# 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  
The mechanism of identifying the cluster-centering odor perception descriptors is illustrated in Figure 3. Specifically, these two clustering algorithms, K-means and GMM, are conducted to obtain clustering centers. Then, CCOPDs are derived from the clustering centers by an updating strategy based on a correlation threshold (CT) to prevent highly correlated odor perception descriptors from being selected as the CCOPDs. K-Means and GMM Clustering Algorithms  
For K-means, Euclidean distance is used as a similarity metric. The cost function is:  
\[J(\mu_{1},\mu_{2},\ldots,\mu_{k})=\sum_{j=1}^{K}\sum_{i=1}^{N_{j}}||x_{i}-\mu_{ j}||^{2} \tag{5}\]  
where \(K\) is the number of the clustering centers, \(N_{j}\) is the number of samples belonging to the \(j_{th}\) cluster, \(x_{i}\) is the 144-dimensional odor perception descriptor vector, and \(\mu_{1}\), \(\mu_{2}\),...\(\mu_{K}\) are the clustering centers, respectively. By calculating the partial derivative of the cost function to minimize it, the clustering centers are formularized as follows:  
\[\mu_{j}=\frac{1}{N}\sum_{i=1}^{N_{j}}x_{i} \tag{6}\]  
where \(N_{j}\) and \(x_{i}\) are defined as above, and \(N\) is the total number of odor perception descriptor vectors. Another disjunctive model is based on GMM. The GMM distribution is written in the form of Gaussian linear superposition:  
\[p(x)=\sum_{i=1}^{K}\pi_{i}\mathcal{N}\big{(}x|\mu_{i},\Sigma_{i}\big{)}, \tag{7}\]  
where \(\mathcal{N}(. )\) represents the Gaussian distribution. \(x\) is the 144-dimensional odor perception descriptor vector, \(K\) is the number of the Gaussian distributions. \(\pi_{i}\) is the probability of the  
Figure 3: The identification of the cluster-centering odor perception descriptors by the disjunctive model. The clustering algorithm is conducted to obtain clustering centers. Then, CCOPDs are derived from the clustering centers by an updating strategy under the constraints of frequency and correlation threshold (CT). \(i_{th}\) Gaussian distribution, and \(\mu_{i}\), \(\Sigma_{i}\) are the mean and covariance matrix of the \(i_{th}\) Gaussian distribution, respectively. The negative log-likelihood functions of GMM is given by:  
\[L_{\pi,\mu,\Sigma}(x)=\sum\limits_{j=1}^{N}log\big{(}\sum\limits_{i=1}^{K} \mathcal{N}\big{(}x_{j}|u_{i}\Sigma_{i}\big{)}\big{)}, \tag{8}\]  
where \(x\), \(K\), \(\pi_{i}\), \(\mu_{i}\), \(\Sigma_{i}\) are defined as above, and \(N\) is the total number of samples, and the value is 144. The parameters \(\pi_{i}\), \(\mu_{i}\), \(\Sigma_{i}\) are determined by the Expectation Maximization (EM) algorithm and are given as:  
\[\left\{ \begin{aligned} \mu_{i}=&\frac{1}{\sum\limits_{j=1}^{N} \gamma\big{(}j,i\big{)}}\sum\limits_{j=1}^{N}\gamma\big{(}j,i\big{)}\,x_{j}\\ \Sigma_{i}=&\frac{1}{\sum\limits_{j=1}^{N}\gamma \big{(}j,i\big{)}\sum\limits_{j=1}^{N}\gamma\big{(}j,i\big{)}\big{(}x_{j}-\mu_ {i}\big{)}\big{(}x_{j}-\mu_{i}\big{)}^{T}\\ \pi_{i}=&\frac{\sum\limits_{j=1}^{N}\gamma \big{(}j,i\big{)}}{N}\end{aligned} \right. \tag{9}\]  
where \(x_{j}\) is \(j_{th}\) sample, \(K\), \(N\) are defined as above, and \(\gamma\big{(}j,x\big{)}\) is defined as follows:  
\[\gamma\big{(}i,j\big{)}=\frac{\pi_{i}\mathcal{N}\big{(}x_{j}|u_{i},\Sigma_{i }\big{)}}{\sum\limits_{k=1}^{K}\pi_{k}\mathcal{N}\big{(}x_{j}|u_{k},\Sigma_{k }\big{)}}. \tag{10}\]  
All the 146 odor perception descriptors vectors are used to identify the clustering centers for both clustering algorithms. Considering the size of the Dravnieks dataset, the number of clustering centers ranges from 10 to 100. K is the number of NCCOPDs. Updating Strategy for Obtaining CCOPDs from Clustering Centers  
Due to the random initialization of these clustering algorithms, each iteration's clustering centers are not always the same. Therefore, each clustering algorithm has been implemented many times to improve the stability of the clustering results.