# Sunspot cycles are connected to the Earth and Jupiter

Lauri Jetsu Department of Physics,
P.O. Box 64,
FI-00014, University of Helsinki, Finland

###### Abstract

The sunspot number record covers over three centuries. These numbers measure
the activity of the Sun. This activity follows the solar cycle of about eleven
years. In the dynamo-theory, the interaction between differential rotation and
convection produces the solar magnetic field. On the surface of Sun, this
field concentrates to the sunspots. The dynamo-theory predicts that the
period, the amplitude and the phase of the solar cycle are stochastic. Here we
show that the solar cycle is deterministic, and connected to the orbital
motions of the Earth and Jupiter. This planetary-influence theory allows us to
model the whole sunspot record, as well as the near past and the near future
of sunspot numbers. We may never be able to predict the exact times of
exceptionally strong solar flares, like the catastrophic Carrington event in
September 1859, but we can estimate when such events are more probable. Our
results also indicate that during the next decades the Sun will no longer help
us to cope with the climate change. The inability to find predictability in
some phenomenon does not prove that this phenomenon itself is stochastic.

Earth-moon system(436) — Jupiter (873) — Solar-terrestrial interactions (1473)
– Sunspot number (1652) — The Sun (1693) — Time series analysis (1916) — Solar
dynamo (2001)

††journal: ApJ111Released on Xxx, xst, 20xx

##  1 Introduction

Schwabe (1844) discovered the 10 year cycle in the number of sunspots. Wolf
(1852) revised the cycle period to 11.1 years. This period is not constant. It
varies between 8 and 17 years (Lassen & Friis-Christensen, 1995). The
modulation amplitude of the sunspot number changes follows a cycle of about 80
years (Gleissberg, 1945). The solar surface magnetic field is strongest in the
sunspots. The polarity of this field is reversed during each solar cycle.
Therefore, the field geometry returns to its original state during the Hale
cycle of about 22 years (Hale et al., 1919).

There have been prolonged periods when very few sunspots, or even none at all,
were observed. Usoskin et al. (2007) identified the four recent grand sunspot
minima. They did not classify the most recent Dalton minimum (Komitov &
Kaftan, 2004), as a grand minimum. We give the epochs of these five most
recent activity minima in Table 1. The Earth’s climate temperature rises
during strong solar activity (Van Geel et al., 1999). The current prolonged
solar activity minimum helps us to cope with the currently ongoing climate
change.

The solar magnetic field has been shown to arise from the interaction between
differential rotation and convection (e.g. Brandenburg et al., 2017). This so-
called “dynamo-theory” can reproduce the observed quasi-periodic stochastic
sunspot cycle (e.g. Bhowmik & Nandy, 2018; Korpi-Lagg et al., 2022). It is
widely accepted that this variability is unpredictable beyond one solar cycle
(Petrovay, 2020). Nevertheless, countless sunspot cycle predictions are
constantly published (e.g. Asikainen & Mantere, 2023; Javaraiah, 2023;
Krasheninnikov & Chumakov, 2023). One persistent solar cycle regularity has
been the Gnevyshev–Ohl rule: every odd 11-year cycle has had a higher
amplitude than the preceding even cycle (Nagovitsyn et al., 2009).

Historically, there have been attempts to find a connection between the
planetary movements and the sunspot numbers (e.g. Schuster, 1911). Recently,
Stefani et al. (2019) proposed the existence of a tidally synchronized solar-
dynamo. The tidal forcing of Venus, Earth and Jupiter could modulate the
sunspot cycle (Cionco et al., 2023). The problem with the “planetary-
influence-theory” is that deterministic sunspot cycle predictions fail. Here,
we present successful deterministic sunspot number predictions.

Table 1: Recent periods of weak sunspot activity.  Minimum | Duration | Reference
---|---|---
Dalton | 1790 - 1830 | Komitov & Kaftan (2004)
Maunder | 1640 - 1720 | Usoskin et al. (2007)
Spörer | 1390 - 1550 | Usoskin et al. (2007)
Wolf | 1270 - 1340 | Usoskin et al. (2007)
Oort | 1010 - 1070 | Usoskin et al. (2007)

##  2 Data

We retrieved the sunspot data from the Solar Influences Data Analysis
Center222Source: WDC SILSO, Royal Observatory of Belgium, Brussels in December
2022. The data reductions, and the eight samples drawn from these data, are
described in our Appendix (Sect. A).

##  3 Analysis

We describe the Discrete Chi-square Method (DCM) in our Appendix (Sect. B).
For simplicity, we divide our $g(t)$ models (Eq. B1) into two categories

  * Pure sines ($K_{2}=1$)

  * Double waves ($K_{2}=2$)

The former category signal curves are pure sines. The latter category curves
can deviate from a pure sine.

We formulate the following solar dynamo-theory and planetary-influence-theory
hypotheses

  * $H_{\mathrm{D}}$: “The solar dynamo is a stochastic process. Therefore, all deterministic sunspot number predictions longer than one solar cycle fail.”

  * $H_{\mathrm{P}}$: “Planetary movements are deterministic. Therefore, sunspot number predictions longer than one solar cycle can succeed.”

We define clear criteria for measuring predictability. The data before the
year 2000 are “predictive data”. The data after 2000 are “predicted data”.
These predicted data samples are about two times longer than a typical 11
solar years cycle. The predictive and predicted data combination is called
“combined data”.

DCM analysis of predictive data gives the predicted data test statistic
$z^{\prime}$ value (Eq. B24). The combined data DCM analysis gives the
predicted model mean $g^{\prime\prime}_{\mathrm{mean}}$ value for any chosen
time interval (Eq. B25).

We use the following criteria to estimate predictablity.

  * C1:

The predicted data $z^{\prime}$ values decrease as long as new real signals
are detected in the predictive data. For unreal signals, these $z^{\prime}$
values begin to increase.

  * C2

Some planetary orbital period signals are detected in the predictive and/or
the combined data.

  * C3:

The predictive data signals are also detected in the combined data.

  * C4

The combined data $g^{\prime\prime}_{\mathrm{mean}}$ mean values can predict
some past activity minima in Table 1.

The proponents of $H_{\mathrm{D}}$ have claimed that none of these four
criteria is fulfilled. The proponents of $H_{\mathrm{P}}$ have not been able
to show that any of these criteria is fulfilled.

The DCM analysis results for all eight samples are summarized in Table 2. The
results for individual samples are given in Tables 10 -20.

There are clear connections between the five best periods in Table 2. We give
these one synodic period

$P_{\pm}=[P_{1}^{-1}\pm P_{2}^{-}]^{-1}$ (1)

connections in Table 3.

Table 2: Signals in monthly and yearly sunspot numbers. (1) Rmonthly2000 sample results for pure sines (Table 10: $K_{2}=1$). Strongest $11.0324\pm 0.0048$ year signal has a peak to peak amplitude $A=98.8\\!\\!\pm\\!\\!2.6$. Other five signals are given in order of decreasing $A$. (2-12) Results for other samples with chosen $K_{2}$ values. Double sinusoid signals are denoted with “2×”.  (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
---|---|---|---|---|---|---|---|---|---|---|---
Rmonthly2000 | Rmonthly2000 | Rmonthly | Rmonthly | Cmonthly2000 | Cmonthly2000 | Cmonthly | Cmonthly | Ryearly2000 | Ryearly | Cyearly2000 | Cyearly
$K_{2}=1$ | $K_{2}=2$ | $K_{2}=1$ | $K_{2}=2$ | $K_{2}=1$ | $K_{2}=2$ | $K_{2}=1$ | $K_{2}=2$ | $K_{2}=1$ | $K_{2}=1$ | $K_{2}=1$ | $K_{2}=1$
Table 10 | Table 11 | Table 12 | Table 13 | Table 14 | Table 15 | Table 16 | Table 17 | Table 18 | Table 19 | Table 20 | Table 21
Rank (-) | Rank (-) | Rank (-) | Rank (-) | Rank (-) | Rank (-) | Rank (-) | Rank (-) | Rank (-) | Rank (-) | Rank (-) | Rank (-)
$P$ (y) |  $P$ (y) |  $P$ (y) |  $P$ (y) |  $P$ (y) |  $P$ (y) |  $P$ (y) |  $P$ (y) |  $P$ (y) |  $P$ (y) |  $P$ (y) |  $P$ (y)
$A$ (-) |  $A$ (-) |  $A$ (-) |  $A$ (-) |  $A$ (-) |  $A$ (-) |  $A$ (-) |  $A$ (-) |  $A$ (-) |  $A$ (-) |  $A$ (-) |  $A$ (-)
(1) | (1) | (1) | (1) | (1) | (1) | (1) | (1) | (1) | (1) | (1) | (1)
$11.0324\\!\\!\pm\\!\\!0.0048$ | $11.0234\\!\\!\pm\\!\\!0.0068$ | $11.0033\\!\\!\pm\\!\\!0.0064$ | $10.9878\\!\\!\pm\\!\\!0.0051$ | $10.805\\!\\!\pm\\!\\!0.013$ | $10.807\\!\\!\pm\\!\\!0.012$ | $10.8585\\!\\!\pm\\!\\!0.0048$ |  $21.667\\!\\!\pm\\!\\!0.014$2× | $11.006\\!\\!\pm\\!\\!0.017$ | $10.981\\!\\!\pm\\!\\!0.020$ | $10.813\\!\\!\pm\\!\\!0.029$ | $10.863\\!\\!\pm\\!\\!0.022$
$98.8\\!\\!\pm\\!\\!2.6$ | $102.9\\!\\!\pm\\!\\!2.7$ | $101.5\\!\\!\pm\\!\\!2.5$ | $104.6\\!\\!\pm\\!\\!2.3$ | $118.2\\!\\!\pm\\!\\!3.7$ | $123.5\\!\\!\pm\\!\\!3.0$ | $117.5\\!\\!\pm\\!\\!2.9$ | $132.6\\!\\!\pm\\!\\!3.4$ | $90.4\\!\\!\pm\\!\\!4.7$ | $98.3\\!\\!\pm\\!\\!7.8$ | $119.2\\!\\!\pm\\!\\!8.5$ | $119.3\\!\\!\pm\\!\\!7.5$
(2) | (2) | (2) | (2) | (2) | (2) | (2) |  | (2) | (2) | (2) | (2)
$9.9842\\!\\!\pm\\!\\!0.0075$ | $9.9882\\!\\!\pm\\!\\!0.0061$ | $10.0001\\!\\!\pm\\!\\!0.0081$ |  $20.0062\\!\\!\pm\\!\\!0.0088$2× | $10.196\\!\\!\pm\\!\\!0.027$ | $10.196\\!\\!\pm\\!\\!0.017$ | $10.0658\\!\\!\pm\\!\\!0.0077$ |  | $9.980\\!\\!\pm\\!\\!0.016$ | $9.975\\!\\!\pm\\!\\!0.017$ | $10.184\\!\\!\pm\\!\\!0.064$ | $10.058\\!\\!\pm\\!\\!0.026$
$77.4\\!\\!\pm\\!\\!2.2$ | $78.6\\!\\!\pm\\!\\!2.6$ | $65.5\\!\\!\pm\\!\\!2.1$ | $75.2\\!\\!\pm\\!\\!2.7$ | $61.9\\!\\!\pm\\!\\!3.9$ | $62.1\\!\\!\pm\\!\\!3.7$ | $59.3\\!\\!\pm\\!\\!2.8$ |  | $64.6\\!\\!\pm\\!\\!7.6$ | $62.8\\!\\!\pm\\!\\!6.1$ | $62.3\\!\\!\pm\\!\\!6.8$ | $61.0\\!\\!\pm\\!\\!9.6$
(3) | (4) | (3) | (4) |  |  | (4) |  | (5) | (5) |  | (4)
$11.846\\!\\!\pm\\!\\!0.012$ |  $23.686\\!\\!\pm\\!\\!0.018$2× | $11.807\\!\\!\pm\\!\\!0.012$ | $11.770\\!\\!\pm\\!\\!0.011$ |  |  | $11.863\\!\\!\pm\\!\\!0.021$ |  | $11.862\\!\\!\pm\\!\\!0.34$ | $11.820\\!\\!\pm\\!\\!0.027$ |  | $11.856\\!\\!\pm\\!\\!0.068$
$60.0\\!\\!\pm\\!\\!2.2$ | $69.4\\!\\!\pm\\!\\!2.7$ | $56.4\\!\\!\pm\\!\\!2.4$ | $62.7\\!\\!\pm\\!\\!2.6$ |  |  | $43.3\\!\\!\pm\\!\\!2.0$ |  | $42.6\\!\\!\pm\\!\\!4.1$ | $40.5\\!\\!\pm\\!\\!3.3$ |  | $43.8\\!\\!\pm\\!\\!7.9$
(4) | (3) | (4) | (3) |  |  | (3) |  | (4) | (4) |  | (3)
$96.2\\!\\!\pm\\!\\!1.0$ | $104.16\\!\\!\pm\\!\\!0.59$ | $99.92\\!\\!\pm\\!\\!0.57$ | $104.23\\!\\!\pm\\!\\!0.68$ |  |  | $116.7\\!\\!\pm\\!\\!1.3$ |  | $100.3\\!\\!\pm\\!\\!2.1$ | $101.4\\!\\!\pm\\!\\!2.4$ |  | $115.6\\!\\!\pm\\!\\!3.6$
$50.1\\!\\!\pm\\!\\!2.8$ | $70.4\\!\\!\pm\\!\\!3.6$ | $51.8\\!\\!\pm\\!\\!2.3$ | $68.1\\!\\!\pm\\!\\!2.7$ |  |  | $52.9\\!\\!\pm\\!\\!2.5$ |  | $52.4\\!\\!\pm\\!\\!6.3$ | $51.7\\!\\!\pm\\!\\!4.9$ |  | $53.8\\!\\!\pm\\!\\!5.2$
(5) | (5) | (5) | (5) |  |  |  |  | (3) | (3) |  |
$10.541\\!\\!\pm\\!\\!0.014$ | $10.5372\\!\\!\pm\\!\\!0.0075$ | $10.569\\!\\!\pm\\!\\!0.020$ | $10.5407\\!\\!\pm\\!\\!0.0060$ |  |  |  |  | $10.620\\!\\!\pm\\!\\!0.033$ | $10.659\\!\\!\pm\\!\\!0.031$ |  |
$45.0\\!\\!\pm\\!\\!1.8$ | $51.2\\!\\!\pm\\!\\!1.8$ | $51.0\\!\\!\pm\\!\\!1.8$ | $55.6\\!\\!\pm\\!\\!2.3$ |  |  |  |  | $52.4\\!\\!\pm\\!\\!8.5$ | $59.9\\!\\!\pm\\!\\!8.6$ |  |
|  |  | (6) |  |  |  |  | (7) | (7) |  |
|  $16.781\\!\\!\pm\\!\\!0.034$2× |  |  $16.753\\!\\!\pm\\!\\!0.027$2× |  |  |  |  | $8.460\\!\\!\pm\\!\\!0.027$ | $8.466\\!\\!\pm\\!\\!0.016$ |  |
| $44.9\\!\\!\pm\\!\\!4.0$ |  | $41.2\\!\\!\pm\\!\\!3.1$ |  |  |  |  | $33.0\\!\\!\pm\\!\\!4.4$ | $31.1\\!\\!\pm\\!\\!4.8$ |  |
|  | (6) |  |  |  |  |  | (6) | (6) |  |
|  | $52.66\\!\\!\pm\\!\\!0.27$ |  |  |  |  |  | $53.24\\!\\!\pm\\!\\!0.78$ | $53.83\\!\\!\pm\\!\\!0.76$ |  |
|  | $33.9\\!\\!\pm\\!\\!2.0$ |  |  |  |  |  | $35.3\\!\\!\pm\\!\\!6.3$ | $34.3\\!\\!\pm\\!\\!3.6$ |  |
(6) | (8) | (7) | (8) |  |  | (5) |  |  |  |  | (5)
$8.1054\\!\\!\pm\\!\\!0.0084$ | $8.173\\!\\!\pm\\!\\!0.011$ | $8.1087\\!\\!\pm\\!\\!0.0076$ | $8.169\\!\\!\pm\\!\\!0.014$ |  |  | $8.009\\!\\!\pm\\!\\!0.017$ |  |  |  |  | $8.005\\!\\!\pm\\!\\!0.008$
$36.5\\!\\!\pm\\!\\!1.7$ | $37.6\\!\\!\pm\\!\\!4.8$ | $33.6\\!\\!\pm\\!\\!1.6$ | $35.5\\!\\!\pm\\!\\!3.3$ |  |  | $25.6\\!\\!\pm\\!\\!2.5$ |  |  |  |  | $24.0\\!\\!\pm\\!\\!7.0$
| (7) |  | (7) |  |  |  |  | (8) | (8) |  |
|  $136.43\\!\\!\pm\\!\\!0.99$2× |  |  $143.39\\!\\!\pm\\!\\!0.99$2× |  |  |  |  | $65.4\\!\\!\pm\\!\\!1.3$ | $66.7\\!\\!\pm\\!\\!1.4$ |  |
| $42.3\\!\\!\pm\\!\\!2.2$ |  | $37.3\\!\\!\pm\\!\\!2.7$ |  |  |  |  | $29.0\\!\\!\pm\\!\\!4.7$ | $26.7\\!\\!\pm\\!\\!4.4$ |  |
Table 3: Synodic period connections (Eq. 1). Periods $\sim 10$ and $\sim 11$ connect to $\sim 100$. Periods $\sim 11.86$ and $\sim 100$ connect to $\sim 10.5$. Period $\sim 10.5$ has no direct connection to $\sim 10$ or $\sim 11$.  (1) | (2) | (3) | (4) | (5) | (6)
---|---|---|---|---|---
| $P_{2}=10.5$ | $P_{2}=11.0$ | $P_{2}=11.86$ | $P_{2}=100$ |
$P_{1}=10.0$ | $P_{-}=210$ | $P_{-}=110$ | $P_{-}=63.7$ | $P_{-}=11.1$ |
| $P_{+}=5.12$ | $P_{+}=5.24$ | $P_{+}=5.42$ | $P_{+}=9.09$ |
$P_{1}=10.5$ |  | $P_{-}=231$ | $P_{-}=91.6$ | $P_{-}=11.73$ |
|  | $P_{+}=5.37$ | $P_{+}=5.57$ | $P_{+}=9.50$ |
$P_{1}=11.0$ |  |  | $P_{-}=152$ | $P_{-}=12.36$ |
|  |  | $P_{+}=5.71$ | $P_{+}=9.91$ |
$P_{1}=11.86$ |  |  |  | $P_{-}=13.4$ |
|  |  |  | $P_{+}=10.60$ |
Figure 1: Rmonthly2000 predictions (Table 10). (a) Predicted data $z^{\prime}$
for pure sine models $\mathcal{M}$=1-8. Best model $\mathcal{M}$=5 minimizes
$z^{\prime}$. Units are x-axis [$\mathcal{M}$]=dimensionless and y-axis
$[z^{\prime}]=$ dimensionless. (b) Black dots denote $y$. Blue dots are
residuals offset to level -300. Predictive data ends at vertical red line. Red
continuous line denotes $g(t)$ model $\mathcal{M}$=5. Dotted red lines denote
$\pm 3\sigma$ model error limits for $g(t)$. Green continuous line denotes
$g(t)$ thirty years sliding mean. Dotted green lines show $\pm 3\sigma$ error
limits for this mean. Units are x-axis $[t]=$ years and y-axis $[y]=$
dimensionless. (c) Horizontal black continuous lines outline Dalton and
Maunder minima. Except for a longer time span, notations are as in “b”. (d-f)
Rmonthly2000 predictions (Table 11). Best double wave model $\mathcal{M}$=6
minimizes $z^{\prime}$ (Table 11: $K_{2}=2$). Otherwise as in “a-c”.  Figure
2: Rmonthly predictions (Table 12). (a) Pure sine $\mathcal{M}$=1-8 model
predictions for $g^{\prime\prime}_{\mathrm{mean}}$ (Eq. B25) during Maunder
(red), Sporer (blue), Wolf (green) and Oort(yellow) minima. Dotted black line
shows the mean sunspot number for all $y$ data. Units are x-axis
[$\mathcal{M}$]=dimensionless and y-axis $[g^{\prime\prime}_{\mathrm{mean}}]=$
dimensionless. (b) $\mathcal{M}$=5 model for modern times and its prediction
after year 2022. (c) $\mathcal{M}$=5 model for past and future (Table 12).
(d-f) Double wave model predictions (Table 13). Otherwise as in “abc”.  Figure
3: Cmonthly predictions (Table 16). Pure sine $\mathcal{M}$=1-8 model
predictions for $g^{\prime\prime}_{\mathrm{mean}}$ during Dalton (black),
Maunder (red), Sporer (blue), Wolf (green) and Oort(yellow) minima. Otherwise
as in Fig. 2.  Figure 4: (a-c) Ryearly2000 predictions (Table 18). Otherwise
as in Fig. 1. (d-f) Ryearly predictions (Table 19). Otherwise as in Fig. 2
Figure 5: Cyearly predictions (Table 21). Otherwise as in Fig. 2.

###  3.1 Non-weighted monthly sunspot data

Sample Rmonthly2000 contains the predictive data. The predicted data are the
Rmonthly sample observtions after the year 2000. The whole Rmonthly sample
represents the combined data.

####  3.1.1 Pure sinusoids

We detect six pure sine signals in the predictive data (Table 2: Column 1).
The best predictability is achieved for five signals (Fig. 1a: smallest
$z^{\prime}$). Predictability is better for one signal than for two signals.
Then, it improves for three, four and five signals. For more than five
signals, predictability becomes worse. Furthermore, the seven and eight signal
models are unstable (Table 10: “${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$” models
$\mathcal{M}$=7 and 8). We compute also these unstable models to verify, if
predictability stops improving due to instability. Criterion C1 is fulfilled.

The five signal prediction after 2000 succeeds quite well until 2013 (Fig.
1b). We can reproduce the Dalton minimum because it is inside the predictive
data. The end of the Maunder minimum is also reproduced, but not the entire
Maunder minimum era (Fig. 1c).

The two best predictive data periods are $11.0324\pm 0.0048$ and $9.9842\pm
0.0075$ years (Table 2: Column 1). Within their error limits, these values are
nearly equal to 11 and 10 orbits of the Earth around the Sun. The third best
$11.846\pm 0.012$ years period is equal to the orbital period of Jupiter
($P_{\mathrm{J}}=11.86$ years). Criterion C2 is fulfilled.

The five strongest predictive data signals are also the strongest signals in
the combined data (Table 2, Columns 1 and 3). The two strongest $11.0033\pm
0.0064$ and $10.0001\pm 0.0081$ year signals for the combined data are exactly
equal to 11 and 10 years. The third strongest combined data $11.807\pm 0.012$
year period is close to Jupiter’s period. The fourth and the fifth strongest
signals are not planetary signals. Criterion C3 is also fulfilled.

The combined data Rmonthly sample $g^{\prime\prime}_{\mathrm{mean}}$ (Eq. B25)
predictions for the past activity minima are not convincing (Fig. 2a). Only
the blue and yellow circles denoting Sporer and Oort minima are below the
black horizontal dotted line denoting the mean level all sunspot numbers.
Criterion C4 is not fulfilled.

The combined data prediction for pure sines indicates that the long-term mean
solar activity level has just started to increase (Fig. 2b: green curve). The
long-term past and future predictions are shown in Fig. 2c.

####  3.1.2 Double waves

The eight strongest double wave signals for the predictive data are given in
Table 2 (Column 2). Six signals give the best predictability (Fig. 1d:
smallest $z^{\prime}$). This predictability parameter behaves as expected when
criterion C1 is fulfilled.  The prediction for the data after the year 2000 is
quite good for about twenty years, until a clear deviation takes place in 2020
(Fig. 1e). We can reproduce the Dalton minimum inside Rmonthly2000, as well as
the end of the Maunder minimum outside Rmonthly2000 (Fig. 1f).

The two strongest predictive data periods $11.0234\pm 0.0068$ and $9.9882\pm
0.0061$ are nearly equal to 10 and 11 years. The fourth strongest period
$23.686\pm 0.018$ is two times longer than Jupiter’s period $(2\times
11.86=23.72)$. We will hereafter refer to this type of signals as“double
sinusoids” (Table 2: highlighted with 2×). This double sinusoid of Jupiter is
shown in our Appendix (Fig. 13a). Criterion C2 is fulfilled, although the
third, the fifth and the sixth strongest periods are not planetary signals.

The two strongest $10.9878\pm 0.0051$ and $20.0062\pm 0.0088$ year signals in
the combined data are exactly equal to 11 and 20 years. The latter is a 10
year double sinusoid (Fig. 13b). The fourth strongest $11.770\pm 0.011$ year
period in the combined data deviates slightly from $P_{\mathrm{J}}$. Criterion
C3 is fulfilled because all eight predictive and combined data periods are the
same (Table 2: Columns 2 and 4).

The combined Rmonthly data prediction for the Maunder minimum is promising
(Fig. 2a: red curve). For the first three $\mathcal{M}$=1-3 signals,
$g^{\prime\prime}_{\mathrm{mean}}$ is above the mean level denoted with dotted
black line. For the next three $\mathcal{M}$=4-6 signals,
$g^{\prime\prime}_{\mathrm{mean}}$ falls below this mean level. Finally,
$g^{\prime\prime}_{\mathrm{mean}}$ rises above mean level in $\mathcal{M}$=7
and 8. Maunder minimum is the closest activity minimum before the beginning of
Rmonthly. It may also be a coincidence that the predictive data $z^{\prime}$
indicates the presence of six signals, and the
$g^{\prime\prime}_{\mathrm{mean}}$ changes support this. The
$g^{\prime\prime}_{\mathrm{mean}}$ prediction fails for other activity minima,
which are further away in the past than the Maunder minimum. The decreasing
Oort minimum trend deserves to be mentioned (Fig. 2d: yellow curve). We find
weak evidence for that Criterion C4 may be fulfilled.

The combined data prediction for the double waves also indicates that the mean
level of solar activity begins to rise in the near future (Fig. 2e). The red
prediction curve in this figure shows a turning point at 2029, which probably
does not represent the beginning of a new activity cycle. The long-term past
and future prediction confirms that the Maunder minimum can be at least partly
predicted (Fig. 2f).

###  3.2 Weighted monthly sunspot data

Sample Cmonthly2000 contains the predictive data. Unfortunately, the DCM
method can detect only two signals in this sample (Table 2: Columns 5 and 6).
Both pure sine and double wave models for more than two signals are unstable
(Tables 14 and 15: “${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$”).

For double wave models, we detect only one period in the combined data sample
Cmonthly (Table 2: Column 8). All other models for two or more signals are
unstable (Table 17: “${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$”).

####  3.2.1 Pure sines

Fortunately, we can detect five pure sine model signals in the combined data
sample Cmonthly (Table 2: Column 7). We do not study Criteria C1 and C3
because only two periods are detected in the predictive data.

In Cmonthly, the two strongest $10.8585\pm 0.0048$ and $10.0658\pm 0.0077$
year periods are close to 11 and 10 years. The fourth strongest $11.863\pm
0.021$ year signal is exactly equal to $P_{\mathrm{J}}$. The third strongest
$116.7\pm 1.3$ year signal does not appear to be a planetary one, but the
weakest $8.009\pm 0.017$ year period is exactly 8 orbits of the Earth around
the Sun. Criterion C2 is fulfilled.

The combined data sample Cmonthly yields promising
$g^{\prime\prime}_{\mathrm{mean}}$ predictions (Fig. 3a). All
$g^{\prime\prime}_{\mathrm{mean}}$ curves show a dip at model $\mathcal{M}$=3.
This dip is clearly below the dotted black line denoting the mean of all
monthly sunspot numbers. Except for the green Wolf minimum curve, all curves
are below this mean level. The Dalton minimum
$g^{\prime\prime}_{\mathrm{mean}}$ prediction is excellent, most probably
because this minimum is closest to, but still outside, the beginning of
Cmonthly sample (Fig. 3a: black curve). The $g^{\prime\prime}_{\mathrm{mean}}$
prediction for the second closest Maunder minimum is also excellent (Fig. 3a:
red curve). The C4 criterion is definitely fulfilled.

Table 4: Predicted four next sunspot minima and maxima. (1) Sample and model. (2-5) Minima for cycles 25-28. (6-9) Maxima for cycles 25-28. Lowest line gives weighted averages for five estimates of each column.  Sample: model | Minimum 25 | Minimum 26 | Minimum 27 | Minimum 28 |  | Maximum 25 | Maximum 26 | Maximum 27 | Maximum 28
---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) |  | (6) | (7) | (8) | (9)
Rmonthly: Five pure sines | $2031.57\pm 0.18$ | $2042.82\pm 0.20$ | $2053.76\pm 0.23$ | $2064.50\pm 0.28$ |  | $2025.86\pm 0.16$ | $2037.40\pm 0.20$ | $2048.41\pm 0.22$ | $2059.17\pm 0.25$
Rmonthly: Six double waves | $2029.0\pm 1.9$ | $2043.3\pm 5.0$ | $2054.1\pm 5.4$ | $2065.6\pm 4.8$ |  | $2024.42\pm 0.20$ | $2037.6\pm 4.9$ | $2047.9\pm 5.1$ | $2058.6\pm 5.3$
Cmonthly: Five pure sines | $2031.77\pm 0.28$ | $2042.35\pm 0.14$ | $2052.58\pm 0.18$ | $2064.21\pm 0.13$ |  | $2024.07\pm 0.19$ | $2037.39\pm 0.13$ | $2047.48\pm 0.14$ | $2058.85\pm 0.20$
Ryearly: Six pure sines | $2030.77\pm 0.38$ | $2041.79\pm 0.38$ | $2052.62\pm 0.41$ | $2063.19\pm 0.44$ |  | $2025.28\pm 0.39$ | $2036.62\pm 0.41$ | $2047.40\pm 0.41$ | $2057.88\pm 0.41$
Cyearly: Five pure sine | $2031.8\pm 1.1$ | $2042.3\pm 2.0$ | $2052.7\pm 2.2$ | $2064.1\pm 2.6$ |  | $2024.36\pm 0.78$ | $2037.4\pm 1.3$ | $2047.5\pm 2.0$ | $2058.8\pm 2.4$
Weighted mean | $2031.50\pm 0.36$ | $2042.44\pm 0.29$ | $2052.98\pm 0.56$ | $2064.19\pm 0.29$ |  | $2024.94\pm 0.79$ | $2037.34\pm 0.19$ | $2047.72\pm 0.41$ | $2058.84\pm 0.39$
Figure 6: True of anomalies the Earth and Jupiter at maximum and minimum epochs of Table 5 signals. (a) $\sim 10$ year signals. Large and small circles denote $t_{\mathrm{max}}$ and $t_{\mathrm{min}}$ epochs, respectively (Table 12: Red, Table 13: Blue, Table 16: Green, Table 19: Black, Table 21: Yellow). Note how small and large red circles overlap because period $P=10.0001$ is so close to integer value 10. Units are x-axis $[t]$=years and y-axis $[\nu]=$ degrees. (b) $\sim 11.0$ year signals. Otherwise as in “a”. (c) $\sim 11.86$ year signals. Otherwise as in “a”.  Figure 7: Monthly sunspot data. (a) Upper panel shows distance between the Sun and Jupiter for years 1750-1880. Epochs of maximum and minimum distance are denoted with vertical black continuous and dotted lines, respectively. Units are x-axis $[t]=$ years and y-axis $[r_{\mathrm{J}}]=$ AU. Lower panel shows simultaneous $h_{4}(t)$ pure sine $11.807$ year signal of model $\mathcal{M}$=4 in Table 12. Units are x-axis $[t]=$ years and y-axis $[h_{4}(t)]=$ dimensionless. (b) Same as in “a” after 1880. (c-d) $h_{2}(t)$ double wave $11.770$ year signal of model $\mathcal{M}$=4 in Table 13. Otherwise as in “a-b”. (c-d) $h_{3}(t)$ pure sine $11.863$ year signal of model $\mathcal{M}$=4 in Table 16. Otherwise as in “a-b”.  Figure 8: Yearly sunspot data. (a-b) $h_{2}(t)$ pure sine $11.820$ year signal of model $\mathcal{M}$=8 in Table 19. Otherwise as in Fig. 7a-b. (c-d) $h_{3}(t)$ pure sine $11.856$ year signal of model $\mathcal{M}$=4 in Table 21. Otherwise as in Fig. 7a-b.  Table 5: Epochs for true anomalies in Fig. 6. (1) Table containing signal. (2-4) Period $P$, primary minimum epoch $t_{\mathrm{min,}}$ and primary maximum $t_{\mathrm{min,}}$ epoch for signals close to 10 years. (5-7) Same for signals close to 10.5 years. (8-10) Same for signals close to 11.85 years.  (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) |
---|---|---|---|---|---|---|---|---|---|---
|  $\sim 10$ signals |  $\sim 11$ signals |  $\sim 11.85$ signals |
Table | $P$ | $t_{\mathrm{1,min}}$ | $t_{\mathrm{1,max}}$ | $P$ | $t_{\mathrm{1,min}}$ | $t_{\mathrm{1,max}}$ | $P$ | $t_{\mathrm{1,min}}$ | $t_{\mathrm{1,max}}$ |
12 | 10.0001 | 1754.16 | 1749.16 | 11.0033 | 1755.897 | 1750.40 | 11.807 | 1760.33 | 1754.42 |
13 |  |  |  | 10.9878 | 1756.678 | 1750.262 | 11.770 | 1749.51 | 1754.40 |
16 | 10.0658 | 1823.52 | 1818.48 | 10.8585 | 1823.580 | 1818.150 | 11.863 | 1818.92 | 1824.85 |
19 | 9.975 | 1704.63 | 1709.62 | 10.981 | 1700.86 | 1706.35 | 11.820 | 1701.01 | 1706.91 |
21 | 10.058 | 1823.55 | 1818.52 | 10.863 | 1823.56 | 1828.99 | 11.856 | 1818.95 | 1824.88 |

###  3.3 Non-weighted yearly sunspot data

Sample Ryearly2000 is the predictive data. The predicted data are the Ryearly
observations made after 2000. The combined data sample is Ryearly.

The four and eight signal double wave models have $\eta=21$ and $\eta=21+20$
free parameters, respectively. Sample Ryearly2000 contains only $n=300$
observations. To avoid over-fitting, we analyse these data only with the pure
sine models.

####  3.3.1 Pure sines

The predictive data $z^{\prime}$ curve indicates the presence of six signals
(Fig. 4a). The Ryearly2000 prediction for the data after 2000 is amazingly
accurate (Fig. 4b). Had we applied DCM to the Ryearly2000 sample in the year
2000, we could have predicted the yearly sunspots number for the next two
decades! The six signal model can reproduce the Dalton minimum inside the
predictive data (Fig. 4c). Criterion C1 is fulfilled.

The two strongest $11.006\pm 0.017$ and $9.980\pm 0.016$ year signal periods
are close to integers 10 and 11 (Table 2: Column 9). The fifth strongest
$11.862\pm 0.034$ year signal period is equal to $P_{\mathrm{J}}$. The other
remaining periods show no clear planetary connection. Criterion C2 is
fulfilled.

The same periods are detected in the predictive and the combined data (Table
2: Columns 9 and 10). These periods include the $11.820\pm 0.027$ year period
equal to Jupiter’s period. Criterion C3 is fulfilled.

The $g^{\prime\prime}_{\mathrm{mean}}$ prediction curves of Ryearly do not
support Criterion C4 (Fig. 4d). The prediction for the next decades indicates
rising solar activity (Fig. 4e). The longer past and future predictions are
shown in Fig. 4f.

###  3.4 Weighted yearly sunspot data

The Cyearly sample size is too small for double wave model analysis $(n=203)$
. We detect only two pure sine signals in Cyearly2000 predictive data (Table
2: Column 11). Additional signals give unstable models (Table 21:
“${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$”). For this reason, we can not study
Criteria C1 and C3.

####  3.4.1 Pure sines

We detect five signals in sample Cyearly. The two strongest $10.863\pm 0.022$
and $10.058\pm 0.026$ year periods are close to 11 and 10 years (Table 2:
Column 12). The fourth strongest $11.856\pm 0.068$ year period is exactly the
same as Jupiter’s period. The fifth strongest $8.005\pm 0.008$ year period is
close to an integer 8 value. The remaining $115.6\pm 3.6$ signal does not
appear to be connected to planets. Criterion C2 is fulfilled.

The $g^{\prime\prime}_{\mathrm{mean}}$ predictions for the Dalton and the
Maunder minimum indicate that Criterion C4 is fulfilled (Fig. 5a: Black and
red curves). These two activity minima occurred before Cyearly. These results
are not unexpected, because Dalton and Maunder minima are closest to this
sample.

The Cyearly sample prediction for the next decades also supports rising level
of solar activity (Fig. 5b). The long-term prediction illustrates how well we
can reproduce the Dalton and the Maunder minima (Fig. 5c).

###  3.5 Future sunspot maxima and minima predictions

We give the predictions for the next four sunspot mimima and maxima in Table
4.

The six double wave model predictions for the largest Rmonthly sample are not
very accurate. The probable reason for this inaccuracy is the large number
$\eta=21+10$ of free parameters (Table 13: model $\mathcal{M}$=6). This model
is also the only one having a turning point during the year 2029 (Fig. 2e).

The accuracy of our prediction for the next cycle 25 maximum is much lower
than that for the next maxima. This is a real effect. The predicted peak of
this next cycle 25 maximum is lower, and not so sharp, as the predicted peaks
of the next maxima (Figs. 2b, 2e, 3b, 4e and 5b).

###  3.6 Connections to the Earth and Jupiter orbits

The maximum and minimum epochs of $\sim 10$, $\sim 11$ and $\sim 11.86$ year
signals are given in Table 5. We exclude the $20.0062\pm 0.0088$ year signal
(Table 13, $\mathcal{M}$=4). The period of this double sinusoid signal is
exactly two times 10 years (Fig. 13b). However, the alternative of dividing
this signal into two parts would only confuse our next analysis. We compute
the true anomalies $\nu$ of the Earth and Jupiter at the epochs of Table 5,
and show the results in Fig. 6.

At first sight, it seems that there is no sense in the true anomalies of the
Earth, if they are computed from the $\sim 10$ year signal maximum epochs and
minimum epochs (Fig. 6a: large and small circles). However, this is true only
for the smaller samples (green, black and yellow circles). For the largest
Rmonthly sample, the true anomalies for $\sim 10$ year signal maximum and
minimum epochs coincide at $\nu=57^{\circ}$ (red circles). This means that the
strongest amplification of sunspot activity occurs at at $\nu=57^{\circ}$.
After five years, the strongest damping occurs also at $\nu=57^{\circ}$. One
possible explanation for this regularity could be that the time intervals when
the true anomaly difference between the Earth and Jupiter increases, or
decreases, are equal. If Jupiter were the hour hand, the Earth would be the
minute hand.

When the Earth’s $\nu$ values are computed at the minima and maxima of $\sim
11$ year signal, the results for smaller samples are confusing (Fig. 6b: blue,
green, black and yellow circles). However, the largest Rmonthly sample maxima
and minima show only a minor shift during over 250 years (red circles). The
maximum amplification of sunspots has occurred at anomalies close to
$\nu=150^{\circ}$. After 5.5 years, the strongest damping of sunspots has
taken place at true anomalies close to $\nu=330^{\circ}$. One explanation for
this regularity could be that the synodic period of Jupiter is 398.88 days.
Hence, ten synodic periods are equal to 10.92 years.

Our results for Jupiter’s true anomaly at the $\sim 11.86$ signal minima and
maxima are undeniably impressing (Fig. 6c: all circles of all samples). The
strongest amplification of sunspots occurs close to $\nu=120^{\circ}$. The
damping of sunspots occurs about 6 years (11.86/2) later at $\nu=300^{\circ}$.
Furthermore, the Earth’s $\sim 10$ year signal and Jupiter’s $\sim 11.86$ year
signal true anomaly dependencies resemble each other (Figs. 6b: red circles
compared to Fig. 6c: all circles).

The distance between the Sun and Jupiter modulates the number of sunspots
(Figs. 7 and 8: blue curves). The strongest amplification occurs between
perihelion and aphelion, as already noticed in the study of true anomalies.
Note that all amplitudes of the blue monthly and yearly sunspot modulation
curves are nearly the same (Figs. 7 and 8). This result could not have been
inferred from the true anomaly changes.  In short, the period, the amplitude
and the phase of every $\sim 11.86$ year sunspot signal are synchronized with
the true anomaly and the distance of Jupiter from the Sun.

##  4 Discussion

The results of our analysis strongly support the planetary-influence
hypothesis $H_{\mathrm{P}}$. These results contradict the dynamo-theory
hypothesis $H_{\mathrm{D}}$, because the C1-C4 criteria are fulfilled. Apart
from these criteria, the following results also support the validity of our
analysis.

  1. 1.

The extremely high Fisher-test significance estimates $Q_{F}$ $<\\!10^{-16}$
$\ll\gamma=0.001$ (Eq. B23) confirm that the signal detections are real.333The
highest achievable accuracy for the computational f.cdf subroutine in
scipy.optimize python library is $10^{-16}$. Simple models rarely beat complex
models, like $\mathcal{M}$=7 $\leftarrow$ $\mathcal{M}$=8 in Table 21.

  2. 2.

These signal detections are consistent. If a new signal is detected, the
previously detected signal periods, amplitudes and phases are also re-
detected. Without this consistency, our analysis could be considered
unreliable.

  3. 3.

These signals can predict future data, because the detection of each new
signal improves predictability as long as $z^{\prime}$ increases (Eqs. B25).

  4. 4.

The periods in all eight samples are consistent.

a. We detect the same best periods in the monthly and the yearly sunspot data.

b. We detect the same best periods in the non-weighted and the weighted data.

c. The one synodic period rule connects all five best periods (Table 3: Eq.
1).

  5. 5.

The detected signals can “predict” past activity minima (e.g. Fig. 1c: Dalton
and Maunder).

  6. 6.

We find clear connections between sunspots and the planetary movements, like
the true anomaly and distance of Jupiter (Figs. 6c, 7 and 8).

  7. 7.

Jupiter and Earth synodic period connections (Table 3) tie the movements of
these planets to the sunspots. The mass of Earth-Moon system exceeds the mass
of Mercury, Venus, or Mars. The two largest masses close to the Sun are the
Earth-Moon system and Jupiter

  8. 8.

The results for different samples show these common features:

  9. a

The long-term mean level of the Sun is going to increase for the next decades
(Figs. 2cf, 3c, 4f, 5c). The temperature of Earth’s climate rises when solar
activity increases (Van Geel et al., 1999). The currently ongoing grand solar
activity minimum helps us to cope with the climate change.

  10. b

Our successful predictions for the “near past” (Fig. 4b), and “long past”
(Fig. 5ac), support the idea that our “near future” predictions after the year
2022 are also correct.

The following ideas deserve to be mentioned.

  1. 1.

Solar surface rotation is faster at the equator, and slows down towards poles.
The orbital planes of planets are close to, but do not coincide with, the
solar equatorial plane. If the planets can modify solar differential rotation
and/or convection, the basic ideas of solar dynamo can still work, except that
the sunspot cycle is not stochactic.

  2. 2.

Carrington event has been connected to the strong solar flare on September 1
1959. The geomagnetic storm lasted from 1 to 2 September. At that time, the
true anomalies of Earth and Jupiter were 233o and 84o, respectively. The next
sunspot maximum took place a few months later. All this fits to the idea that
the influence of Jupiter is strongest at anomalies of $\nu\approx
120^{\mathrm{o}}$ (Fig. 6c).

  3. 3.

If the sunspot numbers are predictable, we are better prepared for the strong
solar flares causing hazards like the Carrington events, or for the currently
ongoing climate change.

  4. 4.

The long-lived starspots in close synchronized binaries tend to concentrate on
the line connecting centres of the two members (Jetsu et al., 2017), as
already predicted by Moss & Tuominen (1997). Can the Earth and Jupiter cause a
similar, but a weaker, effect in the Sun?

  5. 5.

We do not detect other planetary signals from the sunspot data. Considering
the cpu-times already spent, we must leave those detections to others.

##  5 Conclusions

No phenomenon is stochastic only because it appears unpredictable. Phenomena
may appear stochastic because we can not yet predict them. Our sunspot
predictions are far too precise to fit to the dynamo-theory hypothesis about a
stochastic sunspot cycle. From the time series point of view, this is a clear
cut case. Everything fits to the planetary-influence hypothesis.

ACKNOWLEDGEMENTS: We thank the Finnish Computing Competence Infrastructure
(FCCI) for supporting this project with computational resources. We thank Juha
Helin, Jani Jaakkola, Sami Maisala and Pasi Vettenranta who helped us to
utilize parallel computation resources in the High Performance Computing (HPC)
platform. This work has made use of NASA’s Astrophysics Data System (ADS)
services.

## References

  * Allen (2004) Allen, M. 2004, Understanding Regression Analysis (Springer US), 113–117.  https://books.google.hn/books?id=XQIjngEACAAJ
  * Asikainen & Mantere (2023) Asikainen, T., & Mantere, J. 2023, Prediction of even and odd sunspot cycles, arXiv e-prints, arXiv:2309.04208, doi: 10.48550/arXiv.2309.04208
  * Bhowmik & Nandy (2018) Bhowmik, P., & Nandy, D. 2018, Prediction of the strength and timing of sunspot cycle 25 reveal decadal-scale space environmental conditions, Nature Communications, 9, 5209, doi: 10.1038/s41467-018-07690-0
  * Brandenburg et al. (2017) Brandenburg, A., Mathur, S., & Metcalfe, T. S. 2017, Evolution of Co-existing Long and Short Period Stellar Activity Cycles, ApJ, 845, 79, doi: 10.3847/1538-4357/aa7cfa
  * Breger et al. (2002) Breger, M., Garrido, R., Handler, G., et al. 2002, 29 frequencies for the $\delta$ Scuti variable BI CMi: the 1997-2000 multisite campaigns, MNRAS, 329, 531, doi: 10.1046/j.1365-8711.2002.04970.x
  * Cionco et al. (2023) Cionco, R. G., Kudryavtsev, S. M., & Soon, W. W. H. 2023, Tidal Forcing on the Sun and the 11-Year Solar-Activity Cycle, Sol. Phys., 298, 70, doi: 10.1007/s11207-023-02167-w
  * Draper & Smith (1998) Draper, N. R., & Smith, H. 1998, Applied Regression Analysis (John Wiley & Sons, Inc.), doi: 10.1002/9781118625590
  * Efron & Tibshirani (1986) Efron, B., & Tibshirani, R. 1986, Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy, Statistical Science, 1, 54
  * Gleissberg (1945) Gleissberg, W. 1945, Evidence for a long solar cycle, The Observatory, 66, 123
  * Hale et al. (1919) Hale, G. E., Ellerman, F., Nicholson, S. B., & Joy, A. H. 1919, The Magnetic Polarity of Sun-Spots, ApJ, 49, 153, doi: 10.1086/142452
  * Handler (2003) Handler, G. 2003, Merging Data from Large and Small Telescopes – Good or Bad? And: How Useful is the Application of Statistical Weights to Time-Series Photometric Measurements?, Baltic Astronomy, 12, 253, doi: 10.1515/astro-2017-0049
  * Javaraiah (2023) Javaraiah, J. 2023, Prediction for the amplitude and second maximum of Solar Cycle 25 and a comparison of the predictions based on strength of polar magnetic field and low-latitude sunspot area, MNRAS, 520, 5586, doi: 10.1093/mnras/stad479
  * Jetsu (2020) Jetsu, L. 2020, Discrete Chi-square Method for Detecting Many Signals, The Open Journal of Astrophysics, 3, 4, doi: 10.21105/astro.2002.03890
  * Jetsu (2021) —. 2021, Say Hello to Algol’s New Companion Candidates, ApJ, 920, 137, doi: 10.3847/1538-4357/ac1351
  * Jetsu et al. (2017) Jetsu, L., Henry, G. W., & Lehtinen, J. 2017, General Model for Light Curves of Chromospherically Active Binary Stars, ApJ, 838, 122, doi: 10.3847/1538-4357/aa65cb
  * Jetsu & Pelt (1999) Jetsu, L., & Pelt, J. 1999, Three stage period analysis and complementary methods, A&AS, 139, 629, doi: 10.1051/aas:1999411
  * Komitov & Kaftan (2004) Komitov, B., & Kaftan, V. 2004, in Multi-Wavelength Investigations of Solar Activity, ed. A. V. Stepanov, E. E. Benevolenskaya, & A. G. Kosovichev, Vol. 223, 113–114, doi: 10.1017/S1743921304005307
  * Korpi-Lagg et al. (2022) Korpi-Lagg, M. J., Korpi-Lagg, A., Olspert, N., & Truong, H. L. 2022, Solar-cycle variation of quiet-Sun magnetism and surface gravity oscillation mode, A&A, 665, A141, doi: 10.1051/0004-6361/202243979
  * Krasheninnikov & Chumakov (2023) Krasheninnikov, I. V., & Chumakov, S. O. 2023, Predicting the Functional Dependence of the Sunspot Number in the Solar Activity Cycle Based on Elman Artificial Neural Network, Geomagnetism and Aeronomy, 63, 215, doi: 10.1134/S0016793222600904
  * Lassen & Friis-Christensen (1995) Lassen, K., & Friis-Christensen, E. 1995, Variability of the solar cycle length during the past five centuries and the apparent association with terrestrial climate., Journal of Atmospheric and Terrestrial Physics, 57, 835, doi: 10.1016/0021-9169(94)00088-6
  * Lehtinen et al. (2011) Lehtinen, J., Jetsu, L., Hackman, T., Kajatkari, P., & Henry, G. W. 2011, The continuous period search method and its application to the young solar analogue HD 116956, A&A, 527, A136, doi: 10.1051/0004-6361/201015454
  * Moss & Tuominen (1997) Moss, D., & Tuominen, I. 1997, Magnetic field generation in close binary systems., A&A, 321, 151
  * Nagovitsyn et al. (2009) Nagovitsyn, Y. A., Nagovitsyna, E. Y., & Makarova, V. V. 2009, The Gnevyshev-Ohl rule for physical parameters of the solar magnetic field: The 400-year interval, Astronomy Letters, 35, 564, doi: 10.1134/S1063773709080064
  * Petrovay (2020) Petrovay, K. 2020, Solar cycle prediction, Living Reviews in Solar Physics, 17, 2, doi: 10.1007/s41116-020-0022-z
  * Rodríguez et al. (2003) Rodríguez, E., Costa, V., Handler, G., & García, J. M. 2003, Simultaneous uvby photometry of the new delta Sct-type variable HD 205, A&A, 399, 253, doi: 10.1051/0004-6361:20021749
  * Schuster (1911) Schuster, A. 1911, The Influence of Planets on the Formation of Sun-Spots, Proceedings of the Royal Society of London Series A, 85, 309, doi: 10.1098/rspa.1911.0046
  * Schwabe (1844) Schwabe, M. 1844, Sonnenbeobachtungen im Jahre 1843. Von Herrn Hofrath Schwabe in Dessau, Astronomische Nachrichten, 21, 233
  * Stefani et al. (2019) Stefani, F., Giesecke, A., & Weier, T. 2019, A Model of a Tidally Synchronized Solar Dynamo, Sol. Phys., 294, 60, doi: 10.1007/s11207-019-1447-1
  * Usoskin et al. (2007) Usoskin, I. G., Solanki, S. K., & Kovaltsov, G. A. 2007, Grand minima and maxima of solar activity: new observational constraints, A&A, 471, 301, doi: 10.1051/0004-6361:20077704
  * Van Geel et al. (1999) Van Geel, B., Raspopov, O., Renssen, H., et al. 1999, The role of solar forcing upon climate change, Quaternary Science Reviews, 18, 331
  * Wolf (1852) Wolf, R. 1852, Bericht über neue Untersuchungen über die Periode der Sonnenflecken und ihrer Bedeutung von Herrn Prof. Wolf, Astronomische Nachrichten, 35, 369, doi: 10.1002/asna.18530352504

##  Appendix A Data

The monthly mean total sunspot number data begin from January 1749 and end to
November 2022 (Table 6: $n=3287$). After January 1818, the monthly mean
standard deviation $(S)$ of the input sunspot numbers from individual stations
are available, as well as the total number of observations $(N)$.

The yearly mean total sunspot number data begin from the year 1700 and end to
2021 (Table 7: $n=322$). The $S$ and $N$ estimates are available after 1818.

The standard errors

$\displaystyle\sigma_{i}=S_{i}/\sqrt{N_{i}}$ (A1)

for these data give the weights $w_{i}=\sigma_{i}^{-2}$. The normalized
weights are

$\displaystyle w_{\mathrm{nor,i}}=nw_{i}/\sum_{i=1}^{n}w_{i}.$ (A2)

If all errors were equal, the normalized weight for every observation would be
one. For all monthly data, the normalized weights show that one observation
out of $n=2458$ observations has the weight $w_{\mathrm{nor,max}}=381$ (Table
8, Line 1). The four most accurate observations, $N_{1/2}=4$, influence the
modelling more than the remaining $n-4=2454$ observations. These monthly data
statistics improve slightly, $N_{1/2}=64$ for sample size $n=2183$, if the
more accurate data after the year 2000 are removed. All yearly data show an
extreme case, where the weight $w_{\mathrm{nor,max}}=135$ of one $N_{1/2}=1$
observation exceeds the weight of all other $n-1=203$ observations. Again, the
statistics improve slightly, if the more accurate data after the year 2000 are
removed.

For these biased normalized weights, the period and the amplitude error
estimates for the weighted data signals would be dramatically larger than the
respective errors for the non-weighted data signals. For example, the
bootstrap reshuffling (Eq. B20) of the four most accurate monthly values, or
the one most accurate yearly value, resembles Russian roulette, where the
majority of remaining other data values do not seem to need to fit to the data
at all. We emphasize that the weighted data itself causes this bias, not our
period analysis method.

Table 6: Monthly $t$ mean total sunspot numbers $y$ $(n=3287)$. The standard
deviation $(S)$ and the total number of estimates $(N)$ are available from
Janaury 1818 onwards. Values -1 indicate cases having no $S$ and $N$ estimate.

$t$ | $y$ | $S$ | $N$
---|---|---|---
(y) | (-) | (-) | (-)
1749.042 | 96.7 | -1 | -1
… | … | … | …
2022.873 | 77.6 | 14.1 | 881

Note. — This table shows only the first and the last lines, because all data
is available in electronic form.

Table 7: Yearly mean total sunspot numbers $(n=322)$, otherwise as in Table 6 $t$ | $y$ | $S$ | $N$
---|---|---|---
(y) | (-) | (-) | (-)
1700.5 | 8.3 | -1 | -1
… | … | … | …
2021.5 | 29.6 | 7.9 | 15233
Table 8: Normalized weights $w_{\mathrm{nor,i}}$ (Eq. A2). (1) Sample. (2) Sample size. (3,4) Maximum and minimum $w_{\mathrm{nor,i}}$. (5) Number of largest $w_{\mathrm{nor,i}}$ having a sum exceeding $n/2$.  Sample | $n$ | $w_{\mathrm{nor,max}}$ | $w_{\mathrm{nor,min}}$ | $N_{\mathrm{1/2}}$
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
Monthly data
All | 2458 | 381 | $2.02\times 10^{-5}$ | 4
Before 2000 | 2183 | 48.9 | $1.83\times 10^{-3}$ | 64
Yearly data
All | 204 | 135 | $1.08\times 10^{-3}$ | 1
Before 2000 | 182 | 21.6 | $1.11\times 10^{-2}$ | 6

We solve this statistical bias of errors $\sigma_{i}$ by using the Sigma-
cutoff weights

$\begin{cases}w_{i}^{\prime}=1,&\text{if $\sigma_{i}\leq K\bar{\sigma}$}\\\
w_{i}^{\prime}=(K\bar{\sigma}/\sigma_{i})^{x},&\text{if
$\sigma_{i}>K\bar{\sigma}$},\end{cases}$ (A3)

where $\bar{\sigma}$ is the mean of all $\sigma_{i}$, and the $K$ and $x$
values can be freely chosen (Handler, 2003; Breger et al., 2002). We use the
same $K=1$ and $x=2$ values as Rodríguez et al. (2003), which gives

$\begin{cases}\sigma_{i}^{\prime}=1,&\text{if $\sigma_{i}\leq\bar{\sigma}$}\\\
\sigma_{i}^{\prime}=\sigma_{i}/\bar{\sigma}>1,&\text{if
$\sigma_{i}>\bar{\sigma}$}.\end{cases}$ (A4)

Our chosen $K$ and $x$ values are reasonable, because they give 55 per cent of
monthly data having full weight 1, and 25 per cent having weight below 1/2.
The respective values for the yearly data are 53 and 24 per cent.

There are four exceptional cases. We use the sigma cutoff weights
$w_{i}^{\prime}=1$ for the three exceptional cases $y=0$, $S=0$ and $N>0$,
which solves the infinite weight $w_{i}=\sigma_{i}^{-2}=\infty$ problem.
Finally, we compute no error estimate for the one exceptional case $S>0$ and
$N=0$.

The four first samples are monthly sunspot data drawn from Table 6. The last
four samples are yearly sunspot data drawn from Table 7. All eight samples are
published only in electronic form.

Table 9: Eight sunspot data samples. (1) Table from where sample is drawn. (2) Sample name. (3-4) Sample first and last observing time. (5) Sample time span. (6) Sample size. (7) Sample electronic file name.  Table | Name | $t_{1}$ | $t_{2}$ | $\Delta T$ | $n$ | File
---|---|---|---|---|---|---
|  | (y) | (y) | (y) | (-) |
(1) | (2) | (3) | (4) | (5) | (6) | (7)
Table 6 | Rmonthly | 1749.0 | 2022.8 | 274 | 3287 | Rmontly.dat
Table 6 | Rmonthly2000 | 1749.0 | 1999.9 | 251 | 3012 | Rmonthly2000.dat
Table 6 | Cmonthly | 1818.0 | 2022.8 | 205 | 2458 | Cmonthly.dat
Table 6 | Cmonthly2000 | 1818.0 | 1999.9 | 182 | 2182 | Cmonthly2000.dat
Table 7 | Ryearly | 1700.5 | 2021.5 | 321 | 322 | Ryearly.dat
Table 7 | Ryearly2000 | 1700.5 | 1999.5 | 300 | 299 | Ryearly2000.dat
Table 7 | Cyearly | 1818.5 | 2021.5 | 203 | 204 | Cyearly.dat
Table 7 | Cyearly2000 | 1818.5 | 1995.5 | 181 | 182 | Cyearly2000.dat

###  A.1 Rmonthly and Rmonthly2000

Rmonthly is our largest sample ($n=3287$, $\Delta T=274^{\mathrm{y}}$). It
contains all $t_{i}$ and $y_{i}$ values from Table 6. Since the error
estimates $\sigma_{i}$ are unknown before January 1818, we use an arbitrary
error $\sigma_{i}=1$ for all data in this sample. We perform a non-weighted
period analysis, which is based on the sum of squared residuals (Eq. B7: $R$).
Hence, the chosen $\sigma_{i}=1$ value has no effect to our analysis results,
because every observation has an equal weight. The particular file name
Rmonthly is used because the period analysis is based on $R$ test statistic.

Rmonthly2000 contains observations from Rmonthly, which were made before the
year 2000 $(n=3012,\Delta T=251^{\mathrm{y}})$. Number “2000” refers to the
omitted data after the year 2000.

###  A.2 Cmonthly and Cmonthly2000

Cmonthly contains all $t_{i}$ and $y_{i}$ observations having an error
estimate $\sigma_{i}^{\prime}$ computed from Eq. A4 $(n=2458,~{}\Delta
T=205^{\mathrm{y}})$. For this sample, we apply the weighted period analysis,
which utilizes the error $\sigma_{i}^{\prime}$ information (Eq. B8:
$\chi^{2}$). The sample name begins with letter “C”, because our analysis is
based on the Chi-square test statistic.

Cmonthly2000 contains those Cmonthly observations, which were made before the
year 2000 $(n=2182,~{}\Delta T=182^{\mathrm{y}})$. The omitted data after the
year 2000 are referred by the number “2000”.

###  A.3 Ryearly and Ryearly2000

Ryearly is our longest sample $(\Delta T=321^{\mathrm{y}})$ containing $n=322$
yearly mean total sunspot number observations over more than three centuries.
Since no error estimates are available for observations before 1818, the value
$\sigma_{i}=1$ is used for all data. We perform a non-weighted period analysis
based on the $R$ test statistic, and therefore the sample name begins with the
letter “R”.

Ryearly2000 contains observations from Ryearly that were made before the year
2000.

###  A.4 Cyearly and Cyearly2000

Cyearly is the smallest sample $(n=204,\Delta T=204^{\mathrm{y}})$. It
contains all yearly mean total sunspot numbers having an error estimate
$\sigma_{i}^{\prime}$ (Eq. A4). We perform a weighted period analysis based on
the $\chi^{2}$ test statistic (Eq. B8), and therefore the sample name begins
with letter “C”.

Cyearly2000 contains Cyearly data before 2000.

##  Appendix B Discrete Chi-Square Method (DCM)

Jetsu (2020,  Paper I) formulated the Discrete Chi-Square Method (DCM). Using
DCM, he discovered the periods of a third and a fourth body from the O-C data
of the eclipsing binary XZ And. An improved DCM version revealed the presence
of numerous new companion candidates in the eclipsing binary Algol (Jetsu,
2021,  Paper II). DCM is designed for detecting many signals superimposed on
an arbitrary trend. In this section, we use the results for model
$\mathcal{M}$=4 in Table 12 to illustrate how DCM works (Figs. 9 and 11-14).

The sunspot number data notations are $y_{i}=y(t_{i})\pm\sigma_{i}$, where
$t_{i}$ are the observing times and $\sigma_{i}$ are the errors
$(i=1,2,...,n)$. The units are $[y_{i}]=$ -, $[\sigma_{i}]=$ \- and $[t_{i}]=$
y. The time span of data is $\Delta T=t_{n}-t_{1}$. The mid point is
$t_{\mathrm{mid}}=t_{1}+\Delta T/2$.

DCM model

$\displaystyle g(t)=g(t,K_{1},K_{2},K_{3})=h(t)+p(t).$ (B1)

is a sum of a periodic function

$\displaystyle h(t)$ $\displaystyle=$ $\displaystyle
h(t,K_{1},K_{2})=\sum_{i=1}^{K_{1}}h_{i}(t)$ (B2) $\displaystyle h_{i}(t)$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{K_{2}}B_{i,j}\cos{(2\pi
jf_{i}t)}+C_{i,j}\sin{(2\pi jf_{i}t)},$ (B3)

and an aperiodic function

$\displaystyle p(t)$ $\displaystyle=$ $\displaystyle
p(t,K_{3})=\sum_{k=0}^{K_{3}}p_{k}(t)$ (B4) $\displaystyle p_{k}(t)$
$\displaystyle=$ $\displaystyle
M_{k}\left[{{2(t-t_{\mathrm{mid}}})\over{\Delta T}}\right]^{k}.$ (B5)

The periodic $h(t)$ function is a sum of $K_{1}$ harmonic $h_{i}(t)$ signals
having frequencies $f_{i}$.

The signal order is $K_{2}$. For simplicity, we refer to $K_{2}=1$ order
models as “pure sine” models, and to $K_{2}=2$ order models as “double wave”
models.

The $h_{i}(t)$ signals are superimposed on the aperiodic $K_{3}$ order
polynomial trend $p(t)$. Function $g(t)$ repeats itself in time, and function
$p(t)$ does not.

DCM model residuals

$\displaystyle\epsilon_{i}=y(t_{i})-g(t_{i})=y_{i}-g_{i}$ (B6)

give the sum of squared residuals

$\displaystyle R=\sum_{i=1}^{n}\epsilon_{i}^{2},$ (B7)

and also the Chi-square

$\displaystyle\chi^{2}=\sum_{i=1}^{n}{{\epsilon_{i}^{2}}\over{\sigma_{i}^{2}}}.$
(B8)

If the data errors $\sigma_{i}$ are known we use $\chi^{2}$ to estimate the
goodness of our model. For unknown errors, we use $R$.

In every sample, $\mathcal{M}$=1-4 models $g(t)$ are computed for the original
$y_{i}$ data (e.g. Table 10:  Rmonthly2000.dat). The next $\mathcal{M}$=5-8
models $g(t)$ are computed for the $\epsilon_{i}$ residuals of $\mathcal{M}$=4
model (e.g. Table 10:  ResidualsRmonthly2000K410R14.dat). For example, the
five signal model for the Rmonthly2000 sample is the sum of $g(t)$ model for
$\mathcal{M}$=4 and $g(t)$ model for $\mathcal{M}$=5. We refer to this five
signal model simply as $\mathcal{M}$=5 model.

Notation for DCM model $g(t)$ having orders $K_{1}$, $K_{2}$ and $K_{3}$ is
“$\mathcal{M}$${}_{K_{1},K_{2},K_{3},R}$” or
“$\mathcal{M}$${}_{K_{1},K_{2},K_{3},\chi^{2}}$”. The last subscripts “$R$” or
“$\chi$” refer to the use of Eq. B7 or B8 in estimating the goodness of our
model.

The free parameters of model $g(t)$ are

$\displaystyle\bar{\beta}=[\beta_{1},\beta_{2},...,\beta_{\eta}]=[B_{1,1},C_{1,1},f_{1},...,B_{K_{1},K_{2}},C_{K_{1},K_{2}},f_{K_{1}},M_{0},...,M_{K_{3}}],$
(B9)

where

$\displaystyle\eta=K_{1}\times(2K_{2}+1)+K_{3}+1$ (B10)

is the number of free parameters. We divide the free parameters $\bar{\beta}$
into two groups

$\displaystyle\bar{\beta}_{I}$ $\displaystyle=$
$\displaystyle[f_{1},...,f_{K_{1}}]$ (B11) $\displaystyle\bar{\beta}_{II}$
$\displaystyle=$
$\displaystyle[B_{1,1}C_{1,1},...,B_{K_{1},K_{2}},C_{K_{1},K_{2}},M_{0},...,M_{K_{3}}]$
(B12)

The first frequency group $\bar{\beta}_{I}$ makes the $g(t)$ model non-linear,
because all free parameters are not eliminated from all partial derivatives
$\partial g/\partial\beta_{i}$. If these $\bar{\beta}_{I}$ frequencies are
fixed to constant known tested numerical values, none of the partial
derivatives $\partial g/\partial\beta_{i}$ contains any free parameters. In
this case, the model becomes linear and the solution for the second group of
free parameters, $\bar{\beta}_{II}$, is unambiguous. Our concepts like “linear
model” and “unambiguous result” refer to this type of models and their free
parameter solutions.

For every tested frequency combination
$\bar{\beta}_{I}=[f_{1},f_{2},...,f_{K_{1}}]$ we compute the DCM test
statistic

$\displaystyle z$ $\displaystyle=$ $\displaystyle
z(f_{1},f_{2},...,f_{K_{1}})=\sqrt{R/n}$ (B13) $\displaystyle z$
$\displaystyle=$ $\displaystyle
z(f_{1},f_{2},...,f_{K_{1}})=\sqrt{\chi^{2}/n}$ (B14)

from a linear model least squares fit. For unknown or known errors
$\sigma_{i}$, we use Eq. B13 or Eq. B14, respectively.

The sum $h(t)$ of signals $h_{1}(t)$ and $h_{2}(t)$ does not depend on the
order in which these signals are added. This causes the symmetry
$z(f_{1},f_{2})=z(f_{2},f_{1})$. The same symmetry applies to any $K_{1}$
number of signals. Therefore, we compute $z$ test statistic only for all
combinations

$\displaystyle f_{\mathrm{min}}\leq f_{1}<f_{2}<...<f_{K_{1}}\leq
f_{\mathrm{max}},$ (B15)

where $f_{\mathrm{min}}$ and $f_{\mathrm{max}}$ are the minimum and the
maximum tested frequencies, respectively. In the long search, we test an
evenly spaced grid of $n_{\mathrm{L}}$ frequencies between $f_{\mathrm{min}}$
and $f_{\mathrm{max}}$. This gives us the best frequency candidates
$f_{\mathrm{1,mid}},...f_{\mathrm{K_{1},mid}}$. In the short search, we test a
denser grid of $n_{\mathrm{S}}$ frequencies within an interval

$\displaystyle[f_{\mathrm{i,mid}}-a,f_{\mathrm{i,mid}}+a],$ (B16)

where $a=c(f_{\mathrm{min}}-f_{\mathrm{max}})/2$. In this study, we use
$c=0.05$ which means that the tested short search frequency intervals
represent 5% of the tested long search interval.

Figure 9: M=4 model in Table 12. Four pure sine signal periodograms for
Rmonthly2000 between $P_{\mathrm{min}}=5$ and $P_{\mathrm{max}}=200$ years. We
test frequency grids having $n_{\mathrm{L}}=100$ and $n_{\mathrm{S}}=30$
tested frequencies. Upper panel shows long search $z_{1}(f_{1})$ (red)
$z_{2}(f_{2})$ (blue) $z_{3}(f_{3})$ (green) and $z_{4}(f_{4})$ (yellow)
periodograms (Eq. B17) for a $n_{\mathrm{L}}=80$ frequency grid. Best
frequencies are marked with diamonds. Units are x-axis $[f]=$ 1/years and
y-axis $[z]=$ dimensionless. Lower panel shows short search periodograms for a
denser $n_{\mathrm{S}}=30$ frequency grid having an interval width
$c=0.05\equiv$ 5% (Eq. B16). Otherwise as in upper panel.

We search for periods between $P_{\mathrm{min}}=1/f_{\mathrm{max}}=5$ years
and $P_{\mathrm{max}}=1/f_{\mathrm{min}}=200$ years.

Figure 10: One pure sine signal $z_{1}(f_{1})$ periodogram for Rmonthly2000
between $P_{\mathrm{min}}=0.01$ and $P_{\mathrm{max}}=20$ years. We test
“overkill” dense frequency grids $n_{L}=1000$ and $n_{S}=500$ to search for
short frequencies, but detect none. Best period is $P_{1}=11.034$ years, which
also given Table 10 ($\mathcal{M}$=1).

The global periodogram minimum

$\displaystyle
z_{\mathrm{min}}=z(f_{\mathrm{1,best}},f_{\mathrm{2,best}},...,f_{\mathrm{K_{1},best}}),$

is at the tested frequencies
$f_{\mathrm{1,best}},f_{\mathrm{2,best}},...,f_{\mathrm{K_{1},best}}$. The
best model for the data has these signal frequencies.

The scalar $z$ periodogram values are computed from $K_{1}$ frequency values.
For example, $K_{1}=2$ two signal periodogram $z(f_{1},f_{2})$ could be
plotted like a map, where $f_{1}$ and $f_{2}$ are the coordinates, and
$z=z(f_{1},f_{2})$ is the height. For three or more signals, such direct
graphical presentation becomes impossible, because it requires more than three
dimensions. We solve this problem by presenting only the following one-
dimensional slices of the full periodogram

$\displaystyle z_{1}(f_{1})$ $\displaystyle=$ $\displaystyle
z(f_{1},f_{\mathrm{2,best}},...,f_{\mathrm{K_{1},best}})$ $\displaystyle
z_{2}(f_{2})$ $\displaystyle=$ $\displaystyle
z(f_{\mathrm{1,best}},f_{2},f_{\mathrm{3,best}},...,f_{\mathrm{K_{1},best}})$
$\displaystyle z_{3}(f_{3})$ $\displaystyle=$ $\displaystyle
z(f_{\mathrm{1,best}},f_{\mathrm{2,best}},f_{3},f_{\mathrm{4,best}},...,f_{\mathrm{K_{1},best}})~{}$
$\displaystyle z_{4}(f_{4})$ $\displaystyle=$ $\displaystyle
z(f_{\mathrm{1,best}},f_{\mathrm{2,best}},f_{\mathrm{3,best}},f_{4},f_{\mathrm{5,best}},f_{\mathrm{K_{1},best}})$
(B17) $\displaystyle z_{5}(f_{5})$ $\displaystyle=$ $\displaystyle
z(f_{\mathrm{1,best}},f_{\mathrm{2,best}},f_{\mathrm{3,best}},f_{\mathrm{4,best}},f_{5},f_{\mathrm{K_{1},best}})$
$\displaystyle z_{6}(f_{6})$ $\displaystyle=$ $\displaystyle
z(f_{\mathrm{1,best}},f_{\mathrm{2,best}},f_{\mathrm{3,best}},f_{\mathrm{4,best}},f_{\mathrm{5,best}},f_{6}).$

Using the above $K_{1}=2$ map analogy, the slice $z_{1}(f_{1})$ represents the
height $z$ at $f_{1}$ coordinate when moving along the constant line
$f_{2}=f_{\mathrm{2,best}}$ that crosses the global minimum
$z_{\mathrm{min}}$.

Figure 11: $\mathcal{M}$=4 model in Table 12. Upper panel shows data $y_{i}$
(black circles), $g(t)$ (continuous black curve) and constant trend $p(t)$
(dotted black curve). Units are x-axis $[t]=$ years and y-axis $[y]=$
dimensionless. Lower panel shows data minus $p(t)$ values (black circles), and
$g(t)-p(t)$ (black curve), $h_{1}(t)$ (red curve), $h_{2}(t)$ (blue curve),
$h_{3}(t)$ (green curve) and $h_{4}(t)$ (yellow curve). Residuals (blue
circles) are offset to $y=-300$ level. Signal periods are $P_{1}=9.9842$,
$P_{2}=11.0234$, $P_{3}=11.846$ and $P_{4}=96.2$ years.

The best frequencies detected in the short search give the initial values for
free parameters $\bar{\beta}_{\mathrm{I,initial}}$ (Eq. B11). The linear model
with these frequencies gives initial values for
$\bar{\beta}_{\mathrm{II,initial}}$ (Eq. B12). The final non-linear iteration
is performed from

$\displaystyle\bar{\beta}_{\mathrm{initial}}=[\bar{\beta}_{\mathrm{I,initial}},\bar{\beta}_{\mathrm{II,initial}}]\rightarrow\bar{\beta}_{\mathrm{final}}$
(B18)

The Rmonthly2000 sample four pure sine signal model periodograms between
$P_{\mathrm{min}}=5$ and $P_{\mathrm{max}}=200$ years are shown in Fig. 9
(Table 10: $\mathcal{M}$=4 model). The one pure sine signal periodogram for
the same Rmonthly2000 sample between $P_{\mathrm{min}}=0.01$ and
$P_{\mathrm{max}}=20$ years shows no signs of short periods (Fig. 10). The
four pure sine signal $g(t)$ model for Rmonthly2000 is shown in Fig. 11.

DCM determines the following parameters for $h_{i}(t)$ signals

  * $P_{i}=1/f_{i}=$ Period

  * $A_{i}=$ Peak to peak amplitude

  * $t_{\mathrm{i,min,1}}=$ Deeper primary minimum epoch

  * $t_{\mathrm{i,min,2}}=$ Secondary minimum epoch (if present)

  * $t_{\mathrm{i,max,1}}=$ Higher primary maximum epoch

  * $t_{\mathrm{i,max,2}}=$ Secondary maximum epoch (if present),

as well the $M_{k}$ parameters of the $p(t)$ trend. For the sunspots, the most
interesting parameters are the signal periods $P_{i}$, the signal amplitudes
$A_{i}$, and signal primary minimum epochs $t_{\mathrm{min,1}}$ (Tables
10-21).

Each $h_{j}(t)$ signal

$\displaystyle y_{i,j}=y_{i}-[g(t_{i})-h_{j}(t_{i})]$ (B19)

of model $\mathcal{M}$=4 in Table 10 is shown separately in Fig. 12.

The DCM model parameter errors are determined with the bootstrap procedure
(Efron & Tibshirani, 1986). We have previously used this same bootstrap
procedure in our TSPA- and CPS-methods (Jetsu & Pelt, 1999; Lehtinen et al.,
2011). A random sample $\bar{\epsilon}^{*}$ is selected from the residuals
$\bar{\epsilon}$ of the DCM model (Eq. B6). Any $\epsilon_{i}$ can be chosen
as many times as the random selection happens to favour it. This random sample
of residuals gives the artificial bootstrap data sample

$\displaystyle y_{i}^{*}=g_{i}+\epsilon_{i}^{*}.$ (B20)

We create numerous such $\bar{y}^{*}$ random samples. DCM model for each
$\bar{y}^{*}$ sample gives one estimate for every model parameter. The error
estimate for each particular model parameter is the standard deviation of all
estimates obtained from all $\bar{y}^{*}$ bootstrap samples.

Figure 12: $\mathcal{M}$=4 model in Table 12. Left panels show four signals
$g_{i,j}$ (Eq. B19) as a function of time. Units are x-axis $[t]=$ years and
y-axis $[h_{i}(t)]=$ dimensionless. Right panels show same signals as function
of phase. Units are x-axis $[\phi]=$ dimensions and y-axis $[h_{i}(t)]=$
dimensionless. Signal periods are $P_{1}=9.9842$, $P_{2}=11.0234$,
$P_{3}=11.846$ and $P_{4}=96.2$ years. Curve colours are as in Fig. 11.
Figure 13: Extracts: (a) Rmonthly2000 double sinusoid $23.686\pm 0.018$ year
signal is exactly equal to two times Jupiter’s 11.86 year period. (b) Rmonthly
double sinusoid $20.0062\pm 0.0082$ year signal is exactly equal to two times
10 years. Numbers and labels for x-axis phases $\phi$ between 0 and 1 are
missing, because these figures are extracts from larger figures, like Fig. 12.
Units are x-axis $[\phi]=$ dimensionless and y-axis $[y]=$ dimensionless.

DCM models are nested. For example, a one signal model is a special case of a
two signal model. As for another example, $\mathcal{M}$=2 model is more
complex than $\mathcal{M}$=1 model in Table 10. DCM uses the Fisher-test to
compare any pair of simple $g_{1}(t)$ and complex $g_{2}(t)$ models. Their
number of free parameters are $\eta_{1}<\eta_{2}$. Their sums of squared
residuals $(R_{1},R_{2})$ and Chi-squares $(\chi_{1},\chi_{2})$ give the
Fisher-test test statistic

$\displaystyle F_{R}$ $\displaystyle=$
$\displaystyle\left({{R_{1}}\over{R_{2}}}-1\right)\left({{n-\eta_{2}-1}\over{\eta_{2}-\eta_{1}}}\right).$
(B21) $\displaystyle F_{\chi}$ $\displaystyle=$
$\displaystyle\left({{\chi_{1}^{2}}\over{\chi_{2}^{2}}}-1\right)\left({{n-\eta_{2}-1}\over{\eta_{2}-\eta_{1}}}\right).$
(B22)

The Fisher-test null hypothesis is

  * $H_{\mathrm{0}}$: “The complex model $g_{2}(t)$ does not provide a significantly better fit to the data than the simple model $g_{\mathrm{1}}(t)$.”

Under $H_{0}$, both test statistic parameters $F_{R}$ and $F_{\chi}$ have an
$F$ distribution with $(\nu_{1},\nu_{2})$ degrees of freedom, where
$\nu_{1}=\eta_{2}-\eta_{1}$ and $\nu_{2}=n-\eta_{2}$ (Draper & Smith, 1998).
The probability for $F_{R}$ or $F_{\chi}$ reaching values higher than $F$ is
called the critical level $Q_{F}=P(F_{R}\geq F)$ or $Q_{F}=P(F_{\chi}\geq F)$.
We reject the $H_{0}$ hypothesis, if

$\displaystyle Q_{F}<\gamma_{F}=0.001,$ (B23)

where $\gamma_{F}$ is the pre-assigned significance level. This $\gamma_{F}$
represents the probability of falsely rejecting $H_{0}$ hypothesis when it is
in fact true. If $H_{0}$ is rejected, we rate the complex $g_{2}(t)$ model
better than the simple $g_{1}(t)$ model.

The basic idea of the Fisher-test is simple. The $H_{0}$ hypothesis rejection
probability increases for larger $F_{R}$ values having smaller $Q_{F}$
critical level. The complex model $R_{2}$ or $\chi^{2}_{2}$ values decrease
when the $\eta_{2}$ number of free parameters increases. This increases the
first $(R_{1}/R_{2}-1)$ and $(\chi^{2}_{1}/\chi^{2}_{2}-1)$ terms in Eqs. B21
and B22. However, the second $(n-\eta_{2}-1)/(\eta_{2}-\eta_{1})$ penalty term
decreases at the same time. This second penalty term prevents over-fitting,
i.e. accepting complex models having too many $\eta_{2}$ free parameters.

DCM model can be used to predict future and past data. The samples in these
predictions are

  * -

Predictive data sample: $t_{i}$, $y_{i}$ and $\sigma_{i}$ contains $n$ values

  * -

Predicted data sample: $t_{i}^{\prime}$, $y_{i}^{\prime}$ and
$\sigma_{i}^{\prime}$ contains $n^{\prime}$ values

  * -

Combined predictive and predicted data sample: $t^{\prime\prime}_{i}$,
$y^{\prime\prime}_{i}$ and $\sigma^{\prime\prime}_{i}$ contains
$n^{\prime\prime}=n+n^{\prime}$ values

We compute the $t_{\mathrm{mid}}$ and $\Delta T$ values from the time points
$t_{i}$ of the predictive data. DCM gives the best model $g(t,\bar{\beta})$
for the predictive data. The “old” predictive data $t_{\mathrm{mid}}$ and
$\Delta T$ values are used to compute the predicted and combined data models

$\displaystyle g^{\prime}(t^{\prime})$ $\displaystyle=$ $\displaystyle
g^{\prime}(t,^{\prime}\bar{\beta})$ $\displaystyle
g^{\prime\prime}(t^{\prime\prime})$ $\displaystyle=$ $\displaystyle
g^{\prime\prime}(t^{\prime\prime},\bar{\beta}).$

In other words, we do not compute “new” $t_{\mathrm{mid}}$ and $\Delta T$
values from the time points $t^{\prime}_{i}$ or $t^{\prime\prime}_{i}$. There
is an obvious reason for this. The predictive data free parameter values
$\bar{\beta}$ give the correct $g^{\prime}(t,^{\prime}\bar{\beta})$ and
$g^{\prime\prime}(t^{\prime\prime},\bar{\beta})$ values only for the “old”
predictive data $t_{\mathrm{mid}}$ and $\Delta T$ values.

The residuals

$\displaystyle\epsilon^{\prime}_{i}=y^{\prime}_{i}-g^{\prime}_{i}$

give

$\displaystyle z^{\prime}=\textnormal{Predicted data test statistic (Eq.
\ref{EqzR} and \ref{EqzChi})}$ (B24)

We emphasize that the predictive data $t_{\mathrm{mid}}$ and $\Delta T$ values
can be used to compute the model value for any arbitrary $t^{\prime}_{i}$ time
point values. This gives parameters, like the value

$\displaystyle
g^{\prime\prime}_{\mathrm{mean}}={{1}\over{n^{\prime\prime}}}\sum_{i=1}^{n^{\prime\prime}}g^{\prime\prime}(t^{\prime\prime}_{i},\bar{\beta}).$
(B25)

of the combined data model mean for any arbitrary time points
$t^{\prime\prime}_{i}$.

Figure 14: $\mathcal{M}$=4 model in Table 12. Left hand panels show bootstrap
signal frequency estimates $(f_{1},f_{2},f_{3},f_{4})$. Units are $[f_{i}]=$
1/years. Right hand panels show respective amplitude estimates
$(A_{1},A_{2},A_{3},A_{4})$. Units are $[A_{i}]=$ dimensionless. Larger blue
circles denote values for original data. Smaller red circles denote values for
bootstrap random samples. Dotted lines denote one, two and three sigma error
limits.

The key ideas of DCM are based on the following robust thoroughly tested
statistical approaches

  1. 1.

DCM model $g(t)$ is non-linear (Eq. B1). This model becomes linear when the
frequencies $f_{1},...,f_{K_{1}}$ are fixed to their tested numerical values.
This linear model gives unambiguous results.

  2. 2.

DCM tests a dense grid of all possible frequency combinations
$f_{\mathrm{min}}\leq f_{1}\\!>\\!f_{2}\\!>\\!...\\!>\\!f_{K_{1}}\leq
f_{\mathrm{max}}$ (Eq. B15). For every frequency combination, the linear model
least squares fit gives the test statistic $z=\sqrt{\chi^{2}/n}$ (errors
$\sigma_{i}$ known) or $z=\sqrt{R/n}$ (errors $\sigma_{i}$ unknown).

  3. 3.

The short search
$f_{\mathrm{1,best}}>f_{\mathrm{1,best}}>...>f_{\mathrm{K_{1},best}}$ grid
combination that minimizes the $z$ test statistic gives the best initial
$\bar{\beta}_{\mathrm{initial}}$ values for the non-linear iteration of Eq.
B18.

  4. 4.

The error estimates for all model parameters are determined with the bootstrap
method (Eq. B20).

  5. 5.

The best model is identified using the Fisher-test, which compares all
different $K_{1}$, $K_{2}$ and $K_{3}$ order nested models (Draper & Smith,
1998; Allen, 2004).

DCM has the following restrictions

  1. 1.

An adequately dense tested frequency grid eliminates the possibility that the
best frequency combination is missed. The restriction is that denser grids
require more computation time. For example, all four signal $z_{1}$, $z_{2}$,
$z_{3}$ and $z_{4}$ periodograms for Rmonthly2000 data are continuous in Fig.
9. No abrupt periodogram jumps occur, because the $z$ values for all close
tested frequencies correlate. The frequencies of the minima of all these
periodograms are accurately determined. There is no need to test an even
denser grid, because this would not alter the final result of the non-linear
iteration of Eq. B18.

  2. 2.

If the long search grid of each tested $f_{1}>f_{2}>...>f_{K_{1}}$ frequencies
uses a value $n_{L}$, the total number of tested frequencies is

$\displaystyle\binom{n_{L}}{K_{1}}={{n_{L}!}\over{K_{1}!(n_{L}-K_{1})!}}$

For example, it took 3 days wall-clock time, and over 300 days of cpu-time,
for a cluster of computers to compute the five signal DCM model shown in Fig.
15.

  3. 3.

Some DCM models are unstable. They are simply wrong models for the data, like
a wrong $p(t)$ trend order $K_{3}$, or a search for too few or too many
$K_{1}$ signals (e.g.  Paper I: Figs. 5-10). This causes model instability. We
denote such unstable models with “${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$” in
Tables 10-21. The signatures of such unstable models are

     * “${\color[rgb]{0,0,1}{\bf\mathrm{IF}}}$” = Intersecting frequencies

     * “${\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$” = Dispersing amplitudes

     * “${\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$” = Leaking periods

  4. Intersecting frequencies “${\color[rgb]{0,0,1}{\bf\mathrm{IF}}}$” occur when the frequencies of two signals are very close to each other. For example, if frequency $f_{1}$ approaches frequency $f_{2}$, the $h_{1}(t)$ and $h_{2}(t)$ signals become essentially one and the same signal. This ruins the least squares fit. It makes no sense to add the same signal twice.

  5. Dispersing amplitudes “${\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$” also occur when two signal frequencies are close to each other, like in the “${\color[rgb]{0,0,1}{\bf\mathrm{IF}}}$” cases. The least squares fit uses two high amplitude signals which cancel out. Hence, the sum of these two high amplitude signals is one low amplitude signal that fits to the data.

  6. We take extra care of the suspected “${\color[rgb]{0,0,1}{\bf\mathrm{IF}}}$” and $"{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$” cases. The reshuffling of bootstrap $\bar{\epsilon}^{\star}$ residuals is a good test for identifying such cases. If we encounter any signs of instability, we test combinations $n_{\mathrm{L}}=100~{}\&~{}c=0.05$, $n_{\mathrm{L}}=100~{}\&~{}c=0.10$, $n_{\mathrm{L}}=120~{}\&~{}c=0.05$ and $n_{\mathrm{L}}=120~{}\&~{}c=0.10$. If there is instability in any of these combinations, we reject the model as unstable $("{\color[rgb]{0,0,1}{\bf\mathrm{UM}}}")$. If there is no instability, we take the $n_{\mathrm{L}}$ and $c$ combination that gives the lowest value for $R$ (non-weighted data) or $\chi^{2}$ (weighted data).

  7. Leaking periods “${\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$” refer to the cases when the period $P=1/f$ given by the detected frequency $f$ is longer than the time span $\Delta T$ of the data.

Figure 15: ”Buying all lottery tickets”. We compute pure five sine signal
model periodograms for Ryearly $(n=322)$. Long search for $n_{\mathrm{L}}=100$
frequency grid requires testing of 79 208 745 models. Short
$n_{\mathrm{S}}=20$ frequency grid requires testing of 2 451 267 models.
Detected signals are 9.977, 10.637, 11.032, 11.815 and 101.339 years. Total
consumed cpu-time exceeds 300 days. Note that error estimates are missing,
because 20 bootstrap samples would require testing of $20\times 2~{}451~{}267$
additional models. Otherwise as in Fig. 9.

We conclude that DCM can test all reasonable alternative non-linear $g(t)$
models for the data. DCM determines the unambiguous results for the best
values of the free parameters of all these alternative $g(t)$ models. DCM uses
a brute numerical approach to find the best $g(t)$ model among all alternative
models. DCM “works like winning a lottery by buying all lottery tickets” (
Paper I). The caption text of Fig. 15 elucidates how such lotteries are won in
practice. In real life, it would make no sense win less money by investing
more money to all lottery tickets. In science, investing enough computing time
to DCM gives the correct answer for free.

###  B.1 DCM improvements

####  B.1.1 Parallel computation

We began the sunspot data analysis in December 2022, and soon discovered the
exact 10 and 11 year periods, as well as the 11.86 period, in the yearly data.
Then we realized that this discovery could be disputed by arguing that the one
year data window produces such spurious periods. We decided to bypass this
problem by analysing the monthly sunspot data. However, the much larger size
of these samples slowed down our analysis. The logical solution was to develop
a parallel DCM computation code.

In the appendix of  Paper I, we gave detailed instructions for using the DCM
python code. That “manual” is adequate for repeating the analysis in this
paper. Here, we provide all necessary information for reproducing our current
DCM analysis of sunspot data.444All necessary files for reproducing our
results are published in Zenodo database: doi xx.xxxx/zenodo.xxxxxxx when the
manuscript is in press. The 24 first control file  dcm.dat parameters are the
same as in  Paper I and  Paper II.

We have added the following three new parameters into the control file
dcm.dat.

  * 25

Parallel=1 activates the “new” parallel computation mode.  Parallel$\neq$ 1
activates the “old” serial computation mode.

  * 26

PoolNumber defines the number of simultaneous processes in parallel
computation.

  * 27

ChunkModels chops the long tested frequency vectors into shorter pieces. In
this way, we can bypass the problem that python slows down when vectors become
very long.

####  B.1.2 No trend

The “ new” control parameter value  K3 $=K_{3}=-1$ means that the model has no
polynomial part, i.e. $p(t)=0$. This alternative is useful in the residual
analysis, where all trends have already been removed. The “old” alternative
K3$=K_{3}=0$ would still remove a constant level trend $p(t)=M_{0}$, and this
could mislead the analysis.

Table 10: Rmonthly2000 analysis for pure sines $(K_{2}=1)$. We use first model $\mathcal{M}$=1 as an example of how to read this table. (1) Model number $\mathcal{M}$=1. (2) Chosen $\mathcal{M}$${}_{K_{1}=1,K_{2}=1,K_{3}=0,R}$ model has $\eta=4$ free parameters, and its sum of squared residuals is $R=1.09\times 10^{7}$. (3) First signal period is $P_{1}=11.0336\pm 0.0074$ years. Dimensionless peak to peak amplitude is $A_{1}=92.2\pm 2.8$. Primary minimum epoch is at year $t_{\mathrm{min,1}}=1755.30\pm 0.12$. (4-6) Periods, amplitudes and primary minima for two, three and four signal models. (7) Every new model is more complex when going down this table because number $\eta$ of free parameters increases (Eq. B10). Fisher-test between simple $\mathcal{M}$=1 model and complex $\mathcal{M}$=2 model gives test statistic $F_{R}=255$ (Eq. B21) reaching a critical level $Q_{F}<10^{-16}$. Criterion $Q_{F}<\gamma=0.001$ is fulfilled (Eq. B23). Therefore, an upward arrow $\uparrow$ indicates that $\mathcal{M}$=2 model is a better model than $\mathcal{M}$=1 model. A leftward arrow $\leftarrow$ would have been used, if model $\mathcal{M}$=1 were better. (8-9) Fisher-tests for models $\mathcal{M}$=3 and 4. (10) Electronic control file  Rmonthly2000K110R14.dat can be used to repeat whole DCM analysis of $\mathcal{M}$=1. Four first signals are detected from original data in electronic file  Rmontly2000.dat. Next four model $\mathcal{M}$=5-8 signals are detected from residuals of model $\mathcal{M}$=4, which are stored to electronic file  RmonthlyK410R14Residuals.dat. Models $\mathcal{M}$=7 and 8 are unstable $({\color[rgb]{0,0,1}{\bf\mathrm{UM}}})$ because they suffer from amplitude dispersion $({\color[rgb]{0,0,1}{\bf\mathrm{AD}}})$. Intersecting frequencies $({\color[rgb]{0,0,1}{\bf\mathrm{IF}}})$ or leaking periods $({\color[rgb]{0,0,1}{\bf\mathrm{LP}}})$ are not encountered in any of these eight $\mathcal{M}$=1-8 models. Models $\mathcal{M}$=4 and 6 give final results summarized in Table 2.  |  | Period analysis |  | Fisher-test |
---|---|---|---|---|---
Data: Original non-weighted data ($n=3012,\Delta T=250.9$:  Rmonthly2000.dat)
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},R}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $R$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$1,1,0,R | $11.0336\\!\\!\pm\\!\\!0.0074$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 4 | $92.2\\!\\!\pm\\!\\!2.8$ | - | - | - |  | 255 | 254 | 231 | Rmonthly2000K110R14.dat
| $1.09\\!\\!\times\\!\\!10^{7}$ | $1755.30\\!\\!\pm\\!\\!0.12$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$2,1,0,R | $9.9998\\!\\!\pm\\!\\!0.0060$ | $11.0200\\!\\!\pm\\!\\!0.0074$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 7 | $76.7\\!\\!\pm\\!\\!2.5$ | $91.6\\!\\!\pm\\!\\!3.2$ | - | - |  | - | 179 | 174 | Rmonthly2000K210R14.dat
| $8.69\\!\\!\times\\!\\!10^{6}$ | $1754.18\\!\\!\pm\\!\\!0.11$ | $1755.65\\!\\!\pm\\!\\!0.11$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$3,1,0,R | $9.9848\\!\\!\pm\\!\\!0.0076$ | $11.0293\\!\\!\pm\\!\\!0.0065$ | $11.845\\!\\!\pm\\!\\!0.012$ | - |  | - | - | $\uparrow$ |
| 10 | $77.0\\!\\!\pm\\!\\!2.4$ | $99.6\\!\\!\pm\\!\\!2.9$ | $60.8\\!\\!\pm\\!\\!1.8$ | - |  | - | - | 144 | Rmonthly2000K310R14.dat
| $7.37\\!\\!\times\\!\\!10^{6}$ | $1754.36\\!\\!\pm\\!\\!0.12$ | $1755.704\\!\\!\pm\\!\\!0.089$ | $1760.24\\!\\!\pm\\!\\!0.16$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$4,1,0,R | $9.9842\\!\\!\pm\\!\\!0.0075$ | $11.0324\\!\\!\pm\\!\\!0.0048$ | $11.846\\!\\!\pm\\!\\!0.012$ | $96.2\\!\\!\pm\\!\\!1.0$ |  | - | - | - |
| 13 | $77.4\\!\\!\pm\\!\\!2.2$ | $98.8\\!\\!\pm\\!\\!2.6$ | $60.0\\!\\!\pm\\!\\!2.2$ | $50.1\\!\\!\pm\\!\\!2.8$ |  | - | - | - | Rmonthly2000K410R14.dat
| $6.44\\!\\!\times\\!\\!10^{6}$ | $1754.363\\!\\!\pm\\!\\!0.096$ | $1755.661\\!\\!\pm\\!\\!0.075$ | $1760.22\\!\\!\pm\\!\\!0.16$ | $1815.3\\!\\!\pm\\!\\!1.0$ |  | - | - | - |
Data: Non-weighted residuals of model $\mathcal{M}$=4 ($n=3012,\Delta
T=250.9$:  Rmonthly2000K410R14Residuals.dat)
|  |  |  |  |  |  |  $\mathcal{M}$=6 |  $\mathcal{M}$=7 |  $\mathcal{M}$=8 |
$\mathcal{M}$=5 |  $\mathcal{M}$1,1,-1,R | $10.547\\!\\!\pm\\!\\!0.012$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 3 | $45.3\\!\\!\pm\\!\\!1.8$ | - | - | - |  | 97 | 92 | 96 | Rmonthly2000K11-1R58.dat
| $5.67\\!\\!\times\\!\\!10^{6}$ | $1753.04\\!\\!\pm\\!\\!0.16$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=6 |  $\mathcal{M}$2,1,-1,R | $8.1054\\!\\!\pm\\!\\!0.0084$ | $10.541\\!\\!\pm\\!\\!0.014$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 6 | $36.5\\!\\!\pm\\!\\!1.7$ | $45.0\\!\\!\pm\\!\\!1.8$ | - | - |  | - | 79 | 87 | Rmonthly2000K21-1R58.dat
| $5.17\\!\\!\times\\!\\!10^{6}$ | $1751.33\\!\\!\pm\\!\\!0.13$ | $1753.13\\!\\!\pm\\!\\!0.20$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=7 |  $\mathcal{M}$3,1,-1,R | $8.174\\!\\!\pm\\!\\!0.033$ | $8.381\\!\\!\pm\\!\\!0.029$ | $105.409\\!\\!\pm\\!\\!0.010$ | - |  | - | - | $\uparrow$ |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 9 | $40\\!\\!\pm\\!\\!57~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | $40\\!\\!\pm\\!\\!57~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | $45.6\\!\\!\pm\\!\\!1.4$ | - |  | - | - | 89 | Rmonthly2000K31-1R58.dat
| $4.79\\!\\!\times\\!\\!10^{6}$ | $1750.23\\!\\!\pm\\!\\!0.58$ | $1750.67\\!\\!\pm\\!\\!0.53$ | $1753.13\\!\\!\pm\\!\\!0.13$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=8 |  $\mathcal{M}$4,1,-1,R | $8.175\\!\\!\pm\\!\\!0.026$ | $8.384\\!\\!\pm\\!\\!0.032$ | $10.543\\!\\!\pm\\!\\!0.010$ | $52.08\\!\\!\pm\\!\\!0.36$ |  | - | - | - |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 12 | $39\\!\\!\pm\\!\\!25~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | $39\\!\\!\pm\\!\\!24~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | $45.7\\!\\!\pm\\!\\!2.1$ | $31.6\\!\\!\pm\\!\\!2.1$ |  | - | - | - | Rmonthly2000K41-1R58.dat
| $4.40\\!\\!\times\\!\\!10^{6}$ | $1750.22\\!\\!\pm\\!\\!0.52$ | $1750.64\\!\\!\pm\\!\\!0.39$ | $1753.11\\!\\!\pm\\!\\!0.13$ | $1760.2\\!\\!\pm\\!\\!1.0$ |  | - | - | - |
Table 11: Rmonthly2000 analysis for double waves $(K_{2}=2)$. Otherwise as in Table 10.  |  | Period analysis |  | Fisher-test |
---|---|---|---|---|---
Data: Non-weighted original data ($n=3012,\Delta T=250.9$:  Rmonthly2000.dat)
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},R}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $R$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$1,2,0,R | $11.0242\\!\\!\pm\\!\\!0.0084$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 6 | $94.7\\!\\!\pm\\!\\!3.6$ | - | - | - |  | 154 | 149 | 164 | Rmonthly2000K120R14.dat
| $1.08\\!\\!\times\\!\\!10^{7}$ | $1755.98\\!\\!\pm\\!\\!0.12$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$2,2,0,R | $11.0118\\!\\!\pm\\!\\!0.0071$ | $20.000\\!\\!\pm\\!\\!0.016$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 11 | $95.0\\!\\!\pm\\!\\!2.8$ | $81.2\\!\\!\pm\\!\\!2.4$ | - | - |  | - | 115 | 135 | Rmonthly2000K220R14.dat
| $8.59\\!\\!\times\\!\\!10^{6}$ | $1756.33\\!\\!\pm\\!\\!0.15$ | $1764.12\\!\\!\pm\\!\\!0.11$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$3,2,0,R | $9.9862\\!\\!\pm\\!\\!0.0068$ | $11.0222\\!\\!\pm\\!\\!0.0062$ | $23.692\\!\\!\pm\\!\\!0.021$ | - |  | - | - | $\uparrow$ |
| 16 | $78.7\\!\\!\pm\\!\\!1.8$ | $103.3\\!\\!\pm\\!\\!2.9$ | $69.1\\!\\!\pm\\!\\!2.4$ | - |  | - | - | 130 | Rmonthly2000K320R14.dat
| $7.21\\!\\!\times\\!\\!10^{6}$ | $1754.72\\!\\!\pm\\!\\!0.18$ | $1756.33\\!\\!\pm\\!\\!0.12$ | $1760.11\\!\\!\pm\\!\\!0.13$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$4,2,0,R | $9.9882\\!\\!\pm\\!\\!0.0061$ | $11.0234\\!\\!\pm\\!\\!0.0068$ | $23.686\\!\\!\pm\\!\\!0.018$ | $104.16\\!\\!\pm\\!\\!0.59$ |  | - | - | - |
| 21 | $78.6\\!\\!\pm\\!\\!2.6$ | $102.9\\!\\!\pm\\!\\!2.7$ | $69.4\\!\\!\pm\\!\\!2.7$ | $70.4\\!\\!\pm\\!\\!3.6$ |  | - | - | - | Rmonthly2000K420R148.dat
| $5.92\\!\\!\times\\!\\!10^{6}$ | $1754.70\\!\\!\pm\\!\\!0.11$ | $1756.295\\!\\!\pm\\!\\!0.099$ | $1760.13\\!\\!\pm\\!\\!0.14$ | $1811.43\\!\\!\pm\\!\\!0.54$ |  | - | - | - |
Data: Non-weighted residuals of $\mathcal{M}$=4 ($n=3012,\Delta T=250.9$:
Rmonthly2000K420R14Residuals.dat)
|  |  |  |  |  |  |  $\mathcal{M}$=6 |  $\mathcal{M}$=7 |  $\mathcal{M}$=8 |
$\mathcal{M}$=5 |  $\mathcal{M}$1,2,-1,R | $10.5400\\!\\!\pm\\!\\!0.0095$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 5 | $51.1\\!\\!\pm\\!\\!2.6$ | - | - | - |  | 68 | 69 | 74 | Rmonthly2000K12-1R58.dat
| $5.04\\!\\!\times\\!\\!10^{6}$ | $1754.34\\!\\!\pm\\!\\!0.19$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=6 |  $\mathcal{M}$2,2,-1,R | $10.5386\\!\\!\pm\\!\\!0.0076$ | $16.215\\!\\!\pm\\!\\!0.014$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 10 | $51.5\\!\\!\pm\\!\\!2.6$ | $41.6\\!\\!\pm\\!\\!3.0$ | - | - |  | - | 63 | 70 | Rmonthly2000K22-1R58.dat
| $4.53\\!\\!\times\\!\\!10^{6}$ | $1754.41\\!\\!\pm\\!\\!0.19$ | $1751.20\\!\\!\pm\\!\\!0.16$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=7 |  $\mathcal{M}$3,2,-1,R | $8.177\\!\\!\pm\\!\\!0.010$ | $10.5400\\!\\!\pm\\!\\!0.0071$ | $16.764\\!\\!\pm\\!\\!0.033$ | - |  | - | - | $\uparrow$ |
| 15 | $39.6\\!\\!\pm\\!\\!4.0$ | $51.7\\!\\!\pm\\!\\!2.2$ | $46.0\\!\\!\pm\\!\\!4.4$ | - |  | - | - | 70 | Rmonthly2000K32-1R58.dat
| $4.10\\!\\!\times\\!\\!10^{6}$ | $1750.46\\!\\!\pm\\!\\!0.60$ | $1754.35\\!\\!\pm\\!\\!0.10$ | $1759.23\\!\\!\pm\\!\\!0.26$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=8 |  $\mathcal{M}$4,2,-1,R | $8.173\\!\\!\pm\\!\\!0.011$ | $10.5372\\!\\!\pm\\!\\!0.0075$ | $16.781\\!\\!\pm\\!\\!0.034$ | $136.43\\!\\!\pm\\!\\!0.99$ |  | - | - | - |
| 20 | $37.6\\!\\!\pm\\!\\!4.8$ | $51.2\\!\\!\pm\\!\\!1.8$ | $44.9\\!\\!\pm\\!\\!4.0$ | $42.3\\!\\!\pm\\!\\!2.2$ |  | - | - | - | Rmonthly2000K42-1R58.dat
| $3.67\\!\\!\times\\!\\!10^{6}$ | $1750.76\\!\\!\pm\\!\\!0.64$ | $1754.38\\!\\!\pm\\!\\!0.12$ | $1759.13\\!\\!\pm\\!\\!0.25$ | $1884.10\\!\\!\pm\\!\\!0.91$ |  | - | - | - |
Table 12: Rmonthly analysis for pure sines $(K_{2}=1)$. Otherwise as in Table
10.  Data: Non-weighted original data ($n=3287,\Delta T=273.8$:  Rmonthly.dat)
---
|  | Period analysis |  | Fisher-test |
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},R}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $R$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$1,1,0,R | $11.0046\\!\\!\pm\\!\\!0.0076$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 4 | $93.8\\!\\!\pm\\!\\!2.3$ | - | - | - |  | 217 | 243 | 237 | RmonthlyK110R14.dat
| $1.14\\!\\!\times\\!\\!10^{7}$ | $1755.59\\!\\!\pm\\!\\!0.11$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$2,1,0,R | $10.618\\!\\!\pm\\!\\!0.010$ | $11.01254\\!\\!\pm\\!\\!0.0067$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 7 | $69.2\\!\\!\pm\\!\\!2.9$ | $94.0\\!\\!\pm\\!\\!2.4$ | - | - |  | - | 224 | 206 | RmonthlyK210R14.dat
| $9.51\\!\\!\times\\!\\!10^{6}$ | $1752.09\\!\\!\pm\\!\\!0.14$ | $1755.363\\!\\!\pm\\!\\!0.084$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$3,1,0,R | $10.0009\\!\\!\pm\\!\\!0.0051$ | $10.9954\\!\\!\pm\\!\\!0.0075$ | $11.780\\!\\!\pm\\!\\!0.012$ | - |  | - | - | $\uparrow$ |
| 10 | $74.6\\!\\!\pm\\!\\!3.2$ | $102.0\\!\\!\pm\\!\\!2.6$ | $61.6\\!\\!\pm\\!\\!2.8$ | - |  | - | - | 156 | RmonthlyK310R14.dat
| $7.89\\!\\!\times\\!\\!10^{6}$ | $1754.302\\!\\!\pm\\!\\!0.095$ | $1756.046\\!\\!\pm\\!\\!0.086$ | $1760.67\\!\\!\pm\\!\\!0.16$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$4,1,0,R | $10.0001\\!\\!\pm\\!\\!0.0081$ | $10.569\\!\\!\pm\\!\\!0.020$ | $11.0033\\!\\!\pm\\!\\!0.0064$ | $11.807\\!\\!\pm\\!\\!0.012$ |  | - | - | - |
| 13 | $65.5\\!\\!\pm\\!\\!2.1$ | $51.0\\!\\!\pm\\!\\!1.8$ | $101.5\\!\\!\pm\\!\\!2.5$ | $56.4\\!\\!\pm\\!\\!2.4$ |  | - | - | - | RmonthlyK410R14.dat
| $6.90\\!\\!\times\\!\\!10^{6}$ | $1754.16\\!\\!\pm\\!\\!0.11$ | $1753.03\\!\\!\pm\\!\\!0.27$ | $1755.897\\!\\!\pm\\!\\!0.076$ | $1760.33\\!\\!\pm\\!\\!0.13$ |  | - | - | - |
Data: Non-weighted residuals of model $\mathcal{M}$=4 ($n=3287,\Delta
T=273.8$:  RmonthlyK410R14Residuals.dat)
|  |  |  |  |  |  |  $\mathcal{M}$=6 |  $\mathcal{M}$=7 |  $\mathcal{M}$=8 |
$\mathcal{M}$=5 |  $\mathcal{M}$1,1,-1,R | $98.6\\!\\!\pm\\!\\!1.1$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 3 | $49.5\\!\\!\pm\\!\\!2.3$ | - | - | - |  | 93 | 103 | 106 | RmonthlyK11-1R58.dat
| $5.84\\!\\!\times\\!\\!10^{6}$ | $1814.2\\!\\!\pm\\!\\!1.5$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=6 |  $\mathcal{M}$2,1,-1,R | $52.67\\!\\!\pm\\!\\!0.29$ | $99.68\\!\\!\pm\\!\\!0.89$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 6 | $34.0\\!\\!\pm\\!\\!2.2$ | $51.5\\!\\!\pm\\!\\!2.0$ | - | - |  | - | 104 | 104 | RmonthlyK21-1R58.dat
| $5.38\\!\\!\times\\!\\!10^{6}$ | $1759.52\\!\\!\pm\\!\\!0.87$ | $1812.24\\!\\!\pm\\!\\!0.75$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=7 |  $\mathcal{M}$3,1,-1,R | $8.1087\\!\\!\pm\\!\\!0.0076$ | $52.66\\!\\!\pm\\!\\!0.27$ | $99.92\\!\\!\pm\\!\\!0.57$ | - |  | - | - | $\uparrow$ |
| 9 | $33.6\\!\\!\pm\\!\\!1.6$ | $33.9\\!\\!\pm\\!\\!2.0$ | $51.8\\!\\!\pm\\!\\!2.3$ | - |  | - | - | 94 | RmonthlyK31-1R58.dat
| $4.91\\!\\!\times\\!\\!10^{6}$ | $1751.29\\!\\!\pm\\!\\!0.14$ | $1759.6\\!\\!\pm\\!\\!0.70$ | $1811.99\\!\\!\pm\\!\\!0.84$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=8 |  $\mathcal{M}$4,1,-1,R | $8.456\\!\\!\pm\\!\\!0.0086$ | $59.6\\!\\!\pm\\!\\!1.2~{}{\color[rgb]{0,0,1}{\bf\mathrm{IF}}}$ | $60.8\\!\\!\pm\\!\\!1.2~{}{\color[rgb]{0,0,1}{\bf\mathrm{IF}}}$ | $103.10\\!\\!\pm\\!\\!0.67$ |  | - | - | - |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 12 | $32.4\\!\\!\pm\\!\\!2.6$ | $274\\!\\!\pm\\!\\!91~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | $274\\!\\!\pm\\!\\!92~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | $54.4\\!\\!\pm\\!\\!1.9$ |  | - | - | - | RmonthlyK41-1R58.dat
| $4.52\\!\\!\times\\!\\!10^{6}$ | $1749.51\\!\\!\pm\\!\\!0.14$ | $1801.1\\!\\!\pm\\!\\!2.5$ | $1768.4\\!\\!\pm\\!\\!3.1$ | $1807.29\\!\\!\pm\\!\\!0.94$ |  | - | - | - |
Table 13:  Rmonthly analysis for double waves $(K_{2}=2)$. Otherwise as in
Table 10.  Data: Original non-weighted data ($n=3287,\Delta T=273.8$:
Rmonthly.dat)
---
|  | Period analysis |  | Fisher-test |
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},R}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $R$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$1,2,0,R | $10.9922\\!\\!\pm\\!\\!0.0064$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 6 | $97.5\\!\\!\pm\\!\\!2.8$ | - | - | - |  | 157 | 155 | 177 | RmonthlyK120R14.dat
| $1.14\\!\\!\times\\!\\!10^{7}$ | $1756.41\\!\\!\pm\\!\\!0.18$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$2,2,0,R | $10.9964\\!\\!\pm\\!\\!0.0090$ | $21.252\\!\\!\pm\\!\\!0.023$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 11 | $97.4\\!\\!\pm\\!\\!3.3$ | $82.8\\!\\!\pm\\!\\!2.8$ | - | - |  | - | 123 | 151 | RmonthlyK220R14.dat
| $9.20\\!\\!\times\\!\\!10^{6}$ | $1756.13\\!\\!\pm\\!\\!0.18$ | $1752.36\\!\\!\pm\\!\\!0.14$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$3,2,0,R | $10.9864\\!\\!\pm\\!\\!0.0052$ | $11.772\\!\\!\pm\\!\\!0.010$ | $19.998\\!\\!\pm\\!\\!0.012$ | - |  | - | - | $\uparrow$ |
| 16 | $106.4\\!\\!\pm\\!\\!2.7$ | $64.2\\!\\!\pm\\!\\!2.3$ | $76.1\\!\\!\pm\\!\\!3.1$ | - |  | - | - | 150 | RmonthlyK320R14.dat
| $7.74\\!\\!\times\\!\\!10^{6}$ | $1756.72\\!\\!\pm\\!\\!0.11$ | $1749.48\\!\\!\pm\\!\\!0.18$ | $1754.36\\!\\!\pm\\!\\!0.11$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$4,2,0,R | $10.9878\\!\\!\pm\\!\\!0.0051$ | $11.770\\!\\!\pm\\!\\!0.011$ | $20.0062\\!\\!\pm\\!\\!0.0088$ | $104.23\\!\\!\pm\\!\\!0.68$ |  | - | - | - |
| 21 | $104.6\\!\\!\pm\\!\\!2.3$ | $62.7\\!\\!\pm\\!\\!2.6$ | $75.2\\!\\!\pm\\!\\!2.7$ | $68.1\\!\\!\pm\\!\\!2.7$ |  | - | - | - | RmonthlyK420K410.dat
| $6.29\\!\\!\times\\!\\!10^{6}$ | $1756.678\\!\\!\pm\\!\\!0.077$ | $1749.51\\!\\!\pm\\!\\!0.15$ | $1754.284\\!\\!\pm\\!\\!0.063$ | $1811.37\\!\\!\pm\\!\\!0.68$ |  | - | - | - |
Data: Non-weighted residuals of model $\mathcal{M}$=4 ($n=3287,\Delta
T=273.8$:  RmonthlyK420R14Residuals.dat)
|  |  |  |  |  |  |  $\mathcal{M}$=6 |  $\mathcal{M}$=7 |  $\mathcal{M}$=8 |
$\mathcal{M}$=5 |  $\mathcal{M}$1,2,-1,R | $10.5486\\!\\!\pm\\!\\!0.0078$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 5 | $53.6\\!\\!\pm\\!\\!2.6$ | - | - | - |  | 70 | 66 | 65 | RmonthlyK12-1R58.dat
| $5.26\\!\\!\times\\!\\!10^{6}$ | $1754.38\\!\\!\pm\\!\\!0.13$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=6 |  $\mathcal{M}$2,2,-1,R | $10.5421\\!\\!\pm\\!\\!0.0072$ | $16.220\\!\\!\pm\\!\\!0.011$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 10 | $54.3\\!\\!\pm\\!\\!2.1$ | $41.0\\!\\!\pm\\!\\!1.8$ | - | - |  | - | 55 | 56 | RmonthlyK22-1R58.dat
| $4.75\\!\\!\times\\!\\!10^{6}$ | $1754.44\\!\\!\pm\\!\\!0.14$ | $1751.167\\!\\!\pm\\!\\!0.099$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=7 |  $\mathcal{M}$3,2,-1,R | $10.5376\\!\\!\pm\\!\\!0.0063$ | $16.217\\!\\!\pm\\!\\!0.013$ | $142.9\\!\\!\pm\\!\\!1.4$ | - |  | - | - | $\uparrow$ |
| 15 | $54.2\\!\\!\pm\\!\\!1.8$ | $40.9\\!\\!\pm\\!\\!2.3$ | $37.5\\!\\!\pm\\!\\!2.4$ | - |  | - | - | 53 | RmonthlyK32-1R58.dat
| $4.38\\!\\!\times\\!\\!10^{6}$ | $1754.50\\!\\!\pm\\!\\!0.13$ | $1751.17\\!\\!\pm\\!\\!0.14$ | $1887.63\\!\\!\pm\\!\\!0.73$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=8 |  $\mathcal{M}$4,2,-1,R | $8.169\\!\\!\pm\\!\\!0.014$ | $10.5407\\!\\!\pm\\!\\!0.0060$ | $16.753\\!\\!\pm\\!\\!0.027$ | $143.39\\!\\!\pm\\!\\!0.99$ |  | - | - | - |
| 20 | $35.5\\!\\!\pm\\!\\!3.3$ | $55.6\\!\\!\pm\\!\\!2.3$ | $41.2\\!\\!\pm\\!\\!3.1$ | $37.3\\!\\!\pm\\!\\!2.7$ |  | - | - | - | RmonthlyK42-1R58.dat
| $4.05\\!\\!\times\\!\\!10^{6}$ | $1750.90\\!\\!\pm\\!\\!0.51$ | $1754.40\\!\\!\pm\\!\\!0.13$ | $1759.19\\!\\!\pm\\!\\!0.23$ | $1887.84\\!\\!\pm\\!\\!0.80$ |  | - | - | - |
Table 14: Cmonthly2000 analysis for pure sines $(K_{2}=1)$. Otherwise as in Table 10.  |  | Period analysis |  | Fisher-test |
---|---|---|---|---|---
Data: Weighted original data before 2000 ($n=2182,\Delta T=181.9$:
Cmonthly2000.dat)
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},\chi^{2}}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $\chi^{2}$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$${}_{1,1,0,\chi^{2}}$ | $10.8338\\!\\!\pm\\!\\!0.0094$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 4 | $121.9\\!\\!\pm\\!\\!3.6$ | - | - | - |  | 203 | 230 | 196 | Cmonthly2000K110R14.dat
| $3.34\\!\\!\times\\!\\!10^{6}$ | $1823.740\\!\\!\pm\\!\\!0.082$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$${}_{2,1,0,\chi^{2}}$ | $10.196\\!\\!\pm\\!\\!0.027$ | $10.805\\!\\!\pm\\!\\!0.013$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 7 | $61.9\\!\\!\pm\\!\\!3.9$ | $118.2\\!\\!\pm\\!\\!3.7$ | - | - |  | - | 202 | 151 | Cmonthly2000K210R14.dat
| $2.61\\!\\!\times\\!\\!10^{6}$ | $1822.12\\!\\!\pm\\!\\!0.23$ | $1823.941\\!\\!\pm\\!\\!0.091$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$${}_{3,1,0,\chi^{2}}$ | $10.265\\!\\!\pm\\!\\!0.026$ | $10.766\\!\\!\pm\\!\\!0.017$ | $267\\!\\!\pm\\!\\!34~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ | - |  | - | - | $\uparrow$ |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 10 | $71.0\\!\\!\pm\\!\\!3.2$ | $118.2\\!\\!\pm\\!\\!3.8$ | $54.6\\!\\!\pm\\!\\!4.9$ | - |  | - | - | 78 | Cmonthly2000K310R14.dat
| $2.04\\!\\!\times\\!\\!10^{6}$ | $1821.51\\!\\!\pm\\!\\!0.17$ | $1824.19\\!\\!\pm\\!\\!0.13$ | $1868.3\\!\\!\pm\\!\\!4.5$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$${}_{4,1,0,\chi^{2}}$ | $10.236\\!\\!\pm\\!\\!0.020$ | $10.770\\!\\!\pm\\!\\!0.010$ | $99.6\\!\\!\pm\\!\\!6.2$ | $8400\\!\\!\pm\\!\\!3800~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ |  | - | - | - |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 13 | $69.0\\!\\!\pm\\!\\!2.8$ | $119.2\\!\\!\pm\\!\\!4.0$ | $34.8\\!\\!\pm\\!\\!4.0$ | $19000\\!\\!\pm\\!\\!8800~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ |  | - | - | - | Cmonthly2000K410R14.dat
| $1.84\\!\\!\times\\!\\!10^{6}$ | $1821.83\\!\\!\pm\\!\\!0.17$ | $1824.192\\!\\!\pm\\!\\!0.079$ | $1903.2\\!\\!\pm\\!\\!1.8$ | $1843\\!\\!\pm\\!\\!24$ |  | - | - | - |
Table 15: Cmonthly2000 analysis for double waves $(K_{2}=2)$ Otherwise as in Table 10.  |  | Period analysis |  | Fisher-test |
---|---|---|---|---|---
Data: Weighted original data before 2000 ($n=2182,\Delta T=181.9$:
Cmonthly2000.dat)
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},\chi^{2}}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $\chi^{2}$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$${}_{1,2,0,\chi^{2}}$ | $10.830\\!\\!\pm\\!\\!0.010$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 6 | $126.1\\!\\!\pm\\!\\!4.2$ | - | - | - |  | 125 | 180 | 160 | Cmonthly2000K120R14.dat
| $3.30\\!\\!\times\\!\\!10^{6}$ | $1824.17\\!\\!\pm\\!\\!0.12$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$${}_{2,2,0,\chi^{2}}$ | $10.196\\!\\!\pm\\!\\!0.017$ | $10.807\\!\\!\pm\\!\\!0.012$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 11 | $62.1\\!\\!\pm\\!\\!3.7$ | $123.5\\!\\!\pm\\!\\!3.0$ | - | - |  | - | 183 | 138 | Cmonthly2000K220R14.dat
| $2.56\\!\\!\times\\!\\!10^{6}$ | $1822.31\\!\\!\pm\\!\\!0.17$ | $1824.34\\!\\!\pm\\!\\!0.10$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$${}_{3,2,0,\chi^{2}}$ | $10.240\\!\\!\pm\\!\\!0.017$ | $10.766\\!\\!\pm\\!\\!0.012$ | $368\\!\\!\pm\\!\\!1730~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ | - |  | - | - | $\uparrow$ |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 16 | $67.5\\!\\!\pm\\!\\!3.2$ | $123.7\\!\\!\pm\\!\\!2.8$ | $330\\!\\!\pm\\!\\!1.9\\!\\!\times\\!\\!10^{7}~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | - |  | - | - | 66 | Cmonthly2000K320R14.dat
| $1.80\\!\\!\times\\!\\!10^{6}$ | $1821.92\\!\\!\pm\\!\\!0.26$ | $1824.63\\!\\!\pm\\!\\!0.17$ | $2097\\!\\!\pm\\!\\!92$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$${}_{4,2,0,\chi^{2}}$ | $12.01\\!\\!\pm\\!\\!0.76$ | $20.23\\!\\!\pm\\!\\!0.54~{}{\color[rgb]{0,0,1}{\bf\mathrm{IF}}}$ | $21.59\\!\\!\pm\\!\\!0.98~{}{\color[rgb]{0,0,1}{\bf\mathrm{IF}}}$ | $2222\\!\\!\pm\\!\\!823~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ |  | - | - | - |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 21 | $39.1\\!\\!\pm\\!\\!9.2$ | $71\\!\\!\pm\\!\\!21~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | $130\\!\\!\pm\\!\\!37~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | $6\\!\\!\times\\!\\!10^{5}\\!\\!\pm\\!\\!3\\!\\!\times\\!\\!10^{5}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ |  | - | - | - | Cmonthly2000K420R14.dat
| $1.56\\!\\!\times\\!\\!10^{6}$ | $1818.4\\!\\!\pm\\!\\!1.8$ | $1833.22\\!\\!\pm\\!\\!0.77$ | $1835.2\\!\\!\pm\\!\\!02.2$ | $3025\\!\\!\pm\\!\\!191$ |  | - | - | - |
Table 16: Cmonthly analysis for pure sines $(K_{2}=1)$. Otherwise as in Table 10.  |  | Period analysis |  | Fisher-test |
---|---|---|---|---|---
Data: Weighted original data ($n=2458,\Delta T=204.8$:  Cmonthly.dat)
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},\chi^{2}}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $\chi^{2}$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$${}_{1,1,0,\chi^{2}}$ | $10.8350\\!\\!\pm\\!\\!0.0089$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 4 | $121.6\\!\\!\pm\\!\\!2.9$ | - | - | - |  | 173 | 197 | 210 | CmonthlyK110R14.dat
| $3.71\\!\\!\times\\!\\!10^{6}$ | $1823.730\\!\\!\pm\\!\\!0.088$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$${}_{2,1,0,\chi^{2}}$ | $10.149\\!\\!\pm\\!\\!0.012$ | $10.8549\\!\\!\pm\\!\\!0.0078$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 7 | $53.8\\!\\!\pm\\!\\!2.5$ | $120.1\\!\\!\pm\\!\\!2.5$ | - | - |  | - | 183 | 189 | CmonthlyK210R14.dat
| $3.06\\!\\!\times\\!\\!10^{6}$ | $1822.40\\!\\!\pm\\!\\!0.16$ | $1823.63\\!\\!\pm\\!\\!0.10$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$${}_{3,1,0,\chi^{2}}$ | $10.113\\!\\!\pm\\!\\!0.010$ | $10.8612\\!\\!\pm\\!\\!0.0083$ | $120.8\\!\\!\pm\\!\\!1.7$ | - |  | - | - | $\uparrow$ |
| 10 | $58.1\\!\\!\pm\\!\\!3.1$ | $122.6\\!\\!\pm\\!\\!2.0$ | $51.1\\!\\!\pm\\!\\!2.8$ | - |  | - | - | 160 | CmonthlyK310R14.dat
| $2.50\\!\\!\times\\!\\!10^{6}$ | $1822.91\\!\\!\pm\\!\\!0.13$ | $1823.516\\!\\!\pm\\!\\!0.086$ | $1908.0\\!\\!\pm\\!\\!1.1$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$${}_{4,1,0,\chi^{2}}$ | $10.0658\\!\\!\pm\\!\\!0.0077$ | $10.8585\\!\\!\pm\\!\\!0.0048$ | $11.863\\!\\!\pm\\!\\!0.021$ | $116.7\\!\\!\pm\\!\\!1.3$ |  | - | - | - |
| 13 | $59.3\\!\\!\pm\\!\\!2.8$ | $117.5\\!\\!\pm\\!\\!2.9$ | $43.3\\!\\!\pm\\!\\!2.0$ | $52.9\\!\\!\pm\\!\\!2.5$ |  | - | - | - | CmonthlyK410R14.dat
| $2.09\\!\\!\times\\!\\!10^{6}$ | $1823.52\\!\\!\pm\\!\\!0.12$ | $1823.580\\!\\!\pm\\!\\!0.053$ | $1818.92\\!\\!\pm\\!\\!0.20$ | $1911.02\\!\\!\pm\\!\\!0.74$ |  | - | - | - |
Data: Non-weighted residuals of $\mathcal{M}$=4 ($n=2458,\Delta T=204.8$:
CmonthlyK410R14Residuals.dat)
|  |  |  |  |  |  |  $\mathcal{M}$=6 |  $\mathcal{M}$=7 |  $\mathcal{M}$=8 |
$\mathcal{M}$=5 |  $\mathcal{M}$${}_{1,1,-1,\chi^{2}}$ | $8.009\\!\\!\pm\\!\\!0.017$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 3 | $25.6\\!\\!\pm\\!\\!2.5$ | - | - | - |  | 64 | 55 | 54 | CmonthlyK11-1R58.dat
| $1.93\\!\\!\times\\!\\!10^{6}$ | $1818.07\\!\\!\pm\\!\\!0.24$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=6 |  $\mathcal{M}$${}_{2,1,-1,\chi^{2}}$ | $8.008\\!\\!\pm\\!\\!0.014$ | $239\\!\\!\pm\\!\\!22~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 6 | $25.0\\!\\!\pm\\!\\!1.9$ | $24.7\\!\\!\pm\\!\\!2.3$ | - | - |  | - | 43 | 46 | CmonthlyK21-1R58.dat
| $1.79\\!\\!\times\\!\\!10^{6}$ | $1818.07\\!\\!\pm\\!\\!0.20$ | $1841\\!\\!\pm\\!\\!11$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=7 |  $\mathcal{M}$${}_{3,1,-1,\chi^{2}}$ | $5.4184\\!\\!\pm\\!\\!0.0064$ | $8.005\\!\\!\pm\\!\\!0.018$ | $239\\!\\!\pm\\!\\!23~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ | - |  | - | - | $\uparrow$ |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 9 | $20.3\\!\\!\pm\\!\\!2.3$ | $24.7\\!\\!\pm\\!\\!2.0$ | $25.4\\!\\!\pm\\!\\!1.5$ | - |  | - | - | 46 | CmonthlyK31-1R58.dat
| $1.70\\!\\!\times\\!\\!10^{6}$ | $1819.89\\!\\!\pm\\!\\!0.14$ | $1818.10\\!\\!\pm\\!\\!0.30$ | $1841\\!\\!\pm\\!\\!10$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=8 |  $\mathcal{M}$${}_{4,1,-1,\chi^{2}}$ | $5.4187\\!\\!\pm\\!\\!0.0075$ | $8.002\\!\\!\pm\\!\\!0.015$ | $38.72\\!\\!\pm\\!\\!0.58$ | $250\\!\\!\pm\\!\\!28~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ |  | - | - | - |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 12 | $20.0\\!\\!\pm\\!\\!1.8$ | $25.7\\!\\!\pm\\!\\!2.2$ | $19.5\\!\\!\pm\\!\\!1.7$ | $24.5\\!\\!\pm\\!\\!2.0$ |  | - | - | - | CmonthlyK41-1R58.dat
| $1.61\\!\\!\times\\!\\!10^{6}$ | $1819.88\\!\\!\pm\\!\\!0.18$ | $1818.16\\!\\!\pm\\!\\!0.26$ | $1856.2\\!\\!\pm\\!\\!1.2$ | $1840\\!\\!\pm\\!\\!12$ |  | - | - | - |
Table 17: Cmonthly analysis for double waves $(K_{2}=2)$. Otherwise as in Table 10.  |  | Period analysis |  | Fisher-test |
---|---|---|---|---|---
Data: Weighted original data ($n=2458,\Delta T=204.8$:  Cmonthly.dat)
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},\chi^{2}}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $\chi^{2}$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$${}_{1,2,0,\chi^{2}}$ | $21.667\\!\\!\pm\\!\\!0.014$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 6 | $132.6\\!\\!\pm\\!\\!3.4$ | - | - | - |  | 140 | 171 | 170 | CmonthlyK120R14.dat
| $3.64\\!\\!\times\\!\\!10^{6}$ | $1823.632\\!\\!\pm\\!\\!0.077$ | - | - | - |  | $<\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$${}_{2,2,0,\chi^{2}}$ | $10.8110\\!\\!\pm\\!\\!0.0092$ | $297\\!\\!\pm\\!\\!68~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 11 | $125.0\\!\\!\pm\\!\\!4.0$ | $152\\!\\!\pm\\!\\!380~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | - | - |  | - | 157 | 144 | CmonthlyK220R14.dat
| $2.83\\!\\!\times\\!\\!10^{6}$ | $1824.50\\!\\!\pm\\!\\!0.13$ | $2067\\!\\!\pm\\!\\!32$ | - | - |  | - | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$${}_{3,2,0,\chi^{2}}$ | $10.136\\!\\!\pm\\!\\!0.0012$ | $21.6733\\!\\!\pm\\!\\!0.0091$ | $323\\!\\!\pm\\!\\!44~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ | - |  | - | - | $\uparrow$ |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 16 | $57.4\\!\\!\pm\\!\\!2.6$ | $128.7\\!\\!\pm\\!\\!4.0$ | $192\\!\\!\pm\\!\\!157~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | - |  | - | - | 98 | CmonthlyK320R14.dat
| $2.14\\!\\!\times\\!\\!10^{6}$ | $1822.74\\!\\!\pm\\!\\!0.24$ | $1834.774\\!\\!\pm\\!\\!0.065$ | $2075\\!\\!\pm\\!\\!21$ | - |  | - | - | $<\\!10^{-16}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$${}_{4,2,0,\chi^{2}}$ | $20.165\\!\\!\pm\\!\\!0.019$ | $21.648\\!\\!\pm\\!\\!0.093$ | $23.895\\!\\!\pm\\!\\!0.037$ | $322\\!\\!\pm\\!\\!59~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ |  | - | - | - |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 21 | $62.4\\!\\!\pm\\!\\!2.6$ | $129.9\\!\\!\pm\\!\\!2.9$ | $45.0\\!\\!\pm\\!\\!3.0$ | $214\\!\\!\pm\\!\\!392~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ |  | - | - | - | CmonthlyK420R14.dat
| $1.78\\!\\!\times\\!\\!10^{6}$ | $1833.39\\!\\!\pm\\!\\!0.11$ | $1834.949\\!\\!\pm\\!\\!0.067$ | $1841.86\\!\\!\pm\\!\\!0.17$ | $2078\\!\\!\pm\\!\\!29$ |  | - | - | - |
Table 18: Ryearly2000 analysis for pure sinusoids $(K_{2}=1)$. Otherwise as in Table 10.  |  | Period analysis |  | Fisher-test |
---|---|---|---|---|---
Data: Non-weighted original data ($n=300,\Delta T=299$:  Ryearly2000.dat)
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},R}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $R$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$1,1,0,R | $11.018\\!\\!\pm\\!\\!0.018$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 4 | $94.7\\!\\!\pm\\!\\!7.9$ | - | - | - |  | 28 | 24 | 26 | Ryearly2000K110R14.dat
| $8.26\\!\\!\times\\!\\!10^{5}$ | $1711.46\\!\\!\pm\\!\\!0.24$ | - | - | - |  | $6.7\\!\\!\times\\!\\!10^{-16}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$2,1,0,R | $10.001\\!\\!\pm\\!\\!0.021$ | $11.046\\!\\!\pm\\!\\!0.020$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 7 | $71.2\\!\\!\pm\\!\\!9.2$ | $90.2\\!\\!\pm\\!\\!5.2$ | - | - |  | - | 15 | 20 | Ryearly2000K210R14.dat
| $6.42\\!\\!\times\\!\\!10^{5}$ | $1704.15\\!\\!\pm\\!\\!0.34$ | $1711.08\\!\\!\pm\\!\\!0.31$ | - | - |  | - | $2.9\\!\\!\times\\!\\!10^{-9}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$3,1,0,R | $9.982\\!\\!\pm\\!\\!0.020$ | $10.624\\!\\!\pm\\!\\!0.029$ | $10.999\\!\\!\pm\\!\\!0.018$ | - |  | - | - | $\uparrow$ |
| 10 | $63.9\\!\\!\pm\\!\\!6.1$ | $53.3\\!\\!\pm\\!\\!8.4$ | $91.2\\!\\!\pm\\!\\!6.7$ | - |  | - | - | 22 | Ryearly2000K310R14.dat
| $5.54\\!\\!\times\\!\\!10^{5}$ | $1704.54\\!\\!\pm\\!\\!0.29$ | $1709.62\\!\\!\pm\\!\\!0.48$ | $1700.71\\!\\!\pm\\!\\!0.25$ | - |  | - | - | $9.8\\!\\!\times\\!\\!10^{-13}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$4,1,0,R | $9.980\\!\\!\pm\\!\\!0.016$ | $10.620\\!\\!\pm\\!\\!0.033$ | $11.006\\!\\!\pm\\!\\!0.017$ | $100.3\\!\\!\pm\\!\\!2.1$ |  | - | - | - |
| 13 | $64.6\\!\\!\pm\\!\\!7.6$ | $52.4\\!\\!\pm\\!\\!8.5$ | $90.4\\!\\!\pm\\!\\!4.7$ | $52.4\\!\\!\pm\\!\\!6.3$ |  | - | - | - | Ryearly2000K410R14.dat
| $4.51\\!\\!\times\\!\\!10^{5}$ | $1704.59\\!\\!\pm\\!\\!0.26$ | $1709.60\\!\\!\pm\\!\\!0.41$ | $1700.62\\!\\!\pm\\!\\!0.26$ | $1711.5\\!\\!\pm\\!\\!3.1$ |  | - | - | - |
Data: Non-weighted residuals of $\mathcal{M}$=4 ($n=300,\Delta T=299$:
Ryearly2000K410R14Residuals.dat)
|  |  |  |  |  |  |  $\mathcal{M}$=6 |  $\mathcal{M}$=7 |  $\mathcal{M}$=8 |
$\mathcal{M}$=5 |  $\mathcal{M}$1,1,-1,R | $11.845\\!\\!\pm\\!\\!0.039$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 3 | $42.1\\!\\!\pm\\!\\!5.2$ | - | - | - |  | 14 | 14 | 14 | Ryearly2000K11-1R58.dat
| $3.84\\!\\!\times\\!\\!10^{5}$ | $1700.78\\!\\!\pm\\!\\!0.54$ | - | - | - |  | $1.6\\!\\!\times\\!\\!10^{-8}$ | $2.4\\!\\!\times\\!\\!10^{-14}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=6 |  $\mathcal{M}$2,1,-1,R | $11.854\\!\\!\pm\\!\\!0.034$ | $52.98\\!\\!\pm\\!\\!0.69$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 6 | $42.5\\!\\!\pm\\!\\!5.5$ | $35.6\\!\\!\pm\\!\\!4.6$ | - | - |  | - | 13 | 13 | Ryearly2000K21-1R58.dat
| $3.36\\!\\!\times\\!\\!10^{5}$ | $1700.64\\!\\!\pm\\!\\!0.53$ | $1705.4\\!\\!\pm\\!\\!2.0$ | - | - |  | - | $5.0\\!\\!\times\\!\\!10^{-8}$ | $7.8\\!\\!\times\\!\\!10^{-13}$ |
$\mathcal{M}$=7 |  $\mathcal{M}$3,1,-1,R | $8.459\\!\\!\pm\\!\\!0.017$ | $11.860\\!\\!\pm\\!\\!0.032$ | $52.98\\!\\!\pm\\!\\!0.67$ | - |  | - | - | $\uparrow$ |
| 9 | $32.9\\!\\!\pm\\!\\!4.3$ | $42.7\\!\\!\pm\\!\\!5.3$ | $35.6\\!\\!\pm\\!\\!4.8$ | - |  | - | - | 11 | Ryearly2000K31-1R58.dat
| $2.96\\!\\!\times\\!\\!10^{5}$ | $1707.15\\!\\!\pm\\!\\!0.45$ | $1700.58\\!\\!\pm\\!\\!0.46$ | $1705.4\\!\\!\pm\\!\\!2.3$ | - |  | - | - | $5.7\\!\\!\times\\!\\!10^{-7}$ |
$\mathcal{M}$=8 |  $\mathcal{M}$4,1,-1,R | $8.460\\!\\!\pm\\!\\!0.027$ | $11.862\\!\\!\pm\\!\\!0.034$ | $53.24\\!\\!\pm\\!\\!0.78$ | $65.4\\!\\!\pm\\!\\!1.3$ |  | - | - | - |
| 12 | $33.0\\!\\!\pm\\!\\!4.4$ | $42.6\\!\\!\pm\\!\\!4.1$ | $35.3\\!\\!\pm\\!\\!6.3$ | $29.0\\!\\!\pm\\!\\!4.7$ |  | - | - | - | Ryearly2000K41-1R58.dat
| $2.65\\!\\!\times\\!\\!10^{5}$ | $1707.15\\!\\!\pm\\!\\!0.40$ | $1700.54\\!\\!\pm\\!\\!0.46$ | $1705.0\\!\\!\pm\\!\\!2.7$ | $1754.3\\!\\!\pm\\!\\!2.5$ |  | - | - | - |
Table 19: Ryearly analysis for pure sinusoids $(K_{2}=1)$. Otherwise as in Table 10.  |  | Period analysis |  | Fisher-test |
---|---|---|---|---|---
Data: Non-weighted original data ($n=322,\Delta T=321$:  Ryearly.dat)
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},R}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
| (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$1,1,0,R | $11.003\\!\\!\pm\\!\\!0.017$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 4 | $95.9\\!\\!\pm\\!\\!7.4$ | - | - | - |  | 26 | 26 | 29 | RyearlyK110R14.dat
| $8.65\\!\\!\times\\!\\!10^{5}$ | $1700.60\\!\\!\pm\\!\\!0.31$ | - | - | - |  | $6.1\\!\\!\times\\!\\!10^{-15}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$2,1,0,R | $10.018\\!\\!\pm\\!\\!0.018$ | $11.024\\!\\!\pm\\!\\!0.011$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 7 | $66.0\\!\\!\pm\\!\\!6.4$ | $94.1\\!\\!\pm\\!\\!5.0$ | - | - |  | - | 21 | 25 | RyearlyK210R14.dat
| $6.94\\!\\!\times\\!\\!10^{5}$ | $1703.97\\!\\!\pm\\!\\!0.38$ | $1711.30\\!\\!\pm\\!\\!0.21$ | - | - |  | - | $2.6\\!\\!\times\\!\\!10^{-12}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$3,1,0,R | $9.974\\!\\!\pm\\!\\!0.017$ | $10.665\\!\\!\pm\\!\\!0.032$ | $10.973\\!\\!\pm\\!\\!0.026$ | - |  | - | - | $\uparrow$ |
| 10 | $62.4\\!\\!\pm\\!\\!6.2$ | $61.8\\!\\!\pm\\!\\!8.8$ | $100.2\\!\\!\pm\\!\\!9.9$ | - |  | - | - | 24 | RyearlyK310R14.dat
| $5.78\\!\\!\times\\!\\!10^{5}$ | $1704.61\\!\\!\pm\\!\\!0.31$ | $1709.27\\!\\!\pm\\!\\!0.44$ | $1700.96\\!\\!\pm\\!\\!0.43$ | - |  | - | - | $3.4\\!\\!\times\\!\\!10^{-14}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$4,1,0,R | $9.975\\!\\!\pm\\!\\!0.017$ | $10.659\\!\\!\pm\\!\\!0.03$ | $10.981\\!\\!\pm\\!\\!0.020$ | $101.4\\!\\!\pm\\!\\!2.4$ |  | - | - | - |
| 13 | $62.8\\!\\!\pm\\!\\!6.1$ | $59.9\\!\\!\pm\\!\\!8.6$ | $98.3\\!\\!\pm\\!\\!7.8$ | $51.7\\!\\!\pm\\!\\!4.9$ |  | - | - | - | RyearlyK410R14.dat
| $4.67\\!\\!\times\\!\\!10^{5}$ | $1704.63\\!\\!\pm\\!\\!0.35$ | $1709.27\\!\\!\pm\\!\\!0.44$ | $1700.86\\!\\!\pm\\!\\!0.24$ | $1710.5\\!\\!\pm\\!\\!3.4$ |  | - | - | - |
Data: Non-weighted residuals of $\mathcal{M}$=4 ($n=322,\Delta T=321$:
RyearlyK410R14Residuals.dat)
|  |  |  |  |  |  |  $\mathcal{M}$=6 |  $\mathcal{M}$=7 |  $\mathcal{M}$=8 |
$\mathcal{M}$=5 |  $\mathcal{M}$1,1,-1,R | $11.814\\!\\!\pm\\!\\!0.034$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 3 | $41.0\\!\\!\pm\\!\\!5.2$ | - | - | - |  | 15 | 15 | 14 | RyearlyK11-1R58.dat
| $3.99\\!\\!\times\\!\\!10^{5}$ | $1701.13\\!\\!\pm\\!\\!0.47$ | - | - | - |  | $3.6\\!\\!\times\\!\\!10^{-9}$ | $5.1\\!\\!\times\\!\\!10^{-15}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=6 |  $\mathcal{M}$2,1,-1,R | $11.826\\!\\!\pm\\!\\!0.028$ | $53.15\\!\\!\pm\\!\\!0.56$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 6 | $40.5\\!\\!\pm\\!\\!5.1$ | $35.1\\!\\!\pm\\!\\!5.2$ | - | - |  | - | 13 | 12 | RyearlyK21-1R58.dat
| $3.49\\!\\!\times\\!\\!10^{5}$ | $1700.95\\!\\!\pm\\!\\!0.44$ | $1705.2\\!\\!\pm\\!\\!2.1$ | - | - |  | - | $4.5\\!\\!\times\\!\\!10^{-8}$ | $2.3\\!\\!\times\\!\\!10^{-12}$ |
$\mathcal{M}$=7 |  $\mathcal{M}$3,1,-1,R | $8.465\\!\\!\pm\\!\\!0.018$ | $11.816\\!\\!\pm\\!\\!0.033$ | $53.19\\!\\!\pm\\!\\!0.50$ | - |  | - | - | $\uparrow$ |
| 9 | $31.2\\!\\!\pm\\!\\!3.6$ | $40.8\\!\\!\pm\\!\\!4.7$ | $35.0\\!\\!\pm\\!\\!5.3$ | - |  | - | - | 10 | RyearlyK31-1R58.dat
| $3.10\\!\\!\times\\!\\!10^{5}$ | $1707.11\\!\\!\pm\\!\\!0.48$ | $1701.06\\!\\!\pm\\!\\!0.44$ | $1705.1\\!\\!\pm\\!\\!1.4$ | - |  | - | - | $1.9\\!\\!\times\\!\\!10^{-6}$ |
$\mathcal{M}$=8 |  $\mathcal{M}$4,1,-1,R | $8.466\\!\\!\pm\\!\\!0.016$ | $11.820\\!\\!\pm\\!\\!0.027$ | $53.83\\!\\!\pm\\!\\!0.76$ | $66.7\\!\\!\pm\\!\\!1.4$ |  | - | - | - |
| 12 | $31.1\\!\\!\pm\\!\\!4.8$ | $40.5\\!\\!\pm\\!\\!3.3$ | $34.3\\!\\!\pm\\!\\!3.6$ | $26.7\\!\\!\pm\\!\\!4.4$ |  | - | - | - | RyearlyK41-1R58.dat
| $2.82\\!\\!\times\\!\\!10^{5}$ | $1707.10\\!\\!\pm\\!\\!0.35$ | $1701.01\\!\\!\pm\\!\\!0.42$ | $1704.5\\!\\!\pm\\!\\!2.7$ | $1753.0\\!\\!\pm\\!\\!3.8$ |  | - | - | - |
Table 20: Cyearly2000 analysis for pure sinusoids $(K_{2}=1)$. Otherwise as in Table 10.  |  | Period analysis |  | Fisher-test |
---|---|---|---|---|---
Data: Weighted original data ($n=182,\Delta T=181$:  Cyearly2000.dat)
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},\chi^{2}}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $\chi^{2}$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$${}_{1,1,0,\chi^{2}}$ | $10.835\\!\\!\pm\\!\\!0.030$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 4 | $123.5\\!\\!\pm\\!\\!8.7$ | - | - | - |  | 21 | 24 | 21 | Cyearly2000K110R14.dat
| $2.30\\!\\!\times\\!\\!10^{5}$ | $1823.75\\!\\!\pm\\!\\!0.31$ | - | - | - |  | $7.4\\!\\!\times\\!\\!10^{-12}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$${}_{2,1,0,\chi^{2}}$ | $10.184\\!\\!\pm\\!\\!0.064$ | $10.813\\!\\!\pm\\!\\!0.029$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 7 | $62.3\\!\\!\pm\\!\\!6.8$ | $119.2\\!\\!\pm\\!\\!8.5$ | - | - |  | - | 20 | 16 | Cyearly2000K210R14.dat
| $1.68\\!\\!\times\\!\\!10^{5}$ | $1822.19\\!\\!\pm\\!\\!0.64$ | $1823.90\\!\\!\pm\\!\\!0.23$ | - | - |  | - | $2.8\\!\\!\times\\!\\!10^{-11}$ | $1.7\\!\\!\times\\!\\!10^{-14}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$${}_{3,1,0,\chi^{2}}$ | $10.244\\!\\!\pm\\!\\!0.059$ | $10.780\\!\\!\pm\\!\\!0.043$ | $248\\!\\!\pm\\!\\!1700~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ | - |  | - | - | $\uparrow$ |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 10 | $70\\!\\!\pm\\!\\!10$ | $118\\!\\!\pm\\!\\!10$ | $51\\!\\!\pm\\!\\!4300~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ | - |  | - | - | 8.9 | Cyearly2000K310R14.dat
| $1.24\\!\\!\times\\!\\!10^{5}$ | $1821.62\\!\\!\pm\\!\\!0.47$ | $1824.12\\!\\!\pm\\!\\!0.33$ | $1870\\!\\!\pm\\!\\!11$ | - |  | - | - | $1.6\\!\\!\times\\!\\!10^{-5}$ |
$\mathcal{M}$=4 |  $\mathcal{M}$${}_{4,1,0,\chi^{2}}$ | $10.217\\!\\!\pm\\!\\!0.050$ | $10.783\\!\\!\pm\\!\\!0.037$ | $99\\!\\!\pm\\!\\!19$ | $8100\\!\\!\pm\\!\\!3600~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ |  | - | - | - |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 13 | $68\\!\\!\pm\\!\\!12$ | $119\\!\\!\pm\\!\\!11$ | $36.2\\!\\!\pm\\!\\!8.0$ | $18000\\!\\!\pm\\!\\!14000~{}{\color[rgb]{0,0,1}{\bf\mathrm{AD}}}$ |  | - | - | - | Cyearly2000K410R14.dat
| $1.07\\!\\!\times\\!\\!10^{5}$ | $1821.93\\!\\!\pm\\!\\!0.48$ | $1824.12\\!\\!\pm\\!\\!0.30$ | $1902.4\\!\\!\pm\\!\\!3.5$ | $1800\\!\\!\pm\\!\\!1300$ |  | - | - | - |
Table 21: Cyearly analysis for pure sinusoids $(K_{2}=1)$. Otherwise as in Table 10.  |  | Period analysis |  | Fisher-test |
---|---|---|---|---|---
Data: Weighted original data ($n=204,\Delta T=203$:  Cyearly.dat)
$\mathcal{M}$ |  $\mathcal{M}$${}_{K_{1},K_{2},K_{3},\chi^{2}}$ |  $P_{1}$ (y) |  $P_{2}$ (y) |  $P_{3}$ (y) |  $P_{4}$ (y) |  |  $\mathcal{M}$=2 |  $\mathcal{M}$=3 |  $\mathcal{M}$=4 |
|  $\eta$ (-) |  $A_{1}$ (-) |  $A_{2}$ (-) |  $A_{3}$ (-) |  $A_{4}$ (-) |  |  $F_{R}$ (-) |  $F_{R}$ (-) |  $F_{R}$ (-) | Control file
|  $\chi^{2}$ (-) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  $t_{\mathrm{min,1}}$ (y) |  |  $Q_{F}$ (-) |  $Q_{F}$ (-) |  $Q_{F}$ (-) |
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10)
$\mathcal{M}$=1 |  $\mathcal{M}$${}_{1,1,0,\chi^{2}}$ | $10.832\\!\\!\pm\\!\\!0.022$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 4 | $123\\!\\!\pm\\!\\!11$ | - | - | - |  | 18 | 22 | 25 | CyearlyK110R14.dat
| $2.55\\!\\!\times\\!\\!10^{5}$ | $1823.76\\!\\!\pm\\!\\!0.29$ | - | - | - |  | $1.5\\!\\!\times\\!\\!10^{-10}$ | $<\\!10^{-16}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=2 |  $\mathcal{M}$${}_{2,1,0,\chi^{2}}$ | $10.150\\!\\!\pm\\!\\!0.042$ | $10.854\\!\\!\pm\\!\\!0.018$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
| 7 | $55.0\\!\\!\pm\\!\\!5.3$ | $121.5\\!\\!\pm\\!\\!9.1$ | - | - |  | - | 20 | 22 | CyearlyK210R14.dat
| $1.99\\!\\!\times\\!\\!10^{5}$ | $1822.24\\!\\!\pm\\!\\!0.50$ | $1823.64\\!\\!\pm\\!\\!0.22$ | - | - |  | - | $1.5\\!\\!\times\\!\\!10^{-11}$ | $<\\!10^{-16}$ |
$\mathcal{M}$=3 |  $\mathcal{M}$${}_{3,1,0,\chi^{2}}$ | $10.107\\!\\!\pm\\!\\!0.031$ | $10.863\\!\\!\pm\\!\\!0.019$ | $119.6\\!\\!\pm\\!\\!5.6$ | - |  | - | - | $\uparrow$ |
| 10 | $59.7\\!\\!\pm\\!\\!7.3$ | $124.2\\!\\!\pm\\!\\!6.5$ | $51.7\\!\\!\pm\\!\\!6.1$ | - |  | - | - | 25 |
| $1.51\\!\\!\times\\!\\!10^{5}$ | $1822.93\\!\\!\pm\\!\\!0.40$ | $1823.51\\!\\!\pm\\!\\!0.23$ | $1907.6\\!\\!\pm\\!\\!3.2$ | - |  | - | - | $<\\!10^{-16}$ | Notes OK$(n_{L}=100,c=0.05)$
$\mathcal{M}$=4 |  $\mathcal{M}$${}_{4,1,0,\chi^{2}}$ | $10.058\\!\\!\pm\\!\\!0.026$ | $10.863\\!\\!\pm\\!\\!0.022$ | $11.856\\!\\!\pm\\!\\!0.068$ | $115.6\\!\\!\pm\\!\\!3.6$ |  | - | - | - |
| 13 | $61.0\\!\\!\pm\\!\\!9.6$ | $119.3\\!\\!\pm\\!\\!7.5$ | $43.8\\!\\!\pm\\!\\!7.9$ | $53.8\\!\\!\pm\\!\\!5.2$ |  | - | - | - |
| $1.17\\!\\!\times\\!\\!10^{5}$ | $1823.55\\!\\!\pm\\!\\!0.25$ | $1823.56\\!\\!\pm\\!\\!0.20$ | $1818.95\\!\\!\pm\\!\\!0.69$ | $1910.4\\!\\!\pm\\!\\!2.1$ |  | - | - | - | Notes OK$(n_{L}=100,c=0.05)$
Data: Weighted residuals of $\mathcal{M}$=4 ($n=204,\Delta T=203$:
CyearlyK410R14Residuals.dat)
|  |  |  |  |  |  |  $\mathcal{M}$=6 |  $\mathcal{M}$=7 |  $\mathcal{M}$=8 |
$\mathcal{M}$=5 |  $\mathcal{M}$${}_{1,1,-1,\chi^{2}}$ | $8.005\\!\\!\pm\\!\\!0.058$ | - | - | - |  | $\uparrow$ | $\uparrow$ | $\uparrow$ |
| 3 | $24.0\\!\\!\pm\\!\\!7.0$ | - | - | - |  | 6.4 | 6.4 | 6.4 | CyearlyK11-1R58.dat
| $1.05\\!\\!\times\\!\\!10^{5}$ | $1826.09\\!\\!\pm\\!\\!0.68$ | - | - | - |  | $3.8\\!\\!\times\\!\\!10^{-4}$ | $3.4\\!\\!\times\\!\\!10^{-6}$ | $6.4\\!\\!\times\\!\\!10^{-8}$ |
$\mathcal{M}$=6 |  $\mathcal{M}$${}_{2,1,-1,\chi^{2}}$ | $8.002\\!\\!\pm\\!\\!0.090$ | $239\\!\\!\pm\\!\\!86~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ | - | - |  | - | $\uparrow$ | $\uparrow$ |
${\color[rgb]{0,0,1}{\bf\mathrm{UM}}}$ | 6 | $23.3\\!\\!\pm\\!\\!5.0$ | $22.5\\!\\!\pm\\!\\!4.2$ | - | - |  | - | 6.0 | 5.9 | CyearlyK21-1R58.dat
| $9.57\\!\\!\times\\!\\!10^{4}$ | $1826.1\\!\\!\pm\\!\\!1.1$ | $1839.4\\!\\!\pm\\!\\!28$ | - | - |  | - | $6.4\\!\\!\times\\!\\!10^{-4}$ | $1.1\\!\\!\times\\!\\!10^{-5}$ |
$\mathcal{M}$=7 |  $\mathcal{M}$${}_{3,1,-1,\chi^{2}}$ | $7.998\\!\\!\pm\\!\\!0.044$ | $38.7\\!\\!\pm\\!\\!2.1$ | $253\\!\\!\pm\\!\\!1700~{}{\color[rgb]{0,0,1}{\bf\mathrm{LP}}}$ | - |  | - | - | $\leftarrow$ |