# Study on the Eye Movement Transfer Characteristics of Drivers Under Different Road Conditions
## 4 Fixation Shift Analysis  
### Description of the Principle  
A Markov chain is a mathematical model for describing stochastic processes, and its core idea is that the future state of a system is determined only by the current state and is independent of the historical state. In the process of modeling the driver's fixation behavior,  
Figure 8: Distribution of drivers’ total fixation duration and frequency in the uphill section. (**a**) Percentage of total fixation duration; (**b**) percentage of total fixation frequency. Figure 7: Distribution of drivers’ total fixation duration and frequency on downhill sections. (**a**) Percentage of total fixation duration; (**b**) percentage of total fixation frequency. the driver's fixation region can be regarded as the state of the Markov chain [23], and the driver's eye-tracking behavior is described by the state transfer probability. The state space of the system is assumed to be, where represents one of the driver's fixation regions, and the probability of state transfer is given by the transfer probability matrix \(\mathrm{P}\), as shown in Equation (1). \[\mathrm{P}= \begin{bmatrix}\mathrm{P}_{11}&\mathrm{P}_{12}&\ldots&\mathrm{P}_{ 1n}\\ \mathrm{P}_{21}&\mathrm{P}_{22}&\ldots&\mathrm{P}_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \mathrm{P}_{n1}&\mathrm{P}_{n2}&\ldots&\mathrm{P}_{nn}\\ \end{bmatrix}  \tag{1}\]  
In equation, \(P_{ij}\) denotes the probability of transferring from state \(s_{i}\) to state \(s_{j}\) and satisfies the following normalization conditions, as shown in Equation (2). \[\sum_{j=1}^{n}P_{ij}=1,\forall i \tag{2}\]  
Among the driver's eye-tracking data under the four road segments, the transition probability matrix \(\mathrm{P}\) is calculated by summing the fixation duration and frequency of each fixation region to calculate the transition probability between each state. The data for each fixation region is summed to determine the transition probability from one state to another. Let \(N_{ij}\) represent the number of times a driver fixates count from fixation region \(s_{i}\) to \(s_{j}\) during the observation period, and \(N_{i}\) represent the total number of times a driver stays in region \(s_{i}\). The one-step transfer probability \(P_{ij}\) is calculated by the following equation, as shown in Equation (3). \[P_{ij}=\frac{N_{ij}}{N_{i}}\,\ \forall i,j \tag{3}\]  
The above equations illustrate that the transfer probability between different fixation regions of a driver can be calculated directly from the experimental data. The driver's eye tracking in different driving scenarios is counted, and the fixation count \(N_{ij}\) between each gaze region is recorded, and then the transfer probability matrix \(P\) can be obtained by normalization. After constructing the transfer probability matrix, the long-term behavior of the system, i.e., the smooth distribution, can be further analyzed. The smooth distribution describes the stable share of each state of the system in the long-term state, i.e., the long-term probability of the driver's fixation in different fixation regions. Assuming that the vector of long-term fixation probabilities of drivers in different states is \(\pi=(\pi_{1},\pi_{2},\ldots,\pi_{n})\), the state distribution satisfies the following equation in the smooth state, as shown in Equation (4). \[\pi P=P \tag{4}\]  
That is, the distribution probability of each fixation region does not change over fixation duration for a driver. The equation shows that the smooth distribution is an eigenvector of the transfer matrix \(P\), corresponding to the eigenvalues. To ensure that \(\pi\) is a valid probability distribution, the normalization condition also needs to be satisfied, as shown in Equation (5). \[\sum_{i=1}^{n}\pi_{i}=1 \tag{5}\]The process of solving for a smooth distribution can be generalized to the problem of solving a system of linear equations, as shown in Equation (6). \[\left\{