# Getting over High-Dimensionality: How Multidimensional Projection Methods Can Assist Data Science
## 3 Ground Theory
### Evaluation of Projected Spaces  
Projection techniques can approximate similar instances and segregate distinct ones, grouping them in the visual space according to some (dis)similarity measure. Nevertheless, as aforementioned, MDPs can generate different levels of distortions in the data neighborhood. Therefore, to assess the distortions and quality of the views created, specialists map real data classes or groups to a color space, allowing them to evaluate the projection results visually. Sadly, this type of assessment can be subjective and complex due to the data cluttering on the screen, and specialists utilize some measures to assess projections' quality. The best-known measure used to evaluate MDPs quality is the stress function [35], which estimates the amount of information lost during the mapping process of data instances. The function is similar to a standard deviation calculation and is defined as  
\[\textit{stress}=\sqrt{\frac{\sum\limits_{i<j}(d_{ij}-\delta_{ij})^{2}}{\sum \limits_{i<j}(\delta_{ij})^{2}}}, \tag{5}\]  
where \(\delta_{ij}\) is the dissimilarity measure in the original space, and \(d_{ij}\) is the dissimilarity in the projected space. The stress function can quantify the distortion degree generated through the mapping process concerning the original space. The results are in the \([0,1]\) range, and the closer to zero, the better the distances preservation, with zero technically indicating a \"perfect\" projection. Although the stress function helps evaluate the generated mappings, it is common to find divergences between stress and visual projection interpretation [9]. Another measure is the neighborhood preservation index [10], which assesses the rate of neighbors that are close in the high dimensional space and remain close in the projected space. Some projection technique generates neighborhood distortions. For example, two instances close to each other (or similar) in the original space can be projected too far, which is a missing neighbor problem. On the other hand, two instances distant from each other (or dissimilar) in the original space can be projected into the same neighborhood, a problem that is named false neighbor (see Figure 1 in [7]). Note these two kinds of distortions that can affect the neighborhood preservation index. False and missing neighbors impact the projection-based analysis in different ways [36] and can generate a degree of uncertainty during the data analysis task. The index is computed using:  
\[NP_{k}(i)=\frac{|kNN(y_{i})\cap kNN(x_{i})|}{|kNN(x_{i})|}, \tag{6}\]  
where \(kNN(\cdot)\) is the set of the \(k\) nearest neighbors of some instance \(x_{i}\) in the original or \(y_{i}\) in the projected space [6]. To evaluate the projection of the entire dataset, one can take the mean of \(NP_{k}(i)\) values in the \([0,1]\) range, with results closer to 1 indicating mappings with better preservation of neighborhoods concerning the original space. The silhouette coefficient [37] is one of the more widely used qualitative measures to check clusters' quality, based on calculating the cohesion and separation between groups of objects. Nonetheless, it has been primarily applied to assess the quality of projected spaces [38; 39]. The cohesion \(a_{i}\) of some instance \(x_{i}\) is calculated by averaging distances between \(x_{i}\) with all other instances belonging to the \(x_{i}\) group. The inter-group separation \(b_{i}\) is defined by the minimum distance between \(x_{i}\) and all the other instances belonging to the other groups [12]. Therefore, the silhouette coefficient of some projected data instance \(x_{i}\) is computed as:  
\[s_{i}=\frac{b_{i}-a_{i}}{\max(a_{i},b_{i})}, \tag{7}\]  
and the average \(Silh\) of all these values determines the average width of the silhouette coefficients of a dataset, considering \(Sillh\in[-1,1]\), and the higher its value, the better the intra-group cohesion and inter-group separations.