## 3 Methodology
### The Proposed VAE Framework  
This study introduces two standard architectures for AE frameworks according to the two possible input formats. Thus, the AE architecture based on a fully connected layer is used for vector-alike inputs while 2-dimensional convolution layers are the basis for the used AE architecture which uses image-alike inputs. The mentioned AE frameworks are analysed _prior_ to their modification into the VAE version since evaluating their efficiency in dimensionality reduction is needed. It is needed since the kernel of VAE architecture is based on the standard AE framework [22]. Furthermore, _Keras_ as the high-level API for the _TensorFlow_, and _scikit-learn_ package within the Python 3.8 environment are used to design and train all investigated models. Furthermore, the Adaptive Moment Estimation (_ADAM_) learning algorithm is used for training all investigated models. The learning rate is set to 0.001. This value is chosen since it is recommended for the used learning algorithm [23]. The hyperparameters for all investigated AE-based models can be seen in Table 1.  
Figure 2: The image-alike representation based on average values of the normalised total trip number between respective origins and destinations for each hour-based subcluster during the investigated year (first column of graphs). Their corresponding analysis concerning maximal values of _drop-off_ and _pick-up_ zones (second row of graphs). The architecture of the AE model is based on four fully connected layers for the encoder and decoder part. Those layers are arranged in a symmetrical structure with 64, 32, 32, and 16 nodes, respectively. The _ReLU_ activation function is used for all nodes except for the layers which encode latent space and output of decoder where the _sigmoid_ activation function is used. The second investigated AE model architecture is based on four 2-dimensional convolution layers. Those layers are arranged in a symmetrical structure. This structure is defined by the distribution of filter numbers for each layer of 2-dimensional convolution layers. Following the distribution of filter numbers 32, 48, 48, and 64 for each of the layers in the encoder and decoder is used. The _strides_ values for all directions is 1, except for the first and third convolution layers where it is set to 3. The convolution layers with increased stride values enable the reduction of the feature maps on the encoder side and feature reconstruction on the decoder side. Additionally, the _kernel_ has a rectangular shape where each edge has a size 17, 4, 10, and 3 for each of the four convolution layers. Those setups of AE models are chosen based on experimental analysis under different architectural parameters. The two investigated VAE model architectures are based on the ones used in the aforementioned AE models. Thus, the fully connected and convolution layers are used in the described symmetrical architectures for the encoder and decoder in both investigated VAE models. These same architectures of AE and VAE models enable their comparison concerning their efficiency in dimensionality reduction. The VAE models introduce regularisation in the latent space by encoding it as a probability distribution over the latent space. Thus, the regularization prevents the decoder model from reconstructing the input data without variations. The regularization prevents overfitting and too large variance while simultaneously enabling the increased generalization of data representation. This improves the VAE's ability to add moderated variances based on generalised dataset patterns for generating diverse data samples. The regularization is performed by introducing a _Kullback-Leibler_ (KL) divergence term into the loss function. Thus, the encoder outputs the mean and log-variance of a _Gaussian_ distribution for sampling the values from latent space. The VAE loss function is based on the calculated KL divergence between the distribution of the learned latent variables and a prior distribution.