# A Novel Multi-Task Performance Prediction Model for Spark
## 3 Proposed Model
### Data Preprocessing
#### 3.2.2 Principal Component Analysis (PCA) Dimensionality Reduction Algorithm  
During the data preprocessing stage, we start by performing One-Hot encoding on categorical variables. However, this method could lead to a significant increase in data dimensions, thereby raising the complexity of model training and potentially increasing the risk of overfitting. On the other hand, we apply standardization to continuous variables,which ensures these variables possess uniform magnitudes during the model training process. This prevents adverse effects caused by magnitude disparities in model training. After processing, we combine the standardized continuous variables with the One-Hot encoded categorical variables through concatenation, forming a unified dataset. Nonetheless, it is worth noting that this dataset's dimensions might be somewhat larger compared to the original data. To solve this problem, we decided to introduce a dimensionality reduction algorithm such as PCA. It is a common data dimensionality reduction method that reduces high-dimensional data to lower dimensions while retaining as much important information as possible from the original data. The main steps of PCA include:  
1. Data standardization: since PCA is sensitive to the variance in the initial variables, all features need to be standardized to have a mean of 0 and a standard deviation of 1. We set the features as a feature matrix \(X\in R^{n\times m}\), as shown in Equation (4). \[X= \begin{pmatrix}x_{11}&\dots&x_{1k}&\dots&x_{1m}\\ \vdots&\ddots&&&\vdots\\ x_{i1}&&x_{ik}&&x_{im}\\ \vdots&&&\ddots&\vdots\\ x_{n1}&\dots&x_{nk}&\dots&x_{mm}\end{pmatrix} _{n\times m}\] (4) where \(i,k,n,m\in\mathbb{Z}^{+}\), \(n\) is the number of data points, \(m\) is the number of features, and \(x_{ik}\) is a feature value of data point i. For each feature \(x_{k}\), we calculate the mean value \(\mu_{x_{k}}\), which is calculated as shown in Equation (5). \[\mu_{x_{k}}=\frac{1}{n}\sum_{i=1}^{n}x_{ik}\] (5) where \(x_{ik}\) represents the original value of data point \(i\) on feature \(k\). The standard deviation \(\sigma_{x_{k}}\) is then calculated, as shown in Equation (6). \[\sigma_{x_{k}}=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_{ik}-\mu_{x_{k}})^{2}}\] (6) Finally, the feature matrix \(X\) is standardized. The standard transformation of each element \(x_{ik}^{{}^{\prime}}\) in matrix \(X\) is shown in Equation (7). \[x_{ik}^{{}^{\prime}}=\frac{x_{ik}-\mu_{x_{k}}}{\sigma_{x_{k}}}\] (7) where \(x_{ik}^{{}^{\prime}}\) is the normalized value. The feature matrix \(X\) becomes \(X^{{}^{\prime}}\). 2. Calculate the covariance matrix: we set the covariance matrix as \(C\in R^{m\times m}\), which represents the covariance between pairs of features. It is calculated based on all data points to quantify and understand how the features in the dataset are related to each other from a global perspective. Finally, we obtain the covariance matrix as shown in Equation (8). \[C= \begin{pmatrix}c_{11}&\dots&c_{1m}\\ \vdots&\ddots&\vdots\\ c_{m1}&\dots&c_{mm}\end{pmatrix} _{m\times m}\] (8) In the covariance matrix \(C\), each element \(c_{jk}\) is computed as shown in Equation (9). \[c_{jk}=\frac{1}{n-1}\sum_{i=1}^{n}(x^{\prime}_{ij}-\mu_{x_{j}})(x^{\prime}_{ik}- \mu_{x_{k}}) \tag{9}\]  
where \(i,j,k,\) and \(n\in\mathbb{Z}^{+}\). Here, \(n\) represents the number of data points and \(i\) denotes any data point index, while \(j\) and \(k\) are indices for any two features of a data point. 3. Calculate the eigenvalues and eigenvectors: we decompose the covariance matrix into its eigenvalues and eigenvectors. The eigenvalues and eigenvectors of the covariance matrix satisfy the conditions of Equation (10). \[C\mathbf{a}=\lambda\mathbf{a}\] (10) where \(\mathbf{a}\) is the eigenvector of matrix \(\mathcal{C}\) and \(\lambda\) is the corresponding eigenvalue. In PCA, the eigenvalues reflect the variability of the data in the direction of the corresponding eigenvectors.