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REVIEW 3 major objections 5 minor 59 references

NonSysId: A nonlinear system identification package with improved model term selection for NARMAX models

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper introduces NonSysId, a MATLAB package that selects NARMAX model terms using iterative orthogonal forward regression, PRESS-statistic cross-validation, and simulation-based stability filtering, allowing parsimonious models to be…

desk verdict NonSysId is a legitimate software contribution with a real gap: the open-source iOFR/PRESS implementation is useful, but the central generalization claim rests on an unproven constant-input stability heuristic that the paper neither justifies nor stress-tests. read the letter →

arxiv 2411.16475 v1 pith:OL3IGDQ7 submitted 2024-11-25 eess.SY cs.LGcs.SY

Rajintha Gunawardena , Zi-Qiang Lang , Fei He This is my paper · ORCID
classification eess.SYcs.LGcs.SY
keywords systemidentificationNARXNARMAXorthogonalforwardregressionPRESSstatisticiterativeOFRsimulation-basedmodelselectionMATLABpackage
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper presents NonSysId, an open-source MATLAB package for identifying polynomial NARX and NARMAX models. Its central claim is that combining iterative Orthogonal Forward Regression (iOFR) with PRESS-statistic term selection and simulation-based stability filtering yields parsimonious models that generalize well without a separate validation dataset. This matters for real-time applications—structural health monitoring, fault diagnosis, and biomedical signal processing—where obtaining consistent validation data is often impractical. The paper backs the claim with two demonstrations: a synthetic DC-motor NARX model excited by white noise and by a multi-tone sinusoid, and a real electro-mechanical system driven by a PRBS. A careful reader would take away that the package targets long-horizon free-run simulation accuracy, not just one-step-ahead fit.

What carries the argument

The load-bearing machinery is the iOFRS procedure: an extension of the iterative OFR algorithm that (step 1) seeds orthogonalization paths from each pre-selected term, (step 2) admits only candidate models whose simulated output for u(t)=0 and u(t)=1 has variance at most ε=$10^{{-2}}$, and (step 3) ranks survivors by BIC computed from simulated mean-squared error. Inside the forward search, PRESS-statistic term selection estimates leave-one-out cross-validation error in closed form from the OLS orthogonalization, replacing ERR-based selection. Complementing the search are four RCT methods that shrink the candidate term set (e.g., via a linear ARX pre-model) and sharpen the initial pre-select set, reducing runtime while preserving the simulation-stability search.

What would settle it

Take a nonlinear ARX system, identify a model with NonSysId using a short non-persistently exciting input, then run the model free on a longer input record drawn from the same distribution; if the simulated output diverges while a model selected by conventional cross-validation on a separate validation set tracks the data, the constant-input stability filter has failed to certify generalization.

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Extended reading notes

Core claim

The central discovery is a concrete algorithmic pipeline, iOFRS, in which model structure is selected by simulation behaviour rather than by one-step-ahead explained variance alone. At each iteration, the algorithm pre-selects candidate terms, follows multiple orthogonalization paths to produce candidate models, filters out models whose free-run output under constant inputs u=0 and u=1 is not bounded with small variance, and then chooses the model with the lowest BIC computed from simulation error variance. Within each path, PRESS-statistic term selection replaces the classical error reduction ratio, giving an efficient leave-one-out cross-validation during forward selection. The paper claims this produces models that are both parsimonious and stable under free-run simulation, without needing a held-out validation set.

Load-bearing premise

The whole approach takes the two constant-input stability tests (u=0 and u=1) as sufficient evidence that a candidate model will simulate stably under the actual, possibly non-persistently exciting input; if that inference fails, the simulation-based BIC selection can pick a model that diverges in free run even though the package judged it stable.

Editorial extensions

If this is right

  • A user can identify a NARX or NARMAX model from a single input-output record, with leave-one-out cross-validation built into term selection rather than requiring a held-out split.
  • Selected models are chosen by free-run simulation error, so they should support long-horizon prediction and simulation-based analysis such as NOFRF frequency-domain studies.
  • The stability filter and PRESS criterion should reduce the redundant-term and overfitting problems reported for OFR-ERR, especially when the input is not persistently exciting.
  • The four RCT methods let practitioners trade a smaller search space for lower computation time in real-time monitoring and diagnosis tasks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The constant-input filter is a heuristic, not a general stability certificate; a polynomial NARX model can be bounded under u=0 and u=1 yet diverge under other bounded inputs, so the 'no validation data needed' claim should be read as conditional on the input class.
  • The paper's own residual plots show residual autocorrelation not fully white in both examples; the claim of robust generalization would be strengthened by reporting out-of-sample free-run error on an unseen segment rather than training-residual variance alone.
  • A natural benchmark, not run in the paper, is a systematic comparison against OFR-ERR and against a standard validation-split approach on systems with weak excitation, measuring both free-run error and model parsimony.
  • The same iOFRS machinery could be lifted to non-polynomial basis functions and to MIMO systems, which the authors list as future work; the stability filter and BIC selection would need re-testing in those settings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents NonSysId, an open-source MATLAB package for identifying polynomial NARX/NARMAX models. It implements an iterative Orthogonal Forward Regression (iOFR) variant with PRESS-statistic-based term selection and a simulation-based model selection step, claiming robust generalization without a separate validation dataset. The manuscript describes the algorithm, four computational-speed-up methods, and two examples: a noise-free synthetic DC-motor NARX model and a real electromechanical system.

Significance. NonSysId addresses a real need: open-source NARMAX toolkits based on plain OFR/ERR can overfit and produce unstable free-run models, and validation data are often scarce. The package integrates known methods (iOFR, PRESS, linear-ARX-based candidate reduction) and provides reproducible code and examples. The claimed contribution—simulation-stable, parsimonious model selection without held-out data—is attractive for online and neurophysiological applications, but the current evidence and the stability-filter justification are insufficient to establish it. If the stability heuristic can be justified or replaced, the package would be a useful methodological contribution.

major comments (3)
  1. [II.A] The stability filter in Section II.A (steps 1-2) tests only u(t)=0 and u(t)=1 with Var(ŷ(t)) ≤ ε, ε=10^-2. Boundedness under two constant inputs is not a stability certificate for polynomial NARX models: for example, y(t)=0.5y(t-1)+4u(t-1)(1-u(t-1))y(t-1)+y(t-1)^2 satisfies the filter for u=0 and u=1 from a zero initial condition, yet for u(t)=0.5 the linear coefficient becomes 1.5 and nonzero initial conditions diverge. Because step 3 (BIC on simulation error) is the only remaining selection mechanism, the headline claim that iOFRS can 'produce models with long-horizon prediction capabilities and simulation stability' is not supported by the stated test. The paper provides no proof, counterexample analysis, or empirical stress test of this heuristic, and the synthetic examples in Section III do not exercise it (they use zero noise and inputs unlike constant signals). Please either remove the strong claim, add a rigorous sufficient condition, or conduct experiments with a diverse set of inputs and initial conditions.
  2. [III.A-B] The experimental evidence does not substantiate 'robust model generalisation without the need for a separate validation dataset.' The synthetic case (Section III.A) is noise-free (residual variances 1.60e-25 and 8.22e-18 versus training variances 0.069 and 0.0581), so it provides no evidence about generalisation under realistic disturbances, and no comparison with the original OFR/ERR algorithm or existing packages is reported. Moreover, the paper's own validation tests fail: the text states that residuals in both cases (a) and (b) 'are not entirely white noise' and that in Fig. 5 the cross-correlation functions 'are not completely within the tolerance bounds', which is explicitly acknowledged as indicating some model bias. Since these tests are the paper's only evidence of model validity on training data, the generalization claim requires either a properly designed noise experiment, an independent held-out record (even if the method does not require one), or a benchmark against an existing validation-based approach.
  3. [II.A step 3 / Algorithm 1] The BIC-based simulation selection in Section II.A (step 3) and Algorithm 1 (line 35) is not specified precisely enough to be reproducible or to assess the parsimony claim. The manuscript states only that BIC is 'calculated based on the simulated error variance (MSSE)' but does not give the formula, the effective number of parameters (including the bias term and any initial-condition parameters), or the initialization rule for the free-run simulation used to compute MSSE. Since the free-run trajectory of a nonlinear ARX model can depend sensitively on initial conditions, and since the central claim is that iOFRS selects models that simulate stably over long horizons, the missing initialization and BIC details make the selection rule ambiguous. Please provide the exact BIC definition and the simulation protocol (initial conditions, burn-in, and the exact training/validation split within the record).
minor comments (5)
  1. [Title] The title header contains a typo: 'im proved' should be 'improved'.
  2. [II.A] Section II.A uses both 'iFRO' and 'iOFR' for the same algorithm; please standardize the notation.
  3. [Algorithm 1] Algorithm 1 (line 8) composes D'' = D' ∪ f(D') and line 10 D''_R = D'_m ∪ f(D'_m), but the function f is never defined; state that f generates all monomials up to degree Np.
  4. [Table I] The ordering of Table I does not match the description in Section III.A that sorting by descending mean squared PRESS error reveals the term-addition sequence; for example, u(t-1) has the largest PRESS value (0.47871) but is listed second, and several entries contain malformed scientific notation such as '1.9004e × 10−5'.
  5. [Figure 6] The Figure 6 caption refers to 'the system given in Eq. (8)', but the real-data example is from [46] and Eq. (8) is the synthetic DC motor; correct the cross-reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the iOFRS/PRESS pipeline is evaluated against external benchmarks and no prediction reduces to a fitted input.

full rationale

The paper's central contribution is an algorithmic identification pipeline (iOFR with PRESS-statistic term selection and simulation-based BIC model choice) implemented in open-source MATLAB and tested on a synthetic NARX system with known true coefficients (Eq. 8) and on real electromechanical data from an external source [46]. No model coefficient or selected term is defined in terms of the headline success metric: PRESS errors, ERR, BIC/MSSE, and correlation-based residual tests are computed from data or from the candidate-term orthogonalization, and the reported held-out simulation segments in Figures 2-3 lie outside the training interval. The iOFRS stability filter in Section II.A steps 1-2, which admits models whose free-run outputs remain bounded with small variance for u=0 and u=1, is an unproven heuristic: for nonlinear ARX models, two constant-input tests do not certify stability under other bounded inputs. That is a correctness and robustness risk, not a circularity, because the filter is not fitted to—or logically identical to—the long-horizon simulation performance it is used to select. Self-citations (e.g., [42] in the NOFRF background) are not load-bearing for the identification claims. No equation in the paper reduces to its own output by construction, and no fitted parameter is renamed as a prediction. Therefore the paper is not circular.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central method relies on four structural assumptions: polynomial NARX representability, informative excitation, a constant-input stability heuristic, and LOO PRESS as a time-series generalization proxy. The main hand-chosen parameter is the stability threshold epsilon=1e-2; training lengths in examples are also chosen to make identification succeed. No new physical entities are introduced.

free parameters (2)
  • Stability variance threshold epsilon = 0.01
    Selected by hand in Section II.A; candidate models are rejected if their free-run variance under u=0 or u=1 exceeds 1e-2. The choice affects which models remain in M, hence the final model; no sensitivity analysis is provided.
  • Training data length for synthetic examples = 60 samples (white noise), 200 samples (multi-tone)
    The paper reports that fewer than 200 samples results in a suboptimal model for the multi-tone input (Section III.A). The training length is chosen to make the example succeed and is not a method parameter.
assumptions (5)
  • domain assumption The true system is representable as a polynomial NARX/NARMAX model (Eq. 4) with bounded degree and lags.
    The whole search space D'' is restricted to polynomial monomials; if the underlying system has non-polynomial dynamics, the identified model is biased by construction (Section I).
  • domain assumption The input must sufficiently excite the system so that correct terms are recoverable.
    The paper relies on informativeness of input; for the less informative multi-tone input it needed 200 training samples and notes fewer samples yields a suboptimal model (Section III.A).
  • ad hoc to paper Boundedness of free-run output under two constant inputs (u=0 and u=1) is a sufficient stability check.
    Section II.A accepts candidate models only if Var(hat_y^[j](t)) <= 1e-2 for j=0,1. No proof is given that this transfers to actual input signals; this is central to the simulation-stability claim.
  • domain assumption Leave-one-out PRESS residuals are a valid proxy for generalization in dependent time-series data.
    Section II.B replaces a validation dataset with PRESS LOO errors. Time-series samples are dependent, so standard LOO can be optimistic; no blocked cross-validation or correction is included.
  • domain assumption The overfitted OFR model used to initialize pre-select terms contains enough correct terms.
    RCT methods 2-4 rely on m''_0 including correct terms (Section II.C); this is inherited from [32] and not re-verified in the PRESS/simulation variant.

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Pith. "Pith review of NonSysId: A nonlinear system identification package with improved model term selection for NARMAX models." pith.science (2026). https://pith.science/paper/OL3IGDQ7

@misc{pith2026241116475,
  author       = {Pith},
  title        = {Pith review of: NonSysId: A nonlinear system identification package with improved model term selection for NARMAX models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OL3IGDQ7}},
  note         = {Machine review of arXiv:2411.16475}
}
read the original abstract

System identification involves constructing mathematical models of dynamic systems using input-output data, enabling analysis and prediction of system behaviour in both time and frequency domains. This approach can model the entire system or capture specific dynamics within it. For meaningful analysis, it is essential for the model to accurately reflect the underlying system's behaviour. This paper introduces NonSysId, an open-sourced MATLAB software package designed for nonlinear system identification, specifically focusing on NARMAX models. The software incorporates an advanced term selection methodology that prioritises on simulation (free-run) accuracy while preserving model parsimony. A key feature is the integration of iterative Orthogonal Forward Regression (iOFR) with Predicted Residual Sum of Squares (PRESS) statistic-based term selection, facilitating robust model generalisation without the need for a separate validation dataset. Furthermore, techniques for reducing computational overheads are implemented. These features make NonSysId particularly suitable for real-time applications such as structural health monitoring, fault diagnosis, and biomedical signal processing, where it is a challenge to capture the signals under consistent conditions, resulting in limited or no validation data.

Figures

Figures reproduced from arXiv: 2411.16475 by the authors.

Figure 1
Figure 1. FIG. 1. This flowchart summarises the procedures for identify [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. and 3 depict the training and testing data alongside the model simulated output for the inputs (a) and (b), respectively. The term ‘testing data’ is used to refer to data not explicitly included during training, as the model is already validated through leave-one-out cross-validation during the identification/training process (see sub-section II B). FIG. 2. Model identification results under input (a). The model sim… view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Model validation results for input (a). The red bound [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Model validation results for input (b). The red bound [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Model validation results for the system in [46]. The r [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

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Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    Centre for Computational Science and Mathematical Model ling, Coventry University, Coventry CV1 5FB, UK

    work page

  2. [2]

    This approach can model the entire system or captur e specific dynamics within it

    School of Electrical and Electronic Engineering, The Uni versity of Sheffield, Western Bank, Sheffield S10 2TN, UK SUMMAR Y System identification involves constructing mathematical models of dynamic systems using input- output data, enabling analysis and prediction of system beh aviour in both time and frequency domains. This approach can model the entire syst...

    work page

  3. [3]

    OFR-ERR may select redundant or incorrect model terms, espe cially in the presence of complex noise structures or certain input signals

    work page

  4. [4]

    The model structures produced from OFR can be sensitive to th e first term selected in the forward selection of model terms

    work page

  5. [5]

    it lacks the informativeness needed to effectively stimulate the system), the resulting model ca n be inappropriate

    If the input does not persistently excite the system under cons ideration (i.e. it lacks the informativeness needed to effectively stimulate the system), the resulting model ca n be inappropriate. This can result in inaccuracies in long-horizon prediction and, in some cases, even uns table models during simulation (free-run or model-predicted output)

    work page

  6. [6]

    Beyond obtaining parsimonious models, the model should generalise w ell to unseen data (validation) that is not used during the learning/training process (i.e

    The ERR focuses solely on explained variance when selecting terms , which can lead to overfitting. Beyond obtaining parsimonious models, the model should generalise w ell to unseen data (validation) that is not used during the learning/training process (i.e. model identificat ion). This is referred to as obtaining a bias- variance trade-off, which can be ach...

    work page

  7. [7]

    Pre-select each term given in P as the first model term and search through p orthogonalization paths using OFR to produce a set of p candidate models ˜M = { ˜m1, . . . , ˜mp}

    work page

  8. [8]

    , m p}, p ≤ p

    From ˜M determine the set of stable candidate models M = {m1, . . . , m p}, p ≤ p

    work page

Show all 59 references
  1. [9]

    From M, based on the simulation error choose the best model m

  2. [10]

    Use the terms in m to form the new set of pre-select terms P

  3. [11]

    In step 2, each model ˜ mi ∈ ˜M, i = 1 ,

    Repeat steps 1-4 and iteratively search through different orth ogonalization paths. In step 2, each model ˜ mi ∈ ˜M, i = 1 , . . . , p , is tested using two inputs: (i) a sequence of 0’s, u[0](t) = 0 ∀t, and (ii) a sequence of 1’s, u[1](t) = 1 ∀t. The corresponding simulated ou...

  4. [12]

    The region shaded in brown represents the ARX model identi fication process, while the blue-shaded region highlights the NARX procedures. NARX models can be analysed in the frequency domain using Nonlinear O utput Frequency Response Func- tions (NOFRFs) [40], which extend class...

  5. [13]

    Ljung, System identification

    L. Ljung, System identification . Springer, 1998

    1998

  6. [14]

    Billings, Nonlinear System Identification: NARMAX Methods In The Time , Frequency, And Spatio-Temporal Do- mains, vol

    S. Billings, Nonlinear System Identification: NARMAX Methods In The Time , Frequency, And Spatio-Temporal Do- mains, vol. 13. Chichester, UK: John Wiley & Sons, Ltd, 2013

    2013

  7. [15]

    Identification of continuous-t ime systems: A tutorial,

    H. Unbehauen and P. Rao, “Identification of continuous-t ime systems: A tutorial,” IF AC Proceedings Volumes, vol. 30, no. 11, pp. 973–999, 1997. IF AC Symposium on System Identific ation (SYSID’97), Kitakyushu, Fukuoka, Japan, 8-11 July 1997

    1997

  8. [16]

    Global Nonlinear Model ing of Gas Turbine Dynamics Using NARMAX Structures ,

    N. Chiras, C. Evans, and D. Rees, “Global Nonlinear Model ing of Gas Turbine Dynamics Using NARMAX Structures ,” Journal of Engineering for Gas Turbines and Power , vol. 124, pp. 817–826, 09 2002

    2002

  9. [17]

    Nonlinear autoregressive exogenous method for structural health monitoring using ultrasonic guided waves,

    K. Wang, J. Zhang, A. J. Croxford, and Y. Yang, “Nonlinear autoregressive exogenous method for structural health monitoring using ultrasonic guided waves,” in Structural Health Monitoring/ Management (SHM) in Aerospa ce Struc- tures (F.-G. Yuan, ed.), Woodhead Publishing Serie...

    2024

  10. [18]

    Estimating the incidence of spontaneous breathing effort o f mechanically ventilated patients using a non-linear auto regressive (narx) model,

    N. M. Zainol, N. S. Damanhuri, N. A. Othman, Y. S. Chiew, M. B. M. Nor, Z. Muhammad, and J. G. Chase, “Estimating the incidence of spontaneous breathing effort o f mechanically ventilated patients using a non-linear auto regressive (narx) model,” Computer Methods and Programs i...

    2022

  11. [19]

    Nonlinear data-driven id entification of polymer electrolyte membrane fuel cells for diagnostic purposes: A volterra series approach,

    D. Ritzberger and S. Jakubek, “Nonlinear data-driven id entification of polymer electrolyte membrane fuel cells for diagnostic purposes: A volterra series approach,” Journal of Power Sources , vol. 361, pp. 144–152, 2017

    2017

  12. [20]

    A narx model-based co ndition monitoring method for rotor systems,

    Y. Gao, C. Yu, Y.-P. Zhu, and Z. Luo, “A narx model-based co ndition monitoring method for rotor systems,” Sensors, vol. 23, no. 15, 2023. 13

    2023

  13. [21]

    Nonlinear interacti ons in the thalamocortical loop in essential tremor: A model-based frequency domain analysis,

    F. He, P. Sarrigiannis, S. Billings, H. Wei, J. Rowe, C. Ro manowski, N. Hoggard, M. Hadjivassilliou, D. Rao, R. Gr¨ unewald, A. Khan, and J. Yianni, “Nonlinear interacti ons in the thalamocortical loop in essential tremor: A model-based frequency domain analysis,” Neuroscienc...

    2016

  14. [22]

    Nonlinear system identification of ne ural systems from neurophysiological signals,

    F. He and Y. Yang, “Nonlinear system identification of ne ural systems from neurophysiological signals,” Neuroscience, vol. 458, pp. 213–228, 2021

    2021

  15. [23]

    A note on the equi valence of narx and rnn,

    J. P. F. Sum, W.-K. Kan, and G. H. Young, “A note on the equi valence of narx and rnn,” Neural Computing & Applications, vol. 8, pp. 33–39, Mar 1999

    1999

  16. [24]

    Sysidentpy: A python package for system identification using narmax models,

    W. R. Lacerda, L. P. C. da Andrade, S. C. P. Oliveira, and S . A. M. Martins, “Sysidentpy: A python package for system identification using narmax models,” Journal of Open Source Software , vol. 5, no. 54, p. 2384, 2020

    2020

  17. [25]

    An r library for nonlinear black-box system identification,

    H. V. H. Ayala, M. C. Gritti, and L. dos Santos Coelho, “An r library for nonlinear black-box system identification,” SoftwareX, vol. 11, p. 100495, 2020

    2020

  18. [26]

    An identification algorith m for polynomial narx models based on simulation error mini- mization,

    L. Piroddi and W. Spinelli, “An identification algorith m for polynomial narx models based on simulation error mini- mization,” International Journal of Control , vol. 76, no. 17, pp. 1767–1781, 2003

    2003

  19. [27]

    Algorithms for minimal mod el structure detection in nonlinear dynamic system identi- fication,

    K. Z. Mao and S. A. Billings, “Algorithms for minimal mod el structure detection in nonlinear dynamic system identi- fication,” International Journal of Control , vol. 68, no. 2, pp. 311–330, 1997

    1997

  20. [28]

    On overparametriz ation of nonlinear discrete systems,

    E. M. A. M. Mendes and S. A. Billings, “On overparametriz ation of nonlinear discrete systems,” International Journal of Bifurcation and Chaos , vol. 08, no. 03, pp. 535–556, 1998

    1998

  21. [29]

    Dynamical effects of over parametrization in nonlinear models,

    L. A. Aguirre and S. Billings, “Dynamical effects of over parametrization in nonlinear models,” Physica D: Nonlinear Phenomena, vol. 80, no. 1, pp. 26–40, 1995

    1995

  22. [30]

    Orthogonal least squa res methods and their application to non-linear system identification,

    S. Chen, S. Billings, and W. Luo, “Orthogonal least squa res methods and their application to non-linear system identification,” Int. J. Control , vol. 50, no. 5, pp. 1873–1896,, 1989

    1989

  23. [31]

    Identification of non-linear output-affine systems using an orthogonal leas t- squares algorithm,

    S. Billings, M. Korenberg, and S. Chen, “Identification of non-linear output-affine systems using an orthogonal leas t- squares algorithm,” 1987

    1987

  24. [32]

    Orth ogonal parameter estimation algorithm for non-linear stoc hastic systems,

    M. Korenberg, S. Billings, Y. Liu, and P. McIlroy, “Orth ogonal parameter estimation algorithm for non-linear stoc hastic systems,” Int. J. Control , vol. 48, no. 1, pp. 193–210,, 1988

    1988

  25. [33]

    Use of press residuals in dynam ic system identification,

    L. Wang and W. R. Cluett, “Use of press residuals in dynam ic system identification,” Automatica, vol. 32, no. 5, pp. 781–784, 1996

    1996

  26. [34]

    Automatic nonline ar predictive model-construction algorithm using forward regression and the press statistic,

    X. Hong, P. Sharkey, and K. Warwick, “Automatic nonline ar predictive model-construction algorithm using forward regression and the press statistic,” IEE Proceedings-Control Theory and Applications , vol. 150, no. 3, pp. 245–254, 2003

    2003

  27. [35]

    Using and under- standing cross-validation strategies. Perspectives on Sa eb et. al.,

    M. A. Little, G. Varoquaux, S. Saeb, L. Lonini, A. Jayara man, D. C. Mohr, and K. P. Kording, “Using and under- standing cross-validation strategies. Perspectives on Sa eb et. al.,” GigaScience, vol. 6, p. gix020, 03 2017

    2017

  28. [36]

    Cross-validatory choice and assessment of s tatistical predictions,

    M. Stone, “Cross-validatory choice and assessment of s tatistical predictions,” Journal of the Royal Statistical Society. Series B (Methodological) , vol. 36, no. 2, pp. 111–147, 1974

    1974

  29. [37]

    A critical review on structur al health monitoring: Definitions, methods, and perspectiv es,

    V. R. Gharehbaghi, E. Noroozinejad Farsangi, M. Noori, T. Y. Yang, S. Li, A. Nguyen, C. M´ alaga-Chuquitaype, P. Gardoni, and S. Mirjalili, “A critical review on structur al health monitoring: Definitions, methods, and perspectiv es,” Archives of Computational Methods in Engine...

    2022

  30. [38]

    New techniques to perfo rm cross-validation for time series models,

    A. Vamsikrishna and E. V. Gijo, “New techniques to perfo rm cross-validation for time series models,” Operations Research Forum, vol. 5, p. 51, Jun 2024

    2024

  31. [39]

    The necessity of leave one subject out (loso) cross v alidation for eeg disease diagnosis,

    S. Kunjan, T. S. Grummett, K. J. Pope, D. M. W. Powers, S. P . Fitzgibbon, T. Bastiampillai, M. Battersby, and T. W. Lewis, “The necessity of leave one subject out (loso) cross v alidation for eeg disease diagnosis,” in Brain Informatics (M. Mahmud, M. S. Kaiser, S. Vassanelli...

    2021

  32. [40]

    Simultaneous whole- head electrophysiological recordings using EEG and OPM-MEG,

    Z. A. Seedat, K. S. Pier, N. Holmes, M. Rea, L. Al-Hilaly, T. M. Tierney, C. M. Embury, R. Pardington, K. J. Mullinger, J. H. Cross, E. Boto, and M. J. Brookes, “Simultaneous whole- head electrophysiological recordings using EEG and OPM-MEG,” Imaging Neuroscience, vol. 2, pp. ...

    2024

  33. [41]

    Dist inct global brain dynamics and spatiotemporal organizatio n of the salience network,

    T. Chen, W. Cai, S. Ryali, K. Supekar, and V. Menon, “Dist inct global brain dynamics and spatiotemporal organizatio n of the salience network,” PLOS Biology , vol. 14, pp. 1–21, 06 2016

    2016

  34. [42]

    Differential contributions of static and time- varying functional connectivity to human behavior,

    A. Eichenbaum, I. Pappas, D. Lurie, J. R. Cohen, and M. D’ Esposito, “Differential contributions of static and time- varying functional connectivity to human behavior,” Network Neuroscience, vol. 5, pp. 145–165, 02 2021

    2021

  35. [43]

    Time in brain: How biological rhythms impact on eeg signals and on eeg-derived brain networks,

    K. Lehnertz, T. Rings, and T. Br¨ ohl, “Time in brain: How biological rhythms impact on eeg signals and on eeg-derived brain networks,” Frontiers in Network Physiology , vol. 1, 2021

    2021

  36. [44]

    An iterative ort hogonal forward regression algorithm,

    S. B. Yuzhu Guo, L.Z. Guo and H.-L. Wei, “An iterative ort hogonal forward regression algorithm,” International Journal of Systems Science , vol. 46, no. 5, pp. 776–789, 2015

    2015

  37. [45]

    Predictio n and simulation errors in parameter estimation for nonline ar systems,

    L. A. Aguirre, B. H. Barbosa, and A. P. Braga, “Predictio n and simulation errors in parameter estimation for nonline ar systems,” Mechanical Systems and Signal Processing , vol. 24, no. 8, pp. 2855–2867, 2010

    2010

  38. [46]

    Structure detection and model validity tests in the identification of nonlinear systems,

    S. Billings and W. Voon, “Structure detection and model validity tests in the identification of nonlinear systems,” IEE Proceedings D (Control Theory and Applications) , vol. 130, pp. 193–199, 1983

    1983

  39. [47]

    Discrete reconstructio n of strange attractors of chua’s circuit,

    L. A. AGUIRRE and S. BILLINGS, “Discrete reconstructio n of strange attractors of chua’s circuit,” International Journal of Bifurcation and Chaos , vol. 04, no. 04, pp. 853–864, 1994

    1994

  40. [48]

    Estimating the Dimension of a Model,

    G. Schwarz, “Estimating the Dimension of a Model,” The Annals of Statistics , vol. 6, no. 2, pp. 461 – 464, 1978. 14

    1978

  41. [49]

    Model-order selection: a revie w of information criterion rules,

    P. Stoica and Y. Selen, “Model-order selection: a revie w of information criterion rules,” IEEE Signal Processing Magazine, vol. 21, no. 4, pp. 36–47, 2004

    2004

  42. [50]

    The relationship between variable select ion and data agumentation and a method for prediction,

    D. M. Allen, “The relationship between variable select ion and data agumentation and a method for prediction,” Technometrics, vol. 16, no. 1, pp. 125–127, 1974

    1974

  43. [51]

    Term and variable selection for non-linear system identification,

    H.-L. Wei, S. A. Billings, and J. Liu, “Term and variable selection for non-linear system identification,” International Journal of Control , vol. 77, no. 1, pp. 86–110, 2004

    2004

  44. [52]

    Energy transfer propertie s of non-linear systems in the frequency domain,

    Z. Q. L. * and S. A. Billings, “Energy transfer propertie s of non-linear systems in the frequency domain,” International Journal of Control , vol. 78, no. 5, pp. 345–362, 2005

    2005

  45. [53]

    The analysis of nonl inear systems in the frequency domain using nonlinear output frequency response functions,

    R. S. Bayma, Y. Zhu, and Z.-Q. Lang, “The analysis of nonl inear systems in the frequency domain using nonlinear output frequency response functions,” Automatica, vol. 94, pp. 452–457, 2018

    2018

  46. [54]

    An effective metho d for evaluation of the nonlinear output frequency response functions from system input-output data,

    S. R. A. S. Gunawardena and Z. Q. Lang, “An effective metho d for evaluation of the nonlinear output frequency response functions from system input-output data,” in 2018 UKACC 12th International Conference on Control (CONTR OL), pp. 134–139, 2018

    2018

  47. [55]

    Nonlinear output frequency response functions: A new evaluation approach and applications to railway and manufacturing sys tems’ condition monitoring,

    Y.-P. Zhu, Z. Lang, H.-L. Mao, and H. Laalej, “Nonlinear output frequency response functions: A new evaluation approach and applications to railway and manufacturing sys tems’ condition monitoring,” Mechanical Systems and Signal Processing, vol. 163, p. 108179, 2022

    2022

  48. [56]

    Nonlinear model st andardization for the analysis and design of nonlinear systems with multiple equilibria,

    Y.-P. Zhu, Z. Q. Lang, and Y.-Z. Guo, “Nonlinear model st andardization for the analysis and design of nonlinear systems with multiple equilibria,” Nonlinear Dynamics , vol. 104, pp. 2553–2571, May 2021

    2021

  49. [57]

    Control of hysteretic systems through an analytical inverse compensation based on a narx model,

    W. R. Lacerda Junior, S. A. M. Martins, E. G. Nepomuceno, and M. J. Lacerda, “Control of hysteretic systems through an analytical inverse compensation based on a narx model,” IEEE Access, vol. 7, pp. 98228–98237, 2019

    2019

  50. [58]

    Identifi ca¸ cao de um motor/gerador cc por meio de modelos polinomiais autorregressivos e redes neurais artificiais,

    W. R. L. Junior, V. M. Almeida, and S. A. Milani, “Identifi ca¸ cao de um motor/gerador cc por meio de modelos polinomiais autorregressivos e redes neurais artificiais, ” in Proc. XIII Simp´ osio Brasileiro De Automa¸ c˜ ao Inteligente, pp. 1–6, 2017

    2017

  51. [59]

    Local regularization assisted orthogonal le ast squares regression,

    S. Chen, “Local regularization assisted orthogonal le ast squares regression,” Neurocomputing, vol. 69, no. 4, pp. 559– 585, 2006

    2006

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.