## 3 Methodology
### Topic Alignment Index  
This research invents a topic alignment index (H-index) to quantify the alignment between NLP topic model assignments and SME classifications. The proposed H-index (H stands for harmonic) comprises two parts. The first part is a _purity_ factor defined as the proportion of documents within an SME category represented by the model's most frequently assigned topic in that category. Hence, the average purity factor \(\rho\) across C SME-assigned categories is  
\[\rho=\frac{1}{C}\sum_{i=1}^{C}\frac{F_{i}^{*}}{D_{i}} \tag{1}\]  
where \(F_{i}^{*}\) is the count of the model's most frequently assigned topic for SME category \(i\) and \(D_{i}\) is the number of documents in SME category \(i\). The maximum purity factor is unity when the most frequently assigned topic in an SME category covers all documents in that category, making it a pure assignment. The minimum purity factor is \(1/C\) where \(F_{i}^{*}=1\), reflecting that no two assigned topics in an SME category were identical. The average purity factor mirrors a homogeneity measure of clustering, which is a global measure of the number of predicted classes that belong to the same cluster ([PERSON] et al., 2022).  
The second part of the proposed H-index is a dispersion factor \(d\) that measures the number of most frequently assigned topics to all SME categories that are unique. Hence, dispersion is  
\[d=\frac{1}{C}|F| \tag{2}\]  
The function \(\mid\cdot\mid\) denotes the cardinality or count of unique values in the set \(F=\{F_{1}^{*}\ldots F_{C}^{*}\}\) of most frequently assigned topics for SME categories 1 through C. Hence, dispersion measures how well model-assigned topics spread across the SME categories to distinguish among them. The maximum dispersion factor is unity when no two most frequent topic assignments across the SME categories are identical. The minimum dispersion factor is \(1/C\) if the model assigned the same topic to all SME categories. Hence, the dispersion factor penalizes repeated topic assignments across the unique SME categories.  
The topic alignment index is a combination of the average purity across SME-assigned topics and the dispersion index. A simple combination of the two indices would be the mean value. Alternatively, this research defines the H-index, denoted \(\psi\), as the harmonic mean of the two indices, thereby reflecting a balanced contribution such that  
\[\psi=2\frac{\rho\times d}{\rho+d}. \tag{3}\]  
Unlike a simple mean, the harmonic mean captures the influence of outliers. For example, if the average purity factor is unity but the dispersion factor approaches zero, the H-index will also approach zero, whereas the mean value of the two components will instead approach 0.5. This situation represents a scenario where the model assigned the same topic to all SME categories. The situation is the same for the other extreme scenario where the average purity factor approaches zero and the dispersion factor is unity. This situation represents a case where the model assigned a unique topic to each document in an SME category and does so for every SME category. Both cases represent the worst possible alignment with an SME classification, representing the worst possible topic modeling performance.