## 2 OMA of Linear Time-Invariant Structures Based on LPP
### OMA of Linear Time-Invariant Structures Based on LPP  
LPP is a dimensionality reduction algorithm that preserves the geometric relations within the data set. A dataset of \(T\) real valued vectors \(\mathbf{X}(t)=\)\((\overset{\rightarrow}{x}_{1}(t),\overset{\rightarrow}{x}_{2}(t),\cdots \overset{\rightarrow}{x}_{k}(t)\cdots,\overset{\rightarrow}{x}_{n}(t))^{T} \in\mathbb{R}^{n\times T}\), \(\overset{\rightarrow}{x}_{k}(t)\in\mathbb{R}^{1\times T}\) in \(\mathbb{R}^{n\times n}\) space is located on a smooth \(d\)-dimensional manifold (\(d\)\(d\)\(=\)\(n\)). LPP calculates the low-dimensional embedding \(\mathbf{S}(t)=\)\(\{\overset{\rightarrow}{s}_{1}(t),\overset{\rightarrow}{s}_{2}(t),\cdots \overset{\rightarrow}{s}_{k}(t)\cdots,\overset{\rightarrow}{s}_{d}(t)\}^{T} \in\mathbb{R}^{d\times T}\), \(\overset{\rightarrow}{s}_{k}(t)\in\mathbb{R}^{1\times T}\) in \(\mathbb{R}^{d\times d}\) space, which has the same geometric properties as \(\mathbf{X}(t)\). There is a transformation matrix \(\mathbf{A}\) that makes \(\overset{\rightarrow}{s}_{i}(t)=\)\(\mathbf{A}^{T}\overset{\rightarrow}{x}_{i}(t)\), \(x_{ij}\in\overset{\rightarrow}{x}_{i}(t)\in\mathbb{R}^{n\times 1}\), \(1\leq i\leq T\), \(1\leq j\leq n\), \(\overset{\rightarrow}{s}_{i}(t)\in\mathbb{R}^{d\times 1}\). LPP calculates the \(d\)-dimensional embedding process of \(n\)-dimensional data as follows. 1. Construct an adjacent-graph \(G\) of \(T\) real valued vectors in \(\mathbb{R}^{n\times n}\) space: \(K\) nearest neighbor points of the node \(\overset{\rightarrow}{x}_{i}(t)(i=1,2,\cdots i\cdots j\cdots,T)\) are obtained by using the K-neighbor algorithm. If \(\overset{\rightarrow}{x}_{j}(t)\) is in the \(K\) nearest neighbor of \(\overset{\rightarrow}{x}_{i}(t)\), a directed edge (\(\overset{\rightarrow}{x}_{i}(t)\), \(\overset{\rightarrow}{x}_{j}(t)\)) is placed. 2. Calculate the weight of the edge: let the matrix \(\mathbf{W}\in\mathbb{R}^{T\times T}\) represents the weight matrix, and \(w_{ij}\) is the weight of the edge \((\overset{\rightarrow}{x}_{i}(t),\overset{\rightarrow}{x}_{j}(t))\). The weights of the connected edges are calculated by heat kernel \(w_{ij}=e^{-\frac{\|\overset{\rightarrow}{x}_{i}(t)-\overset{\rightarrow}{x }_{j}(t)\|^{2}}{\sigma}}\). If the two nodes are not connected, the weight is 0. 3. Calculate transformation matrix \(\mathbf{A}\): the eigenvectors and eigenvalues of the generalized eigenvector problem are calculated as follows. \[\mathbf{X}(t)\mathbf{L}\mathbf{X}^{T}(t)\overset{\rightarrow}{a}=\lambda \mathbf{X}(t)\mathbf{D}\mathbf{X}^{T}(t)\overset{\rightarrow}{a}\] (5) where the diagonal matrix \(\mathbf{D}\in\mathbb{R}^{T\times T}(D_{ii}=\underset{j}{\Sigma}w_{ji})\) is the degree matrix of graph \(G\), \(\mathbf{L}=\mathbf{D}-\mathbf{W}\) is the Laplacian matrix. 4. Let the column vector set \(\left\{\overset{\rightarrow}{a}_{0},\overset{\rightarrow}{a}_{1},\cdots, \overset{\rightarrow}{a}_{d-1}\right\}\) correspond to the eigenvector of Equation (5) to be solved, order according to their eigenvalues \(\lambda_{0}\leq\lambda_{1}\leq\ \cdots\ \leq\lambda_{d-1}\). The low-dimensional embedded vector \(\overset{\rightarrow}{s}_{i}(t)\) of \(\overset{\rightarrow}{x}_{i}(t)\) is represented as follows. \[\overset{\rightarrow}{s}_{i}(t)=\mathbf{A}^{T}\overset{\rightarrow}{x}_{i}(t)\] (6)\[\mathbf{A}=\left\{\overset{\rightarrow}{a}_{0},\overset{\rightarrow}{a}_{1},\, \ldots,\overset{\rightarrow}{a}_{d-1}\right\}\in\mathbb{R}^{n\times d} \tag{7}\]  
where Vector \(\overset{\rightarrow}{a}_{i}(i=1,2,\cdots,d-1)\) satisfies normalized orthogonality. \[\mathbf{A}^{\mathsf{T}}\mathbf{A}=\mathbf{I}_{d\times d} \tag{8}\]  
Then, \(\mathbf{X}(t)\)\(=\)\(\left\{\overset{\rightarrow}{x}_{1}(t),\overset{\rightarrow}{x}_{2}(t),\cdots \overset{\rightarrow}{x}_{k}(t)\cdots,\overset{\rightarrow}{x}_{n}(t) \right\}^{T}\in\mathbb{R}^{n\times T}\) can be decomposed as follows.