## 3 Methods  
### The Brief Introduction of the Static Model  
This part briefly introduces the quasi-static hysteresis model based on probability, and the details can be found in reference [22]. The magnetoresistive element is regarded as a finite number of hysteresis operators. The graph of the hysteresis operator is shown in Figure 1.  
In the figure, the H-axis is the strength of the applied magnetic field, and the M-axis is the value of the hysteresis operator, with only two values of 0 and 1. If the initial value of the hysteresis operator is 0 and the applied magnetic field gradually increases to be greater than or equal to \(\alpha\), then the value of the hysteresis operator becomes 1. Suppose that the initial value of the hysteresis operator is 1 and the applied magnetic field gradually decreases to less than or equal to \(\beta\); the value of the hysteresis operator becomes 0. The expression is shown in Equation (1):  
\[\gamma^{\prime}{}_{\alpha\beta}=\left\{ \begin{array}{ll}1&\text{H}\geq \text{a}\\ \text{Nochange}&\text{Others}\\ 0&\text{H}\leq\text{b}\end{array} \right. \tag{1}\]  
where \(\gamma^{\prime}\) represents the hysteresis operator, and H is the strength of the applied magnetic field. The output of the magneto resistive sensor can be expressed as follows:  
\[\text{f}(\text{u}(t))=\iint_{\beta\leq\alpha}\text{u}(\alpha,\beta)\gamma^{ \prime}{}_{\alpha\beta}\text{U}(t)d\alpha d\beta \tag{2}\]  
where \(\text{u}(\alpha,\beta)\) is the weight function, \(\gamma^{\prime}\) is the hysteresis operator, U(t) is the input, and f(u(t)) is the output of the sensor. Since the value of the hysteresis operator is only 0 and 1, the hysteresis operator with the value of 0 can be ignored during integration, and only the \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Method** & **Literatures** & **Shorteming** \\ \hline Structure & inner & [10; 11; 12; 23; 24] & Cost highest. Difficulty applying. Cost higher. \\  & outside & [13; 25; 26; 27; 28; 29; 30] & Power consumption. Difficulty applying. Specific range. \\ Model & Black-box & Empirical formulas & [14; 15; 16; 31; 32; 33; 34; 35; 36] & Accuracy low. Cost high. Difficulty applying. \\  & & Machine learning & [17; 18; 19; 37; 20; 21; 22; 39] & Accuracy low. \\  & & White-box & [20; 21; 22; 39; 40] & Accuracy low. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the literature. Figure 1: Preisach operator [22]. hysteresis operator with the value of 1 can be integrated. This is equivalent to the number of hysteresis operators with the value of 1. Accordingly, when the intensity of the applied magnetic field is greater than or equal to the positive saturated magnetic field, the value of all hysteresis operators is 1. Divide the output of the sensor by that at positive saturation to obtain the proportion of the hysteresis operator with value 1. This is also the probability of the hysteresis operator satisfying this condition. Moreover, the probabilistic sensor output is as follows:  
\[\mathrm{g(t)}=\frac{f(\mathrm{u(t)})}{f(+\mathrm{Hsat})} \tag{3}\]  
where f(u(t)) is the output of the sensor under input u(t), f(+Hsat) represents the output of the sensor under the positive saturated magnetic field, and g(T) is the probabilistic sensor output. Let A(a) and B(b) represent the random variables of the upper and lower threshold of the hysteresis operator, respectively. P(A) and P(B), respectively, represent the probability that the upper threshold of the hysteresis operator is less than a and the lower threshold is less than B. It has the following formula:  
\[\mathrm{P(A=a)}=\mathrm{g}^{+}(\mathrm{a}) \tag{4}\]  
\[\mathrm{P(B=b)}=\mathrm{g}^{-}(\mathrm{b}) \tag{5}\]  
where g\({}^{+}(\mathrm{a})\) and g\({}^{-}(\mathrm{b})\) are the limit rising curve and limit falling curve, respectively [22; 40]. The model satisfies the wipe-out property and the equal vertical chords property. The wipe-out property is that the model will preserve the influence of the input extreme value.