https://rjlipton.wordpress.com/2014/01/30/global-warming/

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?