The Use of Gaussian Processes in System Identification

Simo S\"arkk\"a

arxiv: 1907.06066 · v1 · pith:LIFLNOK3new · submitted 2019-07-13 · 📊 stat.ML · cs.LG· cs.SY· eess.SY

The Use of Gaussian Processes in System Identification

Simo S\"arkk\"a This is my paper

Pith reviewed 2026-05-24 22:03 UTC · model grok-4.3

classification 📊 stat.ML cs.LGcs.SYeess.SY

keywords Gaussian processessystem identificationNFIR modelsNARX modelsstate-space modelstime seriesmachine learning

0 comments

The pith

Gaussian processes form NFIR, NARX, and state-space models to identify non-linear system dynamics from data.

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

The paper reviews how Gaussian processes learn input-output mappings by placing a prior on the unknown function and conditioning on observations. It shows their use in building non-linear finite-impulse response and autoregressive prediction models, plus state-space forms that separately identify dynamics and measurements. Temporal and spatio-temporal variants can regress directly on time-domain data. A reader would care because the approach supplies a non-parametric, uncertainty-aware way to capture complex behaviors without committing to fixed functional forms upfront.

Core claim

Gaussian processes are used in machine learning to learn input-output mappings from observed data. Gaussian process regression is based on imposing a Gaussian process prior on the unknown regressor function and statistically conditioning it on the observed data. In system identification, Gaussian processes are used to form time series prediction models such as non-linear finite-impulse response (NFIR) models as well as non-linear autoregressive (NARX) models. Gaussian process state-space models (GPSS) can be used to learn the dynamic and measurement models for a state-space representation of the input-output data. Temporal and spatio-temporal Gaussian processes can be directly used to form a

What carries the argument

Gaussian process regression, which imposes a prior on an unknown function and conditions the prior on observed data to produce non-linear mappings for system models.

If this is right

NFIR models capture non-linear effects from past inputs alone without output recursion.
NARX models add autoregressive output terms to improve multi-step prediction accuracy.
GP state-space models allow separate probabilistic identification of transition and observation functions.
Direct temporal GPs enable non-parametric regression over raw time-series observations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Sequential updating of the GP posterior could support online identification without full batch retraining.
Uncertainty estimates from these models could feed directly into robust control design.
Sparse approximations might be needed to scale the same constructions to very high input dimensions.

Load-bearing premise

The NFIR, NARX, GP state-space, and direct temporal GP constructions represent the main directions of Gaussian process use in system identification.

What would settle it

A documented major method or application of Gaussian processes in system identification that falls outside the NFIR, NARX, GPSS, and temporal/spatio-temporal categories listed.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a short review outlining four standard ways to apply Gaussian processes to system identification, with no new results or derivations.

read the letter

The paper's core is a brief sketch of how GPs have been used in system ID. It covers NFIR and NARX models for prediction, GP state-space models for learning dynamics and measurements, and direct temporal or spatio-temporal GPs on the time series. That's the whole contribution: a compact list of existing model classes plus pointers to the literature the author already knows well. For a newcomer from control engineering who needs a one-page map of the intersection, this does the job cleanly and without fluff. Särkkä keeps the language direct and stays within the stated scope of 'briefly outline.' The text does not claim new theorems, new algorithms, or new empirical findings, so there is nothing to verify or falsify beyond the author's scoping decision that these four directions are the main ones. That scoping is reasonable given the citations, but it is not itself a technical claim that needs proof. The main limitation is exactly what the abstract advertises: the piece stays at the level of description. There are no worked examples, no complexity comparisons, no discussion of inference cost or approximation quality, and no attempt to resolve open questions such as how to scale these models to long time series. Readers already active in GP system ID will not find new material here. This paper is useful for students or engineers who want a quick orientation before diving into the primary references. It is not the sort of work that changes research directions or supplies reusable tools. I would not bring it to a reading group unless the group is explicitly collecting short surveys. I would not cite it in my own papers because it contains no result to build upon. A serious editor could still send it to review if the journal wants short, well-organized overviews in the area; the writing is competent and the citations look standard. Otherwise it is a low-priority submission.

Referee Report

0 major / 1 minor

Summary. This manuscript is a short review outlining the use of Gaussian processes in system identification. It describes forming time-series prediction models such as non-linear finite-impulse response (NFIR) and non-linear autoregressive (NARX) models via GPs, using Gaussian process state-space models (GPSS) to learn dynamic and measurement models, and directly applying temporal and spatio-temporal GPs for regression on time-domain data. The stated aim is to briefly outline the main directions in this area.

Significance. As a concise descriptive survey that synthesizes standard model classes already present in the cited literature, the paper could serve as an accessible entry point for researchers new to GP-based system identification. Its value is limited to synthesis rather than new theorems, derivations, empirical results, or falsifiable predictions; no machine-checked proofs or reproducible code are provided.

minor comments (1)

[Abstract] Abstract: the claim that the three listed model classes constitute the 'main directions' is presented without supporting discussion or comparison to other GP applications in system identification; a brief justification paragraph would improve clarity of scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their assessment of the manuscript as a concise survey and for recommending minor revision. The paper is explicitly positioned as a brief outline of existing directions rather than a source of new theoretical results. Below we respond to the significance evaluation.

read point-by-point responses

Referee: As a concise descriptive survey that synthesizes standard model classes already present in the cited literature, the paper could serve as an accessible entry point for researchers new to GP-based system identification. Its value is limited to synthesis rather than new theorems, derivations, empirical results, or falsifiable predictions; no machine-checked proofs or reproducible code are provided.

Authors: We agree with this characterization. The abstract states that the aim is to 'briefly outline the main directions in system identification methods using Gaussian processes,' which aligns with the referee's description of it as a synthesis of standard model classes (NFIR, NARX, GPSS, and temporal GPs). The manuscript does not claim new theorems, derivations, or empirical results, and we view the value of such a short review as providing an accessible entry point, consistent with the referee's positive note on that aspect. No revision is required on this point. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

This is a short descriptive review paper whose sole aim is to outline existing modeling approaches (NFIR/NARX, GP state-space, temporal GPs) already present in the cited literature. No derivations, parameter fits, uniqueness theorems, or predictions are advanced. The statement that these constitute the 'main directions' is an authorial scoping choice, not a load-bearing claim that reduces to self-definition or fitted inputs. No equations or self-citations function as hidden premises for any result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper. No free parameters, axioms, or invented entities are introduced in the provided abstract.

pith-pipeline@v0.9.0 · 5661 in / 971 out tokens · 19612 ms · 2026-05-24T22:03:14.104910+00:00 · methodology

The Use of Gaussian Processes in System Identification

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)