# Cascade Network with Deformable Composite Backbone for Formula Detection in Scanned Document Images
## 4 Experimental Results
### Result and Discussion
#### 4.3.2. Marmot
The reduced number of missed detections in isolated and embedded formulas demonstrates the superiority of the proposed method. Figure 11: Correct detection accuracy achieved on varying IoU threshold ranging from 0.5 to 1.0 on the Marmot dataset. (**a**) depicts the correct detection accuracy on isolated formulas, whereas (**b**) demonstrates the correct detection accuracy on embedded formulas. Figure 10: Performance evaluation of the proposed method on the Marmot dataset in terms of average precision (AP) at an IoU threshold of 0.5 and 0.75. (**a**) represents the the evolution of AP on isolated formulas, whereas (**b**) exhibits the the evolution of AP on embedded formulas. **Lemma A2:** _If \(\textbf{E}\leq\textbf{A}\) the Democratic party wins for sure and picks \(\textbf{E}=\textbf{A}\) and \(\textbf{E}=\textbf{A}\)_  
**Proof:** This follows by observing that for the Democrats win for sure and hence pick their ideal policy. Now define:  
\[-\kappa_{H}=-\kappa_{L}+\frac{\Delta}{\left(1+T_{v}\left(\bar{v}\right)\right)}\;.\]  
**Lemma A3:** _For \(\left[\textbf{E}\in\left(\kappa_{L},\kappa_{H}\right)\right]\), \(v<v_{D}^{\prime}<\overline{v}=v_{B}^{\prime}\)_  
**Proof:** First, we show for all \(\textbf{E}\in\left(\kappa_{L},\kappa_{H}\right)\) the Republicans will pick \(\textbf{E}=\textbf{A}\). To see this, observe that at \(\textbf{E}=\textbf{A}\) and \(\textbf{E}=\textbf{A}\) the change in the payoff of the Republican party from a small increase in \(v\) is:  
\[\left[\frac{1}{2}-\xi\left[-\kappa+\underline{v}-\overline{v} \right]\right]\left(1+T_{v}\left(\bar{v}\right)\right)+\xi\left[\Delta+ \overline{v}-\underline{v}\right] >\] \[\left[\frac{1}{2}-\xi\left[-\kappa_{H}+\underline{v}-\overline{v }\right]\right]\left(1+T_{v}\left(\bar{v}\right)\right)+\xi\Delta = 0\]  
from the definition of \(\textbf{E}\). Moreover, Assumption 1 implies that this inequality holds for all \(\textbf{E}=\textbf{A}\). Second, we show that it is optimal for the Democrats to pick \(\textbf{E}=\textbf{E}\). Suppose not, such that \(\textbf{E}=\textbf{A}\). Then, a small increase in \(\textbf{E}\) alters the Democratic payoff by:  
Each experiment was replicated 2,000 times for the \(\textbf{E}=\textbf{A}\), \(\textbf{A}\) pairs with \(\left[\textbf{A},\textbf{A}=20,30\right]50,100,200\). In each experiment we computed the CCE Mean Group and the CCE Pooled estimator provided by formula (39) and (42), assuming equal weights \(\textbf{E}=\textbf{E}\), \(i=1,...,\textbf{A}\). We further considered a misspecified structure that ignores the presence of common factors and/or spatial correlations, i.e. the fixed effects estimator  
\[\textbf{\hat{b}}_{FE}=\left(\sum_{i=1}^{N}\textbf{X}_{i}^{\prime}\textbf{M}_ {\tau}\textbf{X}_{i}\right)^{-1}\sum_{i=1}^{N}\textbf{X}_{i}^{\prime}\textbf{ M}_{\tau}\textbf{y}_{i}, \tag{44}\]  
where \(\textbf{M}_{\tau}=\textbf{I}_{T}-\tau\left(\tau^{\prime}\tau\right)^{-1} \textbf{A}\) and \(\textbf{A}\) is a vector of ones. To facilitate the interpretation of results, in each experiment we computed a statistic of cross section dependence, the \(CD\) test ([PERSON], 2004), a statistic of local cross section correlation, the \(CD\) and the simple average of pair-wise cross section correlation coefficients of the residuals, \(\textbf{E}\). We have chosen these tests because they do not require the specification of a generating process for the error term. The \(CD\) statistic is  
\[CD=\sqrt{\frac{2T}{N(N-1)}}\left(\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\hat{r}_{ij} \right),\]  
where \(\textbf{E}\) is the sample estimate of the pair-wise correlation of the residuals, specifically  
Figure 9:0 Example of partial and missed detections of embedded formulas in a document image taken from the Marmot dataset. While green color highlights the correct predictions, partial and missed detections are marked with red and blue colors, respectively. Figure 9:1 Instances of correct detections of embedded formulas in a document image taken from the Marmot dataset.