they induce an inaccuracy for the slope.\
$$\begin{aligned}
        \tilde{h}_{poly1}(k+1) &= h(k)  + 1.56 \times 10^{-4}(4.98 Q_{in}  (1 - H_1) + Q_{out} ) \label{eq:poly1_model}, \nonumber \\
        H_{1}(k) &= \begin{cases}
        1, & \text{if } h > h_{max}. \\
        0, & \text{if } h < h_{min}. \\
        H_{1}(k-1), & \text{else}.
        \end{cases} \end{aligned}$$

![image](polydegSimHSVG){width="\linewidth"} \[fig:exp2\_sim\_h\]

Influence of sample rate
------------------------

As a further objective, we analyzed the effect of the sampling rate. As explained in Part \[sec:proximity\], this method cannot localize transitions if the critical points are not included in the given data, which may happen for slow sampling rates. Figure \[fig:exp3\_normal\_sim\_h\] shows, that state transitions are not detected using standard hysterons for slower sampling than $0.02$ s.

![image](samplingrateNormalHystSVG){width="\linewidth"} \[fig:exp3\_normal\_sim\_h\]

As a remedy, we presented the proximity hysteron in Section \[sec:proximity\]. The effect can be seen in Fig. \[fig:exp3\_proxi\_sim\_h\]. With this extension, the algorithm achieves good results for all sampling rates. The best model is shown in Eq. and was built from a library of 23 basis functions. It is equivalent to the model in Eq. , as $1 - H_1 = \bar{H}_1$.\
$$\begin{aligned}
    \tilde{h}_{prox, 0.01}(k+1) &= h(k)  + 1.56 \times 10^{-4}(4.98 Q_{in} \ \bar{H}_1 + Q_{out} ) \label{eq:snr_model}, \nonumber \\
    H_{1}(k) &= \begin{cases}
        