# Rotor Location During Atrial Fibrillation: A Framework Based on Data Fusion and Information Quality
## 2 Materials and Methods
### Statistical Features
#### 2.4.1 Approximate and Sample Entropy  
ApEn(\(m\), \(r\), \(N\)) can be applied to short and noisy time series of clinical data [43]. Series with more frequent and similar sequences lead to lower ApEn values. The ApEn is calculated using the following expression:  
\[ApEn(m,r,N)=\phi^{m}(r)-\phi^{m+1}(r). \tag{3}\]  
The tuning of the parameters \(m\), \(r\), and \(N\) is explained in [43], and the function \(\phi^{m}(r)\) is defined as follows:  
\[\phi^{m}(r)=\frac{\sum_{i=1}^{N-m+1}log[C_{i}^{m}(r)]}{N-m+1}, \tag{4}\]  
where \(C_{i}^{m}(r)\) represents the number of sequences that are close to the \(i\)-th sequence of length \(m\). The closeness is determined by the parameter \(r\). In practice, the ApEn has two major drawbacks. First, the estimation of the ApEn is affected by the length of the signal, and tends to be uniformly lower than the theoretical value for short recordings. Second, it lacks relative consistency. To address these limitations, the authors of [44] proposed the SampEn. SampEn(\(m\), \(r\), \(N\)) is the negative natural logarithm of the conditional probability that, given two similar \(m\)-sample sequences, they remain similar for \(m+1\) samples [44]. \begin{table}
\begin{tabular}{p{42.7 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Level** & **Criteria** & **Metric** & **Equation** \\ \hline
1 & Accuracy & Difference between ground truth and detected rotor tip & \(error=\sum_{i=1}^{n}\|Tip-TIP\|\), where \(Tip\) is the calculated and \(TIP\) is the ground truth. \\ \hline
1 & Precision & Deviation of distances among features (\%) nearest to tip & \(\gamma_{i}=\sum_{j=1}^{n}\|TipFeature_{i}-TipFeature_{j}\|\), where \(n\) is the 5\% of the features nearest to rotor tip. \\ \hline
1, 2 & Reputation &
\begin{tabular}{l} Noise tolerance of the features. This quantifies the noise tolerance \\ \end{tabular} & \(\frac{\sum_{i=1}(\text{MapNoise}_{i}-\text{OriginalMap})}{n-1}\). \\ \hline
2 & Interpretability & \
\begin{tabular}{l} Quality of the map figure for showing the rotor tip location based on the contrast \\ \end{tabular} &
\begin{tabular}{l} \(Inter=|\mu_{Mapa}-\mu_{Tip}|+\frac{1}{\delta_{Mapa}}+\frac{1}{\delta_{Tip}}\), where \(\mu_{Mapa}\) is the mean measure and \(\delta_{Map}\) is the bandwidth of the distribution of the data of the map and \(\mu_{Tip}\) and \(\delta_{Tip}\) (5\% of high values) are the mean and bandwidth of the distribution of the tip. \\ \hline
2, 3 & Objectivity &
\begin{tabular}{l} The objective assessment given by the quality of the information \\ \end{tabular} & \(IQ_{G}=\sum_{j=1}^{n}IQ_{Ln}\), where \(IQ_{G}\) is the Global IQ evaluate until level 2 or 3 depending on the level and \(IQ_{Ln}\) is the IQ of level \(n\). \\ \hline
3 & Accuracy &
\begin{tabular}{l} Accuracy to the prediction made by the expert supported by the system \\ \end{tabular} & \(Accuracy=\frac{SEvents}{TEvents}\), where \(SEvents\) is the number of successfully predicted events and \(TEvents\) is the total events. \\ \hline
0–5 & Efficiency & Computational cost & \(\sum_{n=1}^{N_{Map}}T_{n}\), where \(T_{n}\) is the processing time, given in seconds. \\ \hline
5 & Authority &
\begin{tabular}{l} Experience and effectiveness of the doctor in conducting treatments \\ \end{tabular} & It is assigned by the expert from the interval \([0,1]\). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._Therefore, a low SampEn value indicates higher regularity in the time series. The definition of SampEn is as follows:  
\[SampEn(m,r,N)=-\ln\frac{A^{m}(r)}{B^{m}(r)}, \tag{5}\]  
where \(A^{m}(r)\) and \(B^{m}(r)\) are defined as follows:  
\[A^{m}(r)=\frac{\sum_{i=1}^{N-m-1}A_{i}^{m}(r)}{N-m-1},\qquad B^{m}(r)=\frac{ \sum_{i=1}^{N-m}B_{i}^{m}(r)}{N-m}, \tag{6}\]  
where \(B_{i}^{m}(r)\) is the number of times the sequence \(x_{m}(j)\), \(j=1\dots,N-m\), \(j\
eq i\) is close to \(x_{m}(i)\) according to threshold \(r\); \(A_{i}^{m}(r)\) is similar to \(B_{i}^{m}(r)\) but considers sequences of length \(m+1\).