## 3 Methodology
### Generating Input Variables for VAEs Decoder from Learned Latent Space  
The next step is to define inputs for the VAE decoder which generates new samples. Those inputs represent queries for the VAE decoder which generates new samples that reassemble features of specific clusters (morning, afternoon and evening). They must be sampled from the latent space that spatially defines those clusters. Furthermore, those clusters must be defined in a 2-dimensional latent space by the corresponding VAE encoder. This is possible since the VAE encoder uses a 2-dimensional representation of latent space. Based on centroids of 2D morning, evening and afternoon clusters, it is possible to determine the two latent inputs (latent variables \(z[0]\) and \(z[1]\) ) for the encoder part of VAE.  
The first step in this process is to use subclustering results computed using the Conv2D VAEs framework. Those hourly subclusters are aggregated into three aforementioned clusters. The next step is to compute the centroid of each of those clusters by the \(k\)-means algorithm. Naturally, the \(k\) value represents the number of clusters. A centroid is a data point that denotes the centre of cluster mass and it does not have to be one of its members. Thus, the potential new points of latent space need to be in spatial proximity to those centroids. This approach generates points in latent space which correspond to the adjacent clusters. Thus, new samples generated by the VAE decoder should resemble features of their corresponding clusters. This approach enables the augmentation of learning samples for specific clusters.  
Using the normal random distribution, it is possible to generate an arbitrary number of latent points in quadratic patches with its centre represented by one of the three cluster's centroids. The length and the width of those patches are set to 25% of \(z[0]\) and \(z[1]\) latent axis, respectively. Figure 3 shows the illustration of the concept with 100 latent inputs for the VAEs decoder created in the proximity of each cluster's centroids.  
Those patches can have two different tasks. The first task complements real-world datasets with additional samples within the same cluster boundaries. This can be observed in the case of evening and afternoon clusters. They are compact and have a high level of homogeneity. Thus, this task expands the number of samples for a particular cluster. The second task is related to expanding the surface of the current cluster which is divided into two subclusters. Thus, the position of the synthetic samples in patches between dispersed sections of clusters should improve classification results for that particular cluster using machine learning. It is assumed that reducing the distance between two divided sections of one cluster by imposing the aforementioned patch between them could improve its identification as single through machine learning. This is the case with the morning cluster which is dispersed in two sections even at the level of weekday observations represented in Figure 4 (last row of graphs).  
Figure 3: The representation of centroids for each of three clusters created as the aggregation of corresponding hourly subclusters. Those centroids are computed by the \(k\)-means algorithm. The 100 latent inputs for the VAEs decoder are created in proximity to each cluster’s centroids for the sake of conceptual illustration.