Choosing Principal Components for Visualization

Choosing PCs Is a Purpose-Dependent Decision

Principal component analysis orders directions by the amount of variation they capture. That ordering is useful, but it does not supply a universal stopping point. Choosing components for visualization means deciding how much of the data's structure to retain for a particular display, while avoiding dimensions dominated by unstable or irrelevant variation. A scree plot can suggest where additional components contribute less variance; stability checks and gene-loading patterns help assess what those components represent.

The choice also depends on what “for visualization” means. A plot with two principal-component axes displays only two dimensions. In contrast, a visualization method that constructs a two-dimensional map from a higher-dimensional representation may use a selected set of PCs as its input. In either case, the number of PCs retained is a modeling choice, not a biological fact. It should be supported by evidence and reported clearly.

The eigenvector for a component gives its gene loadings, and projecting cells onto that vector gives their PC scores. A component's sign is arbitrary: multiplying all its loadings and scores by \(-1\) describes the same axis. Consequently, “positive-loading genes” have no intrinsic status apart from their contrast with negative-loading genes.

What a Scree Plot Can and Cannot Tell You

A scree plot places component number on the horizontal axis and component variance, often the eigenvalue or explained-variance proportion, on the vertical axis. A sharp decline followed by a flatter tail can indicate that early components capture substantially more variation than later ones. The apparent bend, or elbow, is a useful summary, but it is not a test that identifies a uniquely correct number of PCs. Some data have a gradual decline with no clear elbow.

Cumulative explained variance answers a different question: how much total variance is represented by the first \(k\) PCs? A target such as 80% or 90% can be a convenient description, but it is not a general rule for single-cell visualization. Many small, meaningful biological effects may be spread across later components; conversely, a component can explain substantial variance because of technical or otherwise unhelpful structure. Explained variance measures variation, not biological importance.

This ratio has a direct justification. The total variance across all component scores is the sum of the eigenvalues, because the trace of the covariance matrix equals the sum of its eigenvalues. Orthogonal PC axes partition that variance: the first \(k\) axes account for \(\lambda_1+\cdots+\lambda_k\). Dividing this retained variance by the total gives \(R_k\). This calculation quantifies variance retained; it does not establish that the retained variation is the variation most useful to display.

Check Whether Components Are Stable

A component that changes substantially under small, reasonable perturbations may be a weak basis for a robust display. Stability can be assessed by repeating the analysis after resampling cells or making modest changes to preprocessing choices, while keeping the comparison meaningful. For example, when the question is sampling stability, keep the gene set and scaling rule fixed and compare fits on overlapping cells. When preprocessing sensitivity is the question, vary one defensible choice at a time.

To compare a component across fits, first match components by their order or by similarity, then compare their loading vectors on the same genes. Since signs can flip, a simple sign-invariant similarity for unit-length loading vectors \(v_a\) and \(v_b\) is \(|v_a^\mathsf{T}v_b|\). Values close to 1 indicate similar directions; values nearer 0 indicate dissimilar directions. This comparison is most interpretable when the component is separated from neighboring eigenvalues.

When adjacent eigenvalues are close, individual PCs can rotate within a nearly equivalent subspace. Two fits may then have different-looking PC loadings but similar combined subspaces. In that situation, compare the subspaces—for example, using principal angles—rather than declaring each component unstable solely from one-to-one loading correlations. Stability is always relative to the perturbations considered; it does not mean a component is biologically valid.

Use Gene Loadings to Interpret, Not Automatically Exclude

For each candidate PC, inspect genes with the largest absolute loadings and the contrast between the two loading directions. Ask whether the pattern is coherent and relevant to the intended visualization. A component whose prominent genes are associated with a known cell program may capture meaningful variation. A component dominated by a technical feature or quality-control pattern may be less useful for the display's purpose. But a gene list alone rarely settles the question: correlated genes can share a signal, and a familiar technical association does not prove that all variation along the component is unwanted.

Loading concentration can be summarized with squared loadings. For a unit-length loading vector, the squared loadings sum to 1, so the sum over a selected gene set gives the fraction of the loading vector's squared magnitude assigned to that set. This is a descriptive diagnostic, not a fraction of biological variance explained by those genes.

A Practical Decision and Its Boundaries

A defensible choice combines evidence rather than applying a single cutoff. Start with the scree shape and cumulative variance to identify a compact candidate range. Check whether candidate components are reproducible under relevant perturbations, taking near-tied eigenvalues into account. Then examine loading patterns and, where appropriate, relationships between component scores and known biological or technical variables. Choose a number that serves the stated display purpose, and record the rationale and sensitivity of the result to nearby choices.

This selection is not automatically the selection for clustering. A visualization may favor a compact representation that makes broad structure legible, while clustering can depend on distinctions that are less prominent in a two-dimensional display. Decisions for clustering require their own evaluation of the representation and the clustering objective; a scree elbow alone does not determine them. Likewise, visible separation in a plot does not by itself establish discrete cell groups.

Common warning signs include a gradual scree curve that tempts an arbitrary elbow, a component whose loading pattern changes greatly across reasonable resamples, or a visualization that changes substantially when one additional PC is included. A further pitfall is to discard a component just because its loadings resemble a technical signature. The correction is to treat these signals as prompts for investigation: check the analysis purpose, stability, score patterns, and relevant biological context. No single diagnostic can certify a PC as either useful or useless.

The main alternative to a combined diagnostic is a fixed rule, such as retaining a preset number of PCs or enough PCs to exceed a chosen cumulative-variance percentage. Fixed rules are simple and reproducible, and can be reasonable when justified for a specific workflow. Their limitation is that they ignore the scree shape, stability, and meaning of the components in a particular dataset. Conversely, choosing components only by inspecting gene names invites subjective selection. A transparent decision records multiple diagnostics and acknowledges that the chosen dimension is purpose-dependent.