Auto-correlation functions of astrophysical processes, and their relation to Gaussian processes - Application to radial velocities of different starspot configurations

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Volume 645 (January 2021)

A&A, 645 (2021) A58

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Table 3

Largest likelihood solution found for every free adjustable parameter averaged over the five simulations of each spot configuration(ONE, POL, RAN, TWO).

Parameter	`ONE`	`POL`	`RAN`	`TWO`	priors	unit
QP: quasi-periodic kernel
h = κ_0,QP	2.94 ± 0.07	3.21 ± 0.17	2.89 ± 0.10	2.89 ± 0.10	[0, 5 σ_v]	m s⁻¹
P	24.96 ± 0.08	24.98 ± 0.10	25.00 ± 0.09	24.92 ± 0.10	[3.4, 50]	day
λ	66.98 ± 4.78	51.86 ± 0.43	51.73 ± 1.34	63.06 ± 5.46	[3.4, 4000]	day
w	0.358 ± 0.011	0.320 ± 0.011	0.282 ± 0.010	0.283 ± 0.013	[0,1]
γ	0.01 ± 0.16	−0.08 ± 0.11	−0.02 ± 0.10	−0.04 ± 0.03	[3 σ_v, +3 σ_v]	m s⁻¹
σ	0.04 ± 0.08	1.37 ± 0.22	0.91 ± 0.11	0.40 ± 0.07	[0, 3 σ_v]	m s⁻¹
	−1178.0 ± 33.9	−1507.0 ± 55.5	−1410.1 ± 14.4	−1287.0 ± 8.8
BIC	2394.1 ± 67.7	3052.1 ± 111.0	2858.3 ± 28.9	2612.2 ± 17.5
SHO: simple harmonic oscillator kernel
C₀	14.78 ± 1.78	11.64 ± 9.42	5.47 ± 0.68	10.68 ± 2.23	[0, 5 σ_v]	m² s⁻²
P	25.00 ± 0.02	17.34 ± 6.02	14.93 ± 4.82	12.49 ± 0.01	[3.4, 50]	day
P_life	1989.4 ± 455.5	89.84 ± 44.69	116.10 ± 15.27	420.8 ± 148.3	[3.4, 4000]	day
γ	0.00 ± 0.00	0.00 ± 0.01	0.00 ± 0.01	0.00 ± 0.01	[3 σ_v, +3 σ_v]	m s⁻¹
σ	2.72 ± 0.15	3.31 ± 1.61	3.60 ± 0.21	2.91 ± 0.11	[0, 3 σ_v]	m s⁻¹
	−1427.7 ± 28.7	−1693.2 ± 55.1	−1604.9 ± 29.6	−1488.1 ± 21.4
BIC	2887.1 ± 57.3	3418.1 ± 110.2	3241.5 ± 59.2	3007.9 ± 42.8
QPC: quasi-periodic and cosine kernel
h₁	3.84 ± 0.11	4.40 ± 0.37	3.86 ± 0.14	3.20 ± 0.15	[0, 5 σ_v]	m s⁻¹
h₂	1.12 ± 0.51	1.97 ± 0.26	1.94 ± 0.16	2.66 ± 0.26	[0, 5 σ_v]	m s⁻¹
P	24.96 ± 0.09	24.99 ± 0.09	24.99 ± 0.06	24.92 ± 0.11	[3.4, 50]	day
λ	135.32 ± 9.75	102.90 ± 1.68	102.60 ± 5.74	117.25 ± 4.31	[3.4, 4000]	day
γ	−0.03 ± 0.05	0.01 ± 0.09	−0.03 ± 0.07	−0.03 ± 0.07	[3 σ_v, +3 σ_v]	m s⁻¹
σ	0.04 ± 0.08	1.35 ± 0.23	0.93 ± 0.11	0.40 ± 0.08	[0, 3 σ_v]	m s⁻¹
	−1179.8 ± 34.1	−1502.6 ± 55.5	−1404.2 ± 15.2	−1266.5 ± 11.0
BIC	2397.7 ± 68.3	3043.4 ± 111.0	2846.6 ± 30.4	2571.1 ± 21.9
Additional parameters
	1.8 ± 0.1	1.9 ± 0.1	1.8 ± 0.1	1.8 ± 0.1		%
	2.3 ± 0.1	2.1 ± 0.8	1.4 ± 0.1	2.0 ± 0.2		%
	2.4 ± 0.1	2.9 ± 0.2	2.6 ± 0.1	2.5 ± 0.1		%
	28.1 ± 11.7	40.8 ± 5.4	45.0 ± 3.2	63.9 ± 4.1		%
	3.84 ± 0.23	3.41 ± 1.38	2.34 ± 0.15	3.27 ± 0.34		m s⁻¹
()	4.00 ± 0.18	4.82 ± 0.35	4.32 ± 0.14	4.17 ± 0.20		m s⁻¹
	−250	−186	−195	−201
	−1.8 ± 2.2	4.4 ± 1.7	5.9 ± 2.6	20.6 ± 4.5
	1.5 ± 2.1	4.6 ± 1.8	7.3 ± 3.0	22.6 ± 4.8
ΔBIC_{QP −SHO}	493	366	383	396
ΔBIC_{QP −QPC}	3.6 ± 4.3	−8.7 ± 3.4	−11.7 ± 5.3	−41.1 ± 9.0
ΔBIC_{QP −QPCw}	3.3 ± 3.8	−2.9 ± 3.0	−8.1 ± 4.6	−38.8 ± 8.2

Notes. We show the fitted parameters (from top to bottom) of the QP, SHO, and QPC kernel and calculate filling factors and likelihood differences.

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