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\newcommand{\thetitle}{Quality \& Privacy in User-generated Big Data: Algorithms \& Techniques}

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% An Optimal Pufferfish Privacy Mechanism for Temporally Correlated Trajectories
% - microdata
% - finite (sequential)
% - batch
% - dependence (temporal)
% - local
% - event
% - differential privacy
% - perturbation (randomized response, Laplace)
\hypertarget{ou2018optimal}{Ou et al.}~\cite{ou2018optimal}

\emph{FGS-Pufferfish}

temporally correlated trajectory data


First, a Laplace noise mechanism is realized through geometric sum of noisy Fourier coefficients of temporally correlated daily trajectories. 

We achieve better data utility for a given privacy budget by solving a constrained optimization problem of the noisy Fourier coefficients via the Lagrange multiplier method.


generation of Laplace noise via the Fourier coefficients' geometric


We present an analytical formula of the optimized Fourier coefficients noise for the constrained optimization problem of achieving a better data utility for a given privacy budget. 

We provide theoretical analysis of the data utility and privacy, as well as the posterior-to-prior knowledge gain of an adversary.





we quantify the temporal correlation in a rigorous mathematics way. 

by adding noise to the Fourier coefficients through geometric sum. 

The discrete Fourier transform transforms a user's daily trajectory

into a set of sine and cosine waves
of different frequencies and corresponding Fourier coefficients 

In this paper, we assume that both the real part and the imaginary part of the Fourier coefficient follow the same Gaussian distribution 



The constrained optimization problem is a strategy of finding the local extrema (maxima and minima) of a function f (b) subject to equality constraint g(b) = 0


The user's mobility pattern can be described by the conditional probability of the next i-th location from the current n-th location


A user's temporal correlation depicts the relation
between two locations at current time slot tn and its following i-th time slots tn+i


the Pufferfish secrets set consists of temporal correlations of all users

A correlation secrets pair consists of two temporal correlations of any two users in the same database

the Fourier coefficients of the temporal correlation are closely related to those of the daily trajectory. 
Thus it is natural to add noise to the Fourier coefficients of the trajectory


we propose to optimize the Fourier coefficients noise for the problem of temporal correlation privacy.

First, we add the noise in the Fourier coefficients

the noisy temporal correlation is obtained from the noisy daily trajectory


we propose to achieve the optimal data utility for a given privacy budget

Here we consider two utilities, including the location utility and the correlation utility

For the location utility, we want the average noisy location deviates from its raw location as small as possible

The correlation utility is the average of the deviation of the noisy correlation from its raw value




Our goal is to prevent an adversary from mining a user's privacy through analyzing the user's temporal correlation based on the adversary’s prior knowledge about the user.



First, we define the constrained optimization problem of achieving a better data utility for a given privacy budget given by the Laplace scale parameter
Next, we solve the constrained optimization problem via the LM method and obtain the optimal obtained Laplace scale parameter for the noisy Fourier coefficients
Then, the FGS-Pufferfish privacy mechanism adds noise to the Fourier coefficients and obtain the noisy Fourier coefficients
At last, we obtain the sanitized daily trajectories with the noisy Fourier coefficients 


we design an algorithm to release temporally correlated trajectories in order to protect individuals' privacy. 
Because our goal is to protect the temporal correlation of a user’s daily trajectory, we first calculate the Fourier coefficients of a daily trajectory which is related to the Fourier coefficients of its temporal correlation.
Then, to achieve the Laplace distribution of the noisy temporal correlation through the Fourier coefficients noise mechanism, i.e., adding noise to Fourier coefficients, with following the geometric distribution.
Furthermore, we obtain the optimal Laplace scale parameters for the noisy Fourier coefficients. Finally, we use IDFT to obtain the noisy loca- tions of the sanitized daily trajectory


We propose a Laplace noise mechanism based on the noisy Fourier coefficients' geometric sum, satisfying Pufferfish privacy, i.e., the FGS-Pufferfish privacy mechanism, to protect the temporal correlation of a user's daily trajectories. 
The optimal noisy Fourier coefficients are obtained by solving the constrained optimization problem via the LM method to achieves a better data utility for a given privacy budget. 
Experiments with both simulated and real-life data show that our FGS-Pufferfish privacy mechanism achieves better data utility and privacy compared to the existing approach.
Although we only deal with daily trajectories with a constant time interval, our proposed mechanism can be readily modified for time-series data with irregular time intervals



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