Contribution magnitude and variance explained AI

Why the size of a component contribution is not an additive percentage of the recorded signal.

The magnitude of a component answers how large its modeled contribution is, regardless of sign. Explained variance asks how much variation in a specified target a model accounts for. These quantities coincide only under particular assumptions; a large component need not account for a correspondingly large share of the observed signal.

Pair the coordinate with its spatial pattern

For a network matrix $W_k$ and coefficient $x_k(t)$, the contribution is $C_k(t)=x_k(t)W_k$. Let $w_k$ contain the unique off-diagonal entries of $W_k$. The reports summarize contribution magnitude as

$$m_k=\left\langle |x_k(t)|\right\rangle\,\|w_k\|_2.$$

This combines expression amplitude with spatial scale. Rescaling $W_k$ by a nonzero scalar and inversely rescaling $x_k$ leaves the contribution and this magnitude unchanged. Comparing coordinate amplitudes alone would not have that property. A signed average addresses a different question: opposing expressions can cancel even when magnitude is substantial.

Two large opposing contribution vectors yield a much smaller combined vector.

Illustrative vectors in the space of matrix entries. Large component magnitudes can coexist with a small combined contribution.

Components need not provide additive shares

For contributions represented as vectors, the combined squared norm includes cross-terms:

$$\left\|\sum_k c_k\right\|^2=\sum_k\|c_k\|^2+2\sum_{k<l}\langle c_k,c_l\rangle.$$

Patterns can reinforce or cancel. Dividing each component magnitude by the sum of magnitudes creates a descriptive allocation, not a decomposition of BOLD variance. The norm here is also measured in the model’s moment space, which is different from the original signal space.

PCA’s explained-variance ratios have a specific centered, orthogonal decomposition behind them (PCA reference). The coefficient matrices of a predictive moment model do not inherit that accounting merely because an SVD appears during training.

To evaluate predictive value, specify a target and score reconstruction or forecast errors on held-out recordings. Removing a component with the other parameters fixed tests a different question from removing it and refitting the model. Either analysis needs its own denominator and interpretation; neither should be presented as a variance percentage derived from magnitude alone.