup_{D_t, D'_t, \pmb{o}_t} \ln \frac{\Pr[\pmb{o}_t | D_t]}{\Pr[\pmb{o}_t | D'_t]}}_{\text{Present privacy loss}\ (\varepsilon_t)}
\end{align}
%
The calculation of BPL (Equation~\ref{eq:bpl-5}) becomes:
\begin{align}
  \label{eq:bpl-local}
  & \adjustbox{max width=0.9\linewidth}{
    $\sup\limits_{D_t, D'_t, \pmb{o}_1, \dots, \pmb{o}_{t - 1}} \ln \cfrac{\sum\limits_{D_{t - 1}} \Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | D_{t - 1}, ] \Pr[D_{t - 1} | D_t]}{\sum\limits_{D'_{t - 1}} \underbrace{\Pr[\pmb{o}_1, \dots, \pmb{o}_{t - 1} | D'_{t - 1}, ]}_{\alpha^B_{t - 1}} \underbrace{\Pr[D'_{t - 1} | D'_t]}_{P^B_{t - 1}}}$
  } \nonumber \\
  & \adjustbox{max width=0.4\linewidth}{
    $+ \underbrace{\sup\limits_{D_t, D'_t, \pmb{o}_t} \ln \cfrac{\Pr[\pmb{o}_t | D_t]}{\Pr[\pmb{o}_t | D'_t]}}_{\varepsilon_t}$
  }
\end{align}
%
The calculation of FPL (Equation~\ref{eq:fpl-2}) becomes:
\begin{align}
  \label{eq:fpl-local}
  & \adjustbox{max width=0.9\linewidth}{
    $\sup\limits_{D_t, D'_t, \pmb{o}_{t + 1}, \dots, \pmb{o}_T} \ln \cfrac{\sum\limits_{D_{t + 1}} \Pr[\pmb{o}_{t + 1}, \dots, \pmb{o}_T | D_{t + 1}] \Pr[D_{t + 1} | D_t]}{\sum\limits_{D'_{t + 1}} \underbrace{\Pr[\pmb{o}_{t + 1}, \dots, \pmb{o}_T | D'_{t + 1}]}_{\alpha^F_{t + 1}} \underbrace{\Pr[D'_{t + 1} | D'_t]}_{P^F_{t + 1}}}$
  } \nonumber \\
  & \adjustbox{max width=0.4\linewidth}{
    $+ \underbrace{\sup\limits_{D_t, D'_t, \pmb{o}_t} \ln \cfrac{\Pr[\pmb{o}_t | D_t]}{\Pr[\pmb{o}_t | D'_t]}}_{\varepsilon_t}$
  }
\end{align}

The authors propose solutions to bound the temporal privacy loss, under the presence of weak to moderate correlations, in both finite and infinite data publishing scenarios.
In the latter case, they try to find a value for $\varepsilon$ for which the backward and forward privacy loss are equal.
In the former, they similarly try to balance the backward and forward privacy loss while they allocate more $\varepsilon$ at the first and last timestamps, since they have higher impact to the privacy loss of the next and previous ones.
This way they achieve an overall constant temporal privacy loss throughout the time series.

According to the technique's intuition, stronger correlations result in higher privacy loss.
However, the loss is less when the dimension of the transition matrix, which is extracted according to the modeling of the correlations (in this work they use Markov chains), is greater due to the fact that larger transition matrices tend to be uniform, resulting in weaker data dependence.
The authors investigate briefly all of the possible privacy levels; however, the solutions that they propose are applied only on the event-level.
Last but not least, the technique requires the calculation of the temporal privacy loss for every individual within the data set that might prove computationally inefficient in real-time scenarios.

When dealing with temporally correlated data, we handle a sequence of $w \leq t \in \mathbb{Z}^+$ mechanisms (indexed by $m \in [1, t]$) as a single entity where each mechanism contributes to the temporal privacy loss depending on its order of application~\cite{cao2017quantifying}.
The first ($m - 1$ if $w \leq 2$ or $m - w + 1$ if $w > 2$) and last ($m$) mechanisms contribute to the backward and forward temporal privacy loss respectively.
When $w$ is greater than $2$, the rest of the mechanisms (between $m - w + 2$ and $m - 1$) contribute only to the privacy loss that is corresponding to the publication of the relevant data.

\begin{theorem}
  [Sequential composition under temporal correlations]
  \label{theor:compo-seq-cor}
  When a set of $w \leq t \in \mathbb{Z}^+$ independent privacy mechanisms, satisfying $\varepsilon_{m \in [1, t]}$-differential privacy, is applied over a sequence of an equal number of temporally correlated data sets, it provides a privacy guarantee equal to:
  $$
    \begin{cases}
      \alpha^B_{m - 1} + \alpha^F_m & \quad w \leq 2 \\
      \alpha^B_{m - w + 1} + \alpha^F_m + \sum_{i = m - w + 2}^{m - 1} \varepsilon_i & \quad w > 2
    \end{cases}
  $$
\end{theorem}

Notice that the estimation of forward privacy loss is only pertinent to a setting under finite observation and moderate correlations.
In different circumstances, it  might be impossible to calculate the upper bound of the temporal privacy loss, and thus only the backward privacy loss would be relevant.

% Notice that---although we refer to it as `sequential'---since Theorem~\ref{theor:compo-seq-cor} refers to the application of a sequence of mechanisms to a respective sequence of disjoint data sets, we would normally expect it to correspond to the parallel composition on independent data (Theorem~\ref{theor:compo-par-ind}).
% However, due to the temporal correlations, the data sets are considered as one single data set; therefore, the application of a sequence of mechanisms can be handled according to the sequential composition on independent data (Theorem~\ref{theor:compo-seq-ind}).


\paragraph{Post-processing}
\label{subsec:p-proc}

Every time a data publisher interacts with (any part of) the original data set, it is mandatory to consume some of the available privacy budget according to the composition theorems~\ref{theor:compo-seq-ind} and~\ref{theor:compo-par-ind}.
However, the \emph{post-processing} of a perturbed data set can be done without using any additional privacy budget.

\begin{theorem}
  [Post-processing]
  \label{theor:post-processing}
  The post-processing of any output of an $\varepsilon$-differential privacy mechanism shall not deteriorate its privacy guarantee.
\end{theorem}

Naturally, using the same (or different) privacy mechanism(s) multiple times to interact with raw data in combination with already perturbed data implies that the privacy guarantee of the final output will be calculated according to Theorem~\ref{theor:compo-seq-ind}.


\begin{example} 
  \label{ex:application}
  To illustrate the usage of the microdata and statistical data techniques for privacy-preserving data publishing, we revisit Example~\ref{ex:continuous}.
  In this example, users continuously interact with an LBS by reporting their status at various locations.
  Then, the reported data are collected by the central service, in order to be protected and then published, either as a whole, or as statistics thereof.
  Notice that in order to showcase the straightforward application of $k$-anonymity and differential privacy, we apply the two methods on each timestamp independently from the previous one, and do not take into account any additional threats imposed by continuity.

  \includetable{scenario-micro}

  First, we anonymize the data set of Table~\ref{tab:continuous-micro} using $k$-anonymity, with $k = 3$.
  This means that any user should not be distinguished from at least $2$ others.
  Status is the sensitive attribute, thus the attribute that we wish to protect.
  We start by suppressing the values of the Name attribute, which is the identifier.
  The Age and Location attributes are the quasi-identifiers, so we proceed to adequately generalize them.
  We turn age values to ranges ($\leq 20$, and $> 20$), and generalize location to city level (Paris).
  Finally, we achieve $3$-anonymity by putting the entries in groups of three, according to the quasi-identifiers.
  Table~\ref{tab:scenario-micro} depicts the results at each timestamp.

  \includetable{scenario-statistical}

  Next, we demonstrate differential privacy.
  We apply an $\varepsilon$-differentially private Laplace mechanism, with $\varepsilon = 1$, taking into account the count query that generated the true counts of Table~\ref{tab:continuous-statistical}.
  The sensitivity of a count query is $1$ since the addition/removal of a tuple from the data set can change the final result of the query by maximum $1$ (tuple).
  Figure~\ref{fig:laplace} shows how the Laplace distribution for the true count in Montmartre at $t_1$ looks like.
  Table~\ref{tab:statistical-noisy} shows all the perturbed counts that are going to be released.

  \begin{figure}[htp]
    \centering
    \includegraphics[width=.7\linewidth]{laplace}
    \caption{A Laplace distribution for location $\mu = 2$ and scale $b = 1$.}
    \label{fig:laplace}
  \end{figure}

\end{example}
