Why Decompose Fitted Variation?
A factor model can produce a compact fitted signal for each omics view, but a single percentage rarely answers every interpretation question. One may want to know how well the model reconstructs each view, how much fitted variation is associated with a particular factor, or how much fitted signal comes from factors designated as shared across views. These quantities have different denominators and different interpretations.
The central complication is that factor contributions need not be orthogonal. If two fitted factors covary across subjects, adding their separate sums of squares does not generally equal the sum of squares of their combined reconstruction. Shared structure also has two distinct summaries: its fitted energy within and across views, and the cross-covariance between fitted signals in different views. The latter can be positive, negative, or zero.
Notation and Definitions
Let \(X_v\) be the \(n\times p_v\) data matrix for omics view \(v\), with subjects in rows and features in columns. For a centered analysis, the columns of \(X_v\) have mean zero. Define the centering matrix \(H=I-\mathbf{1}\mathbf{1}^T/n\), where \(\mathbf{1}\) is the \(n\)-vector of ones. Then \(HX_v\) is the centered data matrix; writing \(H\) explicitly also handles fitted signals that are not centered.
Suppose the fitted matrix in view \(v\) is \(\widehat X_v\). It may be decomposed into shared and view-specific components, \(\widehat X_v=C_v+P_v\), where \(C_v\) is the fitted signal attributed to factors shared across views and \(P_v\) is the fitted signal attributed to view-specific factors. In a factor representation, a component for factor \(k\) in view \(v\) can be written \(A_{vk}=z_k l_{vk}^T\), with subject scores \(z_k\) and feature loadings \(l_{vk}\). A factor is called shared here when its subject-score structure is used across views; its loading vector can differ by view.
This reconstruction fraction is not automatically the same as a proportion of variation explained in a regression sense. In particular, it is an energy ratio, and it can exceed 1 if a fitted signal is not a least-squares projection of the data. If the model is fit by least squares with an intercept and the fitted signal is the orthogonal projection onto the fitted subspace, then the residual is orthogonal to the fitted signal and the usual decomposition of centered sums of squares applies. Otherwise, report the fitted energy and residual error separately rather than assuming that they partition total variation.
Factor Contributions and Their Cross-Terms
For a factor component \(A_{vk}\), its centered fitted energy is \(\|HA_{vk}\|_F^2\). If \(z_k\) is centered, then \(HA_{vk}=z_k l_{vk}^T\), and its energy is \((z_k^Tz_k)(l_{vk}^Tl_{vk})\). This gives a meaningful size for the component, but it does not by itself provide an additive share of the total fitted energy.
For two centered components \(A_{vj}\) and \(A_{vk}\), expansion of the squared Frobenius norm gives
The final term is the cross-term. It is zero when the components are orthogonal under the matrix inner product, but it need not be zero merely because the factors have different labels. In the rank-one representation with centered scores, \(\mathrm{tr}(A_{vj}^T A_{vk})=(z_j^Tz_k)(l_{vj}^Tl_{vk})\). Thus, orthogonal scores are sufficient for a zero cross-term, as are orthogonal loading vectors, but neither condition should be presumed without checking.
Worked Example: Two Nonorthogonal Factors in One View
Consider one centered feature measured on four hypothetical subjects. Let the observed vector be \(x=(3,1,-1,-3)^T\). Suppose the fitted signal is the sum of two centered factor components, \(a_1=(2,1,-1,-2)^T\) and \(a_2=(1,0,0,-1)^T\). The data total sum of squares is
The component energies are \(a_1^Ta_1=4+1+1+4=10\) and \(a_2^Ta_2=1+0+0+1=2\). Their inner product is \(a_1^Ta_2=2+0+0+2=4\). Therefore the combined fitted energy is
The individual component energies are 50% and 10% of the data total sum of squares, but those percentages sum to 60%, not the combined reconstruction fraction of 100%. The missing 40 percentage points are the cross-term, \(2(4)/20=0.40\). Reporting the two component energies is possible, but presenting them as an additive partition would be incorrect.
If centered factor components are mutually orthogonal, their cross-terms vanish, and their energies add exactly to the fitted energy. In that special case, factor-specific percentages of \(T_v\) can be summed to obtain the reconstruction fraction. If orthogonality does not hold, a model may still report component energies, but it must label them as nonadditive. A sequential decomposition can instead orthogonalize each new component against those already included; the resulting increments add, but depend on the chosen order.
Shared Reconstruction Across Views
To summarize shared fitted energy, calculate the energy of \(C_v\) within each view and then pool those energies. A pooled shared reconstruction fraction is
This weights each view according to its total centered sum of squares. A view with more features or larger overall variation can therefore contribute more to the pooled numerator and denominator. The formula does not give each view equal influence. If equal view weighting is the scientific target, first calculate a within-view fraction for each view and then average those fractions, stating that choice explicitly.
The pooled fraction describes how much centered data energy is represented by the fitted signals classified as shared. It is not a cross-view covariance, and it does not imply that every feature in one view is positively associated with every feature in another. The subject-score structure may be shared while the loading patterns have opposite signs or different feature directions.
Worked Example: Pooled Shared Energy and Cross-View Covariance
For three hypothetical subjects, suppose two views each contain one centered observed feature: \(x_1=(2,0,-2)^T\) and \(x_2=(4,0,-4)^T\). Their shared fitted signals are \(c_1=(1,0,-1)^T\) and \(c_2=(2,1,-3)^T\), both centered. The total sums of squares are \(T_1=4+0+4=8\) and \(T_2=16+0+16=32\). The shared fitted energies are \(\|c_1\|^2=2\) and \(\|c_2\|^2=4+1+9=14\). Thus the view-specific shared fractions are \(2/8=0.25\) and \(14/32=0.4375\), while the pooled shared fraction is
The sample cross-covariance between the two fitted signals is a separate quantity. Since both vectors are centered, it is \(c_1^Tc_2/(n-1)=(2+0+3)/2=2.5\), in the product of the two feature units. The pooled fraction is dimensionless; the covariance has units of view 1 times units of view 2. Neither substitutes for the other.
In general, the centered sample cross-covariance matrix between fitted signals \(C_v\) and \(C_w\) is
This centering is essential unless both fitted-signal matrices are already centered. Centering the observed matrices alone does not guarantee that estimated or predicted components have zero column means. For example, fitted signals \((3,2,1)^T\) and \((7,6,2)^T\) have centered versions \((1,0,-1)^T\) and \((2,1,-3)^T\). Their sample covariance is \(2.5\), because the centered cross-product is \(5\) and \(n-1=2\). The uncentered product is \(3(7)+2(6)+1(2)=35\), giving \(17.5\) after division by 2; that is an uncentered cross-product, not the sample covariance.
Worked Example: A Shared Score Can Produce Negative Covariance
Let the centered subject score be \(z=(1,0,-1)^T\), with one loading of 2 in view 1 and one loading of \(-1\) in view 2. The shared fitted signals are \(c_1=2z=(2,0,-2)^T\) and \(c_2=-z=(-1,0,1)^T\). The sample cross-covariance is
The two views use the same subject score, so this fitted structure is shared by construction. Its cross-view covariance is nevertheless negative because the loadings have opposite signs. The covariance has the product of the two view-specific feature units. This example shows why shared fitted energy and cross-view covariance answer different questions.
Assumptions, Diagnostics, and Alternatives
These summaries are descriptive calculations, not tests of whether a factor is biologically real. They require matched subjects across views for a subject-level cross-view covariance, consistent subject ordering, and a clear specification of which fitted components count as shared. Within-view ratios additionally require a nonzero total sum of squares. Calculations across views should use the same subjects; if missingness changes the sample, the sums and covariance should be computed on a clearly defined common set or reported with view-specific sample sizes.
Before interpreting results, check that data and fitted matrices have the same row ordering, confirm whether an intercept or nonzero fitted means are present, and verify that the denominator matches the chosen centering and feature scaling. Inspect factor cross-products or correlations between score columns: large nonzero cross-products signal that raw factor energies will not form an additive partition. Also compare the sum of component energies with the energy of their actual sum; their difference is twice the sum of pairwise cross-terms.
The main alternatives answer somewhat different questions:
| Summary | What it measures | Main qualification |
|---|---|---|
| Combined reconstruction fraction | Energy in the full fitted signal relative to data energy within a view | May exceed 1 unless fitting and projection assumptions support the usual variance partition |
| Individual component energy | Squared size of one fitted factor component | Nonadditive when components have nonzero cross-terms |
| Sequential orthogonal increments | Additive increments after removing earlier fitted directions | Depends on factor order |
| Pooled shared reconstruction fraction | Shared fitted energy summed across views relative to total data energy summed across views | Weights views by their total energy and is not cross-view covariance |
| Fitted cross-covariance | Linear covariation between centered fitted signals in two views | Has feature-product units and can be negative despite shared scores |
A common mistake is to calculate each factor's squared fitted norm, divide by the data total sum of squares, and report the resulting percentages as an additive variance decomposition. That reasoning silently assumes orthogonality. The correction is to calculate the norm of the summed fitted signal, include cross-terms in any expansion, and label nonorthogonal component energies as nonadditive. A second mistake is to use \(C_v^TC_w/(n-1)\) as a covariance without confirming fitted signals are centered; use \(C_v^THC_w/(n-1)\) unless centering is established.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A centered view has total sum of squares 50 and a centered fitted signal with energy 35. What is its reconstruction fraction?
- Two centered factor components have energies 9 and 4, and their matrix inner product is \(-2\). Find the energy of their sum. Do their separate energies add to the total?
- Two views have total sums of squares 20 and 80, and shared fitted energies 10 and 20. Find the pooled shared reconstruction fraction.
- Why can a factor shared through the same subject scores have negative cross-view covariance?
- Fitted-signal matrices have nonzero column means. Which expression gives their sample cross-covariance, and what does the uncentered product represent?
Solutions. (1) The reconstruction fraction is \(35/50=0.7000\), or 70.00%. (2) The summed energy is \(9+4+2(-2)=9\). The separate energies sum to 13, so they are not additive; the cross-term is \(-4\). (3) The pooled fraction is \((10+20)/(20+80)=0.3000\), or 30.00%. (4) The shared scores can have loadings with opposite signs in the two views, producing negative covariance despite shared subject-level structure. (5) Use \(C_v^THC_w/(n-1)\). The uncentered product \(C_v^TC_w/(n-1)\) is an uncentered cross-product divided by \(n-1\), not generally the sample covariance.