# Low-Rate Non-Intrusive Load Disaggregation with Graph Shift Quadratic Form Constraint
## 2 Methodology
### Graph Shift Quadratic Form Constrained Active Power Disaggregation  
As mentioned in [42], one of the key points for designing a MF problem is to find an appropriate application-specific constraint. Unlike previous methods, in this paper, we innovatively integrate the technical idea of the event-based approaches into the MF problem. Rather, we use GSP, an emerging classification technique, to classify the active power variation at each moment by adding graph shift quadratic constraint to the optimization problem in Equation (4). According to the classification results, we can determine whether there is an appliance changing its state and obtain the activation coefficient for each device. That is to say, MF is realized. Once all activation coefficients are calculated, the power consumption at different times of each appliance can be acquired. Hence, our proposed algorithm can be called GSP-based hybrid NILM, which combines the basic ideas of event-based and non-event based NILM. In the following paragraphs, we elaborate on the GSQF-APD algorithm. There are two basic methods to signal processing on graphs [38]. One is based on the spectral graph theory [43], adopting the graph Laplacian matrix that characterizes the underlying relational structure of the data as its fundamental building block. The other is motivated by the algebraic signal processing (ASP) theory [44; 45], using the graph shift operator as its basic building block. We construct a graph \(G=(\mathbf{v},\,\mathbf{E})\), where \(\mathbf{v}=\{v_{1},\cdots,v_{N}\}\) indicates the set of nodes and \(\mathbf{E}\) is a weighted adjacency matrix of the graph. Each node \(v_{i}\in\mathbf{v}\) is associated to one sample \(\Delta P(i)\), \(i=1,\cdots,N\). The similarity between \(\Delta P(i)\) and \(\Delta P(j)\) is embodied in the edge weight between \(v_{i}\) and \(v_{j}\), which is defined by the Gaussian kernel weighting function,  
\[E_{i,j}=\exp\left\{-[\Delta P(i)-\Delta P(j)]^{2}/\sigma^{2}\right\} \tag{5}\]  
where \(\sigma\) is the scaling factor. Using the graph, we refer to the samples as a graph signal \(\mathbf{s}\), where each element \(s(i)\) is indexed by node \(v_{i}\) of the given representation graph \(G=(\mathbf{v},\,\mathbf{E})\). In this paper, \(\mathbf{s}\) is defined as a set of classification labels, in which \(s(i)\) corresponds to the label of the class that element \(\Delta P(i)\) belongs to. The features of a graph signal are very complex, where the smoothness is an important property to reflect the intrinsic structure of the graph signal. The smooth signal model based on the concept of smoothness is widely used in the regularization and semi-supervised learning of graphs. There are two methods to define the total smoothness on graph signals, one using the graph shift matrix (i.e., adjacency matrix) and the other via the graph Laplacian matrix. The graph signal's total variation (i.e., global smoothness) based on the shift matrix is defined as [38],  
\[\left\|\mathbf{s}\right\|_{G}=\sqrt{0.5\|\mathbf{s}-\mathbf{E}^{norm}\mathbf{ s}\|_{2}^{2}} \tag{6}\]  
where \(S_{2G}(2)=0.5\|\mathbf{s}-\mathbf{E}^{norm}\mathbf{s}\|_{2}^{2}\) is the graph shift quadratic form and \(\mathbf{E}^{\text{norm}}=(1/|\lambda_{\text{max}}|)\mathbf{E}\) is the normalized form of graph shift matrix. \(\lambda_{\text{max}}\) indicates the eigenvalue of \(\mathbf{E}\) with the largest magnitude. The graph signal's total variation based on the Laplacian matrix is defined as [37],  
\[\left\|\mathbf{s}\right\|_{L}=\sqrt{\mathbf{s}^{T}\mathbf{Ls}} \tag{7}\]  
where \(S_{2L}(2)=\mathbf{s}^{T}\mathbf{Ls}\) is the graph Laplacian quadratic form. The Laplacian matrix \(\mathbf{L}\) is defined as \(\mathbf{L}=\mathbf{D}-\mathbf{E}\), where \(\mathbf{D}\) is a \(N\times N\) diagonal matrix with elements \(D_{k,k}=\sum_{j=1}^{N}E_{j,k},k=1,\cdots,N\). Large dataset classification usually cannot be performed manually. A common method to address the problem is to use a known structure of the dataset to predict the labels of the remaining elements [38].