## 3 PCAWF- and ARPCAWF-Based OMA Methods and Their Limitations
### Theory of ARPCAWF  
In classical PCA, \(\mathbf{X}^{(i)}\in\mathbb{R}^{N\times i}\) is a data matrix of \(N\) variables and \(i\) samples. This matrix can be decomposed as follows:  
\[\mathbf{X}^{(i)}\approx\mathbf{W}^{(i)}\mathbf{Y}^{(i)} \tag{6}\]  
where \(\mathbf{W}^{(i)}\in\mathbb{R}^{N\times M}\) and \(\mathbf{Y}^{(i)}\in\mathbb{R}^{M\times i}\) are the transformation matrix and PC matrix, respectively, and \(M\) is the number of PCs. The correlation matrix \(\mathbf{C}_{\mathbf{X}^{(i)}\mathbf{X}^{(i)}}\in\mathbb{R}^{N\times N}\) can be written as:  
\[\mathbf{C}_{\mathbf{X}^{(i)}\mathbf{X}^{(i)}}\approx\frac{1}{i-1}\mathbf{X}^{ (i)}\mathbf{X}^{(i)\intercal}=\mathbf{W}^{(i)}\mathbf{\Lambda}^{(i)}\mathbf{W }^{(i)\intercal} \tag{7}\]  
where the columns of \(\mathbf{W}^{(i)}\) are the eigenvectors of \(\mathbf{C}_{\mathbf{X}^{(i)}\mathbf{X}^{(i)}}\) corresponding to the \(M\) largest eigenvalues, and \(\mathbf{\Lambda}^{(i)}\in\mathbb{R}^{M\times M}\) is a real positive diagonal matrix corresponding to the \(M\) largest eigenvalues.  
The classical PCA or PCAWF model is a batch-wise statistical method. When a new sample becomes available, the autocorrelation matrix or transformation matrix and the PC matrix of PCA/PCAWF need to be recalculated. In RPCA methods, the autocorrelation matrix or transformation matrix and PC matrix of PCA/PCAWF are updated recursively. Various recursive methods can be employed for this purpose, such as autocorrelation matrix recursion, eigenvector recursion [32], rank-\(1\) corrected recursion [41], and subspace tracking recursion [42]. In this section, we describe how a weighting factor is introduced into the recursive autocorrelation calculations. The forgetting factor will have a significant effect on the statistical properties of the model output. Let us assume that the row mean vector \(\vec{b}_{\mathbf{X}^{(i)}}\in\mathbb{R}^{N\times 1}\), data matrix \(\mathbf{X}^{(i)}\), and auto-correlation matrix \(\mathbf{C}_{\mathbf{X}^{(i)}\mathbf{X}^{(i)}}\in\mathbb{R}^{N\times N}\) have been calculated using the first \(i\) samples. \(b_{\vec{x}_{j}^{(i)}}\) denotes the \(j\)throw of \(\vec{b}_{\mathbf{X}^{(i)}}\). Thus, the recursive calculations for the mean vector \(\vec{b}_{\mathbf{X}^{(i+1)}}\), standard deviation vector \(\vec{\sigma}_{\mathbf{X}^{(i+1)}}\), and \(\mathbf{C}_{\mathbf{X}^{(i+1)}\mathbf{X}^{(i+1)}}\) are:  
\[b_{\vec{x}_{j}^{(i+1)}}=\mu b_{\vec{x}_{j}^{(i)}}+(1-\mu)x_{j}(i+1),\;j=1,...,N \tag{8}\]  
\[\sigma^{2}_{\vec{x}_{j}^{(i+1)}}=\mu\{\sigma^{2}_{\vec{x}_{j}^{(i)}}+[\triangle b _{j}(i+1)]^{2}\}+(1-\mu)[x_{j}(i+1)-b_{\vec{x}_{j}^{(i+1)}}]^{2} \tag{9}\]  
\[\mathbf{C}_{\mathbf{X}^{(i+1)}\mathbf{X}^{(i+1)}}=\mu\Xi_{i+1}^{-1}[\mathbf{ \Sigma}_{i}\mathbf{C}_{\mathbf{X}^{(i)}\mathbf{X}^{(i)}}\mathbf{\Sigma}_{i}+ \triangle\bar{b}(i+1)\triangle\bar{b}^{T}(i+1)]\Sigma_{i+1}^{-1}+(1-\mu)\vec{x }(i+1)\vec{x}^{T}(i+1) \tag{10}\]  
where \(\triangle b_{j}(i+1)=b_{\vec{x}_{j}^{(i+1)}}-b_{\vec{x}_{j}^{(i)}}\), \(\triangle\bar{b}(i+1)=\vec{b}_{\mathbf{X}^{(i+1)}}-\vec{b}_{\mathbf{X}^{(i)}} \in\mathbb{R}^{N\times 1}\), \(\mathbf{\Sigma}_{i+1}=diag\{\sigma_{\vec{x}_{1}^{(i+1)}},...,\sigma_{\vec{x}_{ N}^{(i+1)}}\}\in\mathbb{R}^{N\times N}\), and \(0<\mu\leq\frac{i}{i+1}<1\) is the forgetting factor. When \(\mu=\frac{i}{i+1}\), the ARPCAWF model reduces to the ARPCA model. Therefore, the transformation matrix can be obtained by decomposing the autocorrelation matrix in Equation (7) and the PCs can be calculated using Equation (6).