## 4 Experiments
### Image Classification  
To evaluate the classification performance for outlier problems of various methodologies, we first divide the images of each database into two parts, namely, non-standard and standard. The former is defined as a group of images under lighting conditions or with occlusions whereas the latter represents normal images with various forms or facial expressions. Let \(N_{i}\), \(i=1,2,\ldots,q\), be the number of images of the \(i\)th individual in the standard part for a particular database with \(q\) represents the number of  
Figure 5: (**a**) JAFFE face database: images with various facial expressions. (**b**) Yale face database: 1 st-3 rd images: normal faces; 4 th–6 th images: faces under various lighting conditions. (**c**) AR face database: 1 st-2 nd images: normal faces; 3 th–4 th images: faces under various lighting conditions; 5 th–6 th images: faces with sunglasses; 7 th–8 th images: faces with scarves. (**d**) COIL-20 image database: 1 st-2 nd images: normal objects; 3 rd–4 th images: objects with artificial blocking. individuals in the respective database and \(r\) be the number of images randomly selected within these \(N_{i}\) images. In the training phase, \(N_{i}-r\) images in the standard part and half of the images in the non-standard part are randomly selected to form the training set whereas the remaining images (i.e., \(r\) images of each individual in the standard part and the other half of the images in the non-standard part) are used for testing. Let \(\Phi\) be any image in the training set and \(\Psi\) be the testing image. Both \(\Phi\) and \(\Psi\) are projected onto \(u\) to obtain the weight vectors \(w_{\Phi}=u^{T}(\Phi-m)\) and \(w_{\Psi}=u^{T}(\Psi-m)\)[58], where \(m\) is the mean of training set. Then the testing image is assigned to a class based on the 1-nearest neighbor classifier, which has been extensively used in image classification problems [24; 28]. To quantify the classification performance of each methodology, we adopt the averaged maximum classification rate (AMCR) and averaged maximum F1-score (AMF1-score) with respect to the optimal number of orthonormal projection vectors. The formula for classification rate is defined as  
\[\text{CR}=\frac{\sum_{\text{c}}\left(\text{TP}(\text{c})+\text{TN}(\text{c}) \right)}{\sum_{\text{c}}\left(\text{TP}(\text{c})+\text{FP}(\text{c})+\text{ FN}(\text{c})+\text{TN}(\text{c})\right)} \tag{17}\]  
where TP(c), FP(c), FN(c) and TN(c) are the true positive, false positive, false negative and true negative for class c respectively. The maximum classification rate is the classification rate with the optimal number of projection vectors. The F1-score is defined as follows:  
\[\text{F1}-\text{score}=\sum_{\text{c}}2\frac{\text{precision}(\text{c}) \times\text{recall}(\text{c})}{\text{precision}(\text{c})+\text{recall}(\text {c})} \tag{18}\]  
Precision and recall are calculated as follows:  
\[\text{precision}(\text{c})=\frac{\text{TP}(\text{c})}{\text{TP}(\text{c})+ \text{FP}(\text{c})} \tag{19}\]  
\[\text{recall}(\text{c})=\frac{\text{TP}(\text{c})}{\text{TP}(\text{c})+\text{ FN}(\text{c})} \tag{20}\]  
Similar to the classification rate, the maximum F1-score is the F1-score with the optimal number of projection vectors. Now, we explain the procedure to obtain the optimal number of projection vectors. For each \(r\) per above random training set, the optimal number of vectors is selected by the holdout set method [59]. First, a pair of holdout sets is created by splitting the above training set into two. 70% of the samples are randomly selected to form the holdout set for training while the rest of them form the holdout set for testing. The optimal number of projection vectors is then the number of dimensions that produces the best classification rate (i.e., CR) with the 1-nearest neighbor being applied to the holdout set for testing 10 times. The averaged maximum classification rate (AMCR), averaged maximum F1-score (AMF1-score) and averaged optimal number of projection vectors (Avg. Dim.)