This forum is about wrong numbers in science, politics and the media. It respects good science and good English.
On the issue of checking whether two time series are correlated or not, my understanding of the basic statistical test to do this is that you calculate a parameter called the "Pearson correlation coefficient". This coefficient is described in this Wikipedia article:
This coefficient is usually defined in terms of a quantity called 'covariance'. However looking at the section 'Mathematical properties' in the Wikipedia article, it appears that after some manipulation, it does tie up with the maths used for the R squared test in curvefitting. So the R in the R squared test is actually Pearson's correlation coefficient.
The Wikipedia article does not state a criterion for what is an acceptable value for R. However there is a criterion used in civil engineering for "statistically independent time histories" that the absolute value of R should be less than 0.3. ['Time history' is the rather ugly engineering equivalent term to 'time series'.} Willis Eschenbach has calculated an R² of 0.13 for the rise of sea level and sunspot numbers time series, which corresponds to an R value of 0.36. Using the civil engineering criterion, the R value for the two time series is not actually low enough to claim they are uncorrelated, but the correlation does appear to be pretty weak.
If you go back to the covariance definition of R, the covariance is itself defined in terms of the mean values of the time series over the time range. That suggests to me that to work out an accurate value for the correlation coefficient you need to calculate accurate mean values of the two time series, which is I think likely to depend on the number of time points used in the digtisation of the graph curves. If I was digitising the graphs myself I would use a lot of time points, and have the time points at equal time intervals, to make sure the means were accurately calculated. Eschenbach may have just used the key points in the graphs at non-uniform time intervals for the digitisation to reduce the amount of work, particularly if he is carrying out the digitisation by hand with the values estimated using a ruler from a printed out copy of the graphs. If I was doing the digitisation myself I would use a free program called "Plot Digitizer", which should be much faster than trying to do it by hand (http://plotdigitizer.sourceforge.net/), and would enable an ample number of time points to be used.
However I think the main problem with the Eschenbach analysis, as JamesV alludes to, is that he is acting as though the correlation coefficient calculation is what is known as a 'robust' statistical test. In the Wikipedia article it is pointed out in the 'Robustness' section that the test is not robust, so you can't really assume it is reliable enough to override a visual inspection of the graphs. There may be a robust statistical test in existence for checking whether a given pair of time series are correlated or not, but if there is I would imagine it would be a much more complicated test and would probably require some qualified statistician to apply it.
I found an old book on the subject of time series analysis from my undergraduate days. It was not on the syllabus, which was all historical engineering rather than the electronic engineering that I hoped to pursue. The book mark in it was a piece of paper, on which I had typed the following:
Time series analysis
Is a compound of fallacies
By subtle catalysis
Causing mental paralysis.
GRRR This fits into our authors poem. The linked video cause my blood pressure to increase. It wouldn't take much to cause me to tantrum. I plot the raw data for the red regions and can't find jack @()#$ in the way of increasing temperatures. There are folks out there with real hats on that are also saying the same thing, but how can these people be so arrogant?
I thought I'd update this thread as I have finally got round to calculating the correlation coefficient myself for the sea level change and sunspot number pair of time series.
The method I used was:
a) digitise the graphs from a screen capure of Fig 2 in the Solheim paper using 'PlotDigitizer'. PlotDigitizer only takes about 5 minutes to produce pretty accurate x, y coordinates for the two graphs, whereas it would take at least several hours if you tried to estimate the coordinate values by scaling from an enlarged printed out copy of Fig 2 using a ruler.
b) interpolate the points at uniform time intervals using a short Fortran program. With this I turned each time series into 911 data points at 0.1 year intervals for the year range 1909 to 2000.
c) run the two 911 point time series through another short Fortran program called 'corrcheck' I wrote about twenty years ago which calculates the absolute value of the correlation coefficient. This program uses the text book definition of R, the same one as in the Wikipedia article.
The value for the correlation coefficient I worked out was 0.35842, or 0.36 to two decimal places. This agrees very well with the value quoted by Willis Eschenbach [I was expecting to prove that he had significantly underestimated it].
So this example does raise an interesting point, two time series which look pretty well correlated by eyeball have quite a poor R value, only just above the acceptance criterion of 0.3 for them being taken as correlated.
Yes. I think your example illustrates the fragility of trying to use just one number to summarise the relationship between two waveforms. Consider a sine and a cosine (possibly derived in a physical system from the same source). The cross-correlation function is clearly periodic, but its average is zero. Even more difficult is the case where there is clearly oscillatory behaviour, but the periodic time is wandering, as happens in many common physical processes. This is clearly visible to the eye, but not to the single number. Even the cross-correlation function might blur the effect to relative invisibility.