Nicolas S wrote:

Now that is pretty much confidence.
And the oil,aviation and automobile industry is working hard on the missing bit.

The tricksters hired by the fossil fuel industry
will just ignore this, and very many other truths.



(Ian Smith) wrote in message . com...
Roger Coppock wrote in message ...
[snip]
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%!
[snip]

I can save you hours of computing time. Use the calculator at
http://members.aol.com/iandjmsmith/FEX.HTM

Set it to calculate 1 - F distribution. With X = 75.92 and Denominator
df = 73 you get
Cumulative probability = 6.429602941527236e-13, so the F value is
1-6.429602941527236e-13

.. . . or better than 12 nines, which is what my software reports.



With X = 276.73745 and Denominator df = 122 you get
Cumulative probability = 3.644936641931845e-33, so the F value is
1-3.644936641931845e-33

I think you'll find lots of calculators can do the calculation this
accurately if they can work with the complement of (i.e. 1 minus) the
F distribution.

Alternatively, if you set the calculator to calculate F distribution,
then enter X = 1/276.73745, Numerator df = 122, Denominator df = 1 and
you'll get the same answer.

Now you can use a most F distribution calculators - even EXCEL gets it
right! It calculates the complement of the F distribution in the first
place so =FDIST(276.73745,1,122) gives 3.644936641694430E-33, a
relative error of approx 6.5e-11.

. . . or better than 32 nines as my software reports.

Thank you Ian! I never thought to use either a
page on the Internet, or a calculator. It looks
like everybody is in agreement. At these extremes,
where we are so close to 100% certain, I'll just
report the number of nines. The Incomplete Beta
function is so nonlinear that it amplifies the
smallest change in its inputs, so reporting any
significant figures would be misleading. The folks
at Heartland, Cato, CO2 'Science,' and other PR
firms hired by the industry are already making more
than enough misleading statements on this issue.
I don't plan to add to the confusion.



Ian Smith

--

"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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