## 2 Load Profiling Mathematical Background
### Clustering Algorithms
#### 2.2.5 Other Clustering Algorithms  
This category refers to algorithms that do not belong to the aforementioned categories. One such algorithm is the Modified FDL, which does not require the initial determination of the number of clusters [58]. The algorithm is iterative, clusters are created in the first iteration and in the rest of the iterations the number of clusters is kept constant and the shifting of patterns to clusters takes place. The number of clusters is determined indirectly by a distance threshold that sets a limit to the maximum distance between patterns and clusters. In the iterations following the 1 st, for each pattern, the modified Euclidean distance is calculated between it and the centroid of the cluster that belongs to. If the distance is greater than the threshold, then the pattern is shifted to the cluster with the minimum distance. The iterative process is terminated when the maximum number of iterations is completed or when there are no shifts of patterns. First, a pattern is selected from the set that defines the original centroid and then compares the distances and the threshold. In addition to the threshold, what determines the function of the algorithm is the choice of the original pattern.  
In [62], the Iterative Self-Organizing Data Analysis Technique Algorithm (ISODATA) is applied to group a large set of load curves. ISODATA is an extension of the K-means, which contains heuristic methods for automatically selecting the number of clusters. The function of the algorithm includes a set of parameters that must be suitably selected, such as the minimum number of members within the cluster, the desired maximum number of clusters, the mean distance between the patterns and the centroid of the cluster and the sum of the largest square distance between the patterns and the centroid of the cluster that they belong to.  
In [74], the authors propose the application of 3 algorithms that are structured upon the between cluster Renyi entropy distance metric. The algorithms are based on a multi-step hierarchical agglomerative operation. Initially, the patterns are treated as singleton clusters. The 3 algorithms differ in terms of the distance metric that is used to measure the similarity. The most similar patterns are merged until 1 cluster that contains all patterns remains.  
The SVC is proposed in [75]. The patterns with dimension \(D\) are projected into a higher dimension space, according to a non-linear transformation, where a Gaussian core is proposed. The new space creates a spherical topology that includes the patterns. Patterns are either within or outside the sphere, or on its surface. Patterns outside the sphere are extreme values, they are isolated from the rest and are considered as the initial centroids. Then, through a process that compares distances between the patterns and centroids, the patterns are split into existing clusters and newly creating ones. The algorithm depends on a parameter that controls the number of extreme values located outside the sphere and from the distance threshold that regulates the distribution of patterns to clusters or the creation of new ones.  
The IRC algorithm is a variant of the modified FDL [80]. In the 1 st step, each pattern is considered a centroid. At the 1 st iteration, Euclidean distances and correlation coefficients between the patterns are calculated. The patterns are sorted in ascending order based on correlation coefficients and the ratio of correlation coefficients to distances is calculated. In the subsequent iterations and after the number of clusters has been determined, the patterns are shifted to clusters.  
The Competitive Leaky Algorithm (CLA) is a generalization of the basic competitive learning algorithm [61]. Contrary to the basic competitive learning, the weight update is held for all neurons, i.e., the winning neuron and all the rest.