# Low-Rate Non-Intrusive Load Disaggregation with Graph Shift Quadratic Form Constraint
## 1 Introduction
### State of the Art in NILM
The non-event-based NILM relies on constructing a model that includes signatures of different devices as its bases. The necessity of user intervention for creating models distinguishes unsupervised from supervised methods. The most common approaches for load disaggregation are Hidden Markov Model (HMM), an unsupervised algorithm, and its variations (see [28; 29; 30; 31] and references therein). The early work on unsupervised NILM is presented in Ref. [28], where four different extensions of HMM are tested for better disaggregation results. However, there are some weaknesses of these algorithms, such as high computational complexity and local convergence. Recently, new methods have been proposed to address the weakness of Ref. [28], for example, Hierarchical Dirichlet ProcessHMM (HDP-HMM) and Hierarchical Dirichlet Process Hidden semi-Markov Model (HDP-HSMM) [31]. In order to improve the performance of standard Factorial HMM (FHMM), [PERSON] et al. [32] presented a novel energy disaggregation model that embeds the information of mutual devices interactions into the FHMM. In [33], the Hierarchical HMM (HHMM) is introduced to enhance the representation power of Markov chain based models, especially for multi-mode appliances. The aforementioned methods have been proved effective to disaggregate simple two state or multi-state loads, yet they heavily rely on clean transition from one state to another, and perform poorly for continuously varying appliances (for example, laptops or any other charger), which do not present a significant step increase in the power. On the other hand, unsupervised methods need manual adjustment of parameters, thus it is challenging to generalize them in practice. For alleviating the aforesaid problems, [PERSON] et al. [34] developed a method based upon a sparse coding algorithm. It uses a dictionary learning approach to construct the model of each appliance without any prior knowledge and does not make any assumptions. Motivated by the success of deep learning, a new deep learning approach, named 'deep sparse coding', is proposed in Ref. [35]. The characteristic is that instead of learning one level of basis/dictionary, it learns multiple layers. Moreover, in order to reduce the complexity of the training/learning process in existing decomposition based algorithms, an innovative time series distance-based approach in the temporal classification domain is proposed [36]. The main strengths are its very short and non-intrusive training period. In this paper, we propose a low-rate hybrid NILM method. It relies on graph signal processing (GSP) theory [37], an emerging data analysis tool which can be applied to data classification by expressing the piecewise smoothness of a graph signal, and is suitable for training data that is not sufficient to establish effective classification models. Assuming that only a subset of data element labels are known, the GSP-based data classifier will predict the missing labels using the structure of the graph. The authors in Ref. [38] propose a definition of total variation on graph signals based on the concept of graph shift, which is demonstrated as useful for frequency analysis and data classification. A supervised classification method based on GSP is proposed in Ref. [39] and is applied to image and document datasets. Recently, some researchers have applied the theory of GSP into NILM, such as Ref. [40; 41], yet they both adopt the graph signal's total Laplacian smoothness as the regularization term, the classification accuracy of which is lower than that of graph shift-based regularization [38].