Spectral Decomposition Report¶
Written decomposition report — instantiates Invariant-Mode Decomposition Design
A written account that turns a raw decomposition into a shared, bounded interpretation — which modes matter, what each may and may not be read to mean, and where independence between them breaks down.
A Spectral Decomposition Report is the document that hands a decomposition to people who did not compute it. Numbers alone — a basis, a spectrum, a residual — do not tell a reader what a mode means or how far that meaning can be trusted; the report supplies exactly that layer. Its defining contribution is not new modes but a bounded interpretation: an explicit contract stating what each mode may and may not be read as, and a register of where the tidy assumption of independent modes actually breaks down. It is the mechanism that stops a decomposition from being over-read — which is why its value lives in its caveats as much as its findings.
Example¶
A sleep lab decomposes a night of multi-channel EEG into spatial-temporal components (via ICA and related methods[n1]) and must hand the result to clinicians who will act on it. The raw output is a set of components with maps and spectra — but several are traps. Component 2 is a clean ~10 Hz rhythm that looks neural yet sits right over the eyes and is actually an eye-movement artifact; components 5 and 6 are not truly independent, splitting one physiological rhythm across two axes because their sources overlap.
The Spectral Decomposition Report is where this gets pinned down. It states, per component, the interpretation contract — "component 1: posterior alpha, read as a marker of drowsy wakefulness; component 2: ocular artifact, do not read as neural; components 8–12: below the noise floor, not interpretable" — and it registers the couplings, flagging which components co-vary and must be read together rather than as separate findings. The clinicians get modes they can act on and the explicit limits on acting.
How it works¶
- Restate the decomposition for a reader. Collect the basis, gains, dominance criterion, and residual from the upstream methods and tests into one legible account — the report curates them; it does not recompute them.
- Write the interpretation contract. For each retained mode, state what it may be read as and, crucially, what it may not: artifact vs. signal, descriptive vs. causal, in-regime vs. extrapolated.
- Register the couplings. Flag modes that are not cleanly independent — near-degenerate pairs, non-orthogonal or non-normal cross-talk — so readers don't treat coupled modes as separate facts.
- Bound the scope. Note the data, the window, and the conditions under which the whole decomposition holds.
Tuning parameters¶
- Audience / register — a specialist appendix versus a decision-maker's summary. The more actionable the audience, the more the interpretation limits must be foregrounded rather than buried.
- Interpretation strictness — how conservative the contract is: label only what is firmly supported, or offer tentative readings clearly marked as such. Strictness trades usefulness against the risk of over-reading.
- Coupling detail — from a single "these modes are entangled" caveat to a full register of pairwise couplings; more detail protects careful readers but can drown a casual one.
- Provenance depth — how much of the method, preprocessing, and residual evidence to carry, so a reader can judge the modes rather than take them on trust.
When it helps, and when it misleads¶
Its strength is that the report is what makes a decomposition shareable and safe to act on — it travels to people who won't inspect the math, and its interpretation contract is often the only thing standing between a modal finding and its misuse. By registering couplings it pre-empts the commonest reading error: treating entangled modes as independent facts.
Its failure mode is that a report can launder uncertainty into authority — a confident write-up makes tentative modes look established, and a mode named persuasively is a mode reified. The classic misuse is the report written to justify a decision already taken, with the interpretation contract quietly relaxed until the modes say what was wanted. The discipline is to state limits before findings, keep the contract's language calibrated to the evidence (and to the residual and out-of-sample checks behind it), and separate what the decomposition shows from what the author believes.
How it implements the components¶
interpretation_scope_contract— the report's core: an explicit, per-mode statement of what may and may not be read into each mode, and over what regime.mode_coupling_register— the report records where modes fail to be independent, so coupled modes are read together rather than mistaken for separate results.
The report produces no modes, gains, or residuals of its own — those come from Principal Component Analysis, Singular Value Decomposition, and the Residual Reconstruction Test; it curates and bounds them. Deciding stability or dominance is likewise upstream, in Modal Stability Analysis and the selection rule.
Related¶
- Instantiates: Invariant-Mode Decomposition Design — the report is the human-facing deliverable that makes a decomposition usable and hard to misread.
- Consumes: the basis and spectrum (from the decomposition methods) and the residual (from the Residual Reconstruction Test).
- Sibling mechanisms: Residual Reconstruction Test · Principal Component Analysis · Singular Value Decomposition · Reduced-Order Model · Spectral Gap Monitor · Modal Stability Analysis
Editorial Notes¶
Form Classification¶
Form family: Representation, Specification & Plan
Rationale: Spectral Decomposition Report operates as a static representation, map, specification, schema, or prospective plan that externalizes information because it a written account that turns a raw decomposition into a shared, bounded interpretation — which modes matter, what each may and may not be read to mean, and where independence between them breaks down.
Independent corroboration: The frozen evidence defines Spectral Decomposition Report as 'A written account that turns a raw decomposition into a shared, bounded interpretation — which modes matter, what each may and may not be read to mean, and where independence between them breaks down', so its operative form is Representation, Specification & Plan.
Nearest alternative: Analysis, Modeling & Optimization — Spectral Decomposition Report includes features of an analytical, modeling, inference, comparison, or optimization procedure that derives insight or a solution, but its defining operation is a static representation, map, specification, schema, or prospective plan that externalizes information.
Review outcome: Independent reviewer agreement; medium confidence.
Origin Attribution¶
Primary origin: Mathematics
Origin pattern: Cross-disciplinary synthesis
Present-day reach: Multi-domain
Rationale: Interpreting modes from an eigen- or spectral decomposition is applied linear algebra.
Related originating lineages:
- Computer Science & Software Engineering — Computer science and software-engineering practice supplies a parallel or contributing lineage for the mechanism's defining operation: a written account that turns a raw decomposition into a shared, bounded interpretation — which modes matter, what each may and may not be read to mean, and where independence….
- Data Science & Analytics — Latent components require bounded interpretive reporting.
- Engineering & Design — Modal analysis informs system design and limits.
- Physics — Normal modes give decomposition physical meaning.
Review resolution: The blind reviewers agree that mathematics is the primary origin and differ only on alternate origin disagreement, origin mode disagreement, domain reach disagreement, encyclopedia synthesis disagreement. I preserve every independently explained alternate from both records rather than imposing a numeric cap. I retain cross_disciplinary_synthesis because the combined evidence shows material contributions from several lineages. The broader reach of multi_domain records portability separately from historical provenance; encyclopedia_synthesis=true preserves the affirmative synthesis judgment where either reviewer identified one.
Encyclopedia synthesis: The exact catalogued form synthesizes established practice rather than reproducing a single standard historical label.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
Because the report only curates upstream results, it goes stale silently: if the decomposition is re-run — new data, a different mode count — the interpretation contract can end up describing modes that no longer exist. Treat the report as versioned against a specific decomposition, not as a standing document.
[n1] Independent component analysis (ICA) — a decomposition that seeks statistically independent sources rather than variance-ranked ones; in EEG it is standard for separating neural rhythms from artifacts such as eye movement and muscle activity. Which components are neural versus artifact is precisely the judgement an interpretation contract must record. ↩