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Nonlinear autoregressive model with exogenous inputs

Model Families AIMech:NonlinearAutoregressiveExogenousModel DRAFT

Aliases: NARX model, Nonlinear autoregressive exogenous model

A discrete-time nonlinear input-output model family whose current output is a function of finite lagged outputs and exogenous inputs, plus an additive current error, without explicit dependence on lagged errors.

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 #finite-lag-predictor

In the strictly causal scalar formulation, y_t=f(y_{t−1},…,y_{t−ny},u_{t−1},…,u_{t−nu})+e_t. The lag orders and function parameterization are model choices. The 1989 OLS paper defines NARX by omitting lagged noise from its general NARMAX form; the current additive noise term remains. Setting that term to zero gives a deterministic recurrence. A neural network is not required by this definition.

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

Mechanism #parameter-estimation-boundary

For a fixed polynomial NARX structure, measured lagged input/output values form a fixed design matrix and coefficient fitting can be posed as linear least squares. Structure selection and coefficient estimation are distinct from the model family. Being linear in coefficients does not make the observed dynamics linear, and a least-squares fit alone establishes neither physical causation nor valid generalization.

supports Source: Orthogonal least squares methods and their application to non-linear system identification · Locator: Section 2, Equations 6–13 and structure-selection discussion, PDF pp. 3–5 / printed pp. 1874–1876; curator exact examples in research/discovery/narx-narmax-boundaries.py

Limitation #local-representation-boundary

A chosen output-history recurrence must be well-defined on its stated domain. In the source example x_next=x+u, y=x+x², solving for x gives two possible branches unless the state is restricted. States x=0 and x=−1 have the same current output 0; with the same input u=1/4 their next outputs are 5/16 and −3/16. Thus those one-step observed arguments alone do not define a global single-valued update across both branches. This example does not rule out other history lengths or restricted-domain representations.

supports Source: Representations of non-linear systems: the NARMAX model · Locator: Section 2.2, Example 1, PDF p. 3 / printed p. 1015; 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

Application #identification-use

The OLS study uses polynomial NARX structure selection to model measured pilot liquid-level input-output data. This is a documented system-identification use, not evidence that all NARX fits recover a physical mechanism or outperform alternative models. The generic family also permits formulations other than the particular polynomial fit.

supports Source: Orthogonal least squares methods and their application to non-linear system identification · Locator: Section 6, Example 1, Figure 2 and Table 1, PDF p. 21 / printed p. 1892

⚠️ 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:AutoregressiveDistributedLag` Autoregressive Distributed Lag	related_match	Both represent dependence on lagged outputs and other inputs. The pinned ARDL definition does not resolve nonlinear function classes or the exclusion of lagged errors. A shared lag structure warrants a related mapping, not identity or an inferred subclass from an underspecified definition.

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/NonlinearAutoregressiveExogenousModel.yaml