PCA Biplot

The PCA Biplot widget projects trials onto the first two principal components of standardized data and overlays the original variables' contributions (loadings) as arrows. It compresses a many-variable design space into one picture: nearby points are similar designs, and each arrow shows which variables drive that direction of spread.

Overview

Principal Component Analysis (PCA) finds the orthogonal directions along which a data set varies the most, and re-expresses each trial as coordinates along those directions (its scores). A biplot overlays two things in the same 2D plane: the trials as points (using the first two PC scores) and the original variables as arrows (their loadings on those same two components). This lets you read both "which trials are similar" and "which variables drive that similarity" from one figure.

Because a study mixes parameters and objectives with unrelated units, PCA here always runs on standardized data, that is, correlation PCA. This prevents a variable from dominating the projection merely because it has a larger numeric scale.

Formula

Let $Z$ be the standardized data matrix, with one row per trial and one column per selected variable:

$$Z_{ij} = \frac{x_{ij} - \mu_j}{\sigma_j}$$

where $\mu_j$ and $\sigma_j$ are the mean and standard deviation of column $j$. This is correlation-matrix PCA rather than covariance-matrix PCA, which is required here because parameters and objectives have different units.

The correlation matrix and its eigendecomposition are

$$R = \frac{1}{n-1} Z^{\top} Z = W \Lambda W^{\top}, \qquad \Lambda = \mathrm{diag}(\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_p)$$

with $W = [w_1, \dots, w_p]$ orthonormal. The scores, the trial positions in the biplot, are the projection of standardized data onto the eigenvectors:

$$T = Z W, \qquad T_{i,k} = \sum_j Z_{ij} w_{jk}$$

The loading for component $k$ is simply the set of coefficients in eigenvector $w_k$. The explained variance ratio of component $k$ is

$$ext{ratio}$$k = \frac{\lambda_k}{\sum{j=1}^{p} \lambda_j}

The denominator sums over all $p$ components, not just the two shown, so the two axis-label percentages do not necessarily add up to a large share of the total variance.

Behavior notes. The correlation matrix is eigendecomposed directly instead of using SVD of $Z$. In exact arithmetic they are equivalent, but direct eigendecomposition is cheaper here because the number of variables is typically much smaller than the number of trials. Zero-variance columns are mapped to the constant $0$ during standardization; they contribute nothing to any component and remain visually inert as zero-length loading arrows. For display, all loading arrows are rescaled by a single global factor so they fit in the same plot range as the trial scores. This preserves the relative lengths and angles between arrows.

Characteristics

• The axes are linear combinations, not original variables. A principal component is a weighted sum of all selected variables, so "high PC1" rarely corresponds to one intuitive quantity. Read the loading arrows before interpreting trial positions.
• The sign of a component is arbitrary. Eigenvectors are defined only up to a sign flip, so a mirrored biplot represents the same structure.
• Watch the explained-variance percentages in the axis labels. When many variables are only weakly correlated, two components may capture only a small fraction of total variance. In that case, distances in the biplot understate dissimilarity in the full variable space.
• Loading arrows are an approximation, not an exact correlation. Because Tunny Dashboard shows the raw eigenvector coefficients, uniformly rescaled only for plotting, an arrow's direction is a reliable indicator of how a variable aligns with the two displayed components, and the angle between arrows approximates the correlation between variables. But arrow-length comparisons remain qualitative.
• Relation to Cluster Scatter 2D / 3D: those widgets do not use a PCA projection. They plot the raw values of the first two or three selected objectives directly. If you want a PCA-plane view of the same trials, use this PCA Biplot widget instead.

References

• Gabriel, K. R. (1971). The biplot graphic display of matrices with application to principal component analysis. Biometrika, 58(3), 453-467. https://doi.org/10.1093/biomet/58.3.453
• Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 24, 417-441. https://doi.org/10.1037/h0070888