## 1 Introduction
### Novel Hybrid UAC-NFC Model
The cluster centers found through the UAC process and specified as data points in the same clusters were used to heuristically allocate the adjustable parameters of the Gaussian membership functions of Equation (2):  
1. The center parameter \(\overline{x}_{i,j}\) of Gaussian membership function \(j\) on universe of discourse \(i\) is initialized by component \(i\) of cluster center \(j\), where \(i=1\), \(2\),..., \(n\); \(n\) is the total number of universes of discourse (input variables); \(j=1\), \(2\),..., \(N_{MF}\); and \(N_{MF}\), which equals \(k\) in the UAC process, is the total number of Gaussian membership functions (cluster centers) on each of universes of discourse. There are \(k\) clusters resulting in \(k\times k\) fuzzy partitions in a feature space as the partition space in fuzzy logic. The partition space containing \(k\times k\) fuzzy partitions is square to the UAC used in this article. The value of \(k\) is identified through the UAC process, nearest-neighbor clustering. 2. The spread/width parameter \(\sigma_{i,j}\) of Gaussian membership function \(j\) on universe of discourse \(i\) is initialized by: \[\sigma_{i,j}=\alpha\cdot(x_{i}^{\prime}-\overline{x}_{i,j})\] (4) where \(x_{i}^{\prime}\) is component \(i\) of a data point that belongs to cluster \(j\) with the computed maximum distance between it and one data point in the same cluster and \(\alpha\), a real number, is a non-zero constant. 3. Based on the results by the UAC process, heuristically allocate the singleton parameters \(\left\{\overline{y}^{l}\right\}_{l=1}^{M}\) of Equation (3) with the desired output of collected input-output data pairs. The output of the NFC used in this article is defined as the class {0, 1, 2,... } of load combinations classified. Each single parameter is assigned a unique class label: class label 0, class label 1, class label 2,..., or class label \(w\). Figure 3: A feed-forward Artificial Neural Network (ANN)-architecture NFC. After the UAC process shown above is complete, the GD process works as follows. Assume that the following input-output data pairs are collected: \(\{(x^{p},y^{p})\}_{p=1}^{N}\), where \(x^{p}\in U\subset R^{n}\) and \(y^{p}\in V\subset R\). The goal of the NFC constructed for load classification in NILM is to fine tune the free parameters of the fuzzy membership functions based on these \(N\) input-output data pairs. In Equation (2), \(\sigma_{i}^{l}\), \(\overline{x}_{i}^{l}\), and \(\overline{y}^{l}\) are automatically constructed through the UAC process. Once these parameters are pre-specified by the UAC, the quasi-optimal structure of the NFC trained by the GD process can be obtained. The GD process is used to update the adjustable parameters of the initialized NFC, according to Equations (5)-(7) at the \(q\)th iteration of the training process with a present input-output data pair \((x^{p},y^{p})\). The training process is performed with \(q=q+1\), until the UAC-NFC is trained and terminates when total error \(\sum|f(\mathbf{x}^{p})-y^{p}|\) with \(p=p+1\) is less than or equal to a pre-specified tolerance is satisfactory. Usually, the training process terminates when the maximum training iteration pre-specified is met. \[\overline{y}^{l}(q+1)=\overline{y}^{l}(q)-\eta[\frac{f(\mathbf{x}^{p})-y^{p}}{ \sum_{l=1}^{M}\prod_{i=1}^{n}\exp(-(\frac{x_{i}-\overline{x}_{i}^{l}}{\sigma_ {i}^{l}})^{2})}][\prod_{i=1}^{n}\exp(-(\frac{x_{i}-\overline{x}_{i}^{l}}{ \sigma_{i}^{l}})^{2})] \tag{5}\]  
where \(\eta\) is a real number pre-specified within (0, 1], which is the training rate. Figure 4: Flowchart of the hybrid UAC-NFC model conducted in the HEMS for NILM and proposed in this article.