Figure 4
From: Cost function for low-dimensional manifold topology assessment
Ranking of different training data preprocessing strategies in their capacity to produce quality low-dimensional projections using PCA. (a) 2D PCA projections generated from an argon plasma dataset. We show projections resulting from various scaling applied to the original data. In the top row, the projections are colored by the electron temperature, \(T_e\), and in the bottom row, by the electron mass fraction, \(Y_e\). (b) The overall cost, \(\mathcal {L}\), computed as the L\(_1\)-norm over the individual costs for the selected dependent variables: temperature of heavy species, \(T_h\), temperature of electrons, \(T_e\), electron mass fraction, \(Y_e\), and argon mass fraction, \(Y_{Ar}\). For comparison, costs are shown for 2D and 3D PCA projections. (c) 3D PCA projections generated from a reacting flow dataset describing combustion of syngas in air. We compare projections resulting from only scaling the data (top row) with projections resulting from scaling combined with feature selection (bottom row). All projections are colored by the temperature variable. (d) The overall cost, \(\mathcal {L}\), computed as the L\(_1\)-norm over the individual costs for the selected dependent variables: temperature and six important chemical species mass fractions. We compare costs corresponding to only scaling the data (circles) versus scaling with feature selection (triangles). We also compare costs corresponding to the 3D projections visualized above (black markers) with the analogous costs of 2D projections (gray markers). The optimal manifold topology corresponding to the lowest \(\mathcal {L}\) for each preprocessing strategy is highlighted with thicker axes in (c). (e) Visualization of 2D projections corresponding to three selected cases of only scaling the original data: \(\langle -1, 1 \rangle\) and Level scaling (corresponding to the two highest \(\mathcal {L}\)) and VAST scaling (corresponding to the lowest \(\mathcal {L}\)).
