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Wiener model

Model Families AIMech:WienerModel DRAFT

A real-valued, single-input single-output, discrete-time deterministic nonlinear dynamical model family formed by a causal linear time-invariant system followed by a memoryless static nonlinearity.

Identity Notes & Concept Scope Compared all 442 pinned AIO IDs, labels and all scoped synonyms and 532 native IDs, names and aliases with both candidate names, expanded cascade descriptions and IDs: no normalized identity match. Ignored- and hidden-inclusive full-text searches of the accessible recovery worktree and readable previous worktree returned four and zero lines; all four were read and concern prior discovery leads or a source example. Original Documents untracked files remain inaccessible and are not claimed re-audited. Read all 55 semantic AIO rows and 62 native name/definition neighbors, nine complete annotated AIO CLASS nodes with incident edges, all nine contextual generated YAMLs, and the complete IteratedWienerFilter, LinearStateSpaceLayer, NARX and NARMAX records. The existing iterated Wiener filter is a fixed-prior image-denoising algorithm. Neural linear state-space layers and NARX/NARMAX recurrences can overlap in representation without defining these ordered two-block model families. GLM specifies a statistical link/variance model; ARDL specifies lag structure; neither specifies this cascade identity. AIO convolutional layers and recurrent networks require neural-component/network scope, and ETS adds smoothing/trend/seasonality structure. These are not exact matches. This assessment is bounded to the inspected pin and accessible records.

Claims & Grounded Evidence

4 Claims

Mechanism #linear-then-static

Write z_t=sum_{j=0}^{t-1} h_j*u_(t-j) for the zero-state linear response and y_t=g(z_t). The linear block first combines inputs across time; the memoryless nonlinearity then transforms that scalar. This differs in general from applying g before filtering: with h=(1,2), inputs (1,2,-1) and g(v)=v², the first three Wiener outputs are (1,16,9), whereas the corresponding Hammerstein outputs are (1,6,9). This is a finite curator example of block-order dependence, not an empirical comparison.

supports Source: Input-Output Parametric Models for Nonlinear Systems. Part II Stochastic Nonlinear Systems · Locator: Section 3, Figure 3.1 and Equations 3.1–3.4, PDF pp. 13–14 / printed pp. 10–11; curator exact calculations in research/discovery/cascade-model-boundaries.py

Mechanism #conditional-output-recurrence

For a finite-dimensional linear block satisfying z_(t+n)=sum_{i=1}^n a_i*z_(t+n-i)+sum_{j=0}^n b_j*u_(t+n-j), an inverse of g on the relevant internal-signal range gives y_(t+n)=g(sum_i a_i*g^(-1)(y_(t+n-i))+sum_j b_j*u_(t+n-j)). If the inverse is only valid on a selected interval, the required internal history must stay on that branch. The convolutional cascade itself does not require an inverse. Finite state order also does not imply a finite impulse-response memory.

supports Source: Input-Output Parametric Models for Nonlinear Systems. Part II Stochastic Nonlinear Systems · Locator: Section 3, Equations 3.2–3.8, PDF pp. 14–15 / printed pp. 11–12

Limitation #noninjective-output-boundary

For the source example x_next=x+u and y=x+x², two reachable internal states x=0 and x=-1 have the same output 0, but with the same current input u=1/4 the next outputs are 5/16 and -3/16. Thus a single-valued update from only current y and u cannot cover both branches. Restricting x>-1/2 gives the inverse branch used in the source. This does not rule out other histories, state tracking or a narrower operating domain.

supports Source: Input-Output Parametric Models for Nonlinear Systems. Part II Stochastic Nonlinear Systems · Locator: Section 2, Example 2.1, Equations 2.6 and 2.22–2.25, PDF pp. 9,11–12 / printed pp. 6,8–9; curator exact calculations in research/discovery/cascade-model-boundaries.py

Limitation #internal-scale-ambiguity

Without a normalization, internal gain can be exchanged between the two blocks. For nonzero c, replacing the zero-state linear operator L by c*L and g(v) by g(v/c) preserves g(Lu) for every admissible input. A c=3 finite example reproduces the same outputs. This algebraic ambiguity means identical external responses alone do not uniquely determine both internal block gains; it does not establish the full set of identification ambiguities.

supports Source: Input-Output Parametric Models for Nonlinear Systems. Part II Stochastic Nonlinear Systems · Locator: Section 3, Equations 3.1–3.4, PDF p. 14 / printed p. 11; direct curator consequence of the cascade equations; curator exact calculations in research/discovery/cascade-model-boundaries.py

⚠️ Curated Limitations & Negative Results

The scoped evidence covers real-valued, single-input single-output, discrete-time deterministic cascades with a causal time-invariant linear block and a memoryless single-valued nonlinearity. Direct feedthrough is permitted. Continuous-time, multivariable, parallel and stochastic variants need explicit additional assumptions.
A model family is distinct from its parameter estimator, internal-state realization, trained instance and physical mechanism. Noise placement, initial state, stability, excitation, identifiability and generalization are separate choices; no universal identification or forecasting guarantee is asserted.
Invertibility assumptions used to derive a particular output-history recurrence are not definition-level requirements for a cascade model. Nonlinear and linear special cases can overlap across model families without making their ordered structures equivalent.
Evidence is the fully read 1984 technical report. The final 1985 journal version and earlier correlation-based identification reports remain follow-up sources. Several displayed equations in the report have scope or transcription problems; the cited cascade definitions and directly checked algebra are used here, not its unchecked global claims.
No model was trained and no empirical result was replicated. Exact finite examples are curator derivations, not author-reported experiments. Both records remain DRAFT; author self-review does not constitute independent scientific approval.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:TimeSeriesAnalysis` Time Series Analysis	related_match	The source develops this ordered input-output dynamical representation for analyzing and identifying nonlinear systems from temporal observations. The pinned target denotes the analysis task; a related model/task mapping preserves that distinction and does not assert that the cascade is a task, neural layer or particular estimator.

Evidence Sources & Primary Literature

1 Source

Source ID	Publication / Document	Type / Access	Reference & Checksum
`leontaritis-billings1984-part2`	Input-Output Parametric Models for Nonlinear Systems. Part II Stochastic Nonlinear Systems I. J. Leontaritis, S. A. Billings	author technical report full_text	ID: https://eprints.whiterose.ac.uk/id/eprint/76841/ sha256:6ef50a90a753…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T13:46:47Z

Updated: 2026-10-12T13:55:47Z

Source File: data/curated/WienerModel.yaml