From Scaled Blocks to a Global Space
In a multi-omics analysis, matched samples are described by several data blocks: for example, one block of gene-expression variables and another of metabolite measurements. Each block has its own patterns of variation. The MFA compromise space gives one joint map of the samples, built from the blocks after each has been scaled by its first principal inertia. Its axes summarize variation in the combined scaled data; they are not simply an average of separately calculated block axes.
The construction assumes that the blocks refer to the same \(n\) samples in the same row order. For this treatment, each block is numerical, has been centered by variable, and has finite entries. Any variable-level preprocessing, such as standardization, must be chosen before calculating the block’s first principal inertia. The global PCA then uses the scaled blocks as provided; it does not by itself resolve missing values or correct poor preprocessing.
Because \(F_j^\mathsf{T}F_j=X_j^\mathsf{T}X_j/\lambda_{1j}\), the largest eigenvalue of \(F_j^\mathsf{T}F_j\) is 1. This is the relevant consequence of first-inertia scaling for this tutorial: each block’s leading direction has unit inertia before the scaled blocks are concatenated. The global PCA is then calculated on \(Z\), not on the original unscaled concatenation.
Let \(Z=UDV^\mathsf{T}\) be a singular value decomposition. The columns of \(V\) are global variable-axis directions, the diagonal entries of \(D\) are nonnegative singular values, and the global sample coordinates are the columns of \(UD\), equivalently \(ZV\). With this convention, the global PCA eigenvalues are the squared singular values, or equivalently the eigenvalues of \(Z^\mathsf{T}Z\). Since the blocks are centered, the global sample coordinates are centered too.
Why the Global Space Is a Compromise
The geometry of the samples makes the compromise interpretation precise. Write \(f_{ij}\) for the row vector of sample \(i\) in scaled block \(F_j\), and \(z_i\) for its row vector in \(Z\). Concatenation gives the following identity for any two samples \(i\) and \(l\):
Thus, squared Euclidean distance in the concatenated data is the sum of squared distances across the scaled blocks. A global PCA map approximates this combined geometry in a low-dimensional space. A pair of samples can be close on one block and far apart on another; their joint separation reflects both, after the specified block scaling. “Compromise” means that the global map is derived from their combined geometry, not that it must satisfy every block equally well in every displayed dimension.
More formally, let \(v\) be a unit vector in the concatenated variable space. The sample coordinates along this direction are \(Zv\). Their sum of squares is \(v^\mathsf{T}Z^\mathsf{T}Zv\), so the first global direction maximizes this quantity subject to \(\lVert v\rVert=1\). The maximizing direction is a leading eigenvector of \(Z^\mathsf{T}Z\). Subsequent PCA directions maximize the same criterion subject to being orthogonal to the earlier directions.
By the Rayleigh quotient result for a symmetric matrix, the maximum of \(v^\mathsf{T}Z^\mathsf{T}Zv\) over unit vectors is the largest eigenvalue of \(Z^\mathsf{T}Z\), attained by a corresponding eigenvector. Applying the same result within the orthogonal complement of the first direction gives the next principal direction. This establishes both the optimization criterion and why the resulting axes belong to the combined, scaled matrix.
This criterion is not the same as selecting the first axis of every block and averaging those axes. A global direction has one coordinate for each column of \(Z\), and the PCA chooses all those coordinates jointly to maximize projected variation in the concatenation. Consequently, how the blocks relate to each other across samples affects the resulting global scores.
Worked Example: Calculating Global Sample Coordinates
Consider three hypothetical samples and two centered, one-variable blocks. Suppose the first-inertia-scaled columns are \(F_1=(1,-1,0)^\mathsf{T}/\sqrt{2}\) and \(F_2=(1,0,-1)^\mathsf{T}/\sqrt{2}\). Each column has squared length \(1\); the corresponding unscaled columns have first inertia \(3\), so scaling by the square root of that inertia gives these columns. The entries of each column sum to zero, as required for centered data.
The concatenated matrix is \(Z=[F_1\ F_2]\). Its cross-product matrix is
The diagonal entries are the squared lengths of the two columns. The off-diagonal entry is their inner product: only the first sample contributes a nonzero product, giving \((1/\sqrt{2})(1/\sqrt{2})=1/2\). The eigenvalues are \(1+1/2=3/2\) and \(1-1/2=1/2\). A unit eigenvector for the leading eigenvalue is \(v_1=(1,1)^\mathsf{T}/\sqrt{2}\).
The global coordinate for each sample on the first axis is the corresponding entry of \(Zv_1=(F_1+F_2)/\sqrt{2}\). Substitution gives
The coordinates sum to zero, and their squared sum is \(1+1/4+1/4=3/2\), matching the leading eigenvalue. Samples 2 and 3 have the same first-axis coordinate, even though their block profiles differ: sample 2 is negative only on the first block, and sample 3 only on the second. The leading global axis captures their combined variation, not a complete account of every block’s pattern.
For completeness, a unit eigenvector for the second eigenvalue is \(v_2=(1,-1)^\mathsf{T}/\sqrt{2}\), giving coordinates \(Zv_2=(0,-1/2,1/2)^\mathsf{T}\). These squared coordinates sum to \(1/2\), the second eigenvalue. This check confirms the score calculation on both axes.
Interpreting Sample Coordinates and Checking the Calculation
A sample’s coordinate on a global axis is its projection onto that axis in the scaled concatenated variable space. Samples with similar coordinates on one axis have similar projections there, but this does not guarantee that they are close in all dimensions or similar in every block. The sign of an axis is arbitrary: replacing an eigenvector and all its scores by their negatives leaves the PCA unchanged. Interpret relative positions, not a positive or negative sign in isolation.
In practical work, check that the global calculation uses the same row order for every block and the scaled matrices—not the unscaled \(X_j\)—in \(Z\). A useful numerical check is that the sum of squared sample coordinates on a component equals its eigenvalue, up to rounding. Also check that every scaled block is centered if the intended analysis is centered PCA. A nonzero column mean can shift the analysis toward mean differences rather than the intended within-sample variation.
Worked Example: Verifying a Global Distance
Use the same hypothetical scaled columns as in the coordinate example. Compare samples 2 and 3. Their block 1 values differ by \(-1/\sqrt{2}-0=-1/\sqrt{2}\), and their block 2 values differ by \(0-(-1/\sqrt{2})=1/\sqrt{2}\). Their squared distance in the concatenation is therefore
The distance is \(1\) in the full two-variable scaled space. In the first-axis coordinate alone, the two samples have identical scores, so their distance on that axis is zero. This is not a contradiction: the second axis separates them by \(1\), and the two-component PCA retains the full distance in this two-variable example. A truncated map can omit some of the combined distance.
Worked Example: Recognizing Non-Unique Global Axes
Consider a different hypothetical pair of centered, first-inertia-scaled block columns for three samples: \(F_1=u\) and \(F_2=w\), where \(u=(1,-1,0)^\mathsf{T}/\sqrt{2}\) and \(w=(1,1,-2)^\mathsf{T}/\sqrt{6}\). Both vectors sum to zero, each has unit length, and \(u^\mathsf{T}w=0\). Thus both columns have squared length \(1\), and their cross-product matrix is
The two eigenvalues are tied at \(1\). For any angle \(\theta\), the unit direction \(v=(\cos\theta,\sin\theta)^\mathsf{T}\) is a leading eigenvector, and its sample coordinates are \(Zv=\cos\theta\,F_1+\sin\theta\,F_2\). For instance, \(v=(1,0)^\mathsf{T}\) gives scores \(F_1\), whereas \(v=(0,1)^\mathsf{T}\) gives scores \(F_2\). Both explain the same inertia. The two-dimensional global subspace is well defined, but a particular first axis within it is not uniquely determined. Reporting one orientation as a uniquely preferred biological pattern would overstate what this calculation establishes.
Alternatives, Limitations, and Common Pitfalls
Separate PCA analyzes each block on its own. It can describe block-specific variation, but it does not provide a single set of global sample coordinates based on the combined scaled geometry. Ordinary PCA of an unscaled concatenation also creates one joint map, but its axes are based on the original block scales rather than the first-inertia-scaled matrices. The MFA global space is therefore a particular joint ordination, defined by a particular preprocessing and weighting choice; it is not a universally optimal map for every scientific question.
The global coordinates are descriptive projections of the supplied data. They do not establish that a pattern is biologically shared, that one omics layer causes variation in another, or that a visually prominent axis is statistically significant. The map can also be sensitive to outliers, variable preprocessing, and noise. If several leading eigenvalues are close, small data changes can rotate the associated axes; if eigenvalues are tied, individual axes may be non-unique, as the example showed.
- Wrong reasoning: “Each block was scaled, so the global PCA must give each block exactly the same representation on every axis.” Scaling sets each block’s first inertia to one before concatenation; it does not impose identical scores or identical contributions on every global axis.
- Correction: Calculate and interpret the PCA of \(Z\). Its axes optimize total projected variation in the scaled concatenation, and the sample map is a joint summary that may represent some block patterns better than others.
- Wrong reasoning: “A sample near another on the first global axis must be similar across all omics blocks.”
- Correction: Closeness on one axis means similar projections on that axis. Differences can remain on other axes or in dimensions not displayed.
The central diagnostic is coherence between the stated construction and the calculation: matched centered rows, the intended first-inertia scaling, and PCA applied to the concatenated scaled matrix. Then verify score centering and eigenvalue checks, and avoid treating a two-dimensional display as the full geometry unless those two components retain it.
Check Your Understanding
Answer each question, then compare with the concise solutions below.
- What matrix is submitted to the global PCA, and what is the largest eigenvalue of each scaled block’s cross-product matrix?
- In precise terms, what does a global sample coordinate on an axis represent?
- Why can two samples have identical first-axis coordinates but still be separated in the full global space?
- If the first two global eigenvalues are tied, what can and cannot be said about the global axes?
- What is one important difference between the MFA global PCA and PCA of the unscaled concatenation?
Solutions. (1) The matrix is \(Z=[F_1\ \cdots\ F_J]\), the column-wise concatenation of the first-inertia-scaled blocks; each scaled block’s largest cross-product eigenvalue is 1. (2) It is the projection of that sample’s row in \(Z\) onto the corresponding global variable-axis direction. (3) They can differ on another global axis or in dimensions omitted from a lower-dimensional display. (4) The tied eigenspace and its explained inertia are defined, but an individual direction within that tied space is not unique. (5) MFA global PCA applies first-inertia scaling to the blocks before concatenation, whereas unscaled concatenated PCA retains the original block scales in its joint analysis.