No statistically significant warming since 1995: a quick mathematical proof

Physicist Luboš Motl of The Reference Frame demonstrates how easy it is to show that there is: No statistically significant warming since 1995

First, since it wasn’t in his original post, here is the UAH data plotted:

UAH_LT_1979_thru_Nov_09

By: Luboš Motl

Because there has been some confusion – and maybe deliberate confusion – among some (alarmist) commenters about the non-existence of a statistically significant warming trend since 1995, i.e. in the last fifteen years, let me dedicate a full article to this issue.

I will use the UAH temperatures whose final 2009 figures are de facto known by now (with a sufficient accuracy) because UAH publishes the daily temperatures, too:

Mathematica can calculate the confidence intervals for the slope (warming trend) by concise commands. But I will calculate the standard error of the slope manually.

x = Table[i, {i, 1995, 2009}]

y = {0.11, 0.02, 0.05, 0.51, 0.04, 0.04, 0.2, 0.31, 0.28, 0.19, 0.34, 0.26, 0.28, 0.05, 0.26};

data = Transpose[{x, y}]

(* *)

n = 15

xAV = Total[x]/n

yAV = Total[y]/n

xmav = x - xAV;

ymav = y - yAV;

lmf = LinearModelFit[data, xvar, xvar];

Normal[lmf]

(* *)

(* http://stattrek.com/AP-Statistics-4/Estimate-Slope.aspx?Tutorial=AP *)

;slopeError = Sqrt[Total[ymav^2]/(n - 2)]/Sqrt[Total[xmav^2]]

The UAH 1995-2009 slope was calculated to be 0.95 °C per century. And the standard deviation of this figure, calculated via the standard formula on this page, is 0.88 °C/century. So this suggests that the positivity of the slope is just a 1-sigma result – a noise. Can we be more rigorous about it? You bet.

Mathematica actually has compact functions that can tell you the confidence intervals for the slope:

lmf = LinearModelFit[data, xvar, xvar, ConfidenceLevel -> .95];

lmf["ParameterConfidenceIntervals"]

The 99% confidence interval is (-1.59, +3.49) in °C/century. Similarly, the 95% confidence interval for the slope is (-0.87, 2.8) in °C/century. On the other hand, the 90% confidence interval is (-0.54, 2.44) in °C/century. All these intervals contain both negative and positive numbers. No conclusion about the slope can be made on either 99%, 95%, and not even 90% confidence level.

Only the 72% confidence interval for the slope touches zero. It means that the probability that the underlying slope is negative equals 1/2 of the rest, i.e. a substantial 14%.

We can only say that it is “somewhat more likely than not” that the underlying trend in 1995-2009 was a warming trend rather than a cooling trend. Saying that the warming since 1995 was “very likely” is already way too ambitious a goal that the data don’t support.

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Jeff L

What’s nice about this post – both the data & methods are posted for everyone to see & verify – wouldn’t it be nice if the CRU & GISS would do the same – like real scientists doing real research would do so that it can be verified (& if decisions are made on the basis of the analysis, those decisions can be made with confidence)

C Shannon

This *should* be filed under things everyone knows. Sadly thanks to climate hysterics very few do know of it because it undermines the “right” policy conclusions.
Whats worse than that though is that an article such as this presenting undeniable facts in a concise and easy to follow manner isn’t likely to make a dent in the problem.
Still the effort is greatly appreciated all the same. The truth will eventually prevail.

Disputin

I am far from expert in statistics, but is this really valid? By including the obvious outlier of 98 (El Nino) the SD is increased so widening the confidence intervals. While I should agree that in the long run the 98 jump is just a part of the variability, over this restricted timescale it is a major anomaly.
But then what do I know?

EdB

Says nothing about what humans might have caused. Totally meaningless imo.
I am betting on galactic cosmic rays, thank you very much.

tallbloke

Heh, love it. Nice one Luboš.

Icarus

The long-term warming trend is around 0.13C per decade according to the entire UAH record. What you should be calculating is whether there is any statistically significant deviation from that warming trend – otherwise you’re just grasping at straws.
http://www.woodfortrees.org/plot/uah/from:1979/to:2009/plot/uah/from:1979/to:2009/trend

Richard M

Seems like I heard it from the AGWers that 15 years without warming would falsify the hypothesis. Of course, they would claim that the greater probability of warming is enough.

Steve Goddard

Whatever the direction, the magnitude is much lower than IPCC forecasts.
http://www.bbc.co.uk/sn/hottopics/climatechange/climate_challenge/aboutgame_2.shtml
If The Nobel Prize winners were correct, temperatures should have risen over half a degree since 1995.

Why start at 1995 instead of 1979?
Why not use monthly data?

Richard

Melly Kalikimaka and a hearty thank you to Anthony and all his merry elvises who do such a great job at this crossroads of knowledge. Thanks for bringing us such nice lumps of carbon rich reading. Wishing you a great and prosperous New Year. Keep up the good work.

Schrodinger's Cat

Good

Henry chance

You are so mean to use facts to go up against emotional arguments. The polar bear extinction alone is over the top for data.

Ah yes, let’s predict the future based on past data.
Firstly you have give us some reasons to believe that your data is meaningful. Where does your satellite data comes from? What sensor does the satellite use? Have satellite readings been calibrated against earth based instruments? Does the satellite read surface temp? Tropospheric temp? Stratospheric temp? Top of the atmosphere temp? Temp at noon? Temp at midnight? Or just temp at any old time? How do you know ? How do the satellite readings change as the satellite sensors age? How many satellites are in your data? One? More? If more than one, how closely do the readings from the various satellites agree with each other?
Then we want to know the average and standard deviation of your data. How much noise (standard deviation is a measure of noise) is in your data? Is the noise level higher than any trend you might be seeing?
Looking at your code, it seems your are asking Mathematica to do a least squares straight line fit to your data. If we believe a straight line is a good approximation , then we can take to slope of the best fit straight line as a trend.
Then you have to explain why you picked 1979 as a start point. Suppose I pick 1998 as a start point? I can get any answer I like if I can pick my start year. If I pick 1998, then I can say temperature has been declining since then. You have to explain your choice of start year.

Eve

My heating fuel usage shows that there has been cooling since 1997, which is the oldest data I have. I will show fuel usage in Litres per year. I do not heat the house when it is warm therefore each year after 1997 must have been cooler. The furnace has a scheduled maintenance each Nov, the same two people live in the same house and the thermostat settings have not changed.
1997-2767.20 Litres
1998-3057.50 Litres
1999-4009.30 Litres
2000-3874.70 Litres
2001-3586.70 Litres
2002-3752.20 Litres
2003-3634.50 Litres
2004-4072.50 Litres
2005-3293.50 Litres
2006-4276.70 Litres
2007-3700 Litres
2008-4476.20 Litres

niphredilflower

How long a period are we blamed for effecting the climate? I heard recently that CO2 is only blamed for the last 20 years… and the 1998 peak was due to an El Nino… If this is the case then surely we are only being blamed for a tenth or so of warming, of which is showing signs of returning.
If we return to temperatures of pre-blame… does that prove the build up of heat is gone? Or can they still claim that we have reduced cooling that would have occurred to a greater degree? – What happens if the heat in the climate system is not measured or has reversed in the next few years?

Bill H

Get rid of the Hadly, CRU, MET, NASA, GISS, NOISE and whala….. Static normal cycle……
Whoda thunk it…?

niphredilflower

– addition: tenth of a degree

Adam from Kansas

They will probably say that this decade is the warmest ever by a significant amount, that is if you cut off the data somewhere after the 1930’s.
Also, Piers Corbyn is talking about a dangerous system developing that could be even worse than the one still rolling in the United States on Dec. 28-30.
http://www.iceagenow.com/Notably%C2%AD_dangerous_warning_for_28-30_Dec.htm
Despite Okla. City getting slammed hard, Wichita got off easy on that system, Weather Underground already has two days in a row with snow chances before New Year’s in their forecast column, we can’t dodge the worst parts of these snow storms forever.

Bill H

And a scientist who shares his methods and equations…
Haven’t seen that from any of those so called scientists who scream the earth is melting…..

KevinB

Hey man, don’t you know that the issue is settled? Al Gore told me so, so it must be true. The fact that he owns a piece of the largest carbon trading firm, and is actively pushing for cap and trade is not a conflict of interest, because he’s thinking of the children (and the polar bears).

PJB

I am constantly dismayed by the “use” of statistical significance and relationship correlations in the media and by the public.
Even a 90% significance is weak by statistical standards. (95% is just enough to be reasonable.)
When a coefficient of correlation (linear regression) is less than 0.9, there is barely more than a “trend” established. When they start to use log plots, almost anything starts to look like a straight line relationship.
Forget about causation. Correlation means that factors demonstrate similar effects. Not that one causes the other.
Forget about cropping and truncating the graphs or picking a range of data that only looks at a short (for the event) time-frame.
When you start to throw in “adjusted” data that mysteriously suits an agenda…
Samuel Clemens had it right. Lies, damn lies and statistics….

Graeme W

Which is why people who want to spin the numbers are now saying that the last decade is the warmest on record, rather than temperatures are getting hotter.
You can spin almost anything using statistics. It’s nice to see them being used properly for a change, though the title of the article is spin because it implies there’s no warming trend. The closing statement of the article isn’t that definite. It merely says that we can’t say that a warming trend is “very likely”.

Kevin

I believe the issue is clouded by earlier analysis that proved there _was_ statistially significant warming for a time interval ending in 2000. Menzie Chin, an economist who publishes on the site Econbrowser replicated that math. He’s something of a Democratic partisan, but surely has command of the math.
Like all of these experiments, the results depend on what data you choose to work with.

kadaka

And what do things look like without the 1998 spike?
How much of the heating of the Pacific Ocean is due to underwater volcanic activity? After all Al Gore informed us it is millions of degrees just a few kilometers down so volcanic activity must be a potentially significant source of heat. The global warming theories are concerned with how CO2, with the positive feedback mechanisms, traps solar energy. Shouldn’t heating from below the surface then be discounted somehow when figuring warming based on atmosphere-based reasons? Thus if El Nino is tied to such from-the-Earth heating, shouldn’t 1998, involving a strong El Nino event, be discounted when calculating the trends?

Rob Vermeulen

The trend is non-significant only because the poster used the average yearly anomaly. Taking every month into account, the trend becomes statistically more significant.

It’s interesting to note that during 1998 the Sun was in an angry mood (high sun spots) as this video shows: http://pathstoknowledge.net/2009/12/20/the-constant-and-never-changing-sun-cant-influence-climate-on-earth
Given that the sun spot maximum occurred near the peak temperature in 1998 one wonders what influence it had (and has). Seems to be potentially correlated.
Would anyone care to expand upon this possible correlation? Luboš Motl? Anthony? Steve?

DirkH

“Graeme W (14:18:37) :
Which is why people who want to spin the numbers are now saying that the last decade is the warmest on record, rather than temperatures are getting hotter.
[…]”
They can say that because the past (according to GISTEMP) is getting colder all the time. Proof here:
http://wattsupwiththat.com/2008/11/14/the-evolution-of-the-giss-temperature-product/

Scarlet Pumpernickel

Is all that snow going to push the chart down, runaway cooling magnified with all that white ice in the northern hemisphere!

But – but – Tamino just a few weeks back posted charts where his linear trend line technique definetly indicated ‘warming’ in the recent past, it seemed to leave ‘no doubt’ in the minds of those who posted that there was positively *no warming*.
He even received *rave* reviews on his site for doing so (albeit from an ever-so select group of sycophants known for their slavish praise) –
Was he just “hiding the decline” in his presentation?
.
.
.

Ano

This kind of thing was inevitable from the moment the EPA was allowed, back in 1993, to get away with using a 90% confidence interval to claim a barely statistically significant risk of lung cancer in non-smokers from environmental tobacco smoke (based on a cherry-picked selection of meta-studies, no less).
When there was no outcry from the statistical or scientific, but instead vast attention from the media, the abuse and manipulation of statistics went into overdrive. Disraeli’s admonition that “There are three kinds of lies: lies, damned lies and statistics”, is now cited as a slightly amusing saying, rather than the caution it was intended as.
I don’t mean to change the subject of this thread to second-hand smoke: just to indicate that this was the first example I can recall of widespread statistical abuse which attracted massive media attention, grants for its purveyors and a government response disproportionate to the problem. The thin edge of the wedge, so-to-speak, and a template followed closely by the AGWers.

1st paragraph should have ended thusly: “that there positively was *warming*.”
.
.
My bad.
.

Lubos,
I did a post that includes your review of “Red Hot Lies”: Prostitution Services.

I’d like to know what happens when you factor in the siginificant autocorrelation in the residuals. That would make the sigmas much wider and the statistical significance smaller.

Bret (13:49:22) :
> Why start at 1995 instead of 1979?
I suspect the short answer is that UAH temperature were going up during that warm PDO phase.
Hmm, tweaking the first few lines from a Robert Woods graph, I get no warming between 1979 and 1995, see http://www.woodfortrees.org/plot/uah/mean:4/plot/uah/from:1979/to:1995/trend/plot/uah/from:1995/trend/plot/uah/from:1997.5/trend/plot/uah/from:1992/to:1999/trend
Unfortunately, WFT doesn’t have a standard deviation function, but you can take its list of data pointd and put them into a spreadsheet. Time for me to make dinner.
Unfortunately, the temperature data stream is so noisy that just looking at temperature doesn’t provide much guidance.

rbateman

0.5C might seem like a lot of change, but convert that to degrees K and what does that now represent in percentage of total?

u.k.(us)

pwl (14:23:55) :
It’s interesting to note that during 1998 the Sun was in an angry mood (high sun spots) as this video shows: http://pathstoknowledge.net/2009/12/20/the-constant-and-never-changing-sun-cant-influence-climate-on-earth
Given that the sun spot maximum occurred near the peak temperature in 1998 one wonders what influence it had (and has). Seems to be potentially correlated.
Would anyone care to expand upon this possible correlation? Luboš Motl? Anthony? Steve?
=====================================
to me, it seems the correlation is clear (probably because i dont know any better).
my question is: whats causing, the suns changes?

Tom in Florida

Eve (14:08:45) : “My heating fuel usage shows that there has been cooling since 1997, which is the oldest data I have. I will show fuel usage in Litres per year. I do not heat the house when it is warm therefore each year after 1997 must have been cooler. The furnace has a scheduled maintenance each Nov, the same two people live in the same house and the thermostat settings have not changed.
1997-2767.20 Litres
1998-3057.50 Litres
1999-4009.30 Litres
2000-3874.70 Litres
2001-3586.70 Litres
2002-3752.20 Litres
2003-3634.50 Litres
2004-4072.50 Litres
2005-3293.50 Litres
2006-4276.70 Litres
2007-3700 Litres

As your house has aged have the windows been checked for air leakage?
How about the doors? What about insulation? What about burner efficiency declining with age? Perhaps the thermstat needs adjusting?
You see one can fail to take into account ALL possible reasons why something happens and come to potentially false conclusions.
Same with the simplistic approach that all warming is CO2 induced.

Although 15 years sounds like an attarctive interval this is cherry picking. 1999 tp 2009 will give you statistical warming at 90% and 1979 gives a warming I think at 95% confidence using UAH I seem to remember froma pot by LUCIA. I confess I prefer Lucia’s 2001 which has the merit of being “when IPCC made its forecast” if we are looking at recent trends. I can’t see the rationale for 1995.

Simon Platt

Hey,
Excluding the El-Nino year the best estimate for the mean rate of temperature increase using Motl’s approach is 1.5 degrees per century, with a 95% confidence interval between 0.2 degrees per century and 2.9 degrees per century. But the linear model is not good, as reflected in the large uncertainty in the slope of the best fit, so probably none of these numbers could be trusted.
So it seems to me that these data tell us nothing – or at least nothing more than the fact that they can’t be modelled by a constant rate of increase. I suppose that’s almost what Motl is saying, although not quite.

Paul

I believe that this posted analysis is flawed. You cannot apply this type of simple linear trend analysis to serially correlated data, since the precision of the parameter estimates is strongly a function of the autocorrelation in temperature data.

Lubos should post his results at Tamino’s place, since Tamino is so much in love with statistical sophistry.

Ric Werme (15:14:24) : “I get no warming between 1979 and 1995”
That’s because there is none, at least for the satellite based series.

photon without a Higgs

…both the data & methods are posted for everyone to see & verify – wouldn’t it be nice if the CRU & GISS…
A wistleblower would be nice too.

A great site to refer to anytime statistically significant trends is mentioned is
Cherry-Pickers Guide to Global Temperature Trends.
“..how global temperature trends look across different datasets and choice of starting dates..”
http://rogerpielkejr.blogspot.com/2009/10/cherry-pickers-guide-to-global.html

ShrNfr

I’ll pick AMO for some warming through 2005 and then for cooling through around 2040. At least in the northern hemisphere.

ShrNfr

Predicted lows for Boston on tuesday are of the order of 8 degrees F. Hey Hansen, want to come up my way? Its only a couple blocks from the Charles River and we could have a nice invigorating dip.

crosspatch

The long-term warming trend is around 0.13C per decade according to the entire UAH record.

Yes, and plot the same since 2002 and you see a rather dramatic cooling trend since that time. If you go all the way back to 0AD we are in a cooling trend. Picking a year when an apparently cycle of natural warming started doesn’t really mean much. Check back in another 30 years when we have a full Pacific cycle in the records.

I have more of an issue with the whole approach.
If at the end of 2009 there is less heat in the climate system (oceans, atmosphere, soil, etc) than at the end of 2008, then the earth has cooled in that year.
It’s not “noise”. It has cooled and there IS a reason. Trying to fit chaotic systems to linear trends or least squares regression or […] is something I don’t really understand.
Across 1,000,000 years the earth has cooled. In 20,000 years the earth has warmed. In 1,000 years possibly cooled (let’s not debate MBH 98 etc). In 150 years warmed. In 10 years cooled/warmed depending on dataset.
Some processes take place over millions of years – mountains being formed and continents moving. Some processes take place over 20,000/100,000 years – changes in the earth’s orbital eccentricity, precession etc. Some processes take place over 1000’s of years – ocean currents and salinity changes. Some take place over years, months and days.
Trying to form a pattern and make something of that pattern? Interesting intellectual exercise but why? And does it mean anything in physical terms? Finding patterns and trying to fit to physical processes makes sense only to help our scientific understanding.
So if the earth cools in a year it has cooled. It’s not a statistical blip. The question is why. “Natural variation?” – why? What is the process?
Check out: http://scienceofdoom.com/2009/12/19/is-climate-more-than-weather-is-weather-just-noise/

Frank K.

One question that I haven’t heard a good answer for is this: Why do people like to attach ** linear ** trends to processes (like the behavior of the climate) that are very ** non-linear **?

docattheautopsy

The reason I’m more convinced by the historical data derived from ice core samples in Greenland and Antarctica is the reach you have with the data. Having a “warming trend” that’s built on data from 1979-1998 is certainly alarming. Taking such a small sample from a geological time scale and drawing worldwide impact of industrial pollution from a mere 100 years of data while discarding data carefully gathered on the past 500,000 years is absurd.
What’s worse is taking computer models of a system that’s not completely understood and applying it to the system as a measure of accurate prediction. I liken it to a man believing Elvis will return in 2012 because four separate psychics agree that Elvis will return in 2012.
Climatology relies on observation, not experimentation, and while we know more about climate, nobody can say with scientific certainty that manmade CO2 emissions are the only cause of warming. Experimentation relies on controlling all variables in the system, and that’s something we can’t reliably replicate.
This statistical analysis is nice, but again, it only works over a small time range. The discussion on confidence intervals is certainly valid, and the conclusion is expected, but we should beware such small data sets in climatological data to prove our points.
Also, I’d like to see 1998 compared to a 1979-2009 range, an 1800-2009 range, and an 1800-2009 range to see if the abnormally warm year can be statistically excluded, as there were certainly warm and cold years in the past 200 years.