# Privacy and Utility of Private Synthetic Data for Medical Data Analyses
## 4 Medical Use Case
### Privacy Challenges  
As mentioned before, the data that are used here can reveal privacy-sensitive information about an individual. For example, the time course of the raw data reveals a person's sleep or activity time. The usual method to mitigate such risks is to use DP, where noise is  
Figure 1: Heart rate of an individual from the Fitbit dataset for one day. added to all data points to make them unlinkable to the raw data. Typical applications for DP include the calculation of average or median values. The concept of DP assumes that the noise added to single data points is canceled out over the entire dataset. This should allow an accurate analysis. For the tachycardia use case described in Section 4.2, average or median values alone are not sufficient. An accurate estimate of the minutes with a heart rate over 100 BPM is required. In an interactive DP setting, this is not easily possible because the raw data must be transformed into one-minute intervals. Therefore, a static data release (or the so-called non-interactive setting) is much better suited for such an analysis [35]. In the following section, we will conduct a plain DP release and use synthetic data technologies to create a private dataset with the same number of entries and individuals. The idea of using synthetic private data in this context is that the statistical properties of the data (e.g., the tachycardia minutes) are preserved, but the activity patterns are disguised. The typical measure for privacy when using DP is \(\epsilon\). While \(\epsilon\) quantifies a mathematical property, choosing an appropriate value to guarantee privacy for a real-world use case is nontrivial to impossible. To motivate the subsequent analyses of this paper, an attacker model is introduced. This attacker has the ability to recognize activity patterns from the heart rate data. The attacker further has the necessary medical knowledge and can understand the heart rate data. Additionally, the attacker has background knowledge about its victim, such background knowledge can be the typical wake-up time of the victim. In combination with the heart rate graph, the attacker can use this knowledge to re-identify the victim and link it to the _anonymized_ heart rate data. These data can then be used to gain additional information about or, in the worst case, to blackmail the victim. **Definition 2** (Re-identification attacker \(\mathcal{A}\)). : _A re-identification attacker \(\mathcal{A}\) combines its medical and victim background knowledge to re-identify a victim using anonymized heart rate data. \(\mathcal{A}\) explicitly uses activity patterns of the heart rate graph for re-identification. Potential background knowledge can be the typical sleep or wake-up time or more precise knowledge about daily routines such as workout times. The attacker is successful if it was possible to link a certain individual to data from a heart rate dataset containing no direct identifiers._  
To measure privacy besides the statistical \(\epsilon\), we define a domain-specific metric called _Tachycardia Privacy (TP)_. As described previously, changes in the heart rate curve are considered as private information because they reveal activity patterns or daily routines. The goal of a private dataset, which was generated using DP or synthetically, should be to have a heart rate curve with a gradient that differs as much as possible from the gradient of the real curve. TP is hence meant to quantify how likely it is that an attacker could use the privatized data to re-identify an individual if they have knowledge of an individual's daily activities. The principle is to measure the similarity of the derivative of the heart rate curve at each point in time for the raw data and the privatized data, respectively. For this purpose, functions \(f^{\prime}_{\textit{r}mv}(t)\) and \(f^{\prime}_{\textit{private}}(t)\) are needed, which represent the derivative of the heart rate curve at a point in time \(t\in T\).