# Using Conceptual Recurrence and Consistency Metrics for Topic Segmentation in Debate
## 6 CSseg Algorithm
### Similarity Cohesion
#### 6.2.1 Sum of Conceptual Similarities  
The sum of the conceptual similarities corresponds to the numerator in the formula for calculating the similarity cohesion. We propose three methods for calculating the sum of the conceptual similarities of a topic segment: Plane, Line, and Point. Through these, we additionally propose a method of combining the properties of the three methods. Method 1: Plane  
The Plane method is a method to sum all conceptual similarities in a topic segment as shown in Figure 7.  
Figure 6: Four small sets of topic segments. Red squares indicate topic segments. Since the way CSseg divides the transcript is by dividing one topic segment into two topic segments, we intend to designate these two adjacent topic segments as a unit, which is called a small set of topic segments. Figure 7: Examples of conceptual similarities used in the Plane method. Each red square represents a topic segment. The light green squares with blue dots inside the red squares indicate the conceptual similarities used in the Plane method. The formula for calculating the sum of the conceptual similarities of the Plane method is as follows:  
\[\text{(sum of conceptual similarities by Plane method)}=\sum_{j=2}^{N}\sum_{i=1}^{j-1} plane(i,j)\cdot S(i,j)\]  
\(i\) means the row index, and \(j\) the column index. \(N\) is the number of utterances in a topic segment. \(S(i,j)\) is a conceptual similarity between the \(i\)-th utterance and the \(j\)-th utterance. \(plane(i,j)\) is a function that assigns a weight to the conceptual similarity between the \(i\)-th and \(j\)-th utterances. The formula for \(plane(i,j)\) is as follows:  
\[plane(i,j)=planeWeight\]  
In the Plane method, all conceptual similarities are added, so the conceptual similarities for all \(i\) and \(j\) are included. \(planeWeight\) means the weight to be applied to the conceptual similarity in the Plane method. Since CSseg is a topic segmentation model through the user interaction from the perspective of visual analytics, the \(planeWeight\) can be specified by the user. Method 2: Line  
The Line method is a method of adding conceptual similarities belonging to the edge of a topic segment's area in the form of \"L\", as shown in Figure 8. We devised the Line method by discovering a pattern of high conceptual similarities between major utterances and other utterances in adjacent time zones. This has the effect of separating a transcript based on major utterances that are similar to other utterances. The formula for calculating the sum of the conceptual similarities of the Line method is as follows:  
\[\text{(sum of conceptual similarities by Line method)}=\sum_{j=2}^{N}\sum_{i=1}^{j-1} line(i,j)\cdot S(i,j)\]  
\(i\) is the row index, and \(j\) the column index. \(N\) is the number of utterances in a topic segment. \(S(i,j)\) is a conceptual similarity between \(i\)-th utterance and \(j\)-th utterance. \(line(i,j)\) is a function that assigns a weight to the conceptual similarity between the \(i\)-th and \(j\)-th utterances. The formula for \(line(i,j)\) is as follows:  
\[line(i,j)= \begin{cases}lineWeight,&\text{if }1\leqq i\leqq N-1\And j=N\\ lineWeight,&\text{if }i=1\And 2\leqq j\leqq N\\ 0,&\text{otherwise}\end{cases} \]  
\(lineWeight\) means the weight to be applied to the conceptual similarity in the Line method. It can be specified by the user. Figure 8: Examples of conceptual similarities used in the Line method. Each red square represents a topic segment. The light green squares with red dots inside the red squares indicate the conceptual similarities used in the Line method. Method 3: Point  
The Point method is a method of selecting one conceptual similarity at the bottom left of a topic segment area, as shown in Figure 9. We devised the Point method by discovering a pattern of high conceptual similarity between major utterances with high similarity to other utterances. It has the effect of separating a transcript based on major utterances that are similar to other utterances.