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.
Your readers have pointed out a fatal flaw in my correlation.
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Then I replied: “Please make a statement to that effect in comments, asking the story be withdrawn.“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
Guest post by R.J. Salvador
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.
http://www.facebook.com/pages/Sunspot-Global-Warming-Correlation/157381154429728
Data sources:
NASA
http://solarscience.msfc.nasa.gov/greenwch/spot_num.txt
NOAA ftp://ftp.ncdc.noaa.gov/pub/data/anomalies/monthly.land_ocean.90S.90N.df_1901-2000mean.dat
Australian Government Bureau of meteorology
http://www.ips.gov.au/Solar/1/6
Related articles
- Current solar cycle data seems to be past the peak (wattsupwiththat.com)
- Paper finds solar influence on climate has been underestimated (oneworldchronicle.com)
Anthony, this is the fourth time Paul Vaughan has posted the above link on this thread. His ravings about thought police and freedom to think independently clearly indicate a personal problem which does not relate to the discussion at hand nor reflect well on the rest of the participants at WUWT.
REPLY: Yes, I’ve noticed this too. I’m going to put him on moderation status. – Anthony
This was rather funny.
Let me just say that I express my support to anyone who figured out for himself from whatever records or whichever method that we are on a cooling curve for the next couple of decades, especially with so many people still “believing” the whole global warming scam….
http://wattsupwiththat.com/2013/05/03/sunspot-cycle-and-the-global-temperature-change-anomaly/#comment-1297004
This the first time I have felt comfortable offering a comment. I retired a few years ago and for most of that time I have bee reading and absorbing the articles and the comments. Unlike other blogs, I fully appreciate the lack of personal attacks and the foul language found at other sites.
I was an engineer in my career and it that career I implemented numerous design improvements on the equipment that I worked on. The underlying bases for those improvements was measured test data and curve fitting with physical parameters.
The curve fitting presented in the article combined with the author’s reply that there was a physical basis that he explained seems to be ignored.
The fact that he did this with a parameter that only involved the sunspot number I also find interesting. This is not the only article posted on this site that suggests it is the sun stupid.
Over the years I have seen the analytical tools available to engineers increase and improve. Maybe some years from now, with further advancements in these tools, we will be able to design equipment without the need for prototypes and test measurements.
For the years that I worked test measurements were treated like gold. That is where the insight for improvement came from.
I think some of the comments offered to the author should be considered to further test what he has presented.
Anyway, either way, I also used two other methods to determine the dates of ca. 1951 and 1995 as turning points on my a-c wave. Which is making me confident in estimating that the next turning point (from cooling to warming) will be around 2040, give or take a few years (of my error)
Which makes my point for me far better than anything I could have said. Thank you, HenryP, for demonstrating the utter emptiness of your curve fitting to more or less arbitrary representations.
rgb
There is often at least some justification, and long record of success, for choosing a low order polynomial base (Taylor’s theorem). Ordinarily, if the function is periodic, there is justification for choosing a harmonic base (Fourier’s theorem). The problem here is that a base which is harmonic in time would be reasonable, but one which is harmonic in SN? What in the world does that physically represent?
A Taylor series historically does a very good job fitting — note well fitting — a smooth function NEAR a selected point where its derivatives are known, as you note. It does a terrible job extrapolating any function that is not, in fact, the Nth order polynomial of the fit as the missing terms in the power series expansion of the actual function soon dominate the low order terms as you depart from the neighborhood of the point. But you know that, I’m explaining for the others.
To emphasize a point Monckton made in the recent “open letter” post, I like to repost the link to Koutsoyiannis:
http://itia.ntua.gr/en/docinfo/673/
This illustrates the problem associated with “unmotivated” curve fitting on some finite data set being used to extrapolate outside of the data set quite nicely. What looks like it is linear on a short time scale, exponential or quadratic on another time scale, turns out to be harmonic on a longer time scale, and that itself can turn out to be something else entirely on a still longer time scale.
No simple function is going to fit the climate record for the last five million years, no simple function can be used to reliably predict e.g. global temperature or anything else. It might work, sure. Or it might not. Only time will tell, and assigning too much belief to the validity of some extrapolation, especially one without a shred of physical justification, is rather a mistake.
rgb
rgb says
Thank you, HenryP, for demonstrating the utter emptiness of your curve fitting to more or less arbitrary representations.
Henry says
Please help me to get it right by showing me where I am wrong?
http://blogs.24.com/henryp/2013/04/29/the-climate-is-changing/
I do not want to make a mistake when I send this out.
richardscourtney says:
May 4, 2013 at 3:15 am
I again ask if anybody can confirm that Leif Svalgaard is alright.
I am fine, thank you. Have been busy. Travelling. Leading workshops, doing science, etc. The postings on WUWT [that I would ordinarily respond to] have been of such low quality lately that further comments from me could hardly lower the level any further
Leif Svalgaard:
Thankyou for confirming you are OK. I am sure I was not alone in getting worried.
Richard
Henry@Leif
welcome back from your travels!
please remember:
we may disagree, but….
we still love you!!!!
rgbatduke says:
May 5, 2013 at 9:16 am
“A Taylor series historically does a very good job fitting — note well fitting — a smooth function NEAR a selected point where its derivatives are known, as you note.”
Yes, a polynomial expansion generally works pretty well for slowly evolving processes, often for at least a short time beyond the confines of the data. Except when it doesn’t.
The main thing I was trying to get across, though, was the unphysical nature of the cos(a*SN) term. What physical basis could we determine for such a term? Given the apparent goodness of the fit R. J. Salvador has stumbled upon, could we perhaps tease apart the terms which go into it into something more physically justifiable?
For example, could we decompose a*SN into a constant plus some small variation, and use the first order expansion
cos(x + dx) := cos(x) – dx*sin(x)
to determine a “dx” which is physically justifiable, with the cos(x) input being transferred to the bias term?
The roadblock I hit on that route is that “a” is so large, and SN should be large, too. So, it seems like this term would only be adding in a bunch of high frequency fuzz. Not having access to R. J. Salvador’s spreadsheet to see exactly how each term contributes to the whole, I can make no more headway in this direction.
But, I would encourage him to do so. His sum is attenuating high frequencies, so if he could isolate the portion of the cos(a*SN) term which is making the major contribution to his result in the low frequency regime, he might come up with something more physically justifiable. Using Vukcevic’s analytical expansion for the SN might produce some analytical insights into doing this.
“””””……rgbatduke says:
May 4, 2013 at 5:43 am
…………………
There are, sadly, a few people who love to fit curves to data and then pretend that the result has extrapolative predictive power. If you’re going to play this game, you might as well not screw around with crippled bases. Build a neural net. At least then your basis is generalized nonlinear function approximator capable of resolving nontrivial multivariate correlations…….”””””
Since we are talking about curve fitting; and in this case using a cosine function to fit some data to (admittedly a finite string of data), and mindful of rgbatduke’s observation that Taylor series; (or izzat simply polynomials) do a fine job of fitting data to a power series, let me offer one of my favorite examples to demonstrate the folly of EXTRAPOLATION beyond the observational data limits.
Suppose I told you that I could construct a table of x-y data pairs, and to keep within the current topic, I shall choose a cosine function form to fit those number pairs to fairly accurately; damn accurately to be precise, in fact they fit exactly, within the boundaries of my table.
BUT !! If you dare to extrapolate the fit one iota beyond the x limits of my tabulated pairs of numbers, you will get a completely wrong prediction for the next data y value; in fact, you never ever will see a predicted value that even lies within the bounds of the values in my table; yet within the boundary of my table, the fit is exact. What is even more interesting, is the fact that my tabulated data series can in fact be approximated by; not a Taylor series, but a power series polynomial; and also the precision of that power series approximation, is also exact, and exact for any real value of x.
What is more, I can make a whole bunch of such tabulated data pairs; the answer is not unique.
So the cosine fit formula, that fits my table pairs is:- y = cos4(arcos (x)) and my x data values lie between -1.0 and + 1.0 ; which is good, because the cosine function magnitude can’t exceed 1.0, and for the same reason, y is also bounded by +/- 1.0
Well I used that formula to generate my tabulated pairs, which is why I know the curve fitting is exact.
My polynomial fit to that same tabulated set is:- y = 1 – 8x^2 + 8x^4 which is also an exact fit between – 1.0 < x < 1.0 but continues to predict y values tending monotonically towards + infinity for values of x outside the bounds of +/- 1.0
So extrapolation can be fatal, and is not to be indulged in on the say so of limited data sets.
Interpolation between experimentally obtained data points, of what are believed to be well behaved continuous functions, can be useful for guessing what one might have been obtained by an observation at the interpolated location.
The cognoscenti, will of course have identified my nuisance function polynomial as the function
T4(x) also known as the fourth order Tchebychev polynomial Tn(x) and the cosine function form as the parametric form of the Tchebychev polynomials; x = cos (a) , y = cos n(a)
Electronics types will also recognize the latter as the Lissajous figure for an (n) to (1) frequency ratio, and zero phase shift; in the example cited, (n) = 4.
So nyet on curve fitting especially when you can't even suggest a physical cause and effect physical reason for a specific mathematical architecture .
And finally, you ignore rgbatduke at your peril !
Janice Moore says:
May 4, 2013 at 7:58 pm
FYI: I would have been convinced more readily and quickly if Nylo’s attitude had not been so arrogantly contemptuous.
You are right and I apologise. I completely lost my temper. I could say that I lost it because, in the same way that I think that James Hansen is ironically the alarmists cause’s worst enemy and the best contributor to people becoming more and more sceptic about CAGW, articles like the one here posted do the opposite for our position. We don’t want bad science supporting that CAGW is a scam. We want to do that with good science, and we know it is entirely possible to do that. And I got a bit carried over, not just because of this being bad science, but because I was reading the other comments and, at that time, they were mostly supportive and greetng the good news. And when we support bad science we hurt our own position. If I had seen any other heavy-weight WUWT user calling this for what it was, I would have been quite more relaxed in my replies. This eventually happened (Willis’ reply was perfect as usual to that effect), and that’s why when it started to happen, I quit.
Henry@george e. smith
looks like you know your stuff!!!
perhaps you can help me?
from a random sample of daily data of maximum temperatures from 47 weather stations I have the following final results:
from 1974 the speed of warming was 0.036 degrees C/ annum
from 1980 the speed of warming was 0.028 degrees C/ annum
from 1990 the speed of warming was 0.015 degrees C/ annum
from 2000 the speed of warming was -0.013 degrees C/ annum
my data set
http://blogs.24.com/henryp/2013/02/21/henrys-pool-tables-on-global-warmingcooling/
only runs until 2012
what would you say is your best fit for these data?
I found this theory on sunspot cycle variation to be particularly interesting:
http://solarcycle24com.proboards.com/thread/324/theory-solar-cycle-www-sibet
Henry@Dan
I cannot figure out the scale of those graphs on the bottom, as far as time is concerned?
Does it correlate to my A-C sine wave for the drop in maxima??
http://blogs.24.com/henryp/2012/10/02/best-sine-wave-fit-for-the-drop-in-global-maximum-temperatures/
I would not be surprised if it did
Okay . . . . so I’ve got a solution regarding the problem of how to;
“lace an average harmonic through the manifold and (either brilliantly or haphazardly, depending on whether it was conscious or not) yank out a crude alias of the shifting anharmonic framework that gets considerably sharpened up (in central limit) by integration”.
Unfortunately, it only works for spherical chickens in a vacuum.
(P.S.; sorry for feeding the trolls, I just couldn’t resist the punch line.)
One final observation: The value of the coefficient a is not unique.
Because the sunspot numbers SN are all single decimal values,
cos( (a + 20*k*pi) *SN) = cos(a*SN),
where k is any positive or negative integer. Thus a could as “meaningfully” be 22.76211, 85.59396 or 211.2577 as the given value of 148.42581. All of the calculations using these different a’s would produce identical results.
RomanM says:
May 6, 2013 at 12:48 pm
Well,
cos( a*SN+ 20*k*pi) = cos(a*SN)
Yes, aliasing is probably involved to produce the low frequency output, and there is a much lower value of a which would put the mean at the same point, but it would scale the other components differently. It would probably be better just to do a polynomial regression in SN and forget about the unphysical cosine.
“Because the sunspot numbers SN are all single decimal values…”
Sorry, missed that. Yes, you are right.
Dear Nylo (re: your gracious comment @ur momisugly 0336 on 5/6/13),
What a fine, gallant, person you are to respond to my complaint with such candor and grace. Of course, there was no need to apologize to me. Perhaps, to the degree that it made their arguments less likely to succeed (with the math-midgets, one of which I am), an apology was owed to your fine colleagues, here. They, most likely, had no problem with your tone. What a kind person you are to do me the courtesy of addressing my concerns!
That you would lose it when fighting apparently alone with such an inane opponent, is perfectly understandable! I lose it ALL THE TIME in other spheres — fortunately for me, knowing so little of the subject matter, on WUWT those opportunities are few. To me, caring enough about the truth to be passionate about it shows you have a good heart (as well as a fine mind).
If ANYONE owes an apology on WUWT, it is I, for all my silly poems and the like and general zaniness. Believe it or not, I’m actually restraining myself quite a bit. Hm, not in this note, though… d’oh!
Thanks again for your gracious response.
With admiration,
Janice
Anthony,
I ask that my story be withdrawn. Thanks to all for your comments. I am not discouraged and when I have something that I think will stand up to your scrutiny,I will try again. It’s been fun.
R J Salvador
R J Salvador says:
May 6, 2013 at 4:50 pm
R. J., thank you for your honesty. I’ve had to do the exact same thing myself, so I know that it is difficult. However, I was able to use my new understanding, provided by my commenters, to get a deeper understanding of the issues.
So keep on keeping on, my friend, well done.
w.
RomanM (May 6, 2013 at 12:48 pm) wrote:
“One final observation: The value of the coefficient a is not unique.
Because the sunspot numbers SN are all single decimal values,
cos( (a + 20*k*pi) *SN) = cos(a*SN),
where k is any positive or negative integer. Thus a could as “meaningfully” be 22.76211, 85.59396 or 211.2577 as the given value of 148.42581. All of the calculations using these different a’s would produce identical results.”
A welcome glimmer of hope & light in the darkness.
Data:
http://solarscience.msfc.nasa.gov/greenwch/sunspot_area.txt
http://solarscience.msfc.nasa.gov/greenwch/sunspot_area_north.txt
http://solarscience.msfc.nasa.gov/greenwch/sunspot_area_south.txt
What they readily confess to undeterred explorers of natural beauty:
http://img13.imageshack.us/img13/5691/911k.gif
May God have mercy on the human race.
I really hope Paul Vaughan is not a teacher. I would feel really sorry for those students with the need to understand what he says, and even more sorry for those who actually managed to understand him!
R.J. Salvador
I think readers here will note your honest behavior by retracting your post in light of doubts that have been highlighted.
I think it shows good character on your part, to be honest 🙂