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Nonlinear autoregressive moving average model with exogenous inputs

Model Families AIMech:NonlinearAutoregressiveMovingAverageExogenousModel DRAFT

Aliases: NARMAX model

A discrete-time nonlinear input-output model family whose current output is a function of finite lagged outputs, exogenous inputs and errors, plus an additive current error.

Identity Notes & Concept Scope Compared all 442 pinned AIO IDs, labels and all scoped synonyms and 528 native IDs, names and aliases with the proposed names, IDs, NARX/NARMAX and expanded variants; no normalized identity match. Ignored- and hidden-inclusive full-text searches covered the recovery worktree and readable previous worktree, returning 22 and zero lines respectively; all hits were inspected and concern the previous OLS record, its saved review, pages and pending source cohort. The original Documents checkout remains inaccessible, so its untracked files are not claimed searched. Read all 76 AIO semantic rows and 98 native name/definition neighbors, 12 complete annotated AIO CLASS nodes with incident edges, four closest generated records and the native linear state-space layer record. The OLS selector is an estimator rather than either model family. A neural autoregressive density estimator models ordered conditional distributions; it does not by its name denote these lagged input-output recurrences. State-space neural layers and architectures maintain different state/parameterization scopes. This assessment is bounded to the inspected pin and accessible corpus. NARMAX is the encompassing notation in the representation paper; the companion OLS paper explicitly distinguishes its NARX subclass. Linear regression, least-squares analysis and forward selection describe estimators or fitting scope. RecurrentNeuralNetwork and EchoStateNetwork require neural structures; VAR requires linear multivariate structure; TAR uses regime changes; autoregressive language models target token distributions. None is an exact identity for this family.

Claims & Grounded Evidence

6 Claims

Mechanism #lagged-error-recurrence

In the strictly causal scalar form, y_t=F(y_{t−1},…,y_{t−ny},u_{t−1},…,u_{t−nu},e_{t−1},…,e_{t−ne})+e_t. Removing explicit lagged-error dependence gives NARX; removing exogenous inputs gives a NARMA form. The moving-average terminology here concerns lagged errors inside a possibly nonlinear function, not a running arithmetic average of observations. Finite-lag recurrence structure does not require a neural network.

supports Source: Orthogonal least squares methods and their application to non-linear system identification · Locator: Section 2, Equations 1–6, PDF p. 3 / printed p. 1874
supports Source: Representations of non-linear systems: the NARMAX model · Locator: Section 6, Equations 69–72, PDF p. 16 / printed p. 1028

Scope #innovation-versus-fitted-residual

In the conditional-mean innovation construction, the prediction is the conditional expectation of the current output given available past information, and the innovation is output minus that expectation. This interpretation requires the relevant moments and information set; it does not imply independent Gaussian errors. The companion OLS formulation separately assumes zero-mean independent noise. For an estimated model with chosen parameters, recursively computed prediction residuals are parameter-dependent and are not automatically the true innovations of the data-generating system.

supports Source: Representations of non-linear systems: the NARMAX model · Locator: Section 6, Equations 58–69 and 77–81, PDF pp. 15–16,18 / printed pp. 1027–1028,1030
supports Source: Orthogonal least squares methods and their application to non-linear system identification · Locator: Section 2, Equations 1–5 and noise definition, PDF p. 3 / printed p. 1874

Mechanism #latent-error-estimation

Polynomial expansion is formally linear in coefficients, but lagged noise variables are not directly observed. Consequently the regressor matrix is not fixed directly from input-output measurements. The OLS paper describes an iterative extension that substitutes prediction residuals, rebuilds noise-containing regressors and refits. This is one estimator for the family; its reported few-iteration behavior is not a universal convergence guarantee or a definition of NARMAX itself.

supports Source: Orthogonal least squares methods and their application to non-linear system identification · Locator: Sections 2 and 5, Equations 83–91, PDF pp. 5,18–20 / printed pp. 1876,1889–1891

Limitation #residual-initialization-and-stability

Prediction with lagged errors requires residual initialization. Model invertibility asks whether residual sequences from different initializations forget that difference for the specified model and input process. Linearity in lagged errors alone is insufficient: in y_t=u_{t−1}²+a*e_{t−1}+e_t, two residual recursions driven by identical observations differ by (−a)^t times their initial difference. With initial difference 1, the difference after eight steps is 1/256 for a=1/2 and 256 for a=2. The source’s particular stochastic sufficient condition and example must not be generalized to arbitrary coefficients or input distributions.

supports Source: Representations of non-linear systems: the NARMAX model · Locator: Section 6, Equations 74–83 and Example 6, PDF pp. 17–19 / printed pp. 1029–1031; curator exact examples in research/discovery/narx-narmax-boundaries.py

Scope #polynomial-scope

A polynomial parameterization expands finite lag variables into monomials and is linear in the monomial coefficients, while it can be nonlinear in the input and output values. The continuous scalar recurrence function on a specified compact finite-dimensional lag domain can be uniformly approximated by some polynomial; this does not prescribe a fixed degree, identify coefficients from finite observations, justify extrapolation, or guarantee stable recursive trajectories. The general model family is not restricted to polynomial parameterizations.

supports Source: Orthogonal least squares methods and their application to non-linear system identification · Locator: Section 2, Equations 6–13, PDF pp. 3–5 / printed pp. 1874–1876
supports Source: Representations of non-linear systems: the NARMAX model · Locator: Section 4, Equations 34–43, PDF pp. 9–11 / printed pp. 1021–1023; curator exact examples in research/discovery/narx-narmax-boundaries.py

Limitation #published-approximation-bound

The source supplies the displayed formulas and stated error bound below; the refutation and trajectory calculation are curator derivations. The representation paper’s particular degree-four approximation example claims error below 0.04 on |y|<=1/4, |u|<=1. Exact substitution in its displayed formulas gives error 5/16 at y=−1/4,u=1, and 2336/15625 at y=−6/25,u=1 (the y coordinate is interior; u is on the boundary). These curator counterexamples reject that numerical bound, not the compact-domain polynomial approximation existence statement. A small one-step function error also need not remain small under repeated simulation: comparing f(y)=2y and g(y)=2y+1/100 from zero gives difference (2^t−1)/100 after t steps.

supports Source: Representations of non-linear systems: the NARMAX model · Locator: Section 4, displayed recurrence, polynomial and claimed error bound after Equation 43, PDF pp. 11–12 / printed pp. 1023–1024; these source inputs support the explicit curator derivations in research/discovery/narx-narmax-boundaries.py, not an author-reported correction

⚠️ Curated Limitations & Negative Results

This record covers finite-lag, real-valued, discrete-time, strictly causal input-output formulations; current-input/feedthrough conventions and broader vector/noise assumptions need explicit specification. The family is not a trained model, dataset or unique estimator.
Local realization, uniqueness, stability, identifiability and predictive accuracy are separate questions. No claim is made that every nonlinear or stochastic system has an exact global representation of this form. Original 1984 reports and 1985 journal proofs are follow-up sources, not audited theorem evidence here.
The 1989 representation paper contains problematic unrestricted inverse formulas and a numerical approximation bound contradicted by exact substitution. Those statements are not adopted. Its explicit recurrence definitions and scoped polynomial construction are distinguishable from these source defects.
No model was trained and no empirical result was replicated. Exact finite curator examples check stated boundaries. The record remains DRAFT; structural validation and author self-review are not 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	This finite-lag input-output model family provides a representation used in time-series analysis and system identification. The pinned target is an analysis task, so this is a model/task relationship rather than equivalence or an is-a assertion.
`AIO:AutoregressiveMovingAverage` Autoregressive Moving Average	related_match	Both use autoregressive and lagged-error structure. The pinned ARMA class describes stationary-series modeling; this record allows nonlinear functions and exogenous inputs and does not imply stationarity. The shared structure supports a related mapping, not an exact or broader-class claim.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`chen-luo1989`	Orthogonal least squares methods and their application to non-linear system identification S. Chen, S. A. Billings, W. Luo	primary paper full_text	ID: https://doi.org/10.1080/00207178908953472 sha256:62199a403df8…
`chen-billings1989`	Representations of non-linear systems: the NARMAX model S. Chen, S. A. Billings	primary paper full_text	ID: https://doi.org/10.1080/00207178908559683 sha256:68f4dc694ad1…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T13:00:32Z

Updated: 2026-10-12T13:07:49Z

Source File: data/curated/NonlinearAutoregressiveMovingAverageExogenousModel.yaml