# Low-Rate Non-Intrusive Load Disaggregation with Graph Shift Quadratic Form Constraint
## 1 Introduction
### State of the Art in NILM  
NILM appeared in the 1980s, since then, its research has been conducted for more than 30 years. Early studies in NILM aim to label the state changing event by identifying distinct electrical features of individual appliances. They can be divided into two groups, steady state and transient state analysis. The common steady-state NILM methods are based on active power and reactive power [7; 8; 9]. Other steady-state approaches rely on harmonics [10], voltage, and current waveform [11], or the voltage-current (V-I) trajectory analysis [12; 13]. Transient analysis at start-up [14; 15; 16; 17; 18] is also a unique and effective method which can provide more distinguishable features. However, most of the aforementioned methods require high sampling frequency in the order of kHz or MHz. For example, it was suggested that the sampling rate should be at least 8 kHz for harmonics-based NILM [19]. Unfortunately, such a sampling rate is not practical in most scenarios and is especially inaccessible with the currently available smart meters. With the large-scale deployment of smart meters in Europe, Australia, and USA [20; 21], low-rate NILM based on the data captured from smart meters has drawn increasing attention. It eliminates the need to install additional acquisition devices, therefore greatly reducing hardware costs. According to the different monitoring strategies of electrical appliance operating state, low-rate NILM methods can be categorized into three main classes. The first group includes techniques for load classification, which are often referred to as event-based NILM. Its goal is to output the state transition of each appliance, such as turning on or off. The basic processing procedure of this kind of method is first detecting events, then extracting the signature related to the load event, and finally labelling step changing events via some classification algorithms. The second group comprises approaches for load disaggregation, which defines the NILM as a problem of optimization decomposition or pattern recognition. Since this type of method is not based on load event analysis, they are called non-event based NILM in general. The third group is a combination of the basic ideas in the former two methods and can be referred to as hybrid NILM [22]. Different classification tools have been used for event-based NILM. The maximum likelihood classifier used in Ref. [14] has been demonstrated to be less sensitive to power grid noise than the \(k\)-mean-based method, but this approach only works with type I appliances, i.e., the appliances with only ON/OFF states. In Ref. [23], the authors develop a systematic approach of NILM using few training examples with two generic classifiers (\(k\)-NN and naive Bayes) for satisfactory accuracy. A method using an Artificial Neural Network (ANN) for training steady-state real power and reactive power signatures is proposed in Ref. [24], but the experiment scene is too simple, which only includes four appliances in the simulation test. In Ref. [25], the authors propose two algorithms: a low-complexity, supervised approach based on Decision Tree (DT), and an unsupervised method based on Dynamic Time Warping (DTW). Experimental results show that these two methods can disaggregate domestic loads even when the training period is very short. In order to enable correct load (i.e., class) prediction of unlabeled instances in NILM, [PERSON] et al. [26] introduced a co-training approach as a semi-supervised machine learning technique. However, most published event-based NILM methods make simplifying assumptions when using the labels of the events to estimate energy use for each appliance. To address this problem, [PERSON] and [PERSON] [27] present a framework for creating appliance models based on classification labels and aggregate power measurements, which can help to relax many of these assumptions. Its results show that the data-driven models outperform the heuristic models used by others.