## 2 The Theory of the Smooth Local Subspace Projection Noise Reduction and PE
### SOD and Data Projection  
SOD [34] aims at obtaining a smooth low dimensional approximation of a high dimensional dynamic process, and it considers the time characteristics and space characteristics, whereas POD only considers the geometrical or topological properties of data, and does not take any temporal characteristics of time series into account. For example, when y is defined as a field of d state variables, assume that this field happens to take samples in \(n\) instantaneous time. The data can be formed into an \(n\times d\) matrix \(\mathbf{Y}\), so \(\mathbf{Y}_{ij}\) is the sample of the \(i\)th moment from the \(j\)th state variable, and the mean value of each row of \(\mathbf{Y}\) matrix is assumed as 0. Then a linear coordinate transformation of this matrix is conducted as shown in Equation (3). \[\mathbf{Q}=\mathbf{Y}\cdot\psi \tag{3}\]  
where each row of \(\mathbf{Q}\in\mathbf{R}^{n\times d}\) is a new smooth orthogonal component, smooth projection modes [29] are decided by the rows of \(\psi\in\mathbf{R}^{d\times d}\). In the process of above change, time coordinates needs to be considered in \(\mathbf{Q}\) which is ranked by the roughness. Then SOD is realized through the following generalized characteristic value problem, as shown in Equation (4). \[\Sigma\psi_{i}=\lambda_{i}\Sigma^{\prime}\psi_{i} \tag{4}\]  
where \(\Sigma=(1/n)\mathbf{Y}^{T}\mathbf{Y}\in\mathbf{R}^{d\times d}\) and \(\Sigma^{\prime}=(1/n)\mathbf{Y}^{T}\mathbf{Y}^{\prime}\in\mathbf{R}^{d\times d}\) are the auto-covariance matrix of \(d\) state variables and their time derivatives. The characteristic value \(\lambda_{i}\) are smooth orthogonal characteristic values. The above Equation (4) can be realized by the following matrix, as shown in Equation (5). \[\mathbf{Y}=\mathbf{UC}\mathbf{\Phi}^{T},\;\mathbf{Y}^{\prime}=\mathbf{VS} \mathbf{\Phi}^{T},\;\mathbf{C}^{T}\mathbf{C}+\mathbf{S}^{T}\mathbf{S}=\mathbf{I} \tag{5}\]  
where smooth, orthogonal modes and smooth, orthogonal components are determined by the rows of \(\mathbf{\Phi}\in\mathbf{R}^{d\times d}\) and \(\mathbf{Q}=\mathbf{UC}\in\mathbf{R}^{n\times d}\) respectively, and smooth, orthogonal characteristic values are determined by the term by term division of the diagonal matrix \(\mathbf{C}^{T}\mathbf{C}\) and \(\mathbf{S}^{T}\mathbf{S}\). The \(k\) (\(k<d\)) dimensional smooth approximation of \(\mathbf{Y}\) can be obtained by retaining \(k\) rows of matrix \(\mathbf{U}\) and \(\mathbf{\Phi}\), meanwhile the matrix \(\mathbf{C}\) is reduced to \(k\times k\) order accordingly. The projection of \(\mathbf{Y}\) in \(k\) dimensional smooth subspace is the purpose. The \(k\) dimensional projection is embedded into \(d\) dimensional space \(\tilde{\mathbf{Y}}\) by using the corresponding matrices, as shown in Equation (6), which involves with three matrix transformations. \[\overline{\mathbf{U}}\in\mathbf{R}^{n\times k},\;\overline{\mathbf{C}}= \mathbf{R}^{\times k},\;\overline{\mathbf{\Phi}}=\mathbf{R}^{d\times k} \tag{6}\]  
Then the \(\tilde{\mathbf{Y}}\) matrix can be obtained by the following Equation (7). \[\overline{\mathbf{Y}}=\overline{\mathbf{UC}\mathbf{\Phi}^{T}} \tag{7}\]  
The main steps of this smooth local subspace projection method are given in Table 1.