# Visualization of a Multidimensional Point Cloud as a 3D Swarm of Avatars
## 3 Methodology
### Paradigm and Semantic Assignment of Variables  
In practical applications, multidimensional datasets often contain both variables that are intuitively meaningful to users (such as sensory ratings and categorical or interpretable parameters) and features that are more technical or abstract (such as chemical coefficients or structural process data). To maximize interpretability, our approach introduces a semantic division of data dimensions into two groups:  
* Intuitive dimensions--readily interpreted by users--are mapped onto visual characteristics of avatars (e.g., face shape, expressions, color). * Technical dimensions--more abstract or difficult to interpret--are projected into spatial coordinates in the observation space, possibly after dimensionality reduction (e.g., PCA). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Meaning** & **Constraints** \\ \hline \multicolumn{3}{c}{Input data parameters} \\ \hline \(N\) & Total number of data points & \\ \(n\) & Total number of data dimensions & \\ \(X\) & Original data matrix & \(X\in\mathbb{R}^{n\times N}\) \\ \hline \multicolumn{3}{c}{Implementation-dependent mapping} \\ \hline \(k\) & Number of navigable display dimensions & \(k\geq 3\) \\  & & (typically 4) \\ \(m\) & Number of available avatar visual features & \(m\leq n-k\) \\  & & (currently 10) \\ \hline \multicolumn{3}{c}{User’s mapping} \\ \hline \(h_{S}\) & Number of human-meaningful dimensions mapped to space & \(h_{S}\leq k\) \\ \(h_{V}\) & Number of human-meaningful dimensions mapped to avatars & \(h_{V}\leq m\) \\ \(h_{A}\) & Number of anonymous dimensions & \(h_{A}\leq n-(h_{S}+h_{V})\) \\ \hline \multicolumn{3}{c}{Projection matrices} \\ \hline \(F\) & Filtering matrix & \(F\in\mathbb{R}^{(k+m)\times n}\) \\  & View matrix: for spatial rotation in \(k\)-dimensional space (\(k\times k\)), & \(V\in\mathbb{R}^{k\times k}\) or \\ \(V\) & or full affine transformation with translation in homogeneous & \(V\in\mathbb{R}^{(k+1)\times(k+1)}\) \\  & coordinates (\((k+1)\times(k+1)\)) & \\ \hline \multicolumn{3}{c}{Derived datasets} \\ \hline \(X_{\text{spatial}}\) & Spatial projection of \(X\) & \(X_{\text{spatial}}\in\mathbb{R}^{k\times N}\) \\ \(X_{\text{visual}}\) & Avatar feature mapping of \(X\) & \(X_{\text{visual}}\in\mathbb{R}^{m\times N}\) \\ \(X_{\text{view}}\) & Spatial data after applying \(V\) & \(X_{\text{view}}\in\mathbb{R}^{k\times N}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols and notation used throughout the paper. The assignment of variables to these groups should be guided by domain knowledge and the analysis's objectives. Features that are critical for individual profiling or human interpretation should be mapped to avatar features, while those essential for discovering structures or clusters may be included in spatial coordinates. Redundancy should be avoided (e.g., highly correlated features mapped to multiple avatar traits), and important variables should not be left out of both avatar and spatial assignment, as this could diminish interpretability. Let the original dataset be represented by a matrix \(X\in\mathbb{R}^{n\times N}\), where \(n\) is the number of original dimensions and \(N\) is the number of data points. To implement the semantic assignment of variables, we construct a filtering matrix \(F\in\mathbb{R}^{(k+m)\times n}\) that separates and processes the dimensions, where \(k\) is the number of spatial dimensions and \(m\) is the number of avatar visual features. The filtered data are given by Equation (1):  
\[X_{\text{filtered}}=F\cdot X. \tag{1}\]  
Each row of \(F\) defines a new output feature (spatial or visual) as a linear combination of the original dimensions of \(X\).