# Process Mining IPTV Customer Eye Gaze Movement Using Discrete-Time Markov Chains +
## 3 Methodology
### Fitting into a DTMC Model
#### 3.4.3 DTMC Modelling Steps
Unfortunately, although the Tobii Software can define an AOI and can provide statistics based on the metrics in relation to those AOIs, such as the mean time to first fixation on each AOI for the user, it does not provide a time-sequential list of the AOIs the user was fixating on for the duration of the recording, which is necessary for the Markov Model. However, the raw Tobii data, which provides X and Y gaze point co-ordinates of the eye, can be used to do this. The X and Y co-ordinates axis are shown in Figure 6 below, where the origin begins in the top left-hand corner; however, the X and Y axis both extend further in the horizontal and vertical directions than shown. Therefore, based on a 1920 \(\times\) 1080 resolution screen in our study, we have extracted the co-ordinates information for each AOI block for both scenarios. A sample of AOIs and co-ordinates of each corner can be seen in Figure 7 below. Each cell (e.g., (349,128)) under the columns shows the x and y axis coordinates, respectively. For the AOI column, E* and F* refer to the AOIs only available on the screen A. Then a Java function was created to convert gaze point coordinates into the block (AOI) labels, as is shown in Figure 8 below. Figure 5: “Content viewing”–Original (**top**) & Layout-only (**bottom**) screens [16]. The function results in an extra column named 'block_letter' in the MySQL table, which we created to store processed gaze movement data (see Figure 9). It indicates which AOI on the screen the current gaze movement falls in. In this way, we have a sequential list of AOIs from the original gaze movement data, which is suitable for a discrete-time Markov model. The full set of all possible states (also known as state space \(S\)) for both scenarios are as follows:  
For \"**content purchase**\" screens:  
* Screen A;
* Screen B;
* \(\bullet\) For \"**content viewing**\" screens:
* \(S\) = {\"A,\" \"B,\" \"C,\" \"D,\" \"E,\"... \"T\"}. Figure 8: Conversion from Gaze Coordinate to Block Label [16]. Figure 6: Tobii Studio Image Co-Ordinate System. Figure 7: AOIs and corner co-ordinates-“content purchase” screens. For a Markov Model, if it exists, \(X_{0}\) refers to the state when \(t=0\), and the initial probability distribution is defined by:  
\[P(X_{0}=s_{i})=\frac{count(X_{0}=s_{i})}{count(X_{0})} \tag{18}\]  
The distribution is also called the initial state vector of the Markov chain (see Equation (7)). For instance, for the '**content viewing**' scenario, among all 14 participants, their eye gaze fell on different areas at first sight. The details are listed below:  
* Two participants looked at AOI-E;
* One participant looked at AOI-I;
* Two participants looked at AOI-L;
* Two participants looked at AOI-M;
* One participant looked at AOI-O;
* Four participants looked at AOI-R;
* Two participants looked at AOI-T. Therefore, the initial state distribution is estimated and displayed in Figure 10 as follows:  
3. Transition matrix  
A transition matrix in the Markov model, in principle, is a reflection of how states change during the whole process. In our study, the transition matrix for a given state space infers how participants move their eyes on the screen; in other words, it illustrates the \"jump\" from the current state to the next state in the Markov Model created for each scenario. In the MySQL database that we use to store eye tracking log data, after data pre-processing (AOI-categorization, non-relevant record filtering out, etc. ), a sequence of states was determined for each participant, combined with the timestamp information and block labels. Furthermore, in the sequence, each pair of states indicates a one-step transition (\(X_{t}\), \(X_{t+1}\)) in the DTMC model. In that regard, the state change pairs can be generated by simply applying an iteration of a length-2 sliding window over each sequence. In this paper, the function was implemented in Java.