Triangular Fuzzy Numbers and the IPCC

Guest Post by Willis Eschenbach

I got to thinking about “triangular fuzzy numbers” regarding the IPCC and their claims about how the climate system works. The IPCC, along with the climate establishment in general, make what to me is a ridiculous claim. This is the idea that in a hugely complex system like the Earth’s climate, the output is a linear function of the input. Or as these savants would have it:

Temperature change (∆T) is equal to climate sensitivity ( λ ) times forcing change (∆F).

Or as an equation,

∆T = λ  ∆F.

The problem is that after thirty years of trying to squeeze the natural world into that straitjacket, they still haven’t been able to get those numbers nailed down. My theory is that this is because there is a theoretical misunderstanding. The error is in the claim that temperature change is some constant times the change in forcing.

Figure 1. The triangular fuzzy number for the number of mammal species [4166,  4629,  5092] is shown by the solid line. The peak is at the best estimate, 4629. The upper and lower limits of expected number of species vary with the membership value. For a membership value of 0.65 (shown in dotted lines), the lower limit is 4,467 species and the upper limit is 4,791 species (IUCN 2000).

So what are triangular fuzzy numbers when they are at home, and how can they help us understand why the IPCC claims are meaningless?

A triangular fuzzy number is composed of three estimates of some unknown value—the lowest, highest, and best estimates. To do calculations involving this uncertain figure, it is useful to use “fuzzy sets.” Traditional set theory includes the idea of exclusively being or not being a member of a set. For example, an animal is either alive or dead. However, for a number of sets, no clear membership can be determined. For example, is a person “old” if they are 55?

While no yes/no answer can be given, we can use fuzzy sets to determine the ranges of these types of values. Instead of the 1 or 0 used to indicate membership in traditional sets, fuzzy sets use a number between 0 and 1 to indicate partial membership in the set.

Fuzzy sets can also be used to establish boundaries around uncertain values. In addition to upper and lower values, these boundaries can include best estimates as well. It is a way to do sensitivity analysis when we have little information about the actual error sources and amounts. At its simplest all we need are the values we think it will be very unlikely to be greater or less than. These lower and upper bounds plus the best estimate make up a triangular number. A triangular number is written as [lowest expected value,  best estimate,  highest expected value].

For example, the number of mammalian species is given by the IUCN Red List folks as 4,629 species. However, this is known to be an estimate subject to error, which is usually quoted as ± 10%.

This range of estimates of the number of mammal species can be represented by a triangular fuzzy number. For the number of species, this is written as [4166,  4629,  5092], to indicate the lower and upper bounds, as well as the best estimate in the middle. Figure 1 shows the representation of the fuzzy number representing the count of all mammal species.

All the normal mathematical operations can be carried out using triangular numbers. The end result of the operation shows the most probable resultant value, along with the expected maximum and minimum values. For the procedures of addition, subtraction and multiplication, the low, best estimate, and high values are simply added, subtracted, or multiplied. Consider two triangular fuzzy numbers, triangular number

T1 = [L1,  B1,  H1]

and triangular number

T2 = [L2,  B2,  H2],

where “L”, “B”, and “H” are the lowest, best and highest estimates.  The rules are:

T1 + T2  =  [L1 + L2,  B1 + B2,  H1 + H2]

T1 – T2  =  [L1 – L2,  B1 – B2,  H1 – H2]  [Incorrect, edited. See below. Posting too fast. -w]

T1 * T2  =  [L1 * L2,  B1 * B2,  H1 * H2]

So that part is easy. For  subtraction and division, it’s a little different. The lowest possible value will be the low estimate in the numerator and the high estimate in the denominator, and vice versa for the highest possible value. So division is done as follows:

T1 / T2 = [L1 / H2,  B1 / B2,  L2 / H1]

And subtraction like this:

T1 – T2  =  [L1 – H2,  B1 – B2,  H1 – L2]

So how can we use triangular fuzzy numbers to see what the IPCC is doing?

Well, climate sensitivity (in °C per W/m2) up there in the IPCC magical formula is made up of two numbers—temperature change expected from a doubling of CO2, and increased forcing expected from a doubling of CO2 . For each of them, we have estimates of the likely range of values.

For the first number, the forcing from a doubling of CO2, the usual IPCC number says that this will give 3.7 W/m2 of additional forcing. The end ranges on that are likely about 3.5 for the lower value, and 4.1 for the upper value (Hansen 2005). This gives the triangular number [3.5,  3.7,  4.0] W/m2 for the forcing change from a doubling of CO2.

The second number, temperature change per doubling of CO2, is given by the IPCC http://news.bbc.co.uk/2/shared/bsp/hi/pdfs/02_02_07_climatereport.pdf as the triangular number [2.0,  3.0,  4.5] °C for the change in temperature from a doubling of CO2.

Dividing the sensitivity per doubling by the change in forcing per doubling gives us a value for the change in temperature (∆T, °C) from a given change in forcing (∆F, watts/metre squared). Again this is a triangular number, and by the rules for division it is:

T1 / T2 = [L1 / H2, B1 / B2, L2 / H1] = [2.0 / 4.0,   3.0 / 3.7,   4.5 / 3.5]

which is a climate sensitivity of [0.5,  0.8,  1.28] °C of temperature change for each W/m2 change in forcing. Note that as expected, the central value is the IPCC canonical value of 3°C per doubling of CO2.

Now, let’s see what this means in the real world. The IPCC is all on about the change in forcing since the “pre-industrial” times, which they take as 1750. For the amount of change in forcing since 1750, ∆F, the IPCC says http://news.bbc.co.uk/2/shared/bsp/hi/pdfs/02_02_07_climatereport.pdf there has been an increase of [0.6,  1.6,  2.4] watts per square metre in forcing.

Multiplying the triangular number for the change in forcing [0.6,  1.6,  2.4] W/m2 by the triangular number for sensitivity [0.5,  0.8,  1.28] °C per W/m-2 gives us the IPCC estimate for the change in temperature that we should have expected since 1750. Of course this is a triangular number as well, calculated as follows:

T1 * T2 = [L1 * L2, B1 * B2, H1 * H2] = [0.5 * 0.6,  0.8 * 1.6,  2.4 *1.28]

The final number, their estimate for the warming since 1750 predicted by their magic equation, is [0.3,  1.3,  3.1] °C of warming.

Let me say that another way, it’s important. For a quarter century now the AGW supporters have put millions of hours and millions of dollars into studies and computer models. In addition, the whole IPCC apparatus has creaked and groaned for fifteen years now, and that’s the best they can tell us for all of that money and all of the studies and all of the models?

The mountain has labored and concluded that since 1750, we should have seen a warming of somewhere between a third of a degree and three degrees … that’s some real impressive detective work there, Lou …

Seriously? That’s the best they can do, after thirty years of study? A warming between a third of a degree and three whole degrees? I cannot imagine a less falsifiable claim. Any warming will be easily encompassed by that interval. No matter what happens they can claim success. And that’s hindcasting, not even forecasting. Yikes!

I say again that the field of climate science took a wrong turn when they swallowed the unsupported claim that a hugely complex system like the climate has a linear relationship between change in input and change in operating conditions. The fundamental equation of the conventional paradigm, ∆T = λ ∆F, that basic claim that the change in temperature is a linear function of the change in forcing, is simply not true.

All the best,

w.