## 1 Introduction
### Novel Hybrid UAC-NFC Model  
NFC [6] is presented first. Suppose that a rule base has the form of fuzzy IF-THEN rules as shown in Equation (1) [22]:  
\[Ru^{(l)}:IF\ x_{1}\ is\ A_{1}^{l}\ and\ldots and\ x_{n}\ is\ A_{n}^{l},THEN\ y\ is\ B^{l}, \tag{1}\]  
where \(A_{l}^{l}\) and \(B^{l}\) are fuzzy sets in \(U_{l}\subset R\) and \(V\subset R\), respectively; \(A_{l}^{l}\) is characterized by normalized Gaussian membership functions; \(B^{l}\) is characterized by fuzzy singleton \(\overline{y}^{l};\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})^{\intercal}\in U\subset R ^{n}\) and \(y\in V\subset R\) are input and output linguistic variables, respectively; and \(M\) is the total number of fuzzy rules; \(l=1,2,\ldots,M\). By adopting a singleton fuzzifier, product inference engine, and center average de-fuzzifier, a fuzzy classifier with the form of fuzzy rules in Equation (1) has the form [22]:  
\[f(\mathbf{x})=\frac{\sum_{l=1}^{M}\overline{y}^{l}[\prod_{i=1}^{n}\exp(-(\frac{x_{ i}-\overline{y}_{i}^{l}}{\sigma_{i}^{l}})^{2})]}{\sum_{l=1}^{M}\left[\prod_{i=1}^{n} \exp(-(\frac{x_{i}-\overline{y}_{i}^{l}}{\sigma_{i}^{l}})^{2})\right]} \tag{2}\]  
where \(f(\mathbf{x})=y\in V\subset R\) is the computed de-fuzzified output with respect to input \(\mathbf{x}\); the symmetric Gaussian membership functions depend on the two factors: \(\sigma_{i}^{l}\in(0,\infty)\) and \(\overline{x}_{i}^{l}\in R\); and \(\overline{y}^{l}\in R\) are the real-valued adjustable parameters of the fuzzy classifier. The NFC of Equation (2) becomes an optimal NFC, where the one fuzzy rule in Equation (1) is used to match one input-output pair in a collected dataset, if each input-output pair in the dataset is viewed as a fuzzy IF-THEN rule. The zero-order Sugeno model, a crisply defined constant in the following, is used in the NFC of Equations (1) and (2) in this article. For a classification problem, the decision making of the NFC constructed and trained is based on:  
\[Class\ label=\operatorname*{argmin}_{c=1,2,\ldots,w}(|f(\mathbf{x})-Class^{c}|) \tag{3}\]  
where \(Class^{c}\), an integer, stands for class label \(c\) and \(w\) is the total number of load classes. The fuzzy classifier in Equation (2) can be viewed as a feed-forward ANN-architecture NFC [6; 22], as shown in Figure 3. In this article, the NFC of Equation (2) is integrated with the UAC, nearest-neighbor clustering, to construct an NFC in an unsupervised and self-organized manner. The hybrid UAC-NFC model proposed in this article was implemented and used to construct the NFC in Equation (2). Also, The Gradient-Descent (GD) algorithm [22; 23; 24; 25; 26], a first-order iterative optimization algorithm for finding the minimum of a given cost function in ANN, is used to train the NFC. In this article, the UAC pilots the NFC to automatically initialize the adjustable network parameters of the constructed NFC. As a result, there is no need to pre-specify the number of clusters to discover data. The flowchart of the proposed hybrid UAC-NFC model is illustrated in Figure 4. In this novel hybrid UAC-NFC model, two stages are involved. In stage 1, a UAC process (nearest-neighbor clustering) is used to automatically cluster input-output data pairs from a given dataset collected prior. With the use of UAC instead of the FCM conducted in [PERSON] et al. [6], there is no need to know the number of clusters to be discovered beforehand. The adjustable parameters of Equation (2) are then heuristically is applied to a collected dataset, discovered data are automatically clustered. Following the UAC process mentioned above, the training process, GD, starts in stage 2. In stage 2, the GD process is used to train the constructed NFC of Equation (2), forming from the adjustable parameters allocated in an unsupervised and self-organized manner and trained from a collected set of data pairs. Finally, the trained UAC-NFC model is applied for load classification in NILM. The constructed hybrid UAC-NFC model was applied on collected input-output data pairs.