## 3 The Proposed Scheme
### Category Discriminative Strength
The degree of choice for word_w\({}_{\mu}\) considering multiple categories is formulated as:  
\[D_{inter}(w_{\mu},c_{\
u})=\frac{CDS\big{(}w_{\mu},c_{\
u}\big{)}}{\sum Q_{p}} =\frac{CDS\big{(}w_{\mu},c_{\
u}\big{)}}{\sum_{j=1}^{m}CDS\big{(}w_{\mu},c_{j }\big{)}} \tag{9}\]  
Here, word_w\({}_{\mu}\)_ is supposed to appear in category_c\({}_{j}\)(\(1\leq j\leq m\)), and \(D_{inter}(w_{\mu},c_{\
u})\) represents the degree of choice of category_c\({}_{\
u}\) for word_w\({}_{\mu}\)_ measuring how much contribution word_w\({}_{\mu}\) makes for category_c\({}_{\
u}\) in predicting the polarity of the tested document. Using only one category for a word maynot be enough because every presence of the word in other categories is also meaningful. Therefore, the intra-category measure of the words in one category also needs to be analyzed in order to precisely characterize the words. Similar to \(Q_{p}\), another queue, \(Q_{q}=CDS(w_{\alpha}c_{v})\geq\ldots\geq CDS(w_{i}\),\(c_{v})\), is built with the sorted values of \(CDS(w_{i}c_{v})\) of word\(w_{i}\) (\(1\leq i\leq m\)) belonging to category\(c_{v}\) in descending order. The degree of intra-category significance of a word is then measured as:  
\[D_{intra}(w_{\mu},c_{v})=\frac{CDS\big{(}w_{\mu},c_{v}\big{)}}{\sum Q_{q}}= \frac{CDS\big{(}w_{\mu},c_{v}\big{)}}{\sum_{i=1}^{m}CDS(w_{i},c_{v})} \tag{10}\]  
With word\(w_{i}\) (\(1\leq i\leq m\)) in category\(c_{v}\), \(D_{intra}(w_{\mu},c_{v})\) measures the intra-category significance of word\(w_{\mu}\). Incorporating \(D_{inter}(w_{\mu},c_{v})\) and \(D_{intra}(w_{\mu},c_{v})\), the discriminability of word\(w_{\mu}\) for category\(c_{v}\) is measured as:  
\[ \begin{array}{ll}\Phi(w_{\mu},c_{\mu})&=D_{intra}\big{(}w_{\mu},c_{v}\big{)} \cdot D_{intra}\big{(}w_{\mu},c_{v}\big{)}\\ &=\frac{CDS\big{(}w_{\mu},c_{v}\big{)}}{\sum_{i=1}^{m}CDS\big{(}w_{\mu},c_{v} \big{)}}\cdot\frac{CDS\big{(}w_{\mu},c_{v}\big{)}}{\sum_{i=1}^{m}CDS(w_{i},c_{ v})}\end{array}  \tag{11}\]  
Based on the relative frequency and discriminability, word\(w_{\mu}\) is represented as a vector, _DisVector_, as depicted in Figure 2. Here, the word is located at the origin, and the pointer that originated from it represents the position of the discriminability of the word. Obviously, \(Length_{i}\) and \(\theta_{i}\) are needed to be computed for constructing _DisVector_ for a word. The relative frequency, \(cf(w_{\mu}\),\(c_{v})\), and discriminability, \(\Phi(w_{\mu}\),\(c_{v})\), are utilized for obtaining them as:  
\[Length_{i}=cf(w_{\mu},c_{\mu}) \tag{12}\]  
\[\theta_{i}=\Phi(w_{\omega},c_{\mu})\cdot\lambda\cdot\frac{\pi}{2} \tag{13}\]  
Here, \(\lambda\) is a linear factor that maps the range of \(\Phi(w_{\omega},c_{\psi})\), [\(\Phi_{min}\), \(\Phi_{max}\)], into [0,\(\pi\)/2]. The value of \(\Phi_{min}\) and \(\Phi_{max}\) are computed based on the training dataset. Refer to Figure 2. The degree of discriminability of word\(\_{i}\), \(D_{i}\), is obtained as:  
\[D_{i}=Length_{i}\cdot\sin\theta_{i} \tag{14}\]  
The discriminability of a word is characterized by _DisVector_ in terms of the orientation of the discriminability and strength of the word.