## 3 Methods
### Introduction to Dynamic Hysteresis Model  
The probability-based dynamic hysteresis model is constructed by introducing time-dependent random variables based on the static hysteresis model. Without losing generality, we will discuss the first-order rotation curve. As shown in Figure 2.  
Figure 2: Schematic diagram of the first-order rotation curve. In the figure, A(a) is the turning point, and B1(b1), B2(b2), and B3(b3) are a series of equidistant points on the first-order rotation curve called standard sampling points; g(a, b1), g(a, b2), and g(a, b3) are the corresponding outputs. When this curve is a static (quasi-static) curve, g(A, Bm) represents the probability that the upper threshold of the hysteresis operators is less than a, and the lower threshold is less than bm. Then, Km = g(A, Bm) - g(A, Bm-1) represents the probability of the operators for which their upper thresholds are less than a and the lower threshold is between bm and bm-1. When the frequency is high, the first-order descending curve of different frequencies is shown in Figure 3.  
In the figure, bm, bm-1, and bm-2 are different inputs, while the three curves are the first-order descent curves at three different frequencies, f1, f2, and f3, and gfn(a,bm) is the corresponding output of the sensor at different frequencies. It can be seen that the amount of change between the same input Bm-1 and Bm is different. This is due to the conversion speed of the operator is inconsistent, so it is necessary to introduce time-dependent variables. The hysteresis operator in which its upper threshold is less than or equal to a is recorded, and the lower threshold is between bm-1 and bm as in K(m). The time distribution of the hysteresis operator at a specific frequency is shown in Figure 4.  
In the figure, a represents the rotation point of the first-order curve. bm-2, bm-1, and bm represents the point on the first-order curve. tm-1 represents the time when bm-2 reaches bm-1, and tm represents the time when bm-1 obtains bm. K(m-1) means the hysteresis operators with the upper threshold less than or equal to a, and the lower threshold is between bm-2 and bm-1. K(m) represents the hysteresis operator with the upper threshold less than or equal to a and the lower threshold between bm-1 and bm. There are many similar hysteresis operators in addition to the drawn hysteresis operators. The left side of K(m) in the figure represents the threshold condition; that is, after the input meets the threshold condition, the value of this hysteresis operator begins to change. The length of K(m) represents the time required for all hysteresis operators contained in K(m) to complete the conversion. The graph of K(m) is shown in Figure 5.  
Figure 3: Schematic diagram of the first-order curve at various frequencies. In the figure, K(m) is the set of the hysteresis operators of segment m, \(t_{\rm m}\) is the time from b\({}_{\rm m-1}\) to b\({}_{\rm m}\). K(m,1) is the hysteresis operator that completes the conversion in the first time segment of the hysteresis operator K(m). K(m,2) is the hysteresis operator that completes the conversion in the second time segment of K(m). K(m,3) is the hysteresis operator that completes the conversion in the third time segment of K(m) and so on. Figure 4: Schematic diagram of the time distribution of the hysteresis operator. Figure 5: Schematic diagram of K(m). At this frequency, the change M\({}_{\rm m}\) of the first-order rotation curve from b\({}_{\rm m-1}\) to b\({}_{\rm m}\) is the sum of the hysteresis operators in this period from K(1) to K(m) in Figure 4. It can be expressed as Equation (6):  
\[\rm M_{m}=g^{\prime}(A,B_{m})-g^{\prime}(A,B_{m-1}) \tag{6}\]  
where g' represents the probability at high frequency, and M\({}_{\rm m}\) is the variable from B\({}_{\rm m-1}\) to B\({}_{\rm m}\).