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arxiv: 2604.11726 · v1 · submitted 2026-04-13 · 🧮 math.OC

Informativity for Data-driven Prediction

Joel Stevens , Jeremy Coulson This is my paper

Pith reviewed 2026-05-10 14:49 UTC · model grok-4.3

classification 🧮 math.OC
keywords data-driven predictioninformativityLTI systemsunique output predictionsystem identificationbehavioral approachoutput trajectorydata conditions
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The pith

Data collected from an unknown linear system can uniquely predict its future outputs under weaker conditions than persistency of excitation.

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

The paper defines informativity for unique prediction as a property of input-output data that guarantees a single possible output sequence for any given future input sequence. This property holds even when multiple different LTI systems remain consistent with the observed data, so full system identification is not required. The authors supply sufficient conditions on the data that enforce this uniqueness and give algorithms that directly compute the predicted trajectory from the data matrices. A numerical demonstration confirms that the output can be recovered uniquely while the underlying dynamics stay non-unique.

Core claim

For linear time-invariant systems, data is informative for unique prediction when the collected trajectories constrain every possible system consistent with the data to produce the identical output response to the future inputs. Under this condition the unique output trajectory can be obtained by solving a set of linear equations derived from the data without first recovering the system matrices.

What carries the argument

Informativity for unique prediction: a set of matrix rank or kernel conditions on the collected data that force the output response to future inputs to be identical for all systems consistent with the data.

If this is right

  • Unique output prediction is possible even when the data-generating system cannot be uniquely identified.
  • Algorithms compute the trajectory by solving linear equations formed from the data and the future inputs.
  • The data requirements are strictly weaker than persistency of excitation when the goal is prediction rather than identification.
  • The same data set can support unique prediction for some input sequences while failing for others.

Where Pith is reading between the lines

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

  • Prediction tasks could succeed with shorter or less rich data records than full identification requires.
  • The distinction between prediction and identification may allow more efficient data collection in control applications.
  • The matrix conditions could be checked numerically before attempting prediction to decide whether the data is sufficient.
  • Similar informativity ideas might be tested on finite-horizon or switched linear systems.

Load-bearing premise

The unknown system is linear and time-invariant and the collected data satisfies the newly defined informativity conditions.

What would settle it

Run the algorithm on a known LTI system with data that meets the informativity conditions; the computed trajectory must match the true output for the chosen future inputs, and any other system fitting the same data must produce the same trajectory.

Figures

Figures reproduced from arXiv: 2604.11726 by Jeremy Coulson, Joel Stevens.

Figure 1
Figure 1. Figure 1: The top plot shows the input data u d . The bottom plot shows the output data y d corresponding to u d . is only a sufficient condition, we in general cannot conclude whether the data is informative for unique prediction. We compute an explaining system of the data, B =     u y  [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: The top plot shows the initial input sequence [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

In this work we examine the problem of data-driven prediction. That is, given a LTI system with unknown dynamics, we wish to use data collected from the system to predict the system's output response to a given sequence of known inputs. Current methods for predicting require strong conditions on the data such as persistency of excitation. We examine this problem with the goal of finding weaker conditions that still enable prediction. We approach the problem from the data informativity perspective and formally define the notion of informativity for unique prediction. We provide sufficient conditions for informativity for unique prediction and design algorithms to compute the unique output trajectory of the unknown system given known inputs. We demonstrate the results with a numerical example showing that unique output prediction is possible without being able to uniquely identify the unknown data-generating system.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript defines a new concept of 'informativity for unique prediction' for linear time-invariant systems. It derives sufficient conditions (claimed to be weaker than persistency of excitation) under which input-output data permit computation of a unique output trajectory for a given future input sequence without requiring unique identification of the unknown system. Algorithms are provided to perform this computation, and a numerical example is used to illustrate that unique prediction is possible even when the data-generating system itself cannot be uniquely recovered.

Significance. If the central derivations hold, the contribution is meaningful for data-driven prediction and control: it relaxes the data richness requirements that currently limit many trajectory-based methods and explicitly separates the tasks of prediction and identification. The provision of concrete algorithms and the numerical demonstration that prediction uniqueness does not imply system uniqueness are practical strengths that could broaden applicability in settings with limited or non-persistently exciting data.

major comments (2)
  1. [§3] §3 (Definition of informativity for unique prediction): the definition is introduced via consistency of output trajectories with the collected data, but it is not immediately clear whether the uniqueness claim remains valid when the system order is unknown or when the data window length is only marginally larger than the minimal required length; a counter-example or explicit rank condition would strengthen the claim.
  2. [§4] §4 (Sufficient conditions and Algorithm 1): the proof that the stated conditions are strictly weaker than persistency of excitation relies on a particular data-matrix rank argument; however, the step from the informativity condition to uniqueness of the predicted output (Eq. (12) or equivalent) appears to assume noise-free data, which contradicts the abstract's mention of noise assumptions and requires clarification on how the algorithm extends to noisy measurements.
minor comments (2)
  1. [§5] The numerical example in §5 would benefit from an explicit statement of the system order, data length, and input sequence used, together with the precise informativity condition that is satisfied.
  2. Notation for the data matrices (e.g., Hankel matrices) should be unified with the most common conventions in the behavioral approach literature to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We have carefully reviewed each point raised and provide detailed responses below. We believe these clarifications will strengthen the paper.

read point-by-point responses

  1. Referee: [§3] §3 (Definition of informativity for unique prediction): the definition is introduced via consistency of output trajectories with the collected data, but it is not immediately clear whether the uniqueness claim remains valid when the system order is unknown or when the data window length is only marginally larger than the minimal required length; a counter-example or explicit rank condition would strengthen the claim.

    Authors: The definition of informativity for unique prediction is formulated without any assumption on knowledge of the system order, as it relies solely on consistency of output trajectories with the observed data for given inputs. This is consistent with the paper's goal of enabling prediction without system identification. The sufficient conditions in §4 impose rank conditions on the data matrices that guarantee uniqueness of the predicted output independently of the unknown order. For the data window length, the conditions require it to be long enough to achieve the necessary rank, which is generally more than marginally larger than the minimal length. We will add a clarifying remark in §3 on the independence from system order and the data-length requirement for the rank condition. revision: partial

  2. Referee: [§4] §4 (Sufficient conditions and Algorithm 1): the proof that the stated conditions are strictly weaker than persistency of excitation relies on a particular data-matrix rank argument; however, the step from the informativity condition to uniqueness of the predicted output (Eq. (12) or equivalent) appears to assume noise-free data, which contradicts the abstract's mention of noise assumptions and requires clarification on how the algorithm extends to noisy measurements.

    Authors: We appreciate this observation. The theoretical results in §4, including the weaker-than-PE proof via the rank argument and the uniqueness step leading to Eq. (12), are derived for the noise-free deterministic case. The abstract references standard assumptions for LTI systems but the presented algorithms and conditions focus on exact data. We will revise the abstract to explicitly note the noise-free setting and add a discussion paragraph clarifying that extensions to noisy data (e.g., via robust or set-valued predictions) are left for future work. This will resolve the inconsistency while preserving the current contributions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from new definitions

full rationale

The paper introduces the definition of informativity for unique prediction for LTI systems, then derives sufficient conditions directly from that definition to enable unique output trajectory computation from input-output data. Algorithms follow from the conditions, and the numerical example illustrates separation between prediction uniqueness and system identification without any reduction of claims to fitted parameters, self-referential equations, or load-bearing self-citations. All steps are independent of the target results and rest on standard LTI trajectory properties plus the newly stated informativity notion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the assumption that the plant is linear time-invariant and that the collected data can be checked against the new informativity definition. No free parameters are mentioned. The key invented concept is the informativity notion itself.

axioms (1)
  • domain assumption The unknown system is linear time-invariant.
    Explicitly stated in the abstract as the setting for the prediction problem.
invented entities (1)
  • Informativity for unique prediction no independent evidence
    purpose: To certify that collected data suffices for unique output forecasting without full system identification.
    Newly defined concept whose sufficient conditions are claimed to enable the algorithms.

pith-pipeline@v0.9.0 · 5417 in / 1253 out tokens · 53732 ms · 2026-05-10T14:49:41.647347+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    From model-based control to data-driven control: Survey, classification and perspective,

    Z.-S. Hou and Z. Wang, “From model-based control to data-driven control: Survey, classification and perspective,”Information Sciences, vol. 235, pp. 3–35, 2013

    work page 2013

  2. [2]

    System identification,

    L. Ljung, “System identification,” inSignal analysis and prediction. Springer, 1998, pp. 163–173

    work page 1998

  3. [3]

    Vapnik,The nature of statistical learning theory

    V . Vapnik,The nature of statistical learning theory. Springer science & business media, 2013. −10 −5 0 5 10 u1 u2 0 5 10 15 20 −10 0 10 yini yf yini Fig. 2: A future prediction is provided by applying Algorithm 2 that is accurate for all explaining systems. The top plot shows the initial input sequenceu ini followed by the future input sequenceu f. The b...

    work page 2013

  4. [4]

    Behavioral systems theory in data-driven analysis, signal processing, and control,

    I. Markovsky and F. D ¨orfler, “Behavioral systems theory in data-driven analysis, signal processing, and control,”Annual Reviews in Control, vol. 52, pp. 42–64, 2021

    work page 2021

  5. [5]

    Data-driven simulation and control,

    I. Markovsky and P. Rapisarda, “Data-driven simulation and control,” International Journal of Control, vol. 81, no. 12, pp. 1946–1959, 2008. [Online]. Available: https://doi.org/10.1080/00207170801942170

    work page doi:10.1080/00207170801942170 1946

  6. [6]

    Data-enabled predictive control: In the shallows of the DeePC,

    J. Coulson, J. Lygeros, and F. D ¨orfler, “Data-enabled predictive control: In the shallows of the DeePC,” in2019 18th European Control Conference (ECC), 2019, pp. 307–312

    work page 2019

  7. [7]

    SPC: Subspace predictive control,

    W. Favoreel, B. De Moor, and M. Gevers, “SPC: Subspace predictive control,”IFAC Proceedings Volumes, vol. 32, no. 2, pp. 4004–4009, 1999

    work page 1999

  8. [8]

    J. W. Polderman and J. C. Willems,Introduction to Mathematical Systems Theory, ser. Texts in Applied Mathematics, J. E. Marsden, L. Sirovich, M. Golubitsky, W. J ¨ager, and F. John, Eds. New York, NY: Springer New York, 1998, vol. 26. [Online]. Available: http://link.springer.com/10.1007/978-1-4757-2953-5

    work page doi:10.1007/978-1-4757-2953-5 1998

  9. [9]

    A note on persistency of excitation,

    J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. De Moor, “A note on persistency of excitation,”Systems & Control Letters, vol. 54, no. 4, pp. 325–329, 2005

    work page 2005

  10. [10]

    The informativity approach: To data-driven analysis and control,

    H. J. Van Waarde, J. Eising, M. K. Camlibel, and H. L. Trentelman, “The informativity approach: To data-driven analysis and control,” IEEE Control Systems Magazine, vol. 43, no. 6, pp. 32–66, 2023

    work page 2023

  11. [11]

    Algorithms for deterministic balanced subspace identification,

    I. Markovsky, J. C. Willems, P. Rapisarda, and B. L. De Moor, “Algorithms for deterministic balanced subspace identification,”Au- tomatica, vol. 41, no. 5, pp. 755–766, May 2005. [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S0005109804003036

    work page 2005

  12. [12]

    Beyond the fundamental lemma: from finite time series to linear system,

    K. Camlibel and P. Rapisarda, “Beyond the fundamental lemma: from finite time series to linear system,” May 2024, arXiv:2405.18962 [math]. [Online]. Available: http://arxiv.org/abs/2405.18962

    work page arXiv 2024

  13. [13]

    P. R. Halmos,Finite-Dimensional Vector Spaces, ser. Undergraduate Texts in Mathematics. Springer, 1993

    work page 1993