Curve fitting and the number of parameters
Guest essay by Antero Ollila
I have written blogs here in WUWT and represented some models of mine, which describe certain physical relationships of climate change. Every time I have received comments that if there is more than one parameter in my model, it has about no value, because using more than four parameters, any model can be adjusted to give wanted results. I think that this opinion originates from the quote of a famous Hungarian-American mathematician, physicist, and computer scientist John von Neumann (1903-1957). I found his statement in the Wikiquote:
“With four parameters I can fit an elephant, and with five I can make him wiggle his trunk”.
I do not know what von Neumann meant with his statement, but I think that this statement can be understood easily in the wrong way. I show an example in which this statement cannot be applied. It is a case of curve fitting. My example is about creating a mathematical relationship between the CO2 concentration and the radiative forcing change (RF). Myhre et al. have published equation (1) in 1988
RF = 5.35 * ln(C/280) (1)
where C is the CO2 concentration in ppm. Somebody might think that clever scientists have proved that this simple equation can be deduced by a pen and paper, but it is impossible. The data points RF versus CO2 concentrations have been calculated using the BBM (Broad Band Model) climate model and thereafter Eq. (1) has been calculated using a curve fitting procedure. There are no data points in the referred paper but only this equation and a graphical presentation. The BBM analysis method is the most inaccurate method of the radiative transfer schemes. The most accurate is the LBL method (Line-B-Line), which I have used in my calculations. Anyway Myhre et al. have shown in their paper that BBM and LBL methods give very closely same results.
I have shown in my paper that I could not reproduce Eq. (1) is my research study: https://wattsupwiththat.com/2017/03/17/on-the-reproducibility-of-the-ipccs-climate-sensitivity/
I carried out my calculations using the temperature, pressure, humidity, and GH gas concentration profiles of the average global atmosphere and the surface temperature of 15 ⁰C utilizing the LBL method. The first step is to calculate the outgoing longwave radiation (OLR) at the top of the atmosphere (TOA) using the CO2 concentration of 280 ppm. Then I increased the CO2 concentration to 393 ppm and calculated the outgoing LW radiation value. It happens in two steps: 1) transmittance or a radiation emitted by the surface and transferred directly to space, and 2) radiation absorbed by the atmosphere and then reradiated to space. The sum of these components shows that the OLR has decreased 1.03 W/m2 due to the increased absorption of the higher CO2 concentration. The other points have been calculated in the same way.
The values of the four data pairs of CO2 (ppm) and RF (W/m2) are: 280/0; 393/1.03; 560/2.165; 1370/5.04. Using a simple curve fitting procedure between the CO2 and the term ln(C/280), I got Eq. (2):
(2) RF = 3.12 * ln(C/280)
The form of equation is the same as in Eq. (1) but the coefficient is different. For this blog I carried out another fitting procedure using the polynomial of the third degree. The result is Eq. (3)
(3) RF = -3.743699 + 0.01690259*C – 1.38886*10-5*C2 + 4.548057*10-9*C3
The results of these fittings are plotted in Fig. 1.
As one can see, there is practically no difference between these fittings. The polynomial fitting is perfect, and the logarithmic fitting gives the coefficient of correlation 0.999888, which means that also mathematically the difference is insignificant. What we learned about this? The number of parameters has no role in the curve fitting, if the fitting is mathematically accurate enough. The logarithmic curve is simple, and it shows the nature of the RF dependency on the CO2 concentration very well. The actual question is this: Are the data points pairs calculated scientifically in the right way. The fitting procedure does not make a physical relationship susceptible and the number of parameters has no role.
So, I challenge those who think that an elephant can be described by a model with four parameters. I think that it is impossible even in two-dimensional world.
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