Correlation-based Hammerstein identification
Algorithms AIMech:CorrelationBasedHammersteinIdentification DRAFT
A two-stage identification algorithm for a real-valued single-input single-output polynomial Hammerstein cascade that estimates a scaled linear response from Gaussian-excited input/output cross-correlation, then fits the input polynomial by least squares on filtered input powers.
Identity Notes & Concept Scope
Compared all 442 pinned AIO IDs, names and scoped synonyms and 534 native IDs, names and aliases with both proposed estimator names, expanded two-stage/correlation names and related Bussgang/linear-approximation terms: no normalized identity match. Ignored- and hidden-inclusive full-text searches of the accessible recovery worktree and readable earlier worktree returned 81 and 68 lines. All distinct matched text was read; it concerns pending Bussgang blind-deconvolution leads and prior cascade-model source context. Blind deconvolution with unknown input is not these observed-input estimators. Original Documents untracked files remain inaccessible and are not claimed re-audited. Read all 36 semantic AIO rows, 58 native name/definition neighbors, eight complete annotated AIO CLASS nodes with incident edges, eight contextual generated YAMLs and six complete native neighbors. Wiener/Hammerstein records denote models, RPE/PLR denote recursive updates, forward OLS selects candidate structure, and the iterated Wiener filter estimates an image under a fixed prior. These are distinct mechanisms. AIO linear/least-squares regression supplies component fitting context, not the ordered two-stage estimator identity. DCC models changing covariances, GLM specifies a statistical link, ARDL specifies lagged model structure, and a convolutional layer is a neural component. No exact mapping is justified. These local descriptive method names do not claim original nomenclature or priority.
Claims & Grounded Evidence
4 Claims
Mechanism
#correlation-separation
The first stage correlates centered output with the centered Gaussian input fluctuation of excitation u=b+x. Under the source’s independent-output-noise and sufficiently white practical-excitation assumptions, the result approximates alpha*h, with alpha=mu*[gamma1+2*b*gamma2+3*gamma3*(variance(x)+b²)+...] and mu the input autocorrelation area. The report explicitly distinguishes finite practical variance from ideal continuous-white variance. For discrete iid unit-variance Gaussian inputs the corresponding factorization is exact at the population level. A nonzero alpha and appropriate moment/linear-response conditions are needed for recovery.
Mechanism
#filtered-input-powers
After estimating the scaled linear response, build a design column for each degree j by filtering u^j through that response. Least-squares fit measured output on these columns. The input u includes the applied mean; x denotes the centered fluctuation only in the correlation stage. The displayed inverse requires full column rank. This differs from raising a filtered input to successive powers as in Wiener identification, and it does not select the polynomial degree or linear order automatically.
Scope
#scale-reparameterization
If the first stage recovers alpha*L, divide every polynomial coefficient by alpha, independent of its degree, to preserve L(g(u)). For h=(1,2), u=(1,2,-1), g(v)=v+v² and scale 3, the two filtered-power columns are (3,12,9) and (3,18,27); coefficients (1/3,1/3) reproduce outputs (2,10,12). These are exact deterministic curator calculations and do not uniquely determine the physical block gains or establish finite-sample estimation performance.
Limitation
#gain-and-variant-boundaries
Even with u_t=1+x_t for iid standard Gaussian x, g(u)=u³-6u has zero centered input/output cross-correlation through a nonzero LTI block: the polynomial contributions cancel. The source’s nonzero-mean recommendation therefore needs a nonzero-gain check. Its separate parallel-feedforward extension uses squared-input correlation, whose gain can also vanish, and generally identifies the combined linear branch h1+omega1*h2 rather than both original paths. Those extra steps are not part of this two-stage Hammerstein algorithm.
⚠️ Curated Limitations & Negative Results
- This record covers the inspected two-stage polynomial identification procedure for a real-valued single-input single-output cascade, observed exogenous input/output, a time-invariant linear block and independent additive measurement noise. The displayed polynomial basis has degrees 1 through k; a constant/intercept, multivariable, closed-loop, colored-input or neural variant needs separate treatment.
- Independence removes additive output noise from correlation with a centered input, even if that noise has a nonzero mean. It does not by itself remove bias from the second-stage raw-output regression without an intercept. Interpreting those fitted coefficients as the system polynomial requires zero-mean noise or separately justified offset treatment; subtracting the entire output mean can also remove a real nonlinear signal mean.
- Gaussian correlation factorization is a population statement under the specified excitation and finite moments. Finite sample correlations, estimated regressors, chosen orders, transients, sampling, noise and conditioning affect fitted results. No unconditional unbiasedness, consistency, unique physical recovery or generalization guarantee is asserted.
- Practical continuous-time white-noise excitation has finite bandwidth. A delta-correlation approximation does not establish exact impulse recovery for arbitrary colored inputs. A literal ideal continuous-white signal cannot be passed pointwise through an unrestricted polynomial; discrete iid Gaussian examples here are well-defined curator calculations.
- Internal gain requires a normalization or external information. A nonzero excitation mean alone does not guarantee a nonzero correlation gain; cancellation can leave the first stage uninformative even for a nonconstant nonlinear system.
- All pages of both original reports were visually read. Later journal versions were identified from publisher/registration metadata but not compared in full. The broader 1980 report and original correlation-theorem, iterative-estimation and parallel-system sources remain follow-up evidence. No training, empirical replication or paid provider was run. Records remain DRAFT; author self-review is not independent approval.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:LeastsquaresAnalysis
Least-squares Analysis
|
related_match | Both stages use residual-squared parameter fitting after the source’s correlation-based separation. The pinned target describes a general fitting analysis, whereas this record specifies an ordered identification procedure with excitation and block-structure assumptions. A related mapping reflects its component mechanism without claiming exact identity or an asserted subclass. |
AIO:TimeSeriesAnalysis
Time Series Analysis
|
related_match | The procedure estimates nonlinear dynamical structure from temporal input/output observations and lagged correlations. The pinned target is the analysis task; related mapping preserves the algorithm/task distinction. |
Evidence Sources & Primary Literature
2 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
billings-fakhouri1978-hammerstein
|
S. A. Billings, S. Y. Fakhouri
|
author technical report
full_text
|
ID: https://eprints.whiterose.ac.uk/id/eprint/75422/
sha256:6d50fec13e59…
|
billings-fakhouri1977-wiener
|
S. A. Billings, S. Y. Fakhouri
|
author technical report
full_text
|
ID: https://eprints.whiterose.ac.uk/id/eprint/75410/
sha256:4520676002d2…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-12T14:10:31Z
Updated: 2026-10-12T14:17:21Z