# Getting over High-Dimensionality: How Multidimensional Projection Methods Can Assist Data Science
## 3 Ground Theory
### Multidimensional Projection
A multidimensional projection technique can be described as a function \(f:X\to Y\) that aims to make \(|\delta(x_{i},x_{j})-d(f(x_{i}),f(x_{j}))|\) as close as possible to zero, \(\forall\ x_{i},x_{j}\in X\)._  
Similarity and dissimilarity represent somewhat vague concepts, as they do not yet have a well-established definition in the literature and may present different interpretations depending on the application domain [7]. In the context of MDP, and to prevent any inaccuracy, we will adopt a geometric definition of (dis)similarity. Therefore, we established that dissimilarity is a numerical measure used to indicate the \"proximity\" (distance) between two data objects. The higher the dissimilarity value, the more distant and distinct the objects are, according to some criterion or comparison function [27]. Inversely, a similarity measure indicates how similar two objects are, with high values of similarity denoting close and similar objects. In this context, the concept of distance has a central role in multidimensional data comparison and projection. An interesting set of dissimilarity metrics is the Minkowski distances calculated as:  
\[d(x,y)=\left(\sum_{i=1}^{n}|x_{i}-y_{i}|^{k}\right)^{\frac{1}{k}},\ \ \ \ k=1,\ldots,\infty, \tag{1}\]  
being \(k\) a modification parameter. The changing of \(k\) generates well-known functions, such as Manhattan distance (\(k=1\)), Euclidean distance or \(L_{2}\) norm (\(k=2\)), and Maximum distance or \(L_{max}\) norm (\(k=\infty\)). Several MDP techniques utilize Euclidean distance as the dissimilarity metric, such as the ones proposed by [PERSON] and [PERSON] [28], [PERSON] et al. [12], and [PERSON] et al. [29]. Nevertheless, depending on the nature of the data and the application domain under analysis, Euclidean distance may not be the best way to express similarities or differences between data objects. A measure is considered a metric if it satisfies the metric space postulates [30], and some measures do not satisfy them. Therefore, it may be convenient to apply measures that do not meet the metric space properties in some cases. A deep analysis of the different types of existing measures is beyond the scope of this paper. Calculation of (dis)similarities can generate distortions owing to (i) significant differences between vectors Euclidean norms, and (ii) features with values much larger or much smaller than others can lead to projection results and force it to ignore the importance of other features [31]. According to [PERSON] et al. [12], distortions are inevitable in the multidimensional projection process, but it is possible to keep them as small as possible. To avoid these situations, a scaling process is applied to each vector (vector normalization) or data feature (features normalization or standardization). Scaling is one of the more important pre-processing steps of projection techniques because it can unify the feature space by forcing the unitary Euclidean norm of vectors or forcing features to satisfy a specific statistical property [32]. Figure 1 illustrates the application of normalization before checking the similarities between two data vectors. In this example, two sequences, \(A\) and \(B\), have different graphical curves in terms of range and magnitude (horizontal and vertical shifting). However, the distance between \(A\) and \(B\) is proportional to the shifting, which can hide information about the sequences' similarity. In other words, if a proper normalization is not applied before the projection task, the projection method may overestimate the distances, which leads to distorted information regarding the correct amount of (dis)similarities among the data vectors (indicated by the shaded area between the curves in Figure 1, right side). After a vector normalization, one can determine the true similarities between the distinct curve profiles [33].