To continue our excursion into natural variation models of global temperature: What do they predict?

Here are a couple of different models fit with data up to the year 1990. This was in order to compare their projections with out-of-sample reality after 1990. The year 1990 is also the start of the major IPCC projections from the TAR WG1 available here.

The upper panel shows the entire HadCRUT global temperature in black up to 1990, the linear models are in red, while the IPCC projections are the grey triangle fanning out from 1990. Shown are two linear combinations, in red. The first is a regression containing linear and sinusoidal 21 and 63 year terms. The next higher red line contains a quadratic ‘acceleration’ term as well. The temperatures excluded from the model are in blue.

The bottom panel is a closeup of the period from 1980 to 2020. Which model predicted the temperatures the best? I would say that the middle one, containing quadratic and sinusoidal terms was a ‘remarkably’ good predictor of global temperatures. It remains to be seen if it stays that way, and the confidence intervals just have to be too narrow, although the dashed lines already represent a 99.9% level. (Yes I know, autocorrelation, I have yet to work that out). The region is so narrow because the significance of the terms is so high.

`Call:`

lm(formula = y~sin(2*pi*x/P[1])+cos(2* pi*x/P[1])+sin(2*pi*x/P[2])+cos(2*pi*x/P[2])+x+I(x^2))

```
```Residuals:

Min 1Q Median 3Q Max

-0.581099 -0.083164 0.003191 0.091657 0.514905

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 1.544e+02 9.292e+00 16.619 < 2e-16 ***

sin(2*pi*x/P[1]) -1.011e-01 5.074e-03 -19.921 < 2e-16 ***

cos(2*pi*x/P[1]) 3.839e-02 5.059e-03 7.588 5.35e-14 ***

sin(2 *pi*x/P[2]) 2.522e-02 4.727e-03 5.334 1.09e-07 ***

cos(2*pi*x/P[2]) -4.283e-02 4.802e-03 -8.920 < 2e-16 ***

x -1.647e-01 9.682e-03 -17.015 < 2e-16 ***

I(x^2) 4.383e-05 2.521e-06 17.383 < 2e-16 ***

---

Signif. codes: 0 â€˜***â€™ 0.001 â€˜**â€™ 0.01 â€˜*â€™ 0.05 â€˜.â€™ 0.1 â€˜ â€™ 1

`Residual standard error: 0.1371 on 1674 degrees of freedom`

Multiple R-squared: 0.5752, Adjusted R-squared: 0.5736

F-statistic: 377.7 on 6 and 1674 DF, p-value: < 2.2e-16

What is remarkable to me is how well the decadal features were projected, including the shape of the last 10 years or so.

The model with the acceleration term, potentially attributable to AGW, arrives at 2100 right outside the low end of the IPCC projection.

As you say, the Z&X paper has not been cited except by sceptics. There’s good reason – it’s wrong for the same reason as is your very similar analysis. It fits similar oscillatory functions, but for AGW uses a straight line from 1880.

Now the AGW effect claimed is not a uniform rise, but rather the prominent rise over the last few decades. The straight line just can’t fit that – if it did, there would be big errors further back. So it has to be fitted by the periodic functions. That sort of works, but leads to the false claim that the AGW effect is periodic and about to dive downwards. This comes because of an arbitrary function chosen to represent it.

So again I did a test. I did your fit with an added term, but where you added a quadratic, I added the function 1/(2040-t), t time in years AD. In the range up to the present, this actually better represents the usual understanding of AGW. This is reflected in generally better fit statistics, although the residual SS was just slightly more than for the quadratic. My figures weren’t exactly the same as yours – I think you must have smoothed the raw monthly data.

Anyway, here’s the result, shown in black with your fit (quadratic) shown in red. Up to now, the curves are very similar, and track the green Hadcrut3. But of course, the future is quite different, with the black curve heading for infinity in 2040.

Now you may say, that’s what happens when you fit a silly curve. But it agrees with the data better than the quadratic, which also has no justification as truly representing AGW. The fact is that none of this means anything. That is why the IPCC goes to a lot of trouble to develop scenarios for future GHG, rather than relying on arbitrary fitting function choices.

Nick,

“In the range up to the present, this actually better represents the usual understanding of AGW.”

and

“But of course, the future is quite different, with the black curve heading for infinity in 2040.”

Couldn’t agree with you more!!!

Thanks Nick, That is a good example of why a universe of models needs to be explored, and goodness of fit is not always a guide, particularly when it comes to extrapolation. No doubt there are a priori arguments that could be concocted for all of these choices, if the main interest is in the relative proportion of warming in a specific period like post-1950, and using the assumed forcing by CO2 is probably the strongest argument for it, though it cant be used for predictions, unless a scenario is used.

BTW if I am not mistaken the presumed trend component in EMD is actually a long period sinusoidal. This differs from SSH which is a moving average I believe. And of course, the objections you raise have not prevented these methods from being used in climate science (had to get that one in).

One issue that concerns me about the issue of how much warming since 1950 is from AGW is that it is mostly of rhetorical value. Despite the huge prominance given it in the SPM as so on, its probably of little research interest. What does it matter of the figure is 49% or 51% depending on your assumptions. What does the word “most” mean? Is it falsified by less than 50%, less than 80%, less than 90% or what. Does anybody care?

Nick, The R2 above was for up to 1990 so that is also less than the whole. Also, ‘silly curves’ can often be eliminated from the universe by out-of-sample testing, so it is not as arbitrary as you make out I believe.

The R2 (and other fit criteria) I mentioned were from a calc I did to 2008 (I just swapped my fn for the quad). As I say, my figures didn’t quite match yours, possibly because of smoothing.

As you say, the Z&X paper has not been cited except by sceptics. There's good reason – it's wrong for the same reason as is your very similar analysis. It fits similar oscillatory functions, but for AGW uses a straight line from 1880.Now the AGW effect claimed is not a uniform rise, but rather the prominent rise over the last few decades. The straight line just can't fit that – if it did, there would be big errors further back. So it has to be fitted by the periodic functions. That sort of works, but leads to the false claim that the AGW effect is periodic and about to dive downwards. This comes because of an arbitrary function chosen to represent it.So again I did a test. I did your fit with an added term, but where you added a quadratic, I added the function 1/(2040-t), t time in years AD. In the range up to the present, this actually better represents the usual understanding of AGW. This is reflected in generally better fit statistics, although the residual SS was just slightly more than for the quadratic. My figures weren't exactly the same as yours – I think you must have smoothed the raw monthly data.Anyway, here's the result, shown in black with your fit (quadratic) shown in red. Up to now, the curves are very similar, and track the green Hadcrut3. But of course, the future is quite different, with the black curve heading for infinity in 2040.Now you may say, that's what happens when you fit a silly curve. But it agrees with the data better than the quadratic, which also has no justification as truly representing AGW. The fact is that none of this means anything. That is why the IPCC goes to a lot of trouble to develop scenarios for future GHG, rather than relying on arbitrary fitting function choices.

Nick,”In the range up to the present, this actually better represents the usual understanding of AGW.”and”But of course, the future is quite different, with the black curve heading for infinity in 2040.”Couldn't agree with you more!!!

Thanks Nick, That is a good example of why a universe of models needs to be explored, and goodness of fit is not always a guide, particularly when it comes to extrapolation. No doubt there are a priori arguments that could be concocted for all of these choices, if the main interest is in the relative proportion of warming in a specific period like post-1950, and using the assumed forcing by CO2 is probably the strongest argument for it, though it cant be used for predictions, unless a scenario is used. BTW if I am not mistaken the presumed trend component in EMD is actually a long period sinusoidal. This differs from SSH which is a moving average I believe. And of course, the objections you raise have not prevented these methods from being used in climate science (had to get that one in).One issue that concerns me about the issue of how much warming since 1950 is from AGW is that it is mostly of rhetorical value. Despite the huge prominance given it in the SPM as so on, its probably of little research interest. What does it matter of the figure is 49% or 51% depending on your assumptions. What does the word “most” mean? Is it falsified by less than 50%, less than 80%, less than 90% or what. Does anybody care?

Nick, The R2 above was for up to 1990 so that is also less than the whole. Also, 'silly curves' can often be eliminated from the universe by out-of-sample testing, so it is not as arbitrary as you make out I believe.

The R2 (and other fit criteria) I mentioned were from a calc I did to 2008 (I just swapped my fn for the quad). As I say, my figures didn't quite match yours, possibly because of smoothing.

As you say, the Z&X paper has not been cited except by sceptics. There's good reason – it's wrong for the same reason as is your very similar analysis. It fits similar oscillatory functions, but for AGW uses a straight line from 1880.Now the AGW effect claimed is not a uniform rise, but rather the prominent rise over the last few decades. The straight line just can't fit that – if it did, there would be big errors further back. So it has to be fitted by the periodic functions. That sort of works, but leads to the false claim that the AGW effect is periodic and about to dive downwards. This comes because of an arbitrary function chosen to represent it.So again I did a test. I did your fit with an added term, but where you added a quadratic, I added the function 1/(2040-t), t time in years AD. In the range up to the present, this actually better represents the usual understanding of AGW. This is reflected in generally better fit statistics, although the residual SS was just slightly more than for the quadratic. My figures weren't exactly the same as yours – I think you must have smoothed the raw monthly data.Anyway, here's the result, shown in black with your fit (quadratic) shown in red. Up to now, the curves are very similar, and track the green Hadcrut3. But of course, the future is quite different, with the black curve heading for infinity in 2040.Now you may say, that's what happens when you fit a silly curve. But it agrees with the data better than the quadratic, which also has no justification as truly representing AGW. The fact is that none of this means anything. That is why the IPCC goes to a lot of trouble to develop scenarios for future GHG, rather than relying on arbitrary fitting function choices.

Nick,”In the range up to the present, this actually better represents the usual understanding of AGW.”and”But of course, the future is quite different, with the black curve heading for infinity in 2040.”Couldn't agree with you more!!!

Thanks Nick, That is a good example of why a universe of models needs to be explored, and goodness of fit is not always a guide, particularly when it comes to extrapolation. No doubt there are a priori arguments that could be concocted for all of these choices, if the main interest is in the relative proportion of warming in a specific period like post-1950, and using the assumed forcing by CO2 is probably the strongest argument for it, though it cant be used for predictions, unless a scenario is used. BTW if I am not mistaken the presumed trend component in EMD is actually a long period sinusoidal. This differs from SSH which is a moving average I believe. And of course, the objections you raise have not prevented these methods from being used in climate science (had to get that one in).One issue that concerns me about the issue of how much warming since 1950 is from AGW is that it is mostly of rhetorical value. Despite the huge prominance given it in the SPM as so on, its probably of little research interest. What does it matter of the figure is 49% or 51% depending on your assumptions. What does the word “most” mean? Is it falsified by less than 50%, less than 80%, less than 90% or what. Does anybody care?

Nick, The R2 above was for up to 1990 so that is also less than the whole. Also, 'silly curves' can often be eliminated from the universe by out-of-sample testing, so it is not as arbitrary as you make out I believe.

The R2 (and other fit criteria) I mentioned were from a calc I did to 2008 (I just swapped my fn for the quad). As I say, my figures didn't quite match yours, possibly because of smoothing.

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