# Metrics-Based Comparison of OWL and XML for Representing and Querying Cognitive Radio Capabilities
## 4 Problem Formulation
### Metrics Formulation
#### 4.2.1 Quality Metrics  
The _query result completeness_ metric, referred to as the _recall metric_ in information retrieval, measures the degree of how many of the expected results are returned by a query. The XML-based metric, \(m_{c}^{\textit{xml}}\), and the OWL-based metric, \(m_{c}^{\textit{xml}}\), are defined in (8) and (9), respectively.  
The _query result soundness_ metric measures the degree of how many query results returned are as expected out of all the returned results. In information retrieval, this metric is called the _precision_ metric. The XML-based metric, \(m_{s}^{\textit{xml}}\), and the OWL-based metric, \(m_{s}^{\textit{owl}}\), are defined in (10) and (11), respectively.  
The _F-Measure_[42] is a combined metric that measures the trade-off between the completeness and the soundness of the query results. The XML-based and OWL-based definitions of this metric are shown in (12) and (13), respectively. In (12), \(m_{c}^{\textit{xml}}()\) is short for \(m_{c}^{\textit{xml}}(f^{\textit{xml}}(D^{\textit{xml}},Q_{i}^{\textit{xml }}),g^{\textit{xml}}(D^{\textit{xml}},Q_{i}^{\textit{xml}}))\) introduced in (8), \(m_{s}^{\textit{xml}}()\) is short for \(m_{s}^{\textit{xml}}(f^{\textit{xml}}(D^{\textit{xml}},Q_{i}^{\textit{xml }}),g^{\textit{xml}}(D^{\textit{xml}},Q_{i}^{\textit{xml}}))\) introduced in (10). In (13), \(m_{c}^{\textit{xml}}()\) is short for \(m_{c}^{\textit{xml}}(f^{\textit{owl}}(D^{\textit{owl}},B_{j},Q_{i}^{\textit{ xml}}),g^{\textit{owl}}(D^{\textit{owl}},Q_{i}^{\textit{owl}}))\) introduced in (9), \(m_{s}^{\textit{aval}}()\) is short for \(m_{s}^{\textit{aval}}(f^{\textit{owl}}(D^{\textit{owl}},B_{j},Q_{i}^{\textit{ val}}),g^{\textit{owl}}(D^{\textit{owl}},Q_{i}^{\textit{val}}))\) introduced in (11). In both equations, \(\beta\in[0,\infty]\) is a parameter that controls the balance between _precision_ and _recall_[42]. In this paper, \(\beta\) was set to 1 since we equally weighed the importance levels of _precision_ and _recall_.  
\[\begin{split} m_{c}^{\textit{xml}}(f^{\textit{xml}}(D^{\textit{ xml}},Q_{i}^{\textit{xml}}),\\ g^{\textit{xml}}(D^{\textit{xml}},Q_{i}^{\textit{xml}}))=\left\{ \begin{array}{l}\frac{|f^{\textit{xml}}(D^{\textit{xml}},Q_{i}^{\textit{ xml}})\cap g^{\textit{xml}}(D^{\textit{xml}},Q_{i}^{\textit{xml}})|}{|g^{ \textit{xml}}(D^{\textit{xml}},Q_{i}^  
\[\begin{split} m_{s}^{out}(f^{outl}(D^{outl},B_{j},Q_{i}^{outl}))&,\begin{cases}\dfrac{|f^{outl}(D^{outl},B_{j},Q_{i}^{outl})\cap g^{outl}(D^{outl},Q _{i}^{outl})|}{|f^{outl}(D^{outl},B_{j},Q_{i}^{outl})|},\text{if }f^{outl}(D^{outl},B_{j},Q_{i}^{outl})\
eq \emptyset\\ g^{outl}(D^{outl},Q_{i}^{outl}))&=\\ 1,\text{if }f^{outl}(D^{outl},B_{j},Q_{i}^{outl})=g^{outl}(D^{outl},Q _{i}^{outl})=\emptyset\\ 0,\text{Others}\end{split} \tag{11}\]  
\[\begin{split} m_{f}^{xml}(m_{c}^{xml}(),m_{s}^{xml}())& =\begin{cases}\dfrac{(\beta^{2}+1)\cdot m_{c}^{xml}()\cdot m_{s}^{xml }()}{\beta^{2}\cdot m_{c}^{xml}()+m_{s}^{xml}()},\text{ if }m_{c}^{xml}()\
eq 0 \lor m_{f}^{xml}()\
eq 0\\ 0,\text{ if }m_{c}^{xml}()=m_{f}^{xml}()=0\end{cases}\\ m_{f}^{outl}(m_{c}^{outl})()&=\\ \dfrac{(\beta^{2}+1)\cdot m_{c}^{outl}()\cdot m_{s}^{owl}()}{\beta^{2}\cdot m _{c}^{outl}()+m_{s}^{outl}()},\text{ if }m_{c}^{outl}()\
eq 0\lor m_{f}^{outl}() \
eq 0\\ 0,\text{ if }m_{c}^{outl}()=m_{f}^{outl}()=0\end{cases}\end{split} \tag{13}\]