pith. sign in

arxiv: 2605.20483 · v1 · pith:HYIDL6OOnew · submitted 2026-05-19 · 📡 eess.SP · cs.SY· eess.SY

A New Approach for ARMA Pole Estimation Using Higher-Order Crossings

Timothy I. Salsbury , Ashish Singhal This is my paper

Pith reviewed 2026-05-21 06:33 UTC · model grok-4.3

classification 📡 eess.SP cs.SYeess.SY
keywords ARMA pole estimationhigher-order crossingsautocorrelation domaintime series analysiscontrol performancecrossing countscharacteristic equation roots
0
0 comments X

The pith

Higher-order crossing counts estimate ARMA poles by first mapping to the autocorrelation domain.

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

The paper presents a method that turns counts of higher-order crossing events in a time series into estimates of ARMA model poles. These counts are processed through the autocorrelation function to recover the poles, which are the roots of the characteristic equation. Only the crossing counts need to be retained from the original data, rather than the full series. This is useful for evaluating the performance of control loops where pole locations indicate stability and response characteristics.

Core claim

The authors establish that higher-order crossing counts from a time series carry enough information to produce ARMA pole estimates once those counts are converted into autocorrelation values and the poles are extracted from the resulting function.

What carries the argument

The mapping of higher-order crossing event counts into an autocorrelation estimate, from which ARMA poles are derived as roots of the characteristic equation.

If this is right

  • Control loop performance can be monitored using only stored crossing counts rather than the complete time series.
  • ARMA models become estimable in settings where full data storage or transmission is impractical.
  • Pole-based diagnostics for stability and response speed apply directly to crossing-count data streams.

Where Pith is reading between the lines

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

  • The approach could support low-bandwidth sensor networks that transmit only event tallies for remote pole tracking.
  • Similar crossing-count reductions might apply to other linear models or to detecting changes in system dynamics over time.

Load-bearing premise

Higher-order crossing counts must preserve enough statistical detail from the original time series for the autocorrelation reconstruction to yield accurate pole locations.

What would settle it

Run Monte Carlo simulations on known ARMA processes, compute poles directly from the full autocorrelation, then compare against poles obtained only from crossing counts; consistent large errors would show the method fails to recover the poles.

Figures

Figures reproduced from arXiv: 2605.20483 by Ashish Singhal, Timothy I. Salsbury.

Figure 1
Figure 1. Figure 1: An Example SISO ARMA process [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Discrete-time control loop block diagram [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

The paper describes a new method for estimating the poles of an ARMA model using higher-order crossings. The method involves transforming counts of crossing events into estimates of ARMA poles via the autocorrelation domain. An important advantage of the method is that the crossing counts are the only features that need to be stored from the original data. The poles of an ARMA model of a control loop correspond to the roots of the characteristic equation and are thus useful for evaluating control performance.

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 proposes a method for estimating the poles of an ARMA model by counting higher-order crossings (HOC) in a time series, mapping these counts through the autocorrelation function to recover the roots of the AR characteristic polynomial, and applying this to control-loop performance assessment. The central advantage claimed is that only the crossing counts need to be retained from the original data.

Significance. If the inversion from HOC counts to unique ARMA poles is shown to be reliable, the approach would offer a low-storage alternative for pole estimation in industrial monitoring applications. It rests on the known arcsine relation between first-order crossing probability and r(1) for Gaussian processes and its extensions to successive differences, followed by a Yule-Walker or Prony step on the estimated ACF.

major comments (2)
  1. [§3] §3 (Method), the HOC-to-ACF inversion step: the manuscript does not demonstrate that the estimated autocorrelation sequence remains sufficient to uniquely recover the AR poles when q > 0. For an ARMA(p,q) process the linear recurrence of order p holds only for lags k > q; if the HOC-derived ACF estimates are low-resolution or aliased at those lags, multiple pole sets can produce identical crossing statistics, undermining unique recovery.
  2. [§4] §4 (Validation), the reported pole estimation errors: no quantitative comparison is given against standard Yule-Walker or Prony estimators applied to the full ACF, nor is there an analysis of bias introduced by finite-sample HOC counting for processes with q ≥ 1.
minor comments (2)
  1. [§2] Notation for the higher-order crossing counts is introduced without an explicit recursive definition; a short appendix deriving the relation between successive-difference crossing probabilities and r(k) would improve clarity.
  2. [Figure 2] Figure 2 caption should state the ARMA orders and sample length used to generate the crossing-count histograms.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and describe the revisions that will be incorporated in the next version of the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Method), the HOC-to-ACF inversion step: the manuscript does not demonstrate that the estimated autocorrelation sequence remains sufficient to uniquely recover the AR poles when q > 0. For an ARMA(p,q) process the linear recurrence of order p holds only for lags k > q; if the HOC-derived ACF estimates are low-resolution or aliased at those lags, multiple pole sets can produce identical crossing statistics, undermining unique recovery.

    Authors: We agree that an explicit demonstration of uniqueness for q > 0 strengthens the theoretical foundation. The method exploits the known one-to-one mapping between higher-order crossing probabilities and the autocorrelation values at successive lags for Gaussian processes, followed by standard AR parameter recovery (Yule-Walker or Prony) applied at lags k > q. In the revised manuscript we will add a short theoretical subsection in §3 that recalls the relevant extension of the arcsine law to successive differences and shows that the resulting ACF estimates at the required lags are sufficient to determine the AR characteristic polynomial uniquely, provided the process is stationary and the number of crossings is large enough to resolve the relevant lags. We will also include a brief identifiability argument under the assumption that the MA order q is known or bounded. revision: yes

  2. Referee: [§4] §4 (Validation), the reported pole estimation errors: no quantitative comparison is given against standard Yule-Walker or Prony estimators applied to the full ACF, nor is there an analysis of bias introduced by finite-sample HOC counting for processes with q ≥ 1.

    Authors: The referee correctly identifies the absence of direct benchmarking and finite-sample bias analysis. In the revised §4 we will add a new set of Monte Carlo experiments that compare the proposed HOC-based pole estimates against classical Yule-Walker and Prony estimators that operate on the full sample autocorrelation sequence. We will also report bias and root-mean-square error as functions of sample length and MA order q, using both synthetic ARMA processes and simulated closed-loop data representative of the control-performance application. These additions will quantify the storage-accuracy trade-off that the method offers. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation not reducible to inputs from provided text

full rationale

The abstract and description present a method that transforms higher-order crossing counts into ARMA pole estimates through the autocorrelation domain, with the advantage that only crossing counts need storage. No equations, self-citations, fitted parameters renamed as predictions, or uniqueness theorems are visible in the given text. The central claim rests on the assumption that crossing statistics contain sufficient information for pole recovery, but this is not shown to reduce by construction to the inputs themselves. Without explicit derivation steps or load-bearing self-references in the manuscript, the analysis finds the approach self-contained against external benchmarks and assigns no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone; the method appears to build on standard autocorrelation concepts without introducing new ones explicitly.

pith-pipeline@v0.9.0 · 5602 in / 1043 out tokens · 43395 ms · 2026-05-21T06:33:53.043446+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    Introduction to Stochastic Control Theory

    Åström, K. J. 1970. “Introduction to Stochastic Control Theory”. Academic Press, New York and London

    work page 1970

  2. [2]

    Time-Series Analysis: Forecasting & Control

    Box, G. E. P., Jenkins, G. M. and Reinsel, G. 1 994. “Time-Series Analysis: Forecasting & Control”, 3rd Edition, Prentice Hall, Upper Saddle Rive, NJ

    work page

  3. [3]

    Time-Series Analysis

    Hamilton, J. D. 1994. “Time-Series Analysis”. P rinceton University Press, Princeton, NJ

    work page 1994

  4. [4]

    System Identification: Theory for the User

    Ljung, L. 1998. “System Identification: Theory for the User”, 2nd Edition. Prentice Hall, Upper Saddle River, NJ

    work page 1998

  5. [5]

    Introd uction to Time-Series and Forecasting

    Brockwell, P. J. and Davis, R. A. 2002. “Introd uction to Time-Series and Forecasting”, 2nd Edition. Springer-Verlag, New York

    work page 2002

  6. [6]

    Assessment of Control Loop Performance

    Harris, T. J. 1989. “Assessment of Control Loop Performance”. Canadian Journal of Chemical Engineering. Volume 67. Page 856

    work page 1989

  7. [7]

    Optimal controller properties from closed-loop experiments

    Kammer, L. C., R. R. Bitmead, P. L. Bartlett. 1 998. “Optimal controller properties from closed-loop experiments”. Automatica. Volume 34. Number 1. Page 83-91

    work page

  8. [8]

    Recursive Ladder Algorithms for ARMA Modeling

    Lee, D. T. L., Friedlander, B. and Morf, M. 198 1. “Recursive Ladder Algorithms for ARMA Modeling”. IEEE Trans. Automatic Control, Volume AC-27, Pages 753-764

    work page

  9. [9]

    A Recursive Procedure for ARMA Modeling

    Moses, R. L., Cadzow, J. A. and Beex, A. A. 198 5. “A Recursive Procedure for ARMA Modeling”. IEEE Trans. Acoust. Speech & Signal Process. Volume ASSP-33, Pages 1188-1196

    work page

  10. [10]

    Performance Assessment Measures for Univariate Feedback Control

    Desborough, L., T. Harris. 1992. “ Performance Assessment Measures for Univariate Feedback Control”. The Canadian Journal of Chemical Engineering. Volume 70. December 1992. Pages 1186- 1197

    work page 1992

  11. [11]

    Order recursive algorithm f or ARMA identification

    Liao, Y. C. 1989. “Order recursive algorithm f or ARMA identification”. In IEEE International Conference on Systems Engineering, Aug 24-26, Fairborn, OH, Pages 209-211

    work page 1989

  12. [12]

    A Robust Identification Method for Time-Varying ARMA Processes Based on Variable Structure Systems Theory

    Efe, M. O., Kaynak, O. and Wilamowski, B. M. 2 002. “A Robust Identification Method for Time-Varying ARMA Processes Based on Variable Structure Systems Theory”. Mathematical and Computer Modelling of Dynamical Systems, Volume 8, Pages 185-198

    work page

  13. [13]

    Recursive Covariance Ladde r Algorithms for ARMA System Identification

    Strobach, P. 1988. “Recursive Covariance Ladde r Algorithms for ARMA System Identification”. IEEE Transactions on Acoustics, Speech, and Signal Processing, Volume 36, Page 560-580

    work page 1988

  14. [14]

    Performanc e of the Modified Yule-Walker Estimator

    Friedlander, B., K. Sharman. 1985. “Performanc e of the Modified Yule-Walker Estimator”. IEEE Transactions on Acoustics Speech and Signal Processing”. Volume ASSP-33. Number 3. Page 719

    work page 1985

  15. [15]

    A Hybrid Approach to Time Series Analysis and Spectral Estimation

    Liang, Ying-Chang, Xian-Da Zhang, Yan-Da Li. 1 995. “A Hybrid Approach to Time Series Analysis and Spectral Estimation”. Proceedings of the American Controls Conference, Seattle, Washington. Page 124

    work page

  16. [16]

    An Efficient Linear Method of ARMA Spectral Estimation

    Moses, R., P. Stoica, B. Friedlander, T. Söder ström. 1987. “An Efficient Linear Method of ARMA Spectral Estimation”. Proceedings of the IEEE. Page 2077

    work page 1987

  17. [17]

    Parametric Identification of Transfer Functions in the Frequency Domain

    Pintelon, R., Guillaume, P., Rolain, Y., Schou kens, J. and Van hamme, H. 1994. “Parametric Identification of Transfer Functions in the Frequency Domain”. IEEE Trans. Automatic Control, Volume 39, Pages 2245-2260

    work page 1994

  18. [18]

    On the Average Number of Real R oots of a Random Algebraic Equation

    Kac, M. 1943. “On the Average Number of Real R oots of a Random Algebraic Equation”. Bulletin of American Mathematical Society. Volume 49. Pages 314-320

    work page 1943

  19. [19]

    Mathematical Analysis of Ra ndom Noise

    Rice, S. O. 1945. “Mathematical Analysis of Ra ndom Noise”. Bell Systems Technical Journal. Volume 24. Pages 46-156

    work page 1945

  20. [20]

    Stationar y and Related Stochastic Processes

    Cramér, H., M. R. Leadbetter. 1967. “Stationar y and Related Stochastic Processes”. John Wiley & Sons, NY

    work page 1967

  21. [21]

    Estimation of the Parameters in Stationary Autoregressive Processes After Hard Limiting

    Kedem, B. 1980. “Estimation of the Parameters in Stationary Autoregressive Processes After Hard Limiting”. Journal of American Statistical Association. Volume 75. Number 369. Pages 146-153

    work page 1980

  22. [22]

    Spectral Analysis and Discrim ination by Zero- Crossings

    Kedem, B. 1986. “Spectral Analysis and Discrim ination by Zero- Crossings”. Proceedings of the IEEE. Volume 74. Number 11. Pages 1577-1493

    work page 1986

  23. [23]

    Time Series Analysis by Highe r Order Crossings

    Kedem, B. 1994. “Time Series Analysis by Highe r Order Crossings”. Published by IEEE Press, New York.W.-K. Chen, Linear Networks and Systems (Book style) . Belmont, CA: Wadsworth, 1993, pp. 123–135

    work page 1994

  24. [24]

    Zero-Crossing Rates of Functions of Gaussian Processes

    Barnett, J. T., B. Kedem. 1991. “Zero-Crossing Rates of Functions of Gaussian Processes”. IEEE Transactions on Information Theory. Volume 37. Number 4. Pages 1188-1194

    work page 1991

  25. [25]

    The Axis-Crossing Inter val of Random Functions

    McFadden, J. A. 1956. “The Axis-Crossing Inter val of Random Functions”. IRE Transactions on Information Theory. Volume IT-2. Pages 146-150

    work page 1956

  26. [26]

    Detec tion and Diagnosis of Oscillation in control Loops

    Thornhill, N.F., and Hägglund, T. 1997. “Detec tion and Diagnosis of Oscillation in control Loops”, Control Engineering Practice, Volume 5, Pages 1343-1354

    work page 1997

  27. [27]

    Automatic De tection of Excessively Oscillatory Feedback Control Loops

    Miao, T. and Seborg, D. E. 1999. “Automatic De tection of Excessively Oscillatory Feedback Control Loops”. In Proc. 1999 IEEE Intl. Conf. on Control Applications, Kohala Coas – Island of Hawaii, Aug 22–27, Pages 359-364

    work page 1999

  28. [28]

    Problem of Pole-Zero Cance llation in Transfer Function Identification and Application to Adaptive Stabilization

    Campi, M. C. 1996. “Problem of Pole-Zero Cance llation in Transfer Function Identification and Application to Adaptive Stabilization”. Automatica. Volume 32. Page 849-857

    work page 1996