# Sunspot Cycle and the Global Temperature Change Anomaly

UPDATE: The author writes:

Thank you for posting my story on Sunspots and The Global Temperature Anomaly.

I was pleasantly surprised when I saw it and the amount of constructive feedback I was given.

In the interests of preventing the misuse of my flawed correlation please withdraw the story.

Then I replied: Please make a statement to that effect in comments, asking the story be withdrawn.

To which he replied:

After further reflection, I have concluded that the objection to the cosine function as having no physical meaning is not valid.

I have posted my response this morning and stand by my correlation.

Personally, I think the readers have it right. While interesting, this is little more than an exercise in curve fitting. – Anthony

I have made an 82% correlation between the sunspot cycle and the Global Temperature Anomaly. The correlation is obtained through a non linear time series summation of NASA monthly sunspot data to the NOAA monthly Global Temperature Anomaly.

This correlation is made without, averaging, filtering, or discarding any temperature or sunspot data.

Anyone familiar with using an Excel spread sheet can easily verify the correlation.

The equation, with its parameters, and the web sites for the Sunspot and Global temperature data used in the correlation are provided below for those who wish to do temperature predictions.

The correlation and the NOAA Global Mean Temperature graph are remarkably similar.

For those who like averages, the yearly average from 1880 to 2013 reported by NOAA and the yearly averages calculated by the correlation have an r^2 of 0.91.

The model for the correlation is empirical. However the model shows that the magnitude, the asymmetrical shape, the length and the oscillation of each sunspot cycle appear to be the factors controlling Global temperature changes. These factors have been identified before and here they are correlated by an equation to predict the Temperature Anomaly trend by month.

The graph below shows the behavior of the correlation to the actual anomaly during a heating (1986 to 1996) and cooling (1902 to 1913) sunspot cycles. The next photo provides some obvious conclusion about these same two Sunspot cycles.

In the graph above the correlation predicted start temperature for these same two solar cycles has been reset to zero to make the comparison easier to see.

High sustained sunspot peak number with short cycle transitions into the next cycle correlate with temperature increases.

Low sunspot peak numbers with long cycle transitions into the next cycle correlate with temperature decreases.

Oscillations in the Sunspot number, which are chaotic, can cause increases or decreases in temperature depending where they occur in the cycle.

The correlation equation contains just two terms. The first, a temperature forcing term, is a constant times the Sunspot number for the month raised to a power. [b*SN^c]

The second term, a stochastic term, is the cosine of the Sunspot number times a constant. [cos(a*SN)] This term is used to model those random chaotic events having a cyclical association with the magnitude of the sunspot number. No doubt this is a controversial term as its frequency is very high. There is a very large degree of noise in the temperature anomaly but the term finds a pattern related to the Sunspot number.

Each term is calculated by month and added to the prior month’s calculation. The summation stores the history of previous temperature changes and this sum approximates a straight line relationship to the actual Global Temperature Anomaly by month which is correlated by the constants d and e. The resulting equation is:

Where TA= the predicted Temperature Anomaly

Cos = the cosine in radians

* = multiplication

^ = exponent operator

Σ = summation

a,b,c,d,e = constants

TA= d*[Σcos(a*SN)-Σb*SN^c]+e from month 1 to the present

The calculation starts in January of 1880.

The correlation was made using a non-linear time series least squares optimization over the entire data range from January of 1880 to February of 2013. The Proportion of variance explained (R^2) = 0.8212 (82.12%)

The Parameters for the equation are:

a= 148.425811533409

b= 0.00022670169089817989

c= 1.3299372454954419

e= -0.011857962851469542

f= -0.25878555224841393

The summations were made over 1598 data months therefore use all the digits in the constants to ensure the correlation is maintained over the data set.

The correlation can be used to predict future temperature changes and reconstruct past temperature fluctuations outside the correlated data set if monthly sunspot numbers are provided as input.

If the sunspot number is zero in a month the correlation predicts that the Global Temperature Anomaly trend will decrease at 0.0118 degree centigrade per month. If there were no sunspots for a year the temperature would decline 0.141 degrees. If there were no Sunspots for 50 years we would be entering an ice age with a 7 degree centigrade decline. While this is unlikely to happen, it may have in the past. The correlation implies that we live a precarious existence.

The correlation was used to reconstruct what the global temperature change was during the Dalton minimum in sunspot from 1793 to 1830. The correlation estimates a 0.8 degree decline over the 37 years.

Australian scientists have made a prediction of sunspots by month out to 2019. The correlation estimates a decline of 0.1 degree from 2013 to 2019 using the scientists’ data.

The Global temperature anomaly has already stopped rising since 1997.

The formation of sunspots is a chaotic event and we can not know with any certainty the exact future value for a sunspot number in any month. There are limits that can be assumed for the Sunspot number as the sunspot number appears to take a random walk around the basic beta type curve that forms a solar cycle. The cosine term in the modeling equation attempts to evaluate the chaotic nature of sunspot formation and models the temperature effect from the statistical nature of the timing of their appearance.

Some believe we are entering a Dalton type minimum. The prediction in this graph makes two assumptions.

First : the Australian prediction is valid to 2019.

Second: that from 2020 to 2045, the a replay of Dalton minimum will have the same sunspot numbers in each month as from may 1798 to may 1823. This of course won’t happen, but it gives an approximation of what the future trend of the Global Anomaly could be.

If we entered another Dalton type minimum post 2019, the present positive Global Temperature Anomaly would be completely eliminated.

See the following web page for future posts on this correlation.

Data sources:

NASA

http://solarscience.msfc.nasa.gov/greenwch/spot_num.txt

Australian Government Bureau of meteorology

http://www.ips.gov.au/Solar/1/6

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steveta_uk

This also shows that the sun doesn’t repond to volcanoes. Which I guess is no surprise.

steveta_uk

Why is there any noise in the predicted paths? I would expect a perfect curve, unless someone has added random noise to an expected SN for 2032.

thingodonta

You can’t tax sunspots.

Nylo

This looks like a nice example of curve fitting and bad science. FIVE empirically adjusted parameters without any physical explanation to them? I’m particularly baffled by the cosine, if a~148 and you expect the angle in radians! What the hell is that? A minuscule change in parameter a would lead to entirely different results, No surprise that you need give it with a precision of 12 decimal positions. Same for the others. This is not science, it is a joke.

johnmarshall

Interesting. The sunspot numbers will affect other solar outputs like magnetic strength. All solar outputs will affect climate on our planet.
A better correlation than a trace gas vital for life.

R.J. Salvador: For future presentations, if you smooth the global surface temperature data with a 5-year (or 61-month) running mean filter, you can minimize the ENSO-related wiggles. This would help to show the agreement between your sunspot model and observations.
Regards

tabloidjournalist

It was the sun wot won it.

Thomas

What result do you get if you use the first half of the data to estimate all your parameters and the second half as a control?

Espen

To see if your reconstruction has any merit, I compared it to BEST, which extends longer back in time, and also other well known long temperature series (e.g. CET, Prague) – and it looks like your reconstruction fails terribly. For instance, both BEST and CET have a peak near 1830 where you have the lowest values.
Please have a look at the advice of Thomas: Try curve-fitting on half of your temperature series and see if your estimated time series for the other half looks anything like the real temperature data.

Kurt in Switzerland

Data fitting is one thing. Let’s see how the model performs going forward, in competition with other models.
Kurt in Switzerland

Nylo

This means that a month with a SN of 76 contributes approximately +0.00626C to TA (warming), whereas if the SN=77, it contributes -0.01C (cooling), and if it is SN=78 it contributes +0.01C (warming)…
It varies wildly even in the sign of the contribution without any logical explanation, and does this even for the smallest variations of SN possible. This blog post should be retracted.

Stephen Wilde

Now link those charts to changes in global cloudiness, the latitudinal positions of the climate zones and the degree of zonality / meridionality of the jets.
I would be surprised if there were not to be a clear correlation.

The big disadvantage of this technique is that it does not require a multimillion dollar computer. It will put the climate modelers out of business. This technique also has the problem of not being able to predict the future as the sunspot cycles are not very predictable.
If this models holds up for the next 20 or 30 years, there is still a lot of science required to explain why?

son of mulder

“Nylo says:
May 3, 2013 at 3:52 am
….parameters without any physical explanation to them”
And what was Newton’s physical explanation for the parameter ^2 in the gravitational inverse square law? If he’d been able to generalise to the n-body problem beginning with n=3 in the lab he’d have been laughed at for curve fitting.
“A minuscule change in parameter a would lead to entirely different results, No surprise that you need give it with a precision of 12 decimal positions. Same for the others.”
Sounds like chaotic behaviour to me. All you can do is curve fit since there is no known physical mechanism for predicting sun spots in the hypothesis. Similarly any pretense that climate models will work is itself “not science, it is a joke” if they contain or don’t account for any unknown physical mechanisms like say aerosols, clouds, cosmic rays, solar variations etc.
If the curve fitting shows good correlation over the next 100 years then the author may be onto something.
However I won’t be holding my breath.

Nylo

steveta_uk says:
May 3, 2013 at 3:43 am
Why is there any noise in the predicted paths? I would expect a perfect curve, unless someone has added random noise to an expected SN for 2032.
Because of the absurd cosine in the formula. It creates that seemingly random noise even if you consider a perfectly smoothed curve in the prediction of the SN. The cosine varies wildly for tiny changes in SN.

wsbriggs

Curve fitting is interesting, but curve fitting that requires lots of decimal places to fit data which has only a couple of places of significance is suspect from the word go.
A good model would get the correlation without lots of digits, e.g. Willis’ tropical thunderstorm analyses are classic. The raw data, correctly presented, shows the salient information without lots of curve fitting necessary.
Good try, but a little more science and a little less numeracy.

Espen says: “For instance, both BEST and CET have a peak near 1830 where you have the lowest values.”
While I’m not agreeing with or disagreeing with the model presented by R.J. Salvador, your comment presents two datasets that do not represent global temperatures. In 1830, BEST covers part of Northern Hemisphere land surface air temperatures:
http://oi41.tinypic.com/v773fd.jpg

higley7

However, there is the problem of the real 1998 warm peak NOT being higher than the 1938 peak. The current higher 1998 peak is due to bias in the data from UHI, rurual and high altitude site dropout in the 1990s, and unfounded adjustments to the data, accentuating warming. The correlation would still be good with the solar has matching warm peaks and such but the actual temps would be lower. Good job, but I do not like the temp data to pretend that we are warmer now than in the 1930s.

An interesting approach, I fear it might be susceptible to the ‘von Neumann’s elephant’ syndrome diagnosis.

An interesting approach, I fear it might be susceptible to the ‘von Neumann’s elephant’ syndrome diagnosis.
The Parameters for the equation are:
a= 148.425811533409
b= 0.00022670169089817989
c= 1.3299372454954419
e= -0.011857962851469542
f= -0.25878555224841393

David L. Hagen

How well does your method do in taking half the data and hindcasting/forecasting the other half of the series?
e.g. compare with Nicola Scafetta 2012.. See Scafetta’s graph of his predictions since 2000 vs IPCC (bottom page).
May I recommend comparing temperatures with the integral of the solar cycle. See David Stockwell’s solar accumulative theory at Niche Modeling. Note especially the phase lag between temperature and solar forcing.
I look forward to your further developments.

Geckko

Where does the parameter “f” fit in. you give a value, but it does not appear in your specification.

Allen63

A few years ago, I too modeled the earth’s land temperature as a function of “sunspots”. I made a very simple physical model that involved earth “heat capacity”, radiation, etc. It had no adjustable parameters that were not physically based. I used raw temperature data from a couple sites that had constant thermometer measurements over the last couple hundred years or more and were less likely to be in “heat islands”.
Result, temperature over the last 250 years tracked sunspots — with an offset. It was as though sunspots up-and-down was like a fire on a gas stove being raised and lowered beneath a pot of water — the temperature of the water had “inertia”.
I did not hypothesize a “mechanism”. Rather, I tentatively-concluded “some natural process moving in concert with sunspot number” seemed to be “strongly influencing” the Earth’s temperature cycles.
Just a way of saying, I agree with the subject post — in principle.
Since then, there have been many articles on WUWT proposing mechanisms.

My results suggest that by 2040 we will be back to where we were in 1950,
more or less..
It is of course possible that certain factors, like an incidental extraordinary large shift of cloud formation more towards the equator simultaneous with an extraordinary coverage of large areas with snow, mostly NH, at the end of the cooling period, could trap us, amplifying the cooling due to a particularly low amount of insolation. In 1940/1 there was a particular large amount of snow in Europe.
However, I am counting on mankind using its ingenuity to be able to reverse that trap, should such a situation occur (usually what would happen is that there would be no spring or summer)
btw
can I ask you all here a big favor? Could anyone of you please have a look at the above quoted log?
I want to use this as a communication to all (specifically) religious (e.g Christian & Judaic) media
(which is why I added some biblical references – never mind those, I just added that in as an aside)
but I would prefer to first hear all WUWT opinions about it.
It would be much appreciated if I could have your (honest) opinion about it.
Thanks!

Village Idiot

Amazing breakthrough in climatology! How did the world science community miss that one? Handy that the sunspots co-operate with one of our basic tenets of faith; the ‘warming’ of the planet is a mirage based on malplaced weather stations, dodgy averaging over wide open spaces and sloppy science.

Kasuha

A nice proof of the classical statement that with five parameters, you can fit an elephant and wiggle its trunk.
The first thing with which I disagree is your statement that you don’t do any “averaging, filtering, or discarding any temperature or sunspot data”. You don’t discard anything but trying to suggest that your expression is not any kind of filtering or averaging is simply not true.
The only purpose of the cosine function is that it likely battles the 11-year sunspot cycle through some kind of aliasing (or introduction of strong enough noise) as otherwise that would be easily visible in the result. I don’t think there is anything physically sound on that cosine.
Value at any point is sum of all previous values; this means sunspot number at any given time is supposed to generate an absolute offset from previous value which will never fall off. I don’t see any physical backing for this, either.

beng

More spinning wheels going nowhere…

alex

Just another cheap bet for the future.
Having many parameters you fit a curve.
So what? Who cares?
Why did you choose sunspots and not Dow Jones? The Dow would be easier to fit.

Nylo

The author indeed does “averaging” of data (he is using sunspots monthly values, not daily values, in his calculations) and also rounding of data (he uses sunspot monthly data as provided by NASA, and NASA rounds it to 1 decimal position). To verify the robustness of the “reconstruction”, I have added random noise to the monthly sunspot number. A noise of between -0.05 and +0.05 to the monthly SSN, so that it would be hidden in NASA’s rounding of the SSN that gets published. And then I’ve recalculated the whole graphic several times. I get enormous differences with every plot (every recalculation of the random noise for all months). I insist, this “study”, this curve fitting exercise, cannot be called anything but crap.

I am just struck by a gigantic idea….
What if the difference in anomaly of 0.5 between 1902 and 1986
as indicated in the 4th graph,
(= 84 years which equals almost 1 Gleissberg solar/weather cycle)
is due to the improvement since 1902 in accuracy of thermometers and the improvement of recording, e.g. by versus by machine?

henry says
e.g. by versus by machine?
e.g. by eye versus by machine

Steve from Rockwood

If you can obtain a high positive correlation you can also obtain a high negative correlation. You would have to conclude that less sun spots correlate with either higher or lower temperatures. No?

jlurtz

A simple discussion on randomness [chaotic ] behavior. If one takes two perfectly balanced dice [had to come by], the long term average, per throw, will be 7. The chaotic behavior of 16 snake eyes in a row, also happens. In this case, we have a “perfect” model. And yet chaotic behavior occurs in the model at each time slice [throw of the dice].
Can the “perfect” model predict the future? NO. But the statistical average over the “perfect” model predicts a 7. Why can’t the “perfect” model predict the future? It doesn’t have enough information!!!! If we know the initial dice position in the hand, the exact physical makeup/balance of the dice, the initial rotations, the initial lateral speed, the characteristics of the table, the wind speed, the temperature, etc., a “giant” computer could predict the trajectories of the dice and almost perfectly predict the outcome.
The Sun is “chaotic” in the short time slice, but in the average it is fairly predictable. We know that approximately every 11 years a Sunspot peak occurs. That is not “chaotic” behavior! As we get more information about how the Sun actually operates, and the “big computer”, prediction of each Sunspot position, magnitude, time length could be know.
Today, we are still trying to decide if the Sun’s output is constant, nearly constant, variable, highly variable, etc. Our time slices are months. The true climate time slices are 100s of years. We didn’t even know that in Indonesia the Pacific Ocean level is 1 to 2 meters higher [trade winds] than Western South America until we had satellites.
Is the Sun “chaotic”? You decide.

Paul Vaughan

Refreshing article.
David L. Hagen (May 3, 2013 at 4:54 am) wrote:
“R.J. Salvador […] May I recommend comparing temperatures with the integral of the solar cycle. See David Stockwell’s solar accumulative theory […]”

Salvador is integrating — note the sigmas in the formula.
It’s a nonlinear integral; Salvador’s more than a few steps ahead of Stockwell.
Btw Salvador’s calculations are effortlessly verified (takes literally only a minute) and the residuals are nothing more than familiar interannual variations (as Bob Tisdale has effortlessly noticed).
There are 2 typos in the parameter list.
“e= -0.011857962851469542
f= -0.25878555224841393”

d= -0.011857962851469542
e= -0.25878555224841393

Nylo

Please guys, this is the result of adding a purely random value of between +0.05 and -0.05 to the value of the monthly Sunspot Number, obviously a different random value every month. NASA gives this monthly SSN value rounded to the 1st digit, so this noise is small enough not to even affect what NASA would have published about the SSN). I have recalculated the noise seven times to get seven different results. You can see them all here:
http://www.elsideron.com/wuwt_ssn_study
As you can see, despite the noise added to the SSN is so low that ti would not even affect the published monthly SSN values, doing the reconstruction with the new SSN values instead of the strictly published provides strikingly different results. That’s why this reconstruction has absolutely zero value.

Paul Vaughan

Allen63 (May 3, 2013 at 5:02 am) wrote:
“A few years ago, I too modeled the earth’s land temperature as a function of “sunspots”. […]
[…]
Result, temperature over the last 250 years tracked sunspots — with an offset. It was as though sunspots up-and-down was like a fire on a gas stove being raised and lowered beneath a pot of water — the temperature of the water had “inertia”.
I did not hypothesize a “mechanism”. Rather, I tentatively-concluded “some natural process moving in concert with sunspot number” seemed to be “strongly influencing” the Earth’s temperature cycles.”

Sensible.

graphicconception

“Anyone familiar with using an Excel spread sheet can easily verify the correlation.”
That rules out Phil Jones, then!

Espen

Bob Tisdale says:
May 3, 2013 at 4:34 am
In 1830, BEST covers part of Northern Hemisphere land surface air temperatures:
thanks for pointing that out, I thought there were at least a couple of SH stations!

SMS

You need to try and develop correlations to the temperatures of rural communities that have not had their temperature records adjusted. Without fitting to the “real” temperature history, your work is a bit useless.

Waiting for Lief… to point out that the sunspot numbers used are (no doubt) not the real sunspot numbers.
Most of my objections to this are similar to those already offered. Without a physical basis, it is curve fitting, numerology, fitting an elephant and making it wiggle its trunk. Also, any fit where one needs to present more than 3 digits of a parameter (because the model fails with fewer digits) is not likely to be robust or meaningful — I would take points off of any work done by a student that presented this many digits given that the DATA ITSELF is far more uncertain than this, indeed (waiting for Lief indeed:-) may be egregiously incorrect.
With all that said, IF the model hindcasts the Dalton minimum correctly (without being tweaked or tuned until it does so, as that is “cheating” in the predictive modeling game) that is something to think about. It is one thing to fit an elephant, quite another to predict a rhinoceros from the fit to the elephant.
Further remarks:
* The magnitude of the a parameter is indeed worrisome, especially given the size of SN and the fact that cosine is periodic modulus 2\pi. In fact, I’d have to say this is pretty meaningless — this parameter has to sum to nearly zero because the odds are that any given month will produce a positive cosine or negative cosine essentially randomly by the time one forms $a S_N$ modulus $2\pi$. Again, remember that the sunspot numbers cannot POSSIBLY be accurate to within a single spot, and a single spot alters the angle by 149 radians! Indeed, if one forms the modulus of the a parameter with $2\pi$ one learns that for integer sunspot numbers one might as well use $a = 3.9125$ as this will produce the same cosine per sunspot number.
I would strongly recommend leaving off the cosine term as if it DOESN’T sum to zero (within noise) it is an accident — it is clearly irrelevant to the fit.
* The constant is also somewhat worrisome. It is pure fudge factor as stated. The only thing that I can think of that it might represent is large scale secular trends, e.g. Milankovitch variation. In this case it should be a RATE, not an additive constant, indicating something like a secular cooling of a quarter of a degree over the fit interval. This can be managed within the sum format by summing (-0.259/# months) from the beginning of the series, or fitting the SUM of a constant, not a constant (it matters!). However, this is not the only contribution to this term, and one needs to separate out the rest of the variation.
* Finally, one can fit essentially the same function to make the terms more meaningful. I would suggest something like:
$T_A = \sum a (\frac{S_N - }{})^b + c$
The terms now have a POSSIBLE physical meaning. $c$ is, as noted, a term representing the cumulated secular trend from e.g. orbital variation, basically a fit to the “slow” variation of post-Holocene-optimum temperatures linearized around the present. $latex$ is the mean sunspot number over a sufficiently long baseline, so that $\frac{S_N - }{}$ is a normalized “sunspot anomaly” of sorts. Converting this dimensionless anomaly to some power $b$ to a rate by multiplying by $a$, this sum is then basically the discretized integral of a differential equation where one is hypothesizing a gain term related to a normalized sunspot anomaly against a slowly varying background rate. This still isn’t “science” — I have no idea how one might predict $a$ or $b$, although $c$ might be estimatable — but at least it can be stated as a coherent hypothesis (“the normalized, dimensionless sunspot anomaly is a parameter in a nonlinear rate equation for global temperature”) and has only three parameters so that the resulting fit can’t quite manage an elephant.
No matter what, the cosine term has to go. If one (for example) shifts every sunspot number by +1 or -1 according to a coin flip, one had better get the same answer, and I promise you that $\sum \cos(148*(S_N \pm 1) \approx \sum \cos(148*(S_N) \approx 0$ for almost all possible $S_N$ as a function of time.
rgb

rgbatduke

Grrr. Goddamn WordPress latex interface. OK, I’ll present the equations it screwed up above without the latex translation attempt.
T_a = \sum [ a (\frac{S_N – }{}^b + c ]
is the proposed fit without the cosine term, using a normalized, dimensionless sunspot anomaly as the single variable in the rate equation, not the absolute number. is the mean sunspot number averaged of the entire interval. One expects this to be roughly halfway between 0 and the sunspot peak, so normalizing with respect to it converts the sunspot number from being something in the tens to hundreds to something ranging from roughly -1 to +1, with rare excursions above +1. One could just renormalize S_N/ to accomplish the same thing and make the range 0 to 2+, but then one would almost certainly alter part of the possible meaning of c. Given the sorta-Gaussian shape of the solar cycle, there are probably even better transforms — deviation from the a local “mean” solar cycle, for example — but this is worth trying and should accomplish most of what the “a” parameter is accomplishing now in the fit up above.
rgb

rgbatduke

http://www.elsideron.com/wuwt_ssn_study
As you can see, despite the noise added to the SSN is so low that ti would not even affect the published monthly SSN values, doing the reconstruction with the new SSN values instead of the strictly published provides strikingly different results. That’s why this reconstruction has absolutely zero value.

Broken link, Nylo. I agree, mostly, but I have to say I’m surprised. The sum of the cosine of a random number should rapidly converge to zero, and d is a small number to start with. It should make the fit bounce around a bit initially and by the 100th month cease to be relevant, much less than 10% of the total, should it not?
Either way, I agree that the term is utterly meaningless and needs to go because it is not robust to the error in the data — it is just a transform of noise. Not that the whole fit isn’t Nikolai and Zeller style numerology in the first place… but there are terms (as I showed) that one COULD turn into a POSSIBLY reasonable model.
rgb

Ken

You should have Climate Audit / Steve McIntyre analyze & critique this analysis.
Wouldn’t THAT be a nice change of pace in an us-versus-them arena in which “sides” are so clearly defined & maintained with great effort….
…that is…seeing people on the ‘same side’ actively work to ensure the validity & objectivity of the analyses & conclusions they reach rather than just apply the analytical criterion of, “Oh! That agrees with my outlook, so I’ll accept it without any objective critical evaluation.”
E.G., note “wsbriggs” observation May 3 at 4:28 am, above.

Rod

Once again, the great quality of the comments is on display at WUWT. Thanks to all who’ve convinced me to take the original post with less than a grain of salt.
Suggestion: Instead of removing the post, add an update to it indicating that those readers of this blog who have a good understanding of the math involved have found serious fault with the analysis. That would serve two purposes: 1) It would confirm the policy of not deleting posts/comments, and 2) It’s retention, along with the update, would serve as a cautionary note for others who are considering posting articles under the new policy, since their efforts will be on display for the duration of WUWT, hopefully a very long time.
Besides, in its own way, the article, taken together with the comments, was educational, just not in the manner the author intended.

Russ R.

Cosine? Really?

AndyG55

So, somehow it fits to the highly manipulated global temperature series.
When I see something that matches the un-adjusted raw temp records rather than the UN-adjusted temp record I might get interested.
As soon as I see data with the average global temps from 1930-40 way below current, I just ignore .

Thomas says:
May 3, 2013 at 3:57 am
What result do you get if you use the first half of the data to estimate all your parameters and the second half as a control?
You’ll end up with garbage
#########################################
even if you didnt the deeper problem with this and with all solar analysis that works with
“spot numbers” is that they are diemensionaly incorrect.
our climate system doesnt “see” spots, it sees watts.
So if you have temperature on the left hand side you better have units on the right hand side that can be related to temperature via known laws of physics, otherwise, even if your curve fit is perfect, it’s meaningless.

Nylo

rgbatduke, try plotting the formula proposed by the author, instead of against the SSN values published by NASA, against a SSN varying in 0.1 steps, between 0 and 30, ignoring the sumatories and “e”. What you will get is what the nonsensical formula says that a particular month with that SSN contributes to the trend in temperatures. I have plotted it for you:
http://www.elsideron.com/wuwt_ssn_study/Monthly_increment_by_SSN.PNG
As you can see, the SSN “magnitude” doesn’t really matter. I plotted until SSN=30 but you can continue to 200 if you want, very little changes. A month with SSN=0.1 can contribute much more (increasing temperatures by 0.007C) than a month with SSN=29.8 (which reduces them by -0.011C). And the reverse is true for sunspot numbers 0.3 and 29.4.
So how come the author gets a result that comes close to what has happened with the temperatures? Just by extremely carefully playing with “a” parameter until, by absolutely pure chance, the results of the cosine of its multiplication by the published SSN more or less matches the ups and downs of real temperatures. It is not so difficult to achieve if you use an “a” really big as he does and start changing it in tiny ammounts until you get what you want, which he does. He probably tried several thousands of possible values of “a” until he found one that produced a result which was “good enough” for him. Then he adds the other terms which just create an escalator roughly matching the overall trend and voilà. See the escalator below in green (his formula except for removing the cosine (cosine=0):
http://www.elsideron.com/wuwt_ssn_study/No_cosine.PNG

Paul Vaughan

A caution for careful, sensible readers:
In their haste to ignorantly &/or deceptively (and falsely) paint the cosine integral as noise, some commentators are failing to recognize that the term is simply pointing to scaling changepoints in the sunspot record. An infinite number of other summaries can be designed to capture the same nonrandom changepoints. A subset of such summaries are physically meaningful. I suggest more sobriety. Salvador’s cosine term may be physically meaningless, but there’s an important learning opportunity here for anyone patient & careful enough to deeply understand and appreciate exactly why the cosine integral’s changepoints are timed as they are.
__
A light-humored peripheral note for those concerned about the number of model parameters: