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On the other hand, the spherical autoregressive model (SAR) considers the Fisher-Rao metric.<ref>{{Cite journal|last1=Zhu|first1=C.|last2=Müller|first2=H.-G.|date=2023|title=Spherical autoregressive models, with application to distributional and compositional time series|journal=Journal of Econometrics|volume=239 |issue=2 |doi=10.1016/j.jeconom.2022.12.008 |doi-access=free|arxiv=2203.12783}}</ref> Following the settings of [[##Tests for the intrinsic mean]], let <math>x_t \in \mathcal{X}</math> with Fréchet mean <math>\mu_x</math>. Let <math>\theta = \arccos(\langle x_t, \mu_x \rangle )</math>, which is the geodesic distance between <math>x_t</math> and <math>\mu_x</math>. Define a rotation operator <math>Q_{x_t, \mu_x}</math> that rotates <math>x_t</math> to <math>\mu_x</math>. The spherical difference between <math>x_t</math> and <math>\mu_x</math> is represented as <math>R_t = x_t \ominus \mu_x = \theta Q_{x_t, \mu_x}</math>. Assume that <math>R_t</math> is a stationary sequence with the Fréchet mean <math>\mu_R</math>, then <math>SAR(1)</math> is defined as
<math display="block">
R_t - \mu_R = \beta (R_{t-1} - \mu_R) + \epsilon_t,
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