to a slow sampling frequency or noisy data it may happen that these critical points are not even included in the data at all. In order to improve robustness of the relay hysterons, we developed a “proximity-hysteron", which utilizes an $\epsilon$-range around the actual threshold value for switches between states. The evaluation is done according to $$\label{eq:proxHysteron}
H_{\epsilon}(k)_{\alpha, \beta} = \begin{cases}
1, & \text{if $I_{\epsilon, \beta}  = 1$.} \\
0, & \text{if $I_{\epsilon, \alpha} = 1$.} \\
H_{\epsilon}(k-1), & \text{for $I_{\epsilon, \beta} = I_{\epsilon, \alpha} = 0$.}
\end{cases}$$

with the proximity indicator functions $$\begin{aligned}
\begin{split}
I_{\epsilon_\beta}(x) &= \begin{cases}
1,  & \text{if $x = \min\{\max( Z_{x_\beta}(t) ), \beta\} $.} \\
0,  & \text{otherwise.} 
\end{cases} \\
I_{\epsilon_\alpha}(x) &= \begin{cases}
1,  & \text{if $x = \max\{\min( Z_{x_\alpha}(t) ), \alpha\} $.} \\
0,  & \text{otherwise.} 
\end{cases}
\end{split}\end{aligned}$$

and the connected sets and\
$Z_{x_\alpha}(t) = \{x \mid x < \alpha + \epsilon_\alpha\}$, with $\epsilon_\beta,\epsilon_\alpha \in \mathbb{R} > 0$, which include $x(t)$.

The proximity indicator functions define an $\epsilon$-range, which moves the threshold for the transition in the corresponding direction (see Fig.\[fig:proximity\_hysteron\]). However, the transition between states does not happen as soon as the shifted threshold is reached, but only when the extremum within the connected 