2. The Validation of the Framework
3. Results and Discussion
- Drawing an imaginary circle and lay the start point of the velocity vectors of each share on the center of the circle.
- Dividing the circle into 12 equal slices and assigning a number to each of them, so each velocity vector is located in one slice.
- Mapping each share to one slice number. For example, the velocity vector of i-th share in the division l is shown as , where t denotes time. l can only accept integer values and it returns the number of the slice that velocity vector is located, and is the degree of each slice which in this research assumed 30 degrees.
- Coupling the joint shares. For example, if share i is in slice l and share j is in slice m, the return of the joint l-m or m-l will be considered. l-m and m-l are considered as one possible combination of joint shares. Therefore, the joint probability of both i and j to be placed in l and m divisions equals 1.
- Counting the possible combinations of joint shares . For example, 0-0, 0-1, 0-2, ..., 10-11, 11-11. It can be a maximum of 78 combinations of joint shares.
- The velocity vector of i-th share at time t is shown by .
- The alignment of the velocity vectors of two separate shares can be calculated by Formula (12). The basis of this calculation is the cosine of the angle between the two velocity vectors.
- The stocks normally are correlated to each other. Thus, the change in one stock can affect the other stocks. This effect does not happen at once. Thus, we considered a delay for the calculations.
- The alignment matrix has been constructed using calculated in Equation (13). The alignment matrix has been generated using dendrogram to present the stocks which are grouped together.
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