Concatenated PCA as a Joint Ordination

From Matched Blocks to One Joint Ordination

When several omics measurements are recorded on the same individuals, concatenated PCA places the measured variables side by side and finds principal directions in that combined feature space. Its sample map gives each individual one set of coordinates, derived jointly from every included block. The map is therefore a function of the chosen variables and their scales: a block with greater total variation can have greater influence, and a block with more variables can contribute more variation in total.

The earlier material on representing matched omics blocks established the basic data layout. Here, let \(K\) blocks be measured on the same \(n\) individuals, in the same row order. Block \(g\) is an \(n \times p_g\) matrix \(X_g\), where \(p_g\) is its number of variables. Center each variable by subtracting its mean across individuals, then concatenate the blocks horizontally:

This tutorial uses centered data. If the columns are not divided by their standard deviations, the procedure is covariance PCA: variables with larger measurement scales or variances can have more influence. If each column is standardized first, it is correlation PCA; that changes the relative influence of individual variables, but does not automatically give each block equal influence.

Here, \(D\) is diagonal, with nonnegative singular values in descending order; \(U\) and \(V\) have orthonormal columns for the nonzero singular values. The sample variance of component \(j\)'s scores is \(d_j^2/(n-1)\), where \(d_j\) is its singular value. Equivalently, \(v_j\), the \(j\)th column of \(V\), maximizes the projected sum of squares \(v^\mathsf{T}X^\mathsf{T}Xv\) over unit vectors \(v\), subject to being orthogonal to earlier directions.

Why the Joint Map Can Be Dominated

A key identity expresses the sample-space matrix of concatenated PCA as a sum of block contributions:

The left singular vectors of \(X\) are eigenvectors of \(XX^\mathsf{T}\). Thus the joint sample directions are determined by the sum of the blocks’ sample-space matrices. A block’s total inertia, meaning its total centered sum of squares, is

A block with larger total inertia can contribute more to the combined matrix. A wider block can also have greater total inertia because it adds more variables’ squared deviations. Width alone does not guarantee dominance: the variables’ scales, variances, and dependence structure matter too. For example, a block of many nearly constant variables may contribute less than a small block with highly variable measurements.

The leading direction maximizes total projected sum of squares across the concatenated variables, not a score that gives each block equal weight. Consequently, a strong direction in one block can determine much of the joint map even if other blocks contain distinct biological patterns. The ordinary PCA calculation itself requires no probability distribution for the measurements. Its interpretation as a representation of a broader population, however, depends on how the individuals were sampled and how the data were measured.

Scaling Choices and Their Consequences

Whether to leave variables in their original units or standardize them is an analytical choice. Covariance PCA preserves differences in the original scales, which may be appropriate when those scales are meaningfully comparable. Correlation PCA gives each nonconstant variable unit sample variance before concatenation; this can prevent a large numerical scale from dominating, but may amplify noisy or low-reliability variables. Neither choice, by itself, ensures equal influence for blocks of different sizes.

Derivation, Diagnostics, and Alternatives

The connection between the loading and score calculations follows directly from the singular value decomposition. If \(X=UDV^\mathsf{T}\), then

Thus the loading directions are eigenvectors of the feature-space matrix \(X^\mathsf{T}X\), while the sample directions are eigenvectors of \(XX^\mathsf{T}\). For any unit loading vector \(v\), the projected sum of squares is

The spectral theorem implies that this quantity is maximized by a leading eigenvector of \(X^\mathsf{T}X\); imposing orthogonality to previous loading directions gives subsequent components. The singular value decomposition connects each such direction to its sample scores. This also justifies the block-dominance interpretation: since \(XX^\mathsf{T}\) is a sum of block matrices, every block adds to the objective that determines the sample map.

Before interpreting a joint ordination, inspect the analysis choices and the resulting representation. Useful checks include:

• Confirm that rows refer to the same individuals in the same order across blocks, and that the column centering and any standardization are as intended.
• Compute each block’s total inertia and compare it with the combined total. Also inspect individual variable variances; a large total may reflect block width, high scales, or both.
• Review the leading loading directions and score plots. If the first axis is dominated by one block’s variables, describe it as such rather than assuming it represents a common signal.
• Check for missingness, outliers, near-constant variables, and duplicated or highly correlated columns. These can alter the matrix being decomposed or make particular directions unstable.

Concatenated PCA is not the same as separate PCA of each block: separate analyses produce different coordinate systems and do not, on their own, give one joint sample map. It is also not CCA or PLS: those methods target relationships between blocks, whereas concatenated PCA maximizes variance across the combined variables. These approaches answer different questions. Concatenated PCA is useful for a direct, exploratory joint ordination, but a leading component is not proof that all blocks share a biological mechanism, nor does it establish an association as causal.

The result can also be sensitive to the feature set. Adding variables can change total block inertia, the leading directions, and the apparent distances between individuals. If measurements are on incomparable scales, unscaled covariance PCA may be hard to interpret; if variables are standardized without considering reliability, small-variance noise may be promoted. Scaling blocks to address their relative influence is a separate decision and should be made explicitly rather than inferred from the concatenated PCA output.