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arxiv: 2605.21884 · v1 · pith:FZSDIIWPnew · submitted 2026-05-21 · 📊 stat.ME

Trend and seasonality estimation for point-process time series

Daniel Gervini , Simon A. Kopischke This is my paper

Pith reviewed 2026-05-22 04:54 UTC · model grok-4.3

classification 📊 stat.ME
keywords point processestrend estimationseasonalityM-estimatorsdoubly-stochastic Poissonlog-Gaussian intensitytime series
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The pith

M-estimators recover trend and seasonality from point process time series under a log-Gaussian intensity model.

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

The paper introduces estimators for trend and seasonality in time series of point processes. It assumes the processes follow a temporal or spatial doubly-stochastic Poisson model with log-Gaussian intensity functions. The estimators are simple M-estimators whose asymptotic distributions are derived and whose finite-sample behavior is checked by simulation. The method is applied to bike demand patterns in Chicago's Divvy system. A sympathetic reader would care because many real event streams, from arrivals to spatial occurrences, carry superimposed long-term drifts and repeating cycles that are otherwise difficult to isolate directly.

Core claim

Under the doubly-stochastic Poisson model with log-Gaussian intensity, trend and seasonality parameters can be estimated by computationally simple M-estimators that possess explicit asymptotic normality and exhibit reliable performance in finite samples, as demonstrated both by simulation and by analysis of Chicago bike-sharing event data.

What carries the argument

M-estimators for the parameters of the log-Gaussian intensity function inside the doubly-stochastic Poisson framework

Load-bearing premise

The point processes follow a temporal or spatial doubly-stochastic Poisson model with log-Gaussian intensity functions.

What would settle it

Generate point-process data from an intensity function that is not log-Gaussian and check whether the M-estimators produce large bias or invalid asymptotic confidence intervals for the trend and seasonality components.

Figures

Figures reproduced from arXiv: 2605.21884 by Daniel Gervini, Simon A. Kopischke.

Figure 1
Figure 1. Figure 1: Divvy data analysis, Ashland & Wrightwood station. (a) Exponentiated trend, [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Divvy data analysis, Lasalle & Washington station. (a) Exponentiated trend, [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Divvy data analysis, Shedd Acquarium station. (a) Exponentiated trend, (b) [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
read the original abstract

This article introduces estimators of trend and seasonality for time series of point processes. We assume the point processes follow a temporal or spatial doubly-stochastic Poisson model with log-Gaussian intensity functions. The proposed estimators are computationally simple M-estimators. Their asymptotic distribution is derived, and their finite-sample performance is studied by simulation. As an example of real-data application, we study the patterns of bike demand in the Divvy bike-sharing system of the city of Chicago.

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 computationally simple M-estimators for trend and seasonality parameters in temporal or spatial point-process time series. The data-generating process is assumed to be a doubly-stochastic Poisson process whose log-intensity is a Gaussian process (or random field). Asymptotic distributions for the estimators are derived under this model, finite-sample behavior is examined via simulation, and the method is illustrated on Divvy bike-sharing counts in Chicago.

Significance. If the derivations are correct, the work supplies a tractable estimation procedure together with limiting normal distributions for a class of point-process models that arise in transportation, ecology, and epidemiology. The emphasis on computational simplicity and the inclusion of both simulation evidence and a real-data example are positive features. The approach could be adopted where event times are observed and a log-Gaussian Cox process is a reasonable working model.

major comments (2)
  1. [§3] §3 (Asymptotic theory): The central limit theorem and the explicit form of the asymptotic variance are derived under the exact assumption that the log-intensity is a Gaussian process. This assumption is load-bearing; the paper should state whether consistency and normality continue to hold under weaker conditions on the intensity (e.g., continuous but non-Gaussian random fields) or provide a counter-example showing failure.
  2. [Simulation study] Simulation section: All Monte Carlo experiments are generated from the assumed log-Gaussian doubly-stochastic Poisson model. A modest misspecification study (e.g., replacing the Gaussian field by a t-distributed or skewed random field with the same mean and covariance) would directly test whether the reported coverage and bias properties degrade outside the model.
minor comments (2)
  1. [§2] The objective function minimized by the M-estimators should be written explicitly (rather than described only as 'M-estimators') so that readers can verify the score and Hessian calculations used in the asymptotics.
  2. [Real-data example] In the Chicago application, the estimated trend and seasonal components should be accompanied by pointwise confidence bands derived from the asymptotic variance; this would make the practical output more informative.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive comments. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Asymptotic theory): The central limit theorem and the explicit form of the asymptotic variance are derived under the exact assumption that the log-intensity is a Gaussian process. This assumption is load-bearing; the paper should state whether consistency and normality continue to hold under weaker conditions on the intensity (e.g., continuous but non-Gaussian random fields) or provide a counter-example showing failure.

    Authors: The derivations in Section 3 rely on the Gaussianity of the log-intensity to establish the joint limiting normal distribution and the explicit form of the asymptotic variance via properties of Gaussian processes. Consistency of the M-estimators may extend to a broader class of ergodic intensity processes satisfying suitable moment conditions, but the central limit theorem and variance expression as stated are specific to the log-Gaussian case. We have revised the opening paragraph of Section 3 to make this dependence explicit and to note that extensions to non-Gaussian random fields remain an open question. revision: partial

  2. Referee: [Simulation study] Simulation section: All Monte Carlo experiments are generated from the assumed log-Gaussian doubly-stochastic Poisson model. A modest misspecification study (e.g., replacing the Gaussian field by a t-distributed or skewed random field with the same mean and covariance) would directly test whether the reported coverage and bias properties degrade outside the model.

    Authors: The Monte Carlo experiments are constructed to verify the finite-sample behavior of the estimators under the exact model assumptions used for the asymptotic theory. We have added a clarifying sentence at the end of the simulation section stating that all reported results assume the log-Gaussian intensity and that the robustness of the procedure under misspecification is left for future study. revision: partial

standing simulated objections not resolved
  • Providing a concrete counter-example demonstrating failure of asymptotic normality for non-Gaussian continuous random fields with the same mean and covariance structure.

Circularity Check

0 steps flagged

No circularity: derivations follow from stated model assumptions

full rationale

The paper assumes a doubly-stochastic Poisson process with log-Gaussian intensity, defines M-estimators for trend and seasonality, and derives their asymptotic distribution directly from the model properties (e.g., compensator and Gaussian process structure). Simulations are performed under the same assumed model, which is standard practice and does not reduce any prediction or result to a fitted input by construction. No self-citations, self-definitional steps, or renamings of known results are evident in the provided abstract or description. The central claims remain independent of the target quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests primarily on the domain assumption of a doubly-stochastic Poisson process with log-Gaussian intensity; no explicit free parameters or invented entities are named in the abstract, though M-estimators may implicitly involve choice of objective function.

axioms (1)
  • domain assumption Point processes follow a temporal or spatial doubly-stochastic Poisson model with log-Gaussian intensity functions
    Explicitly stated in the abstract as the modeling framework required for the estimators to apply.

pith-pipeline@v0.9.0 · 5591 in / 1285 out tokens · 69822 ms · 2026-05-22T04:54:39.465828+00:00 · methodology

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Baddeley, A. (2007). Spatial point processes and their applications. InStochastic Geom- etry,Lecture Notes in Mathematics18921–75. Springer, New York

    work page 2007

  2. [2]

    Begu, B., Panzeri, S., Arnone, E., Carey, M., and Sangalli, L.M. (2024). A nonparametric 16 penalized likelihood approach to density estimation of space–time point patterns. Spatial Statistics61100824

    work page 2024

  3. [3]

    Bouzas, P.R., Valderrama, M., Aguilera, A.M., and Ruiz-Fuentes, N. (2006). Modelling the mean of a doubly stochastic Poisson process by functional data analysis.Com- putational Statistics and Data Analysis502655–2667

    work page 2006

  4. [4]

    Bouzas, P.R., Ruiz-Fuentes, N., and Oca˜ na, F.M. (2007). Functional approach to the random mean of a compound Cox process.Computational Statistics22467–479

    work page 2007

  5. [5]

    (2006).Time Series: Theory and Methods (2nd ed.)

    Brockwell, P.J., and Davis, R.A. (2006).Time Series: Theory and Methods (2nd ed.)

    work page 2006

  6. [6]

    (1980).Point Processes.Chapman and Hall/CRC, Boca Raton

    Cox, D.R., and Isham, V. (1980).Point Processes.Chapman and Hall/CRC, Boca Raton

    work page 1980

  7. [7]

    (2013).Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, Third Edition.Chapman and Hall/CRC, Boca Raton

    Diggle, P.J. (2013).Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, Third Edition.Chapman and Hall/CRC, Boca Raton

    work page 2013

  8. [8]

    and Su, T.-l

    Diggle, P., Rowlingson, B. and Su, T.-l. (2005). Point process methodology for on-line spatio-temporal disease surveillance.Environmetrics16423–434

    work page 2005

  9. [9]

    J., Moraga, P., Rowlingson, B., and Taylor, B

    Diggle, P. J., Moraga, P., Rowlingson, B., and Taylor, B. M. (2013). Spatial and spatio- temporal log-Gaussian Cox processes: extending the geostatistical paradigm.Statis- tical Science28542–563. Fern´ andez-Alcal´ a, R.M., Navarro-Moreno, J., and Ruiz-Molina, J.C. (2012). On the esti- mation problem for the intensity of a DSMPP.Methodology and Computing ...

    work page 2013

  10. [10]

    Gervini, D. (2016). Independent component models for replicated point processes.Spatial Statistics18474–488

    work page 2016

  11. [11]

    and Khanal, M

    Gervini, D. and Khanal, M. (2019). Exploring patterns of demand in bike sharing systems via replicated point process models.Journal of the Royal Statistical Society Series C: Applied Statistics68585–602

    work page 2019

  12. [12]

    and Baur, T.J

    Gervini, D. and Baur, T.J. (2020). Joint models for grid point and response processes in longitudinal and functional data.Statistica Sinica301905–1924

    work page 2020

  13. [13]

    Gervini, D. (2025). Autocorrelation functions for point-process time series.Journal of Time Series Analysis. https://doi.org/10.1111/jtsa.70018. Horv´ ath, L., and Kokoszka, P. (2012).Inference for Functional Data with Applications

    work page doi:10.1111/jtsa.70018 2025

  14. [14]

    Huang, K., Chen, X., Guan, Y., and Li, Y. (2025). Partially linear single-index models and functional principal component analysis of spatially and temporally indexed point processes.Journal of Business and Economic Statistics431077–1091

    work page 2025

  15. [15]

    (2009).Robust Statistics (2nd ed.)John Wiley & Sons,

    Huber, P.J., and Ronchetti, E.M. (2009).Robust Statistics (2nd ed.)John Wiley & Sons,

    work page 2009

  16. [16]

    (2021).Object Oriented Data Analysis (1st ed.)

    Marron, J.S., and Dryden, I.L. (2021).Object Oriented Data Analysis (1st ed.). Chapman and Hall/CRC. Møller, J., Syversveen, A.R., and Waagepetersen, R.P. (1998). Log Gaussian Cox pro- cesses.Scandinavian Journal of Statistics25451–482. Møller, J., and Waagepetersen, R.P. (2004).Statistical Inference and Simulation for Spatial Point Processes. Chapman and...

    work page 2021

  17. [17]

    Nair, R., and Miller-Hooks, E. (2011). Fleet management for vehicle sharing operations. Transportation Science45524–540

    work page 2011

  18. [18]

    (1984).Convergence of Stochastic Processes.Springer, New York

    Pollard, D. (1984).Convergence of Stochastic Processes.Springer, New York

    work page 1984

  19. [19]

    Qiu, J., Dai, X., & Zhu, Z. (2024). Nonparametric estimation of repeated densities with heterogeneous sample sizes.Journal of the American Statistical Association119 176–188

    work page 2024

  20. [20]

    (2005).Functional Data Analysis (2nd ed.).Springer, New York

    Ramsay, J.O., and Silverman, B.W. (2005).Functional Data Analysis (2nd ed.).Springer, New York

    work page 2005

  21. [21]

    Ratcliffe, J. (2010). Crime mapping: spatial and temporal challenges. InHandbook of Quantitative Criminology, A.R. Piquero and D. Weisburd (eds.), pp. 5–24. Springer, New York. 18

    work page 2010

  22. [22]

    Shaheen, S., Guzman, S., and Zhang, H. (2010). Bike sharing in Europe, the Americas and Asia: Past, present and future.Transportation Research Record: Journal of the Transportation Research Board2143159–167

    work page 2010

  23. [23]

    (1991).Random Point Processes in Time and Space

    Snyder, D.L., and Miller, M.I. (1991).Random Point Processes in Time and Space

    work page 1991

  24. [24]

    (2010).Poisson Point Processes: Imaging, Tracking, and Sensing.Springer, New York

    Streit, R.L. (2010).Poisson Point Processes: Imaging, Tracking, and Sensing.Springer, New York

    work page 2010

  25. [25]

    Wu, S., M¨ uller, H.-G., and Zhang, Z. (2013). Functional data analysis for point processes with rare events.Statistica Sinica231–23

    work page 2013