# Identifying the Primary Odor Perception Descriptors by Multi-Output Linear Regression Models
## 2 Materials and Methods
### POPDs and the Identification Frameworks
#### 2.2.1 Disjunctive Models
In each execution of the clustering algorithms, after obtaining the clustering centers by K-means or GMM algorithms, the odor perception descriptor vectors closest to these clustering centers are updated to be the candidate CCOPDs. The frequencies of these candidate CCOPDs are recorded to identify the final CCOPDs. The frequency refers to the number of times an odor perception descriptor is selected as the candidate clustering-center odor perception during the executions. It should be noted that the number of candidate CCOPDs is always larger than the specified number of cluster centers. For example, if the specified number of clustering centers K is 10, the number of candidate CCOPDs could reach up to 50 due to the clustering algorithms' randomness. The frequencies of the candidate CCOPDs are quite different. For instance, the frequency of \"fragrant\" selected as the candidate CCOPDs is 894, while the frequency of \"pear\" is only 1. Considering that some odor perception vectors are highly correlated, they could be selected as the candidate CCOPDs by a similar probability. Hence, it is not reasonable to use only the frequencies as the metric for determining the CCOPDs from the candidate ones. It would make the selected CCOPDs concentrate within some local vicinity in odor perceptual space. The complementary information among different clusters could not be captured, and the information is essential for identifying the POPDs. To this end, a CT is set as a constraint. To be specific, the candidate clustering-center odor perception descriptor with the highest frequency is selected as the first clustering-center odor perception descriptor. The other clustering-center ones are chosen as those with high frequencies and satisfy the CT constraint; that is, the correlation between them and all the selected CCOPDs are below the CT. In a word, the top K candidate clustering-odor perception descriptors with the highest frequency, whose correlation coefficients with other candidate ones are lower than the CT, are identified as the final CCOPDs. For these two disjunctive models based on clustering algorithms, the Pearson CT or the upper bound is searched in set {0.5, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 1.0}. Although the number of clustering centers ranges from 10 to 100. It should be emphasized that when the CT is low, the number of clustering-centers odor perception descriptors could not reach up to 100. For example, the maximum number of CCOPDs obtained from GMM with the CT of 0.6 is 57. It is because the candidate CCOPDs could not meet the constraint of the CT. The procedures of the disjunctive model are summarized as follows:  
1. the clustering algorithms are implemented to determine the clustering centers;
2. then the odor perception descriptors closest to these clustering centers are chosen as the candidate CCOPDs;
3. Finally, both the frequency and CT are set as the constraints to select the CCOPDs from the candidate ones. The updating strategy for identifying the CCOPDs is illustrated in Figure 4.  
Besides, to qualify the coverage ability of the set of the CCOPDs with different CTs, the coverage coefficient (CC) and the augmented coverage coefficient (ACC) are introduced. CC is defined as follows:  
\[CC=\frac{\sum\limits_{i=1}^{m_{c}}n_{i}}{\sum\limits_{j=1}^{m}n_{j}}, \tag{11}\]  
where \(n_{i}\), \(n_{j}\) are the frequencies for \(i_{thr}\), \(i_{th}\) cluster-center odor perception descriptors selected in these executions, respectively. The maximum value for \(n_{i}\) and \(n_{j}\) is 1000. \(m_{c}\) is the number of the clusters specified. \(m\) is the total number of all the candidate clustering-centers odor  
Figure 4: The updating strategy for identifying the CCOPDs from the clustering centers. After obtaining the clustering centers by the clustering algorithm, the odor perception descriptors closest to these clustering centers are chosen as the candidate CCOPDs. Then, under the constraints of frequency and CTs, the CCOPDs are identified from the candidate CCOPDs. perception descriptors.