Here, out-of-sample tests are used to test the robustness of the linear regression models of natural variation in global temperature. Previous models were developed on the whole data set. Here we develop them on partial data sets and examine how well they predict temperatures on the other part. These are also called independent tests.

The models that do well on the unseen data are in some sense more robust, reliable, and it gives you a feel for the constraints the data are placing on the models. You can see what conditions are needed to give certain results.

The results are placed in the animated gif above, where the blue temperatures are the out-of-sample values.

A number of models are shown.

Red – Linear
Green – Sinusoidals (21 and 63 year) and acceleration
Blue – Sinusoidals (21 and 63 year) and linear
Cyan – Sinusoidals (21, 63 and 1000)
Magenta – Sinusoidals (21 and 63 year) and exponential
Gray triangle – IPCC TAR climate model projections

As data is cut off from the front end, there is quite good accuracy in the cyan, magenta and green models, even back to 1950. That is, models incorporating natural variation, and either exponential, acceleration, or a long term (1000yr) sinusoidals predict temperature up to 2009 quite well. The linear, and linear+sigmoids are poor.

As data is cut off from the beginning, all the models hindcast reasonably well, except the linear and sigmoidals.

The models really struggle when you look only at the period since 1975, as they are clearly unconstrained. Models developed on this period are the only ones that drive the linear model into the IPCC projected gray zone.

In all reasonable cases, the models with sinusoids and non-linear terms project slightly higher temperatures than the linear models, but still well below the IPCC projected regions.

The conclusions would be:

1. While there is evidence of an acceleration in temperatures, the temperatures projected are lower than the lower limit of the IPCC projections.

2. These models, with acceleration and the ~20 and ~60 year natural variation, are very robust predictors of temperature, even up to 60 years in the future.

3. The confidence intervals of these predictors are far narrower than the corresponding CI’s of climate models.

These predictions are only based on the limited 150 year temperatures. As these natural variations are ‘quasi-periodic’ they can change in intensity quite suddenly, and different modes switch in and out.

It should be noted that the high rates of warming are only obtained in the absence of data before 1950 or 1975. It could be argued that only the recent warming is relevant to AGW. But given the fit and predictions of recent temperatures using the whole of the data, this argument does not hold water.

These analyses show that focusing on this portion of the record, in order to highlight high rates of warming, as often done by people such as Rahmstorf, could be regarded as an example of, to quote RealClimate, ‘an intellectually bankrupt nature of the scientific argument’.

These results support the conservative view, shown in many empirical studies, that when viewed in the proper context of natural variation, a large part of the recent increase in temperatures is natural, and the underlying rate of increase is lower than the lowest projection of the IPCC climate models.

Here is the figure showing observational estimates and IPCC projections to 2100 from Roy Spencer again.

The linear regression estimates fall right on the estimates of Spencer, Douglass, Forster and others.