Predicting asteroid lightcurves using ARIMA models

&amp;lt;p&amp;gt;The asteroids&amp;amp;#8217; brightness is not generally constant, but presents time variations that depend on the asteroid shape, albedo, rotation period, and other factors. In 1901, Von Oppolzer first noted changes in the brightness of (433) Eros (von Oppolzer 1901), and the first correct period for this asteroid was published in Bailey &amp;amp; Pickering (1913). Today, lightcurves, time-series data for brightness variations, are available for more than 10,000 asteroids, and periods are known for more than 8,300 bodies (Warner et al. 2009). Data from asteroid lightcurves can be used, through lightcurve inversion methods, to infer information on the asteroid shapes, rotation periods, and spin axis directions. The Database of Asteroid Models from Inversion Techniques (DAMIT, (Durech et al. 2010)) currently has models for 3303 asteroids. Asteroid photometric lightcurves in a &amp;amp;#8220;FITS-like&amp;amp;#8221;, self-describing, style are publicly available for several bodies at the Asteroid Lightcurve Data Exchange Format (ALCDEF), website (http://alcdef.org (Stephens &amp;amp; Warner 2018; Warner et al. 2011; Stephens et al. 2010)).&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;Asteroid lightcurves data are, in essence, a time series, and, as such, may be considered part of the discipline that Griffin et al. (2012) call &amp;amp;#8220;time-domain astronomy&amp;amp;#8221;. In statistical literature, dominant methods for studying time series are time-domain parametric modelling. Popular introductory (Brockwell &amp;amp; Davis 2016) and graduate-level (Shumway &amp;amp; Stoffer 2017) textbooks mostly focus on auto-regressive time-domain models, with only brief mentions of Fourier frequency-domain methods. Yet, while the latter methods have been used for several decades to study asteroid lightcurves and obtain asteroid periods (see for instance Lenz &amp;amp; Breger (2005); Szabo&amp;amp;#769; et al. (2016) and references therein), auto-regressive models, like the ARIMA methods, have not received as much attention. ARIMA stands for Auto-regressive (AR), integrated (I), moving average (MA). The AR part indicates that the variable of interest is regressed on its own lagged values. The MA indicates that the regression error is a linear combination of error terms that occurred at present and various past times. The I means that the variable values have been replaced by differences in their values and the previous ones.&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;In this work, we attempt to use ARIMA methods to model observed lightcurves of main-belt asteroids, as available at the ALCDEF database. The ideal lightcurve shape would display &amp;amp;#8220;smooth, double-waved, quasi-sinusoidal&amp;amp;#8221; form Cellino et al. (1989). This kind of perfect shape is, however, very rare. Some features deforming the lightcurve are very common, and, for the same asteroid, can be caused by changes of the aspect angle, the angle between the asteroid spin axis and the direction of observation of the body from Earth, at different observing epochs. Cellino et al. (1989) introduced four categories for classifying irregularities in the shape of lightcurves:&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;(i) Irregular morphology of one or more extrema&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;(ii) Different levels of the extrema&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;(iii) Different time intervals between the extrema&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;(iv) More or less than two maxima or minima per cycle&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;More recently, Harris et al. (2014) identified cases of very complex lightcurves with six maxima, for asteroids (5404) Uemura and 2010 RC130, which were later also retrieved by Szabo&amp;amp;#769; et al. (2016) for the asteroids (29628) 1998 TX30 and (37201) 2000WS94 observed during the K2 Kepler space telescope mission. For this work, we will define three categories of lightcurves:&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;(i) Regular lightcurves, with no prominent irregularities as defined by Cellino et al. (1989). Examples of asteroids with these kinds of lightcurves in the ALCDEF database are (8) Flora, (16) Psyche, and (593) Titania.&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;(ii) Complex lightcurves, with at least one shape irregularity. Examples are (21) Lutetia, (39) Laetitia, and (43) Ariadne.&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;(iii) Very complex lightcurves, as defined by Harris et al. (2014). Examples are (5404) Uemura and (37201) 2000WS94.&amp;lt;/p&amp;gt; &amp;lt;p&amp;gt;&amp;lt;img src=&amp;quot;data:image/jpeg;base64, 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