The Simplex Projection: Lossless Visualization of 4D Compositional Data on a 2D Canvas

TL;DR

The simplex projection expands the capabilities of simplex plots to achieve a lossless visualization of 4D compositional data on a 2D canvas and offers rigorous proofs that support its extension to compositional data of any (finite) dimensionality.

Abstract

The simplex projection expands the capabilities of simplex plots (also known as ternary plots) to achieve a lossless visualization of 4D compositional data on a 2D canvas. Previously, this was only possible for 3D compositional data. We demonstrate how our approach can be applied to individual data points, point clouds, and continuous probability density functions on simplices. While we showcase our visualization technique specifically for 4D compositional data, we offer rigorous proofs that support its extension to compositional data of any (finite) dimensionality.

Figures (11)

• Figure 1: Projection of $x=(\pi_1, \pi_2, \pi_3)$ onto the facet (edge) $\sigma_{-1}=\overline{v_2v_3}$ in a triangle (2-simplex). It can be seen clearly that the perspective projection $\psi_1$ does not change the points' coordinate ratios, $\pi_2/\pi_3 = \tilde{\pi}_2 / \tilde{\pi}_3$.
• Figure 2: Not all sets of simplices are simplicial complexes. In (a), three $2$-simplices form a (pure) simplicial complex. In (b), the dashed intersection of the 2-simplices is neither empty nor a face of the two simplices.
• Figure 3: Existing visualization techniques for compositional data. The data are equal for all three visualization methods and they stem from a Bayesian model comparison experiment: For $i=1,\ldots,100$, we have posterior model probabilities $p(M_1\,|\, y^{(i)}), p(M_2\,|\, y^{(i)}), p(M_3\,|\, y^{(i)})$ which are, per construction, non-negative and sum to one for each $i$. (a) Parallel coordinates convey the dispersion for each compartment $M_j$. Here, $M_3$ has less variance across simulation indices $i$. (b) Stacked plots illustrate the dependency on a third variable. Here, there is no systematic dependency on the simulation index $i$. (c) Simplex plots visualize the relation between compartments. Here, we observe a banana-shaped relation where some combinations of compartments are more likely than others.
• Figure 4: Illustration of the simplex projection $\phi=(\psi_1, \psi_2, \psi_3, \psi_4)$ for a single point $x$ in the tetrahedron $\Delta^3$.
• Figure 5: Illustration of the simplex projection $\Phi$ for a set of points $\{x^{(l)}\}$. The labels of the projections onto the facets in (c) are lost: We do not know which points across facets originate from the same original point in the preimage. Yet, \ref{['theo:preimage-unlabeled']} shows that the original points in (a) can still be recovered from the unlabeled projections in (c)

Theorems & Definitions (4)

• Theorem 1: Bijective simplex projection for labeled points
• Theorem 2: Bijective simplex projection for sets of points
• proof
• proof