Table 1 Contributions and scores of various text comparison measures according to the word shift framework. The word contribution \(\delta \Phi _{\tau }\) indicates how an individual word impacts a measure, and each contribution is expressed as a difference in weighted averages so that it can be easily identified with the components of the word shift framework
Measure | Notation | Word contribution \(\delta \Phi _{\tau }= p_{\tau }^{(2)}\phi _{\tau }^{(2)} - p_{\tau }^{(1)} \phi _{\tau }^{(1)}\) |
|---|---|---|
Relative Frequency | \(P^{(i)}\) | \(p_{\tau }^{(2)} - p_{\tau }^{(1)}\) |
Shannon Entropy | \(H (P^{(i)} )\) | \(- p_{\tau }^{(2)} \log p_{\tau }^{(2)} + p_{\tau }^{(1)} \log p_{\tau }^{(1)}\) |
Generalized Entropy | \(H_{\alpha } (P^{(i)} )\) | \(-p_{\tau }^{(2)} [ \frac{ (p_{\tau }^{(2)} )^{\alpha -1}}{\alpha -1} ] + p_{\tau }^{(1)} [ \frac{ (p_{\tau }^{(1)} )^{\alpha -1}}{\alpha -1} ]\) |
Kullback–Leibler Divergence | \(D^{(\text{KL})} (P^{(2)} \parallel P^{(1)} )\) | \(- p_{\tau }^{(2)} \log p_{\tau }^{(1)} + p_{\tau }^{(2)} \log p_{\tau }^{(2)}\) |
Jensen–Shannon Divergence | \(D^{(\text{JS})} (P^{(1)} \parallel P^{(2)} )\) | \(p_{\tau }^{(2)} \pi _{2} \log \frac{p_{\tau }^{(2)}}{m_{\tau }} - p_{\tau }^{(1)} \pi _{1} \log \frac{m_{\tau }}{p_{\tau }^{(1)}}\) |
Generalized Jensen–Shannon Divergence | \(D_{\alpha }^{(\text{JS})} (P^{(1)} \parallel P^{(2)} )\) | \(p_{\tau }^{(2)} \pi _{2} [ \frac{ (p_{\tau }^{(2)} )^{\alpha -1} - m_{\tau }^{\alpha -1}}{\alpha -1} ] - p_{\tau }^{(1)} \pi _{1} [\frac{m_{\tau }^{\alpha -1} - (p_{\tau }^{(1)} )^{\alpha -1}}{\alpha -1} ]\) |