FENRIR: A statistical model of stellar variability - I. A physics-based, fast Gaussian process model to represent stellar activity and perform statistical Doppler imaging

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Volume 696 (April 2025)

A&A, 696 (2025) A141

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Table 3

Cross-validation tests for the active-regions component of the model trained on Sun RV + photometry.

Active regions					Magnetic cycle kernel	Cross-validation score	RMS (m/s)

Model type	Modes	Period (d)	$\bar{i} (deg)$ $\bar i({\rm{deg}})$	Kernel			Train set	Test set
data-driven	2	${27.04}_{- 0.11}^{+ 0.11}$ ${27.04_{ - 0.11}^{ + 0.11}}$	–	M1/2	M1/2	$- 221.53 (- {223.69}_{- 2.50}^{+ 2.23})$ ${ - 221.53\left( { - 223.69_{ - 2.50}^{ + 2.23}} \right)}$	0.54 ± 0.02	1.06 ± 0.03
data-driven	2	${27.05}_{- 0.12}^{+ 0.12}$ ${27.05_{ - 0.12}^{ + 0.12}}$	–	M1/2	M3/2	$- 222.11 (- {224.47}_{- 2.61}^{+ 2.28})$ ${ - 222.11\left( { - 224.47_{ - 2.61}^{ + 2.28}} \right)}$	0.54 ± 0.02	1.08 ± 0.03
physical	sep., lat. dist.	${26.83}_{- 0.07}^{+ 0.07}$ ${26.83_{ - 0.07}^{ + 0.07}}$	${1.84}_{- 1.21}^{+ 1.83}$ ${1.84_{ - 1.21}^{ + 1.83}}$	M1/2	M1/2	$- 225.62 (- {227.00}_{- 1.76}^{+ 1.75})$ ${ - 225.62\left( { - 227.00_{ - 1.76}^{ + 1.75}} \right)}$	0.57 ± 0.02	1.04 ± 0.02
physical	sep., opp. lat.	${26.84}_{- 0.07}^{+ 0.07}$ ${26.84_{ - 0.07}^{ + 0.07}}$	${3.62}_{- 2.57}^{+ 4.10}$ ${3.62_{ - 2.57}^{ + 4.10}}$	M1/2	M1/2	$- 225.64 (- {227.02}_{- 1.82}^{+ 1.78})$ ${ - 225.64\left( { - 227.02_{ - 1.82}^{ + 1.78}} \right)}$	0.57 ± 0.02	1.04 ± 0.02
physical	sep., lat. dist.	${26.83}_{- 0.08}^{+ 0.07}$ ${26.83_{ - 0.08}^{ + 0.07}}$	${2.19}_{- 1.55}^{+ 2.52}$ ${2.19_{ - 1.55}^{ + 2.52}}$	M1/2	M3/2	$- 225.65 (- {227.05}_{- 1.94}^{+ 1.75})$ ${ - 225.65\left( { - 227.05_{ - 1.94}^{ + 1.75}} \right)}$	0.57 ± 0.02	1.04 ± 0.02
physical	sep., opp. lat.	${26.84}_{- 0.08}^{+ 0.07}$ ${26.84_{ - 0.08}^{ + 0.07}}$	${3.75}_{- 2.68}^{+ 4.48}$ ${3.75_{ - 2.68}^{ + 4.48}}$	M1/2	M3/2	$- 225.92 (- {227.22}_{- 1.79}^{+ 1.78})$ ${ - 225.92\left( { - 227.22_{ - 1.79}^{ + 1.78}} \right)}$	0.57 ± 0.02	1.04 ± 0.02
physical	sep., lat. dist.	${26.54}_{- 0.17}^{+ 0.15}$ ${26.54_{ - 0.17}^{ + 0.15}}$	${2.99}_{- 2.06}^{+ 3.73}$ ${2.99_{ - 2.06}^{ + 3.73}}$	M3/2	M1/2	$- 226.40 (- {227.58}_{- 1.51}^{+ 1.51})$ ${ - 226.40\left( { - 227.58_{ - 1.51}^{ + 1.51}} \right)}%>$	0.62 ± 0.02	1.06 ± 0.02
data-driven	2	${26.32}_{- 0.24}^{+ 0.25}$ ${26.32_{ - 0.24}^{ + 0.25}}$	–	M3/2	M1/2	$- 227.31 (- {230.16}_{- 3.23}^{+ 2.66})$ ${ - 227.31\left( { - 230.16_{ - 3.23}^{ + 2.66}} \right)}$	0.57 ± 0.02	1.20 ± 0.04
data-driven	2	${26.29}_{- 0.24}^{+ 0.24}$ ${26.29_{ - 0.24}^{ + 0.24}}$	–	M3/2	M3/2	$- 230.55 (- {234.06}_{- 3.62}^{+ 3.15})$ ${ - 230.55\left( { - 234.06_{ - 3.62}^{ + 3.15}} \right)}$	0.57 ± 0.02	1.28 ± 0.04
latent	2	${26.98}_{- 0.20}^{+ 0.22}, {22.40}_{- 0.41}^{+ 0.44}$ $26.98_{ - 0.20}^{ + 0.22},22.40_{ - 0.41}^{ + 0.44}$	–	ESP	M3/2	$- 233.43 (- {235.75}_{- 2.27}^{+ 2.13})$ $- 233.43\left( { - 235.75_{ - 2.27}^{ + 2.13}} \right)$	0.59 ± 0.02	1.19 ± 0.03
data-driven	1	${26.39}_{- 0.27}^{+ 0.29}$ ${26.39_{ - 0.27}^{ + 0.29}}$	–	M3/2	M1/2	$- 235.25 (- {238.99}_{- 3.33}^{+ 3.17})$ $- 235.25\left( { - 238.99_{ - 3.33}^{ + 3.17}} \right)$	0.60 ± 0.02	1.32 ± 0.04
physical	sep., opp. lat.	${20.41}_{- 0.16}^{+ 0.15}$ ${20.41_{ - 0.16}^{ + 0.15}}$	${6.75}_{- 4.83}^{+ 7.87}$ ${6.75_{ - 4.83}^{ + 7.87}}$	M3/2	M1/2	$- 238.13 (- {241.24}_{- 3.57}^{+ 2.89})$ ${ - 238.13\left( { - 241.24_{ - 3.57}^{ + 2.89}} \right)}$	0.56 ± 0.02	1.34 ± 0.03
data-driven	1	${27.06}_{- 0.12}^{+ 0.12}$ ${27.06_{ - 0.12}^{ + 0.12}}$	–	M1/2	M1/2	$- 238.46 (- {243.56}_{- 4.13}^{+ 3.59})$ $- 238.46\left( { - 243.56_{ - 4.13}^{ + 3.59}} \right)$	0.57 ± 0.04	1.31 ± 0.04
physical	mix., lat. dist.	${27.08}_{- 0.12}^{+ 0.12}$ ${27.08_{ - 0.12}^{ + 0.12}}$	${29.03}_{- 2.13}^{+ 2.27}$ ${29.03_{ - 2.13}^{ + 2.27}}$	M1/2	M1/2	$- 239.64 (- {241.29}_{- 2.12}^{+ 1.96})$ ${ - 239.64\left( { - 241.29_{ - 2.12}^{ + 1.96}} \right)}$	0.52 ± 0.02	1.33 ± 0.03
physical	mix., lat. dist.	${27.09}_{- 0.12}^{+ 0.11}$ ${27.09_{ - 0.12}^{ + 0.11}}$	${29.24}_{- 2.18}^{+ 2.09}$ ${29.24_{ - 2.18}^{ + 2.09}}$	M1/2	M3/2	$- 239.77 (- {241.14}_{- 1.94}^{+ 1.78})$ ${ - 239.77\left( { - 241.14_{ - 1.94}^{ + 1.78}} \right)}$	0.52 ± 0.02	1.33 ± 0.03
data-driven	1	${26.46}_{- 0.29}^{+0 .30}$ ${26.46_{ - 0.29}^{{\rm{ + 0}}{\rm{.30}}}}$	–	M3/2	M3/2	$- 240.61 (- {244.68}_{- 3.77}^{+ 3.27})$ $- 240.61\left( { - 244.68_{ - 3.77}^{ + 3.27}} \right)$	0.60 ± 0.02	1.42 ± 0.04
physical	sep., lat. dist.	${25.01}_{- 0.13}^{+ 0.09}$ ${25.01_{ - 0.13}^{ + 0.09}}$	${24.92}_{- 2.34}^{+ 3.26}$ ${24.92_{ - 2.34}^{ + 3.26}}$	M3/2	M3/2	$- 242.37 (- {247.13}_{- 3.45}^{+ 3.52})$ ${ - 242.37\left( { - 247.13_{ - 3.45}^{ + 3.52}} \right)}$	0.61 ± 0.01	1.39 ± 0.04
latent	1	${24.84}_{- 0.58}^{+ 0.58}$ ${24.84_{ - 0.58}^{ + 0.58}}$	–	ESP	M1/2	$- 242.58 (- {245.95}_{- 3.76}^{+ 3.03})$ ${ - 242.58\left( { - 245.95_{ - 3.76}^{ + 3.03}} \right)}$	0.52 ± 0.06	1.33 ± 0.02
latent	2	${34.35}_{- 0.17}^{+ 0.14}, {24.66}_{- 0.56}^{+ 0.60}$ ${34.35_{ - 0.17}^{ + 0.14},24.66_{ - 0.56}^{ + 0.60}}$	–	ESP	M1/2	$- 243.15 (- {246.32}_{- 3.48}^{+ 2.96})$ ${ - 243.15\left( { - 246.32_{ - 3.48}^{ + 2.96}} \right)}%>$	0.51 ± 0.06	1.36 ± 0.03
physical	sep., opp. lat.	${25.24}_{- 0.28}^{+ 0.31}$ ${25.24_{ - 0.28}^{ + 0.31}}$	${42.70}_{- 2.37}^{+ 2.06}$ ${42.70_{ - 2.37}^{ + 2.06}}$	M3/2	M3/2	$- 243.53 (- {249.07}_{- 3.13}^{+ 4.13})$ ${ - 243.53\left( { - 249.07_{ - 3.13}^{ + 4.13}} \right)}$	0.62 ± 0.02	1.41 ± 0.04
physical	mix., opp. lat.	${26.96}_{- 0.10}^{+ 0.10}$ ${26.96_{ - 0.10}^{ + 0.10}}$	${31.02}_{- 2.20}^{+ 2.19}$ ${31.02_{ - 2.20}^{ + 2.19}}$	M1/2	M3/2	$- 244.33 (- {245.94}_{- 2.13}^{+ 1.94})$ ${ - 244.33\left( { - 245.94_{ - 2.13}^{ + 1.94}} \right)}$	0.54 ± 0.02	1.41 ± 0.03
physical	mix., opp. lat.	${26.96}_{- 0.11}^{+ 0.11}$ ${26.96_{ - 0.11}^{ + 0.11}}$	${30.98}_{- 2.18}^{+ 2.32}$ ${30.98_{ - 2.18}^{ + 2.32}}$	M1/2	M1/2	$- 244.46 (- {246.21}_{- 2.15}^{+ 2.02})$ ${ - 244.46\left( { - 246.21_{ - 2.15}^{ + 2.02}} \right)}$	0.54 ± 0.03	1.41 ± 0.04
physical	mix., lat. dist.	${26.42}_{- 0.24}^{+ 0.24}$ ${26.42_{ - 0.24}^{ + 0.24}}$	${26.85}_{- 2.13}^{+ 2.42}$ ${26.85_{ - 2.13}^{ + 2.42}}$	M3/2	M3/2	$- 245.91 (- {246.99}_{- 1.77}^{+ 1.54})$ ${ - 245.91\left( { - 246.99_{ - 1.77}^{ + 1.54}} \right)}$	0.56 ± 0.01	1.49 ± 0.02
physical	mix., lat. dist.	${26.41}_{- 0.24}^{+ 0.25}$ ${26.41_{ - 0.24}^{ + 0.25}}$	${26.91}_{- 2.42}^{+ 2.66}$ ${26.91_{ - 2.42}^{ + 2.66}}$	M3/2	M1/2	$- 246.00 (- {247.30}_{- 1.89}^{+ 1.71})$ ${ - 246.00\left( { - 247.30_{ - 1.89}^{ + 1.71}} \right)}$	0.56 ± 0.01	1.50 ± 0.03
physical	mix., opp. lat.	${25.93}_{- 0.21}^{+ 0.21}$ $25.93_{ - 0.21}^{ + 0.21}$	${29.24}_{- 3.16}^{+ 3.48}$ $29.24_{ - 3.16}^{ + 3.48}$	M3/2	M3/2	$- 246.56 (- {247.63}_{- 1.81}^{+ 1.56})$ $- 246.56\left( { - 247.63_{ - 1.81}^{ + 1.56}} \right)$	0.58 ± 0.01	1.54 ± 0.03
physical	mix., opp. lat.	${25.94}_{- 0.22}^{+ 0.23}$ ${25.94_{ - 0.22}^{ + 0.23}}$	${29.28}_{- 3.15}^{+ 3.62}$ ${29.28_{ - 3.15}^{ + 3.62}}$	M3/2	M1/2	$- 246.69 (- {247.82}_{- 1.81}^{+ 1.62})$ ${ - 246.69\left( { - 247.82_{ - 1.81}^{ + 1.62}} \right)}$	0.58 ± 0.01	1.55 ± 0.03
data-driven	1	${27.08}_{- 0.12}^{+ 0.12}$ ${27.08_{ - 0.12}^{ + 0.12}}$	–	M1/2	M3/2	$- 248.03 (- {253.88}_{- 4.02}^{+ 3.83})$ ${ - 248.03\left( { - 253.88_{ - 4.02}^{ + 3.83}} \right)}$	0.56 ± 0.05	1.51 ± 0.07
latent	1	${24.72}_{- 0.56}^{+ 0.58}$ ${24.72_{ - 0.56}^{ + 0.58}}$	–	ESP	M3/2	$- 275.46 (- {277.60}_{- 2.63}^{+ 2.29})$ ${ - 275.46\left( { - 277.60_{ - 2.63}^{ + 2.29}} \right)}$	1.07 ± 0.02	1.20 ± 0.02

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