# A Knowledge and Semantic Fusion Method for Automatic Geometry Problem Understanding
## 6 Experiment and Application
### Experiment and Analysis
#### 6.1.2 Application Verification Test and Error Analysis
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Relationship Type** & **Total** & **Accuracy** & **Failure Causes** \\ \hline Unary Geo.Relationship & 182 & 97.80\% & 1, 3 \\ Binary Geo.Relationship & 1027 & 96.11\% & \(1,2,3,4\) \\ Ternary Geo.Relationship & 534 & 93.26\% & 1 \\ Numerical Relationship & 89 & 91.01\% & 2 \\ Total Relationships & 1832 & 95.23\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Statistics on failure text-level ERE from the target relationship type dimension. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**RMT** & **Total** & **Accuracy** & **Failure Causes** \\ \hline SC-B & 458 & 98.25\% & \(1,2,3,4\) \\ SC-T & 367 & 96.46\% & 3 \\ SI-B & 183 & 94.54\% & \(1,2,3,4\) \\ SI-T & 277 & 92.06\% & 1, 3 \\ MC-S & 329 & 90.58\% & \(1,3,4\) \\ MC-D & 146 & 87.67\% & \(1,3,4\) \\ MT & 72 & 83.33\% & \(1,2,3,4\) \\ Total Relationships & 1832 & 95.23\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Statistics on failure text-level ERE from the RMT dimension. parsing of other atomic relationships, for example, the sentence excerpt from a problem text: \"through \(\triangle\)ABH's circumcenter E draw EF\(\|\)GC\". Although the Circumcenter relationship is not processed, the handled PointOnLine relationship (E-pointOnLine-EF) and the Parallel relationship (EF-parallel-GC) can still be correctly identified and extracted. Furthermore, even in more complex cases such as \"Trought \(\triangle\)ABH's circumcenter E draw AD's perpendicular bisector line EF\(\|\)GC intersects PM at point Q\", the unprocessed circumcenter relationship (E-circumcenter-\(\triangle\)ABH) does not affect the extraction of the processed relationships: E-pointOnLine-AD, EF-perpendicularBisector-AD, EF-parallel-GC, and EF-intersect-PM. Further, to verify the scope of error impact, the following experiment was conducted. Experimental Setup: From Dataset B, 209 problems that did not contain any extraction errors were selected. In the current SGKO, one relationship type was manually masked each time, and the accuracy of relationship extraction was tested to assess its impact on the extraction of other relationships. Three relationship types were chosen for masking, PerpendicularBisector and Parallel, representing the least and most frequently mentioned binary relationship types in the dataset, respectively, and Perpendicular, a kind of ternary relationship type whose mention may include a perpendicular foot, with the core relationship type Perpendicular marked and its attached relationship type foot left unmarked. As shown in Table 16, among a total of 1705 relationships, the masked group with PerpendicularBisector, Parallel, and Perpendicular, respectively, achieved extraction accuracies of 99.06%, 95.78%, and 92.79%, with all errors confined to the masked relationship type. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Experimental Group** & **Impacted Relationships** & **Accuracy** \\ \hline Total & 1705 & - \\ \hline Unmasked & 0 & 100\% \\ \hline PerpendicularBisector-masked & 16 & 99.06\% \\ \hline Parallel-masked & 72 & 95.78\% \\ \hline Perpendicular-masked & 123 (45) & 92.79\% \\ \hline \hline \end{tabular}
\end{table} 
Table 16: The impact of restricted error propagation at the atomic relationship level on accuracy. \begin{table}
\begin{tabular}{l l l} \hline \hline
\begin{tabular}{l} Unable to detect and extract \\ entities. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Error causes and handling in entity and relationship extraction. This indicates two conclusions: First, error propagation was effectively limited to the atomic relationship. Second, the impact of failures caused by unprocessed relationship types was minimal, correlating only with the frequency of relationship mentions. Additionally, tests on 123 Perpendicular relationships showed that attached relationships were not extracted when their relationship type was not marked. Their correct processing depended on the extraction of their relative core relationships.