A non-linear solar cycle exploration
Preamble, do not take this work too literally. the intent is food for thought.
Recent very long free thinking discussion (paged comments) on Tallbloke’s Talkshop evolved into trying to create a model of astronomical effect on the sun of planetary parameters, a longstanding tantalising problem where “ought to” and reality throw buns. Probably fits the tales of the farmer of old leaning on the farm gate chewing straw watching fools trying to get the cart under the bridge. Won’t fit but watching is fun. Lets hope that one day more is learnt.
I can play too, know a bit about the local bridge.
Figure 1 is a result of some novel work but explaining what I have done is going to be tough. (spreadsheet supplied for the brave)
I have written some unique software which approximates a non-discrete Fourier Transform or similar where the driving input is a discrete dataset. The output can produce datapoints for any time and therefore produce a new discrete time series on any scale. Validity is a different matter, why we have brains, computers just try and fit carts under bridges, can’t know it won’t go.
In this case I have played some further tricks since I know sunspot data is more than just cantankerous. (and is known inaccurate anyway)
The data is transformed (preprocessed) from SIDC published monthly sunspot count into the form shown figure 2 and this was the input to the software.
Data has been roughly turned into bipolar more akin to solar polar magnetic data but not offset to the apt 90 degree phase shift. (irrelevant here). In addition root 1.4 has been applied, pre-distorting the wave, perhaps into a more friendly shape for analysis. Power 1.4 is sometimes mentioned as a possible sunspot parameter. 
For exploitative analysis KISS (keep it simple, stupid) is generally the most likely to lead to insight so in this case I restricted the software to two terms but selected complicated versions, variable amplitude modulation, adding two more parameters plus modulation depth.
The software was left on automatic defaults. (apart from setup, turnkey)
A final twist was almost hide part of the input data from the software, for which there is a good reason.
Those familiar with sunspot data know there is a severe problem circa year 1790 when the sun seems to have behaved very strangely, although this is uncertain, disputed but either way the data is questionable. We are not even certain whether an additional short sunspot cycle was present around 1895. There is also cloud over post 1800.
I used the software “confidence” function to make the merit function almost ignore year 1780 through 1829. Engineers have a rule: if it looks hard, run away very fast. Lets see what the software assumes happened.
Analysis ran normally and complete promptly, a good sign.
One version of the output can be dropped into some spreadsheet software and implements an interactive model in the spreadsheet. This is used.
For this instance the preprocessing transform has to be reversed, output raised to power 1.4, spreadsheet can do this.
Figure 1 is one result. R2 of 0.57 but that includes the ignore section of data. I consider this quite good given no attempt is made to closely model. (could with many more terms be raised very high but only over the input period) In this case I am not interested in party tricks.
I won’t discuss the detail results, can leave that for any blog comments, you have eyes.
I’ve also produced a second result on a much longer timescale. Tallbloke is particularly interested in whether any models match the Maunder minimum.
Figure 3 shows a longer timescale running far outside the input data period. Treat with extreme caution because it is wrong. Illuminating, you decide.
These are machine output, no input or guidance from me. (which could be done)
Read as column associated.
The software chose the above parameters.
22 year might be related to solar system magnetics, only some planets are magnetically active, dominated by Jupiter. 132 year makes no sense to me.
19.9 year suggests gravitational sum of force on the sun. In this case the modulation period is I think very interesting, circa 45 years is strongly present in some solar asymmetry data and in the past I have contended a pseudo 45 year in various terrestrial datasets. (note: there is no pure frequency, is an approximation)
The modulation depth (figure “aux”), is present relatively strongly, zero is none, one is full and can exceed one. Wikidpedia might help a little, assume partially suppressed carier.
A web search might help further or discuss.
1. A paper which might be of interest, by a veteran,
Title: Three-halves law in sunspot cycle shape
Authors: Bracewell, R. N.
Journal: Monthly Notices of the Royal Astronomical Society (ISSN 0035-8711), vol. 230, Feb. 15, 1988, p. 535-550.
Bibliographic Code: 1988MNRAS.230..535B
2. Spreadsheet in OpenOffice .ods format inside a zip, this is large and may pose speed problems with LibreOffice. Plots are unlikely to survive to your software. Ask if you need help or other file versions.
Reason for update, apparently when Excel imports ods it imports formula as literal values, supposed to be a live spreadsheet (no macros), zip now contains an additional simple xls file which does import formula. I’ve also added the analysis remainder data.
1.4M 2.7M hosted at Tallbloke’s where paid for extra of zip storage is present, part of what donations provide)