# Getting over High-Dimensionality: How Multidimensional Projection Methods Can Assist Data Science
## 4 Multidimensional Projection Approaches and Domains
### SHAP Clustering  
A fundamental challenge in the multidimensional projection domain is related to metric spaces. As we have seen through this section, most projection methods rely on the concept of distance to build similarities relationships between instances that will embed the high-dimensional space in the visual one. However, in projection and clustering tasks, we often need to handle data with features bearing very different types, i.e., features may be unitless scores, grams, meters, and others. Using features with different types as dimensions in a single multidimensional space forces any distance metric to compare the relative relationship in different units (grams vs. meters, for example) [90].  
To tackle this issue, [PERSON] et al. [90] applied SHAP values to convert the input features into importance values with the same units. SHAP (SHapley Additive exPlanations) is a computationally efficient way to verify the relationship between input features and the output of supervised machine learning methods. Specifically, SHAP is a feature importance model-agnostic method applied in the context of Explainable Artificial Intelligence (XAI) [91] to explain machine learning predictions through effect measurements reflecting the importance of input features. It is based on Shapley values [92] cost-sharing problem from classical cooperative game theory to share and allocate each feature a fair importance value for a particular prediction. The explicit definition is:  
\[\phi_{i}=\sum_{S\subseteq m\setminus\{i\}}\frac{|S|!(m-|S|-1)!}{m!}(f(S\cup\{ i\})-f(S)) \tag{36}\]  
where \(S\subseteq\{1,\ldots,m\}\) is a feature subset of all \(m\) features, \(f(S\cup\{i\})\) and \(f(S)\) represents a mapping function (a trained machine learning model, for example) with the feature \(i\) present and withheld, respectively. Thus, the Shapley value of a feature \(i\) can be understood as a weighted average of \(i\)'s marginal contributions to every possible subset of features [93], which means Shapley values exact computation is an NP-hard problem unfeasible to be applied over high-dimensional data. Then, [PERSON] and [PERSON] [94] proposed SHAP as an approximation of classical Shapley values based on statistical samplings. SHAP is currently a state-of-the-art XAI method used to explain machine learning predictions in the more variate domains, ranging from aerospace to medicine and social sciences [95; 96; 97]. [PERSON] et al. [90] extended SHAP to unsupervised learning context presenting SHAP Clustering.  
SHAP Clustering converts all input features into SHAP values, unitless measures of feature importance. Moreover, even when all original inputs are in the same units, often some of them are more important than others. SHAP Clustering also introduces the concept of \"supervised clustering\" as a consequence of converting input features into SHAP values to obtain their effect information. In other words, SHAP values are log-odds attributing the impact of each feature to a machine learning model, then fluctuations in the feature values only affect the clustering if those fluctuations have significance to the outcome [90].  
SHAP Clustering differs significantly from everything in previous multidimensional projection literature by applying concepts from game theory, positioning itself as an interesting and little-explored alternative. Moreover, SHAP Clustering handles one challenging issue in the multidimensional projection domain allowing direct comparisons between the relative importance of features bearing the more different units. Similarly, a recent approach used Shapley values as a tool to provide explanations regarding important features of the clustering process of multidimensional projection methods [98].  
In the next section, we will discuss a comparative summary of the approaches reviewed in this section.