I'll be adding a new feature to my monthly world temperature
analysis, an evaluation of the confidence of nonzero correlation.
This "F ratio" inferential statistic tests whether the slope
of the correlation is zero. This procedure is well known. Even
"CO2 Scientists," use it. Consult any good statistics text for
more information.

(You may also find information on, or may already be aware of, a
"t test," which evaluates whether the slope of the correlation is
positive or negative. I would have used the "t test" to test for
a positive warming slope, but to avoid confusing some people I
chose to follow the lead of the "CO2 Scientists" and use the
"F ratio." (Actually, "F" and "t" are largely the same, both use
the Incomplete Beta Function, but with different arguments.))

I would have added this feature earlier, but the 124-year record
shows warming so very strongly that special attention is required
to evaluate the "F distribution function" correctly. In most cases,
a simple table listing "F" values for 95% and 99% confidence will
do. (For an example, see this table from the "CO2 Science" site:
http://co2science.org/ushcn/ftable.htm) However, the positive
slope regression lines of global warming in temperature data series
have been more than 99% certain for over a half-century now.
(The positive correlation from 1880 to 1954 in the GISS land and
sea data set is about 99.9999999999% certain, with 73 degrees of
freedom and F = 75.92.)

I have therefore used the "BETAI" routine from "Numerical Recipes"
by William H. Press, Brian P. Flannery, Saul A. Teukolsky, and
William T. Vetterling in 1986. My results pass the test routines
provided by the authors, and they match the table from "CO2 Science"
as well. I've also checked my more extreme results, with their long
series of "9"s, against Wolfram Research's "Mathematica" version 3.0
for Macintosh. I used variations on these three commands:
Statistics'ContinuousDistributions'
$MaxExtraPrecision = 2000
N[CDF[FRatioDistribution[1, insertdegreesfreedomhere], Fratiogoeshere], 500]
This computation uses symbolic representations for numbers to simulate
many digits of precision, and is therefore a very slow process. A
single evaluation by "Mathematica," like the one needed to test the
case below, can take hours on my 266 MHz Mac G3 Power PC. In most
cases I tested, "Mathematica" and my implementation of "BETAI" agreed
on the number of "9"s; occasionally, they differed by a single 9 digit.


Here are the results for 124 years of GISS global land and sea data:
Rxy 0.833087 Rxy^2 0.694034
TEMP = 13.666145 + (0.004797 * (YEAR-1879))
Degrees of Freedom = 122 F = 276.73745
Confidence of nonzero correlation = approximately
0.99999999999999999999999999999999 (32 nines), which is darn close to 100%!



These globally averaged yearly temperature data come from NASA:
http://www.giss.nasa.gov/data/update...LB.Ts+dSST.txt
They represent the results of hundreds of millions of readings
taken at thousands of stations around the globe over the last
124 years. Yes, the land data are corrected for urban heat
island effect. The sea data do not need to be. There are few
urban centers in the sea.


--

"One who joyfully guards his mind
And fears his own confusion
Can not fall.
He has found his way to peace."

-- Buddha, in the "Pali Dhammapada,"
~5th century BCE


-.-. --.- Roger Coppock )


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