Stable spline identification of linear systems under missing data

Alessandro Chiuso; Gianluigi Pillonetto; Giuseppe De Nicolao

Stable spline identification of linear systems under missing data

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper establishes that stable spline kernels induce a new family of radial basis function kernels describing filtered white noise, so missing output data can be imputed in closed form and the resulting models can outperform full-data…

desk verdict Solid extension of stable spline identification to missing data; the kernel derivation is clean and the central claims hold, with a real but non-fatal caveat about the Gaussian approximation in the marginal likelihood. read the letter →

arxiv 1908.03913 v2 pith:3N7X5TJS submitted 2019-08-11 eess.SY cs.SY

Gianluigi Pillonetto , Alessandro Chiuso , Giuseppe De Nicolao This is my paper · ORCID

classification eess.SYcs.SY MSC 93E12

keywords linearsystemidentificationmissingdataGaussianprocesseskernel-basedregularizationstablesplinekernelsradialbasisfunctionsimputation

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 tries to establish a fully nonparametric cure for missing output data in linear system identification. Its key move is to show that the stable spline kernel, previously used to model impulse responses as Gaussian processes, also yields a new family of radial basis function kernels that describe the disturbance term of a linear model, the convolution of white noise with a stable impulse response. With both kernels in hand, the paper derives the minimum variance linear estimator of the missing outputs in closed form, parameterized only by a few hyperparameters estimated by marginal likelihood. Numerical experiments on ARMAX systems indicate that this stable spline imputation procedure is more predictive than parametric Prediction Error Methods, even when the latter see the full data set and use an oracle for model order selection.

What carries the argument

The load-bearing object is the RBF kernel family defined by Eq. (13), $h(x) = \int_0^\infty K(y, y+|x|)\,dy$, obtained by integrating the stable spline kernel $K$ along the diagonal; it expresses the second-order statistics of the noise component $\xi$ as a function of time lag only. For $q=1$ it reduces to the Laplace kernel $h(x) = e^{-\beta|x|}/\beta$, and for $q=2$ to $h(x) = (3e^{-2\beta|x|} - e^{-3\beta|x|})/(18\beta)$. This kernel, together with the stable spline output kernel $P$, enters the minimum variance linear estimator (24) and the marginal-likelihood criterion (26), which is optimized over only two or four hyperparameters before all missing outputs are returned in closed form.

What would settle it

Run a Monte Carlo with an ARMAX system whose noise-path pole is much faster than the system's least-damped pole, so the shared-$\beta$ RBF kernel is misspecified; if the closed-form imputer's mean-squared error on missing outputs is not close to that of the best linear estimator computed with the true noise covariance, the shared-$\beta$ assumption is load-bearing. A simpler check is to estimate the residual autocovariance from a long simulation and compare its shape with Eq. (15).

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Extended reading notes

Core claim

The central discovery is the covariance identity in Eq. (13): if the noise-path impulse response $f_p$ is a zero-mean Gaussian process with stable spline covariance $\lambda K$, then the disturbance $\xi(t) = (e \otimes f_p)(t)$ is a zero-mean stationary process whose covariance is the translation-invariant radial basis function $R(s,t) = h(s-t)$, where $h(x) = \int_0^\infty K(y, y+|x|)\,dy$. For the classic first- and second-order stable spline kernels this yields the explicit Laplace-like forms (14) and (15). Combined with the stable spline kernel for the system impulse responses, this makes the best linear predictor of the missing outputs in Eq. (24) computable in closed form; the paper calls the resulting procedure stable spline imputation. In Monte Carlo tests on ARMAX models with 25% output data missing, the imputation-then-identification pipeline produces models whose $k$-step-ahead prediction accuracy is better than that of PEM with oracle order selection in almost 87% of runs, and comparable to or better than full-data estimators.

Load-bearing premise

The estimator is the minimum variance linear one only when the disturbance process $\xi(t)$ is zero-mean stationary with covariance exactly $\lambda R$, which holds if the noise-path impulse response $f_p$ is drawn from a Gaussian process with stable spline covariance using the same stability decay $\beta$ as the system's impulse responses.

Editorial extensions

If this is right

If the covariance identity holds, missing outputs can be imputed in closed form with no model-order search, just a two- or four-dimensional optimization for the hyperparameters.
The estimator extends to continuous-time systems without re-derivation, since the kernel identities are stated for both discrete and continuous time.
The approach can enrich the stable spline kernel with a low-order parametric component, so it can represent ARMAX systems with a small parametric part while retaining the closed-form imputer.
In the experiments, imputation plus stable spline estimation matches or beats the full-data stable spline estimator and full-data PEM, suggesting that the imputed values do not degrade subsequent model quality.
Because the estimate does not depend on the scale factor $\lambda$, imputation only requires relative hyperparameters, reducing the tuning burden.

Reading between the lines

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

The covariance identity suggests a direct diagnostic: given a long record from a known system, the empirical autocorrelation of the residuals after removing the deterministic input could be compared against the predicted $h(x)$; a mismatch would indicate either a wrong $\beta$ or a noise path not well modeled by the stable spline prior.
The same closed-form imputer could be adapted to missing inputs rather than outputs by symmetry of the linear model, though the paper only treats output gaps.
Because the RBF kernel arises from a convolution, a similar construction may yield kernels for other Gaussian-process priors, such as Mat\'ern or periodic kernels, extending the family of tractable imputation problems.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear [Name],

Read the Pillonetto–Chiuso–De Nicolao paper on stable spline imputation with missing data. The headline: this is a genuine extension of the stable spline framework to missing outputs, and the central mechanism—deriving a new RBF kernel family from stable spline priors—checks out. The math is clean; I verified the q=1 and q=2 discrete-time formulas in (18)–(19), and the cross-covariance argument behind the closed-form estimator (24) is standard and correct.

What is new: the family of RBF kernels in (13)/(17), the discrete-time specializations, and the application to imputation via a minimum-variance linear estimator with hyperparameters estimated by marginal likelihood. The q=1 case reduces to the Laplace kernel, a useful sanity check, but the general family is not in prior work. The experiments are thorough: 500 Monte Carlo runs, missing-data PEM with oracle order selection, and full-data baselines. The claim that imputation plus a stable spline estimator is competitive with, and often better than, oracle PEM on missing data is credible—the oracle has genuine extra information, so the reported 87% figure is not a straw man.

Soft spots, in proportion. The Gaussian approximation in the marginal likelihood (Eq. 26) is real and untested. It is a Bayesian modeling choice, not an error, but it means the hyperparameter estimate is not the exact marginal likelihood, and the minimum-variance guarantee is conditional on the prior being right. The MCAR assumption is implicit throughout; if missingness depends on the output, the method would need a different justification. No code or data are shipped, so the numerical results are not independently reproducible yet. The proof of Proposition 1 is omitted with a note that it uses simple computations—I re-derived it and it is fine, but a referee should ask for the details.

None of these are load-bearing. The central argument holds. This is a solid within-subfield contribution that people working on nonparametric system identification will want to read and cite. The self-citations are appropriate given that the stable spline framework is the authors' own. I would send this to peer review without hesitation, and would accept after minor revisions if the authors supply the missing proof details and comment on the Gaussian approximation.

Referee Report

0 major / 6 minor

Summary. The paper proposes a nonparametric Bayesian approach to identifying linear systems from output measurements with missing samples. The authors model the unknown impulse responses as zero-mean Gaussian processes with stable spline covariances and show that the disturbance component xi(t) = (e \otimes f_p)(t) is stationary with a covariance that is a new RBF kernel obtained by integrating the stable spline kernel along its diagonal (Proposition 1, Eq. (13), with discrete-time versions in Eqs. (17)-(19)). Using this kernel together with the stable spline kernel, they derive a closed-form linear minimum variance estimator for missing outputs (Proposition 3, Eq. (24)) and an algorithm, stable spline imputation (Algorithm 4), in which hyperparameters are estimated by marginal likelihood under a Gaussian approximation (Eq. (26)). Numerical experiments on ARMAX models compare the method against PEM baselines, including an oracle that selects model order on the test target, and report competitive or better predictive performance even when the benchmarks use the full data set.

Significance. If the results are correct, the paper makes a useful contribution to kernel-based system identification. The main theoretical content, the derivation of RBF kernels from stable spline priors, is transparent and checkable: the q=1 case reduces to the Laplace kernel, and the q=2 closed form is consistent with direct summation. The estimator in Proposition 3 is the standard linear minimum variance expression, and the independence structure of the Bayesian network is clearly specified. The experimental design is strong: the PEM baselines include oracle order selection on the test target, which gives the parametric competitors an advantage, and the conclusion that the nonparametric imputation method is competitive is carefully worded. The paper is also clearly organized, and its scope covers both discrete- and continuous-time settings, which is broader than most existing missing-data solutions. The acknowledged Gaussian approximation in the marginal likelihood is a limitation but not an internal inconsistency. Overall this is a solid, publishable contribution.

minor comments (6)

[Section 2.3, Eqs. (6)-(7)] The notation for R_m is inconsistent: the text calls it a covariance matrix, but the displayed definition in Eq. (6) is E[\xi_m \xi^T], which is a cross-covariance, and Eq. (7) uses it in exactly that cross-covariance role. Please define R_{mo} = E[\xi_m \xi^T] explicitly and use a separate symbol for the auto-covariance of \xi_m, or state clearly that R_m denotes the cross-covariance.
[Section 4.2, Proposition 1] The proof of Proposition 1 is omitted with the note that it relies on simple computations. Since this proposition is the main theoretical novelty of the paper, please include a derivation in the appendix; this would also clarify the integration over R_+ and the role of |x|.
[Section 5.3, Eq. (26)] The Gaussian approximation for the disturbance is introduced but not discussed. Please add a sentence or two explaining the nature of the approximation and its expected effect on hyperparameter estimates; a brief empirical check would strengthen the paper, although the current experiments do not indicate a problem.
[Section 6.5, bottom panel] The text says 'boxplots of the 1000 values of COD5' but the Monte Carlo study has 500 runs; this appears to be a typo and should be corrected to 500.
[Section 6.4, top panel] The sentence reporting 'almost 87% of the cases' should clarify whether 'cases' refers to the 2000 individual COD_k values from the first 100 runs or to the number of Monte Carlo runs; the current wording is ambiguous.
[Section 5.1, Eq. (21)] The definition of h_x(t) in the output kernel P(x,y) is hard to parse. Please rewrite it with a more explicit convolution formula, such as P(x,y) = \sum_l \int\int u_l(y-s) u_l(x-r) K(s,r) ds dr, or state clearly that h_x(t) is the function t \mapsto (u_l(\cdot) \otimes K(t,\cdot))(x).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new RBF kernel class is derived from the stable spline prior, and the imputation estimator is a standard Bayesian/linear-minimum-variance construction with an acknowledged Gaussian approximation.

full rationale

I walked the paper's derivation chain. Proposition 1 (Eq. 13) computes the covariance of the disturbance xi(t) = (e ⊗ f_p)(t) by integrating the assumed stable spline covariance K(y, y+|x|) over y; the closed forms (14)-(15) in continuous time and (18)-(19) in discrete time follow by direct integration or summation from the stated q=1 and q=2 stable spline kernels. These are genuine derived quantities, not renamed inputs, and they match direct computation, providing an external check. The estimator in Proposition 3 (Eqs. 24-25) is the standard linear minimum variance estimator: V(y_o) = lambda(P+R) and the cross-covariance V(y(t), y_o) = lambda(P(t,·)+R(t,·)) follow from the stated mutual independence of e and the f_l's, with no fitted parameter secretly renamed as a prediction. Hyperparameter estimation via marginal likelihood (Eq. 26) uses only the observed vector y_o, while the imputation target y_m is not used to fit the hyperparameters; the Gaussian approximation for the disturbance is explicitly acknowledged in the text as an approximation, which is a modeling limitation rather than a circular step. The paper does cite the authors' prior stable spline framework, but that framework is used as the prior assumption from which the new RBF class is derived; the cited results do not contain the target RBF/imputation result, and the numerical comparisons against PEM baselines, including an oracle-order PEM that sees the full data, provide independent evidence for the performance claim. No equation in the paper reduces to its own inputs by construction, and no fitted quantity is presented as an independent prediction in a way that would constitute circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The method takes the stable spline prior from prior work (self-cited) and derives the RBF kernel for disturbances without additional free parameters. The hyperparameters β and λ are estimated from data. The paper postulates no new physical entities.

free parameters (3)

β (stable spline decay) = estimated from data via marginal likelihood (Eq. 26)
Controls the exponential decay of the stable spline and RBF kernels; shared between system and disturbance in the experiments.
λ (scale factor) = estimated from data via marginal likelihood (Eq. 26)
Overall variance scale of the impulse response and disturbance; cancels in the estimator (24) but appears in the marginal likelihood.
φ, ϕ (ARMAX pole parameters) = estimated from data when the enriched kernel is used
Describe two high-frequency poles added to the stable spline kernel in the ARMAX experiments (Section 4.4).

assumptions (5)

domain assumption Each impulse response f_l is a zero-mean Gaussian process with covariance λK_l, where K_l is the stable spline kernel (or its enriched version).
Section 3, Eq. (8). This is the prior that defines the nonparametric model.
domain assumption The innovation e is zero-mean white noise with unit variance and is independent of all f_l.
Section 3 and Fig. 1. This independence is used to derive the covariance of ξ and the estimator.
domain assumption The disturbance ξ(t) = (e ⊗ f_p)(t) is a stationary process with finite second-order moments, and the convolution is well defined.
Section 4.2, Eqs. (12)-(13). Standard for LTI systems with stable impulse responses.
ad hoc to paper The marginal likelihood for hyperparameter estimation is computed under a Gaussian approximation of the output distribution.
Section 5.3, Eq. (26). The output is non-Gaussian because ξ is a product of a Gaussian process and white noise; the approximation is not analyzed.
domain assumption Missing output data are missing at random (MCAR).
Not explicitly stated; the experiments discard outputs independently with probability 0.25 (Section 6.1). The estimator is derived under the assumption that missingness does not depend on unobserved values.

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Pith. "Pith review of Stable spline identification of linear systems under missing data." pith.science (2026). https://pith.science/paper/3N7X5TJS

@misc{pith2026190803913,
  author       = {Pith},
  title        = {Pith review of: Stable spline identification of linear systems under missing data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3N7X5TJS}},
  note         = {Machine review of arXiv:1908.03913}
}

read the original abstract

A different route to identification of time-invariant linear systems has been recently proposed which does not require committing to a specific parametric model structure. Impulse responses are described in a nonparametric Bayesian framework as zero-mean Gaussian processes. Their covariances are given by the so-called stable spline kernels encoding information on regularity and BIBO stability. In this paper, we demonstrate that these kernels also lead to a new family of radial basis functions kernels suitable to model system components subject to disturbances given by filtered white noise. This novel class, in cooperation with the stable spline kernels, paves the way to a new approach to solve missing data problems in both discrete and continuous-time settings. Numerical experiments show that the new technique may return models more predictive than those obtained by standard parametric Prediction Error Methods, also when these latter exploit the full data set.

Figures

Figures reproduced from arXiv: 1908.03913 by the authors.

**Figure 2.** ARMAX identification (subsection 6.4). Top: assessment of models prediction capability. Values of {CODk} 20 k=1 returned by PEM+Oracle (missing) (x-axis) vs those returned by SS imputation+SS (y-axis) after the first 100 runs. Bottom: assessment of quality in missing data reconstruction. MATLAB boxplots of the values of CODmiss obtained after the 500 Monte Carlo runs. In particular, each transfer function is that o… view at source ↗

**Figure 3.** ARMAX identification (subsection 6.5). Top: CODk , i.e. average coefficient of determination relative to k-step ahead prediction, using PEM+Oracle (full) (•), Stable Spline (full) (◦), SS imputation+SS (✸), PEM+BIC (full) (∗) and PEM+AICC (full) (4). Bottom: boxplots of the 1000 values of COD5. Recall that all the estimators, except SS imputation+SS, exploit all the identification data. The new technique has been … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Variants of the ARMAX identification experiment using either a lowpass input in place of white noise (left) or inserting a couple of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Ljung, System Identiﬁcation - Theory For the User

L. Ljung, System Identiﬁcation - Theory For the User . Prentice Hall, 1999

work page 1999

[2] [2]

A new look at the statistical model identiﬁcation,

H. Akaike, “A new look at the statistical model identiﬁcation,” IEEE Transactions on Automatic Control , vol. 19, pp. 716–723, 1974

work page 1974

[3] [3]

A new kernel-based approach for linear system identiﬁcation,

G. Pillonetto and G. De Nicolao, “A new kernel-based approach for linear system identiﬁcation,” Automatica, vol. 46, pp. 81–93, 2010

work page 2010

[4] [4]

Prediction error identiﬁcation of linear systems: A nonparametric Gaussian regression approach,

G. Pillonetto, A. Chiuso, and G. De Nicolao, “Prediction error identiﬁcation of linear systems: A nonparametric Gaussian regression approach,” Automatica, vol. 47, pp. 291–305, 2011

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G. Pillonetto, A. Chiuso, and G. De Nicolao, “Regularized estimation of sums of exponentials in spaces generated by stable spline kernels,” in Proceedings of the 2010 American Control Conference, Baltimora, 2010

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T. Chen, H. Ohlsson, and L. Ljung, “On the estimation of transfer functions, regularizations and Gaussian processes - revisited,” Automatica, vol. 48, no. 8, pp. 1525–1535, 2012

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Kernel methods in system identiﬁcation, machine learning and function estimation: a survey,

G. Pillonetto, F. Dinuzzo, T. Chen, G. D. Nicolao, and L. Ljung, “Kernel methods in system identiﬁcation, machine learning and function estimation: a survey,” Automatica, vol. 50, no. 3, pp. 657– 682, 2014

work page 2014