# Getting over High-Dimensionality: How Multidimensional Projection Methods Can Assist Data Science
## 4 Multidimensional Projection Approaches and Domains
### Hierarchical Approaches  
Hierarchical Point Placement Strategy (HiPP) is a projection technique built to preserve the clustering and segregation of a dataset [10]. It generates a hierarchical structure that allows exploration in several levels of detail. With this asset, HiPP can be applied to larger datasets than several other projections. The authors reported that all process has a computational complexity of \(O(n\sqrt{n})\), being \(n\) the total number of data instances. The authors divided the HiPP process into three stages. In the first one, HiPP creates a tree that has all data instances as children of the root node. With _Bisecting k-means_[74], the children's level is split into \(k\) clusters. One node is inserted as a child of the ancestor node representing each cluster created. After this, all cluster items are connected with their related group node. This process is recursively applied to each new node until a global threshold of items inside each node is reached. [PERSON] and [PERSON] [10] used a threshold equal to the square root of the dataset instances, \(\sqrt{n}\), and a local \(k=\sqrt{m}\), with \(m\) being the number of items inside the node (group) being divided. In the second stage, the authors applied LSP to project tree nodes. They represented groups as circles, being its center point related to the group centers, and the size of the circle proportional to the number of items in the group. Thus, the first level is projected with these attributes, placing the circles on the plane. Next, internal data are projected when one interacts with a circle (group), zooming in and out on the structure. In the third stage, a spreading algorithm removes overlapping in the visual layout. A vector between every two nodes is calculated and used to spread them. The nodes are dislocated in the vector orientation but opposite senses. Furthermore, the authors used HiPP to explore text datasets. Thus, they present topic terms as labels of each group and colored circles based on topics. The technique also allows split and reassemble groups interactively. HiPP is suitable when one needs to tailor clustering. On the other hand, it does not allow for flexibility of clustering and projection techniques or data types. To enhance HiPP capabilities, [PERSON] and [PERSON] [75] proposed the _eXtend Hierarchical Point Placement Strategy_ (xHiPP). First and foremost, the authors use three cluster techniques in the tree construction process: _k-means_, _k-medoids_, and a basic _Hierarchical Agglomerative Clustering_. Instead of using \(\sqrt{n}\) to decide how many groups will be generated, the authors used Sturges' Rule [76], which defines \(k=1+3.3\times log(n)\). It results in a smaller number of clusters than the previous technique, which is better when one works with a large dataset. In addition, the user also can modify this parameter. In the second change proposed, the user can choose which projection better fit the data under analysis. It is possible to choose among LSP MDS, Force Scheme, t-SNE, PLMP, LAMP, or the PCA, all previously described in this review. Force Scheme is always used to project internal data to improve performance. [PERSON] and [PERSON] [75] allowed xHiPP to cluster data before the projection stage (such as the original approach) and also the projection stage, followed by the clustering stage. It takes advantage of the projection techniques' ability of data points that have come together in projected space and partition the dataset with high precision. Beyond text exploration tools implemented in the original HiPP, the authors added word clouds to summarize the content of both groups and individual documents. Authors also added functionalities to facilitate the exploration of image datasets, audios, and general data. The groups' medoids are used to represent image sets and spectrograms of audio ones. Heatmap images are employed to map attributes' distribution of general data.