# Privacy and Utility of Private Synthetic Data for Medical Data Analyses
## 4 Medical Use Case
### Privacy Challenges
\(T\) is the normalized time for all heart rate curves, where the earliest time is 1 and the latest \(t_{n}\). In this work, one day is considered as a time period, thus \(1\leq t\leq 1440=t_{n}\wedge t\in\mathbb{N}\) (\(t\) can be any minute of one day). In addition, we use the so-called _Privacy Impact Factor (PIF)_ in the TP to weight the impact of the relative difference between the two derivatives. **Definition 3** (Tachycardia Privacy (TP)). : \[TP=\frac{\sum_{t\in T}min\left(\frac{|f^{\prime}_{\textit{r}mv}(t)-f^{\prime} _{\textit{private}}(t)|}{f^{\prime}_{\textit{r}mv}(t)},PIF\right)}{|T|\times PIF}\]The core idea of Tachycardia Privacy (TP) is to measure the difference between the derivatives of the raw and privatized heart frequency curves. From a privacy perspective, a maximal different derivative leads to a maximal different heart rate curve. With this, our previously described attacker has no way to make use of their background knowledge to re-identify an individual. To accomplish this, we use the absolute difference between the gradient of the raw and privatized heart frequency and put it in to relation to the raw gradient. The Privacy Impact Factor (PIF) defines the maximum difference between the raw and synthetic curve which is considered. A \(\Pi F=1\) means that every difference larger than \(100\%\) will not have a greater impact on the value because of the minimum function. For our experiments, we saw that a \(\Pi F=3\) gives the best range of values for the privacy results. With this, relative differences of up to \(3\)x are considered in the calculation. We do this calculation now for all time points in our heart frequency series. To arrive at the final TP between 0 and 1, the summed value is divided through the number of entries times PIF. In summary, this means the closer the TP is to one, the better the privatization, because the courses of the synthetic and raw heart rates differ more, making re-identification less likely. While privacy is an important factor and should be a fundamental principle in the processing of any health data, the data must also be usable. For this, a definition of utility is required, which we call _Tachycardia Utility (TU)_. To measure utility for the tachycardia use case, the number of minutes with a heart rate above 100 BPM (hereafter referred to as _tachycardia minutes_) must be compared with the tachycardia minutes of the generated data. **Definition 4** (Tachycardia Utility (TU)). : \[TU=1-min(1,\frac{|\text{private minutes}-\text{raw minutes}|}{\text{raw minutes}})\]  
The Tachycardia Utility (TU) measures the deviation between the tachycardia minutes of the privatized and the raw data. To that end, we use the absolute difference between the private and raw minutes and divide it by the raw value to put this difference in relation to the size of the raw value. We also define that a difference of more than \(100\%\) is the worst possible and that such a privatized value has no more utility. Therefore, we limit the result to \(1\) using the minimum function. A TU of \(1\) is considered a perfect utility, since the number of minutes is the same. The closer the value is to \(0\), the worse the utility is. Finally, these metrics allow us to measure the vulnerability and utility of the privatized data in the forthcoming experiments.