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arxiv: 1906.11158 · v1 · pith:XXGG7LJAnew · submitted 2019-06-26 · 🌌 astro-ph.IM · stat.AP

Discrete-time autoregressive model for unequally spaced time-series observations

Felipe Elorrieta , Susana Eyheramendy , Wilfredo Palma This is my paper

Pith reviewed 2026-05-25 15:11 UTC · model grok-4.3

classification 🌌 astro-ph.IM stat.AP
keywords irregular time seriesautoregressive modelstate-space representationvariable starslight curvesKalman filtermaximum likelihoodforecasting
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The pith

A discrete-time autoregressive model for unequally spaced observations is weakly stationary and admits an exact state-space representation.

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

The paper proposes the complex irregular autoregressive (CIAR) model as a direct discrete-time process for time series observed at irregular intervals. It establishes that the model is weakly stationary and possesses a state-space form that permits maximum-likelihood estimation through Kalman recursions together with forecasting of missing values. Monte Carlo simulations confirm that parameter estimates are accurate in finite samples. The approach is illustrated on variable-star light curves, where it identifies inadequate harmonic fits arising from incorrect periods or multiperiodicity. Readers would care because many scientific domains collect data with large gaps yet still require reliable dependence modeling without solving continuous-time differential equations.

Core claim

The CIAR model is a novel discrete-time autoregressive process for unequally spaced observations that is weakly stationary, admits an exact state-space representation, supports accurate maximum-likelihood estimation via Kalman recursions, and can be used to detect poor harmonic fits in variable-star light curves while enabling forecasting.

What carries the argument

The complex irregular autoregressive (CIAR) process, a discrete-time autoregressive structure whose coefficients depend on the observed time intervals and are chosen to preserve weak stationarity.

If this is right

  • Maximum-likelihood estimates of the model parameters remain accurate even when observations are spaced irregularly.
  • The state-space representation allows efficient computation of the likelihood and direct forecasting of unobserved values.
  • The model functions as a diagnostic check on the adequacy of harmonic fits to periodic signals in gapped astronomical data.
  • It supplies a practical alternative to continuous-time CARMA processes when the gaps between observations are large.

Where Pith is reading between the lines

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

  • If CIAR recovers parameters reliably on real data with very large gaps, discrete formulations may become preferable to continuous-time models in astronomy.
  • The same state-space construction could be extended to higher-order or moving-average versions while retaining exact Kalman recursions.
  • The method could be tested on irregular series from finance or climatology to check whether the complex-coefficient construction generalizes.

Load-bearing premise

A discrete-time autoregressive structure with the specific complex-coefficient form proposed for CIAR adequately captures the dependence structure of astronomical time series when time gaps are large.

What would settle it

A real or simulated light curve with known large gaps on which the CIAR maximum-likelihood estimates yield systematically inaccurate forecasts or fail to flag a deliberately incorrect harmonic period.

Figures

Figures reproduced from arXiv: 1906.11158 by Felipe Elorrieta, Susana Eyheramendy, Wilfredo Palma.

Figure 1
Figure 1. Figure 1: Boxplot of the root mean squared error computed for the fitted models on the 1000 sequences simulated of the real part of the CIAR process. In a) each CIAR process was generated using φ R = 0.99. In b) each CIAR process was generated using φ R = −0.99. The other parameters of the models are defined as φ I = 0, c = 0 and length n = 300. The observational times are generated using a mixture of Exponential di… view at source ↗
Figure 2
Figure 2. Figure 2: Estimated coefficients φ R (y-axis) by the CIAR model in k = 200 harmonic processes generated using frequencies (x-axis) in the interval (0, π). The black line corresponds to the coefficients estimated by the CIAR model. The red line is the theoretical autocorrelation of the pro￾cess yti 4.1. Modeling light curves One of the most important challenges in the analysis of variable stars is to classify them ba… view at source ↗
Figure 4
Figure 4. Figure 4: This result is consistent with those obtained in Section [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Kernel Density of the RMSE computed for the residuals of har￾monic fit in the light curves when the CIAR coefficient is negative. The red density corresponds to the RMSE computed using the CIAR model, and the blue density corresponds to the RMSE computed using the IAR model. lated residuals. However, if the variable star has a shorter period, it could mean the opposite. Therefore, we cannot make decisions … view at source ↗
Figure 5
Figure 5. Figure 5: a) Light curve of a RRc star observed by the HIPPARCOS survey. The continuous blue line is the harmonic best fit. b) Natural logarithm of the absolute value of the estimated parameter φˆR by the CIAR model on the residuals of the harmonic model fitted with different frequencies; x-axis shows percentage variation from the correct frequency, y-axis shows the natural logarithm of φˆR . c) Natural logarithm of… view at source ↗
Figure 6
Figure 6. Figure 6: a) Boxplot of the p-value estimated from the CIAR model in the RR-Lyraes variable stars separated by subclass. b) Boxplot of the p-value estimated from the CIAR model in the Cepheids variable stars separated by subclass. and re-estimate the model to forecast the observation at the following time tj+2. This procedure is repeated iteratively until the remaining 10% of the data is forecasted, obtaining the ve… view at source ↗
Figure 7
Figure 7. Figure 7: a) Normalized K-band MCG-6-30-15 light curve. The red dots are the forecasted values. b) Zoom of the last 10% of the MCG-6-30-15 light curve. The red dots are the forecasted values using the CIAR model and the gray bars are the confidence intervals at the 90% level. positive real part of the complex autocorrelation parameter, the CIAR model becomes the IAR model. The connection between both models is verif… view at source ↗
Figure 8
Figure 8. Figure 8: a) CIAR process with positive autocorrelation generated using parameters φ R = 0.99, φ I = 0, c = 0 and length n = 300 . b) CIAR process with negative autocorrelation generated using parameters φ R = −0.99, φ I = 0, c = 0 and length n = 300. for multiperiodic variable stars will be implemented using this feature. The main aim of this work is to continue developing models for irregularly observed time serie… view at source ↗
read the original abstract

Most time-series models assume that the data come from observations that are equally spaced in time. However, this assumption does not hold in many diverse scientific fields, such as astronomy, finance, and climatology, among others. There are some techniques that fit unequally spaced time series, such as the continuous-time autoregressive moving average (CARMA) processes. These models are defined as the solution of a stochastic differential equation. It is not uncommon in astronomical time series, that the time gaps between observations are large. Therefore, an alternative suitable approach to modeling astronomical time series with large gaps between observations should be based on the solution of a difference equation of a discrete process. In this work we propose a novel model to fit irregular time series called the complex irregular autoregressive (CIAR) model that is represented directly as a discrete-time process. We show that the model is weakly stationary and that it can be represented as a state-space system, allowing efficient maximum likelihood estimation based on the Kalman recursions. Furthermore, we show via Monte Carlo simulations that the finite sample performance of the parameter estimation is accurate. The proposed methodology is applied to light curves from periodic variable stars, illustrating how the model can be implemented to detect poor adjustment of the harmonic model. This can occur when the period has not been accurately estimated or when the variable stars are multiperiodic. Last, we show how the CIAR model, through its state space representation, allows unobserved measurements to be forecast.

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 paper proposes the Complex Irregular Autoregressive (CIAR) model, a novel discrete-time autoregressive process for unequally spaced observations. It asserts that the model is weakly stationary, admits an exact state-space representation permitting maximum-likelihood estimation via Kalman recursions, demonstrates accurate finite-sample parameter recovery in Monte Carlo simulations, applies the model to variable-star light curves to identify poor harmonic fits (arising from inaccurate periods or multiperiodicity), and uses the state-space form for forecasting unobserved values.

Significance. If the central modeling assumption holds, the CIAR process supplies a computationally convenient discrete-time alternative to CARMA models when observation gaps are large. Explicit demonstration of weak stationarity and an exact state-space representation are concrete strengths; the Monte Carlo validation of MLE accuracy and the astronomical application provide direct evidence of practical utility. The work addresses a genuine need in astrostatistics for irregular sampling.

major comments (2)
  1. [Abstract; §3 (model definition and stationarity)] The central claim that the proposed complex-coefficient discrete AR form adequately captures second-order dependence for large irregular gaps (the weakest assumption flagged in the review) is load-bearing for both the methodological novelty and the variable-star application. No explicit derivation or numerical check is supplied showing that the implied covariance (power-law decay in the AR coefficient) matches the continuous-time limit or avoids misspecification when gaps exceed the correlation timescale; a direct comparison to the covariance of a CAR(1) process under the same sampling times would be required to substantiate the claim.
  2. [§4, §5] §4 (Monte Carlo) and §5 (application): the reported simulation performance and light-curve results rest on the unverified covariance behavior for large gaps; if the model is misspecified in that regime, the MLE accuracy and the diagnostic for poor harmonic fits cannot be taken as general evidence of superiority over existing continuous-time methods.
minor comments (2)
  1. [§2] Notation for the complex autoregressive coefficient and its modulus should be introduced with an explicit equation number at first use to avoid ambiguity when the time-difference adjustment is later applied.
  2. [Abstract] The abstract states that the model 'can be represented as a state-space system' but does not indicate whether the transition matrix is derived in closed form or obtained numerically; a brief equation reference would clarify this for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive review of our manuscript. We address each of the major comments in turn below.

read point-by-point responses

  1. Referee: [Abstract; §3 (model definition and stationarity)] The central claim that the proposed complex-coefficient discrete AR form adequately captures second-order dependence for large irregular gaps (the weakest assumption flagged in the review) is load-bearing for both the methodological novelty and the variable-star application. No explicit derivation or numerical check is supplied showing that the implied covariance (power-law decay in the AR coefficient) matches the continuous-time limit or avoids misspecification when gaps exceed the correlation timescale; a direct comparison to the covariance of a CAR(1) process under the same sampling times would be required to substantiate the claim.

    Authors: The CIAR model is formulated as a discrete-time autoregressive process specifically to address situations with large observation gaps, as an alternative to continuous-time models like CARMA. The proof of weak stationarity in §3 follows directly from the discrete definition, showing that the autocovariance depends solely on the time lag via the complex coefficient. We do not assert that the covariance matches that of a CAR(1) process for large gaps, nor do we claim it is an approximation to the continuous-time limit in that regime. The model is designed to be well-defined and stationary for arbitrary spacings under its own assumptions. A comparison to CAR(1) would be relevant for assessing approximation quality but is not required to validate the discrete model. We will add a clarifying statement in the revised manuscript to better articulate the model's scope and distinction from continuous-time approaches. revision: partial

  2. Referee: [§4, §5] §4 (Monte Carlo) and §5 (application): the reported simulation performance and light-curve results rest on the unverified covariance behavior for large gaps; if the model is misspecified in that regime, the MLE accuracy and the diagnostic for poor harmonic fits cannot be taken as general evidence of superiority over existing continuous-time methods.

    Authors: The Monte Carlo experiments in §4 evaluate the accuracy of parameter estimation under the CIAR model with irregularly spaced data, including cases with large gaps, and confirm reliable recovery. The application in §5 demonstrates the model's utility in identifying inadequacies in harmonic fits for stellar light curves. These results are valid within the context of the discrete CIAR framework. We do not claim general superiority over continuous-time methods but rather present CIAR as a practical discrete alternative for large-gap time series. We will revise the text in §§4 and 5 to more clearly delineate the intended scope of the results. revision: partial

Circularity Check

0 steps flagged

No significant circularity: CIAR model defined directly with independent stationarity proof and state-space construction

full rationale

The paper introduces the CIAR process by direct definition as a discrete-time AR model with complex coefficient whose modulus enforces stationarity, then derives weak stationarity and exact Kalman-filterable state-space representation from the defining recurrence and time-gap adjustment. Monte Carlo validation of MLE and the light-curve application are downstream uses, not inputs. No equation reduces to a fitted quantity renamed as prediction, no load-bearing self-citation chain, and no ansatz smuggled via prior work. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition of the CIAR process and its claimed stationarity and state-space properties; these are not expanded in the abstract, so the ledger is necessarily incomplete.

free parameters (1)
  • CIAR autoregressive coefficients
    Model parameters estimated by maximum likelihood; exact form and number not given in abstract.
axioms (1)
  • domain assumption The proposed discrete-time process is weakly stationary under suitable parameter restrictions.
    Stated as shown in the abstract.
invented entities (1)
  • CIAR model no independent evidence
    purpose: Discrete-time autoregressive representation of unequally spaced series.
    New model introduced in the paper.

pith-pipeline@v0.9.0 · 5799 in / 1295 out tokens · 22378 ms · 2026-05-25T15:11:51.300815+00:00 · methodology

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...

    work page

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

    work page

  3. [3]

    & Wainwright, T

    Alder, B. & Wainwright, T. 1970, Physical review A, 1, 18

    work page 1970

  4. [4]

    2017, in 2017 Tenth International Conference Management of Large-Scale System Development (MLSD), 1--4

    Alperovich, Y., Alperovich, M., & Spiro, A. 2017, in 2017 Tenth International Conference Management of Large-Scale System Development (MLSD), 1--4

    work page 2017

  5. [5]

    & Ivanova, K

    Ausloos, M. & Ivanova, K. 2001, Phys. Rev. E, 63, 047201

    work page 2001

  6. [6]

    & Palma, W

    Bondon, P. & Palma, W. 2007, Journal of Time Series Analysis, 28, 261

    work page 2007

  7. [7]

    Box, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. 2015, Time Series Analysis: Forecasting and Control (5th edition) (John Wiley & Sons, Inc.)

    work page 2015

  8. [8]

    & Davis, R

    Brockwell, P. & Davis, R. 2002, Introduction to Time Series and Forecasting (Springer-Verlag New York)

    work page 2002

  9. [9]

    Broersen, P. M. T. 2006, Automatic Autocorrelation and Spectral Analysis (Secaucus, NJ, USA: Springer-Verlag New York, Inc.)

    work page 2006

  10. [10]

    Y., Lo, A

    Campbell, J. Y., Lo, A. W.-C., & MacKinlay, A. C. 1997, The Econometrics of Financial Markets (princeton University press), 632

    work page 1997

  11. [11]

    Carvalho, L. M. V. d., Tsonis, A. A., Jones, C., Rocha, H. R. d., & Polito, P. S. 2007, Nonlinear Processes in Geophysics, 14, 723

    work page 2007

  12. [12]

    Chan, K. S. & Tong, H. 1987, Journal of Time Series Analysis, 8, 277

    work page 1987

  13. [13]

    S., Hameed, A., & Niden, C

    Conrad, J. S., Hameed, A., & Niden, C. 1994, Journal of Finance, 49, 1305

    work page 1994

  14. [14]

    M., Aerts , C., et al

    Debosscher , J., Sarro , L. M., Aerts , C., et al. 2007, , 475, 1159

    work page 2007

  15. [15]

    Dubois, S. R. & Glanz, F. H. 1986, IEEE Trans Pattern Anal Mach Intell, 8, 55

    work page 1986

  16. [16]

    2017, The Astrophysical Journal, 840, 41

    Edelson, R., Gelbord, J., Cackett, E., et al. 2017, The Astrophysical Journal, 840, 41

    work page 2017

  17. [17]

    Edelson , R. A. & Krolik , J. H. 1988, , 333, 646

    work page 1988

  18. [18]

    2016, , 595, A82

    Elorrieta , F., Eyheramendy , S., Jord \'a n , A., et al. 2016, , 595, A82

    work page 2016

  19. [19]

    2018, Monthly Notices of the Royal Astronomical Society, 481, 4311

    Eyheramendy, S., Elorrieta, F., & Palma, W. 2018, Monthly Notices of the Royal Astronomical Society, 481, 4311

    work page 2018

  20. [20]

    D., Babu, G

    Feigelson, E. D., Babu, G. J., & Caceres, G. A. 2018, Frontiers in Physics, 6, 80

    work page 2018

  21. [21]

    2017, The Astronomical Journal, 154, 220

    Foreman-Mackey, D., Agol, E., Ambikasaran, S., & Angus, R. 2017, The Astronomical Journal, 154, 220

    work page 2017

  22. [22]

    2007, Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory, and Beyond (Wiley-Interscience)

    Gao, J., Cao, Y., Tung, W.-w., & Hu, J. 2007, Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory, and Beyond (Wiley-Interscience)

    work page 2007

  23. [23]

    C., Becker, A

    Kelly, B. C., Becker, A. C., Sobolewska, M., Siemiginowska, A., & Uttley, P. 2014, The Astrophysical Journal, 788, 33

    work page 2014

  24. [24]

    2013, Stationary stochastic processes for scientists and engineers (Chapman and Hall)

    Lindgren, G., Rootz \'e n, H., & Sandsten, M. 2013, Stationary stochastic processes for scientists and engineers (Chapman and Hall)

    work page 2013

  25. [25]

    Lira, P., Arévalo, P., Uttley, P., McHardy, I. M. M., & Videla, L. 2015, Monthly Notices of the Royal Astronomical Society, 454, 368

    work page 2015

  26. [26]

    1999, Signal Processing, 77, 139

    Martin, R. 1999, Signal Processing, 77, 139

    work page 1999

  27. [27]

    1974, Complex Stochastic Processes: an Introduction to Theory and Application, Advanced book program (Addison-Wesley Publishing Company, Advanced Book Program)

    Miller, K. 1974, Complex Stochastic Processes: an Introduction to Theory and Application, Advanced book program (Addison-Wesley Publishing Company, Advanced Book Program)

    work page 1974

  28. [28]

    Perryman , M. A. C., Lindegren , L., Kovalevsky , J., et al. 1997, , 323, L49

    work page 1997

  29. [29]

    & Bondon , P

    Picinbono , B. & Bondon , P. 1997, IEEE Transactions on Signal Processing, 45, 411

    work page 1997

  30. [30]

    2011, Nonlinear Processes in Geophysics, 18, 389

    Rehfeld, K., Marwan, N., Heitzig, J., & Kurths, J. 2011, Nonlinear Processes in Geophysics, 18, 389

    work page 2011

  31. [31]

    W., Starr , D

    Richards , J. W., Starr , D. L., Butler , N. R., et al. 2011, , 733, 10

    work page 2011

  32. [32]

    1991, Complex autoregressive model and its properties

    Sekita , I., Kurita , T., & Otsu , N. 1991, Complex autoregressive model and its properties

    work page 1991

  33. [33]

    2011, Characterization of Financial Time Series

    Sewell, M. 2011, Characterization of Financial Time Series

    work page 2011

  34. [34]

    2009, Bernoulli, 15, 178

    Tsai, H. 2009, Bernoulli, 15, 178

    work page 2009

  35. [35]

    1999, , 49, 223

    Udalski , A., Soszynski , I., Szymanski , M., et al. 1999, , 49, 223

    work page 1999

  36. [36]

    Uritskaya, O. Y. & Uritsky, V. M. 2015, Energy Economics, 49, 72

    work page 2015

  37. [37]

    R., Bryant, G., Snook, I

    Williams, S. R., Bryant, G., Snook, I. K., & van Megen, W. 2006, Physical review letters, 96, 087801

    work page 2006

  38. [38]

    & K \"u rster , M

    Zechmeister , M. & K \"u rster , M. 2009, , 496, 577

    work page 2009