Abstract
Real time series sometimes exhibit various types of “irregularities”: missing observations, observations collected not regularly over time for practical reasons, observation times driven by the series itself, or outlying observations. However, the vast majority of methods of time series analysis are designed for regular time series only. A particular case of irregularly spaced time series is that in which the sampling procedure over time depends also on the observed values. In such situations, there is stochastic dependence between the process being modelled and the times of the observations. In this work, we propose a model in which the sampling design depends on all past history of the observed processes. Taking into account the natural temporal order underlying available data represented by a time series, then a modelling approach based on evolutionary processes seems a natural choice. We consider maximum likelihood estimation of the model parameters. Numerical studies with simulated and real data sets are performed to illustrate the benefits of this model-based approach.
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Notes
This point process is stationary under the conditions given by Proposition 6.4. VII. from Daley and Vere-Jones (2003)
We use package yuima, with \(S_0=0\) and a discretization of time domain in 1600 points equally spaced.
Recall \(\omega \) is a reparametrization of \(\sigma \), defined in Eq. (12).
\(\hbox {MAPE}=\frac{1}{R}\sum \nolimits _{r=1}^{R}\frac{\left| t_{54}-{\widehat{t}}_{54,r}\right| }{t_{54}}\times 100\%\)
\(\hbox {MAE}=\frac{1}{R}\sum \nolimits _{r=1}^{R} \left| t_{54}-{\widehat{t}}_{54,r}\right| \)
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Acknowledgements
The authors acknowledge Foundation FCT (Fundação para a Ciência e Tecnologia) as members of the research project PTDC/MAT-STA/28243/2017 and Center for Research & Development in Mathematics and Applications of Aveiro University within project UID/MAT/04106/2019.
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Monteiro, A., Menezes, R. & Silva, M.E. Modelling informative time points: an evolutionary process approach. TEST 30, 364–382 (2021). https://doi.org/10.1007/s11749-020-00722-2
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DOI: https://doi.org/10.1007/s11749-020-00722-2