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Pseudo-linear regression estimation

Algorithms AIMech:PseudoLinearRegressionEstimation DRAFT

Aliases: Pseudo-linear regression algorithm

A family of recursive coefficient estimators that uses the current regressor as a surrogate prediction sensitivity, omitting derivatives through parameter-dependent regressors such as past estimated residuals.

Identity Notes & Concept Scope Normalized comparisons covered all 442 pinned AIO IDs, labels and scoped synonyms and 530 native IDs, names and aliases, including expanded prediction-error and pseudo-linear names. No identity match was found. Ignored- and hidden-inclusive full-text searches covered the recovery worktree and readable previous worktree, returning five and zero lines; all five concern the pending source cohort, not existing estimator records. Original Documents untracked files remain inaccessible and are not claimed re-audited. Read all 50 AIO semantic rows and 159 native name/definition neighbors, eight complete annotated AIO CLASS nodes with incident edges, and the complete SGD, L-BFGS and orthogonal least-squares records. The NARX/NARMAX family records were fully read in the preceding round. Generic optimization and regression are related scopes; they do not supply this estimator identity. NARMAX is a model family, forward OLS selects structure, and recursive neural networks denote architectures. Recursion in a name is not equivalence. These methods overlap in special cases without being identical families. LinearRegression does not specify parameter-dependent residual recursion, ARMA is a model family, and TimeSeriesAnalysis/Forecasting and SupervisedLearning are task contexts. None is an exact match. SGD can overlap with a simple search-direction choice, but its generic sampled-gradient definition does not identify this recursive predictor treatment; L-BFGS uses a different curvature-history mechanism.

Claims & Grounded Evidence

5 Claims

Mechanism #raw-regressor-update

For yhat_t=theta_(t-1)^T*phi_t with regressors that may depend on past estimated residuals, the basic recursive pseudo-linear estimator uses epsilon_t=y_t-yhat_t, R_t=(1-gamma_t)R_(t-1)+gamma_t*phi_t*phi_t^T, and theta_t=theta_(t-1)+gamma_t*R_t^(-1)*phi_t*epsilon_t. It uses phi in place of the prediction-error method’s sensitivity psi. This is the unfiltered form displayed in the source; additional filtering defines further variants.

supports Source: Recursive prediction error parameter estimator for non-linear models · Locator: Section 5, Equations 83–85, PDF pp. 15–16 / printed pp. 582–583

Scope #surrogate-derivative-boundary

When phi depends on theta through earlier residuals, using phi alone omits part of the total derivative. In the fixed-parameter example yhat_t=b*x_t+c*epsilon_(t-1), epsilon_0=0, x=(1,4), y=(1,3), and b=c=1/2, the second regressor is (4,1/2) while the total prediction sensitivity is (7/2,1/2). Starting a single update from the same P_previous=I and lambda=1 gives increments (4/23,1/46) for the raw-regressor rule and (7/36,1/36) for the sensitivity rule. These are curator calculations comparing a common state, not two complete adaptive trajectories or a performance ranking.

supports Source: Recursive prediction error parameter estimator for non-linear models · Locator: Sections 2–3 and 5, Equations 8–12,21,83–85, PDF pp. 3–5,15–16 / printed pp. 570–572,582–583; exact curator calculations in research/discovery/recursive-estimation-boundaries.py

Scope #least-squares-special-cases

The source identifies recursive extended least squares for an ARMAX model as a pseudo-linear regression example. It is a specialized member, not an alias for every pseudo-linear estimator. If regressors truly do not depend on the parameters or their residual history, the omitted derivative terms vanish and the basic recursion reduces to ordinary recursive least-squares fitting. Linear appearance in coefficients with residual-dependent regressors does not by itself make the full prediction-error criterion quadratic in those coefficients.

supports Source: Recursive prediction error parameter estimator for non-linear models · Locator: Sections 2–3 and 5, Equations 8–12,19,83–85, PDF pp. 3–5,15–16 / printed pp. 570–572,582–583

Limitation #invertibility-versus-positive-real-condition

For linear-model RELS, the source’s convergence discussion adds correct model specification and the strict positive-real condition Re[1/C°(exp(i*omega))-1/2]>0 for all frequencies, together with the applicable regularity conditions. Stable inversion alone does not imply that stronger condition: C(q^-1)=(1+0.9*q^-1)^2 has both C* roots at -0.9, while the displayed test at zero frequency equals -161/722. This exact example shows failure of a sufficient condition; it does not prove that a particular RELS run diverges.

supports Source: Recursive prediction error parameter estimator for non-linear models · Locator: Sections 4.1 and 5, Equations 29–30 and 86–88, PDF pp. 6,16 / printed pp. 573,583; exact curator calculations in research/discovery/recursive-estimation-boundaries.py

Limitation #nonlinear-convergence-scope

Extending the recursion to nonlinear polynomial regressors does not automatically transfer linear-model convergence results. The source notes that existence of input/noise second-moment limits alone may be insufficient for the required nonlinear regressor moment limits, and leaves more general data-dependent noise-model analysis for further study. The paper’s Section 6 simulations use the recursive prediction-error estimator; they are not reported comparative evaluations of pseudo-linear regression.

supports Source: Recursive prediction error parameter estimator for non-linear models · Locator: Section 5, Equations 89–91 and closing paragraph, and Section 6, PDF pp. 16–18,22 / printed pp. 583–585,589

⚠️ Curated Limitations & Negative Results

The inspected mechanism evidence is the complete 1989 polynomial-NARMAX treatment. Earlier recursive-identification sources and broader filtered, vector, nonquadratic, rational and output-affine variants remain follow-up scopes; no priority claim is made.
Model structure, initialization, gain or forgetting schedule, numerical conditioning and predictor stability require separate choices. A recursive parameter estimator does not select a correct model structure, establish physical causality or guarantee out-of-sample accuracy.
The source defines m-invertibility by mean-square residual-initialization forgetting. Almost-sure or exponential forgetting is not inferred solely from that definition. Its nonlinear noise examples and convergence extensions require their stated assumptions; no unrestricted theorem is adopted.
All 27 rendered PDF pages were read, including the experiments and references. Exact examples are curator derivations, not replications of the source experiments. No model training, empirical benchmark or paid service was run. DRAFT status and author self-review do not imply independent scientific approval.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:LeastsquaresAnalysis` Least-squares Analysis	related_match	The squared-residual criterion and recursive least-squares algebra directly relate these parameter updates to residual-squared fitting. The pinned target describes an overdetermined fitting analysis, not this recursive estimator; the pseudo-linear approximation also need not minimize the full nonlinear prediction-error objective. A related mapping preserves these differences.
`AIO:RegressionAnalysis` Regression Analysis	related_match	The estimator fits a parameterized relationship between observed outputs and predictive lagged information. The pinned target groups statistical fitting processes and does not specify this update or its residual dependence. This is a contextual algorithm/fitting relationship rather than an exact or inferred subclass assertion.

Evidence Sources & Primary Literature

1 Source

Source ID	Publication / Document	Type / Access	Reference & Checksum
`chen-billings1989`	Recursive prediction error parameter estimator for non-linear models S. Chen, S. A. Billings	primary paper full_text	ID: https://doi.org/10.1080/00207178908559655 sha256:7716a5eca47f…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T13:23:25Z

Updated: 2026-10-12T13:24:22Z

Source File: data/curated/PseudoLinearRegressionEstimation.yaml