It snowed in Atlanta, and we are closed, for the foreseeable future—where is global warming?

Atlanta is frozen. Here is what we looked like the other night.

Today I thought I might talk about the weather in Atlanta, and its connection to mathematical paradoxes.

It has been extremely cold the last few weeks. Our pipes froze and broke in my house, destroying our kitchen. Just Tuesday we had about two inches of snow; the city is paralyzed, and the main interstates are clogged with trucks and cars that cannot move. A baby was born last night in one of those cars. It is a mess.

Where is global warming when we need it? The claim that seems well founded is that the earth’s average temperature is increasing; hence, global warming. But we are freezing, Tech has been closed for the last two days and is closed today too. Is this consistent with global warming?

## Global Temperature

Being stuck in our house gave me time to think about the issue of global warming. I started to read some of the papers on the claim that the global temperature is rising. I believe that CO${_{2}}$ gas is a major problem, am all for better treatment of the Earth, but I started to see some potential issues in how the climate experts compute the global temperature (GT).

Here are two of the issues that a quick search found:

• What does GT even mean?
• Is there potential misuse of averages?

The later is the one that I think is the most interesting, especially for me. It is a mathematical question.

The former question is about physics, so let me just mention one paper. Christopher Essex, Bjarne Andresen, and Ross McKitrick have written a paper called Does a Global Temperature Exist? They do not say GT is not rising, they say it is meaningless. Here is their abstract:

Physical, mathematical and observational grounds are employed to show that there is no physically meaningful global temperature for the Earth in the context of the issue of global warming. While it is always possible to construct statistics for any given set of local temperature data, an infinite range of such statistics is mathematically permissible if physical principles provide no explicit basis for choosing among them. Distinct and equally valid statistical rules can and do show opposite trends when applied to the results of computations from physical models and real data in the atmosphere. A given temperature field can be interpreted as both “warming” and “cooling” simultaneously, making the concept of warming in the context of the issue of global warming physically ill-posed.

This is an attack on their paper—asking if it is a joke? Statements like that disturb me—unless they are really out there—I prefer an attack to remain scientific. They have their defenders—see this for a defense of their paper. My suggestion is glance at the paper and decide for yourself. By the way, the writers Essex et al are from research universities and do not seem to me to just poking fun. They do make a point that I found interesting:

Despite popular thinking otherwise, temperature and energy are not equivalent. Temperatures can be very high at very low energies. While heat is a form of energy, temperature is, fundamentally, a measure of how energy is spread over quantum states. For example, radiation from a small laser powered by flashlight batteries can have temperatures peaking as high as ${\approx 10^{11}}$K. This is higher than many stellar interiors, but one cannot even feel the heat of the beam on one’s hand.

However, this is not my area of expertise.

Let’s leave this debate and move on to discuss the use and mis-use of averages.

## The Mathematics Of Averages

The notion of average of a set of numbers is one of the most basic, most powerful, and most useful notions in elementary mathematics. Averages can be used to prove many theorems in combinatorics—this is one of main tools used in the probabilistic method created mainly by Paul Erdős.

I have been looking out my window to see white, no cars moving in my neighborhood, unplowed roads, and the local news showing jack-knifed trucks all over the city. Thousands of poor people were stuck in their cars all last night, with no food, no water, and little heat.

While I have been watching this I started to think about how averages can be mis-leading. Can we say anything mathematical about the rise of the average temperature of the planet and the freeze we are now experiencing? Indeed. Perhaps we can.

The Simpson Paradox was discussed by Edward Simpson in a 1951 paper. It was clearly known to statisticians earlier: by Karl Pearson in 1899 and Udny Yule in 1903. The paradox was named by Colin Blyth in 1972. It is an instance of Stigler’s law of eponymy, which says that laws are often not named after their discoverers. Many therefore call the effect the Yule—Simpson paradox.

The paradox arises in the following simple case: In the figure the two red lines are above the two blue lines. But when the data is combined the result flips: now the blue is above the red. This is the paradox.

The mathematical version is this: Suppose that

$\displaystyle \begin{array}{rcl} \frac{a_{1}}{b_{1}} &<& \frac{c_{1}}{d_{1}} \\ \frac{a_{2}}{b_{2}} &<& \frac{c_{2}}{d_{2}}. \end{array}$

Then it need not follow that

$\displaystyle \frac{a_{1} + a_{2}}{b_{1} + b_{2}} < \frac{c_{1} + c_{2}}{d_{1} + d_{2}}.$

For example, ${\frac{1}{3} < \frac{34}{100}}$ and ${\frac{66}{100} < \frac{2}{3}}$. But ${ \frac{67}{103} > \frac{36}{103}}$.

## Averages Of Averages

Daniel Lemire has a neat blog where he talked about the danger of taking averages. Another way to think about the Simpson paradox is to look at computing averages. Here is an example of this:

$\displaystyle 3,4,6,5,4.5$

has average ${4.5}$. Now let’s split the list into two:

$\displaystyle 3,4$

and

$\displaystyle 6,5,4.5.$

The average of the former is ${3.5}$ and the latter has average ${5.16666666667\dots}$. However, the average of the averages is

$\displaystyle (5.16666666667 + 3.5)/2 < 4.5.$

In general,

$\displaystyle (\mathsf{avg}(A) + \mathsf{avg}(B))/2 \neq \mathsf{avg}(A \cup B),$

for disjoint sets ${A}$ and ${B}$.

## Connection With Global Warming?

What does Simpson paradox and taking averages of averages have to do with Global Warming. A lot. One of the fundamental problems that is faced is the computation of GT. This is done by repeatedly taking averages and more averages. This sounds on the face of it to be dangerous. Simpson anyone?

NOAA uses this average of average method, at least it appears to. See here for some details. One quote from there is a bit scary:

Different agencies use different methods for calculating a global average.

Do they get the same answer?

## Can We Help?

Let try and be positive. Can we turn the computation of GT into an interesting theory question? I think there is a good chance that we can.

Let ${f:E \times T \rightarrow \mathbb{R}}$ be a map from the earth ${E}$ and a given time interval ${T}$ to the local temps. We can view their averaging strategy as an attempt to compute the integral:

$\displaystyle \int_{E} \int_{T} f(x,t)dxdt.$

They only have values of ${f(x,t)}$ at certain places and at certain times. What is the best method for calculating this? It seems that this is a kind of quadrature problem, which is well studied. Can any of this theory be brought to bear here?

A novelty is perhaps that only certain values of ${f(x,t)}$ are known. These values come from sensors and the location of the sensors is not uniform: that means that getting a value ${f(x,t)}$ for certain ${x \in E}$ is impossible, no matter what the time ${t}$ is. Another issue is that sensors are of course noisy and that needs to be accounted for in any scheme.

Aside from this, can we suggest other measures? Total energy in the atmosphere and oceans? Perhaps a purely kinetic measure? Ken opines that the primary effects of the processes tabbed as responsible for global warming are greater energy and dynamism. Note that southern Australia had record heat that forced stoppages in the Australian Open tennis tournament, while the global warming movie “The Day After Tomorrow” featured a deep freeze in Scotland as an effect. He also notes that the University at Buffalo has been open all week and everything feels normal there.

## Open Problems

For me the main open problem is when will we be able to leave our house? The sun is out in Atlanta. It is a balmy ${20^{\circ}}$ Fahrenheit and it may go up to above freezing today. Time to dig out our car and get some food, since we are running low. When will Atlanta become normal again? For the rest the main question is there any better way to talk about global increase in the average temperatures, so we avoid any paradoxes?

1. January 30, 2014 2:51 pm

It’s that Bi-Polar Vortex, dontcha know …

January 30, 2014 3:20 pm

Maybe the overall ice amount on the earth could be the right measure. I have heard that glaciers are overall melting…

3. January 30, 2014 3:39 pm

It’s probably worth having a look at the XKCD argument too (http://xkcd.com/1321/). Good luck with the shovel, and remember, use your knees, not your back! (From a guy who used to live in Montréal :-) )

January 30, 2014 6:06 pm

Dick Lipton asks  “Can we suggest other measures [of global warming]? Total energy in the atmosphere and oceans?”

A prominent advocate of energy-balance climate-change measures is the celebrated — justly celebrated as it seems to me — planetary climatologist James Hansen (see for example, “Earth’s Energy Imbalance and Implications”, arXiv:1105.1140).

A quick & certain way to be banned from denialist web-sites is to reference Hansen’s work. This illustrates a prominent trait of denialist cognition: skepticism that focuses not upon the strongest climate-change science, but upon the weakest. This too-common denialist practice is evidently (when we think about it) is the opposite of rational skepticism.

Needless to say, this same principle applies broadly in research: the discourse that matters (in the long run) subjects the strongest research to the strongest skepticism; the essence of a collegial research community being that all parties give of their best … and enjoy the resulting vigor of discourse.

January 31, 2014 11:36 am

Proponents of the “mankind is causing global warming” discussion, such as John Sidles, dismiss any “rational skepticism” by marginalizing the folks with differing data and opinions by calling them derogatory names such as “denialist”. What is rational to one person may not necessarily be rational to others.

The global warming folks claim that mankind is responsible for the apparent warming trend. Yet, as little as 20000 years ago, much of the planet was blanketed with ice. Somehow, and I am reasonably certain it was NOT due to the activities of mankind, the planet warmed up sufficiently to melt much of that ice. Some “denialists” recognize that the planet has naturally occurring temperature fluctuations that can cause its climate to change.

Science is all about facts – it is not about plurality. Correlation does not equal causation. Simply because one side of a discussion has more supporters than the other does not necessarily mean that the majority are right. Marshall and Warren proved that h. pylori causes stomach ulcers despite being globally ridiculed for suggesting that bacteria could live in the harsh environment of the stomach.

January 31, 2014 12:08 pm

John

Thanks as always for thoughtful comment

January 31, 2014 2:48 pm

The American Institute of Physics (AIP) maintains a hyperlinked website, authored mainly by Spencer Weart, titled The Discovery of Global Warming.

For students especially, the AIP website’s unique blend of science-and-history creates a wonderful environment for students to learn about climate-science and about human cognition of science.

January 30, 2014 6:14 pm

Here in France, this year we’ve been having a particularly mild winter up to now. Is global warming for europe only?

January 30, 2014 6:32 pm

Spatial interpolation is often solved with “kriging” (aka Gaussian Processes). There are analogous space-time models, e.g. see “Statistics for Spatio-Temporal Data” by Cressie and Wikle. So one could interpolate and then numerically integrate. However it does seem that there should be something more direct and provably optimal given Gaussian assumptions. Interesting question!

7. January 30, 2014 7:49 pm

January 30, 2014 7:57 pm

I live here in Brazil, São Paulo state specifically, and the last two weeks had the higher temperatures ever measured (the measurement started on 1943). For example, today in my city the temperature reached the 35 ºC (95 ºF). I had never experienced such an unpleasant day.
I think global warming is misnamed. I don’t have a better name, but I strongly believe that what really happens are the extreme temperatures, either higher or lower. It should be a challenge for any scientist to explain why some places is getting cooler and others hotter, but it really seems that is happening.

January 31, 2014 6:57 am

Reblogged this on Pink Iguana and commented:
rjl v Al Gore debate in 2014 plz

January 31, 2014 2:22 pm

About your averaging problem: of course we need somehow to restrict the class of functions. One reasonable way, a la Vapnik, is to assume that there is so called reproducing kernel: a function belongs to some Hilbert Space, the values at given points are bounded linear functional there and the integral is also a bounded functional. In other words given vectors
f_0(integral),f_1,…,f_n (evaluations) in some Hilbert Space solve the following quadratic minimization problem: \min_{x_i \geq 0, x_1+…+x_n= 1} ||f_0 – x_1f_1+…+x_nf_n||^2. It is a convex minimization, and essentially what Support Vector Machines are about.