One of the nifty things about Singular Spectrum Analysis is it allows you to decompose your data using basis functions (eigenvectors) which are adapted to the data record. Each eigen vecter/value pair is a direction in Hilbert space which minimizes the variance in that direction. You can look at the contribution from each of these modes individually. The top pane below shows the contribution of the 1st mode (called the trend) where the SSA window length (L=42) was adjusted for maximum amplitude difference between the first and second modes. Note the trend is not linear and in fact looks fairly sinusoidal. It has an apparent period (using the term loosely since it might not be periodic) of about 180 years with an inflection point somewhere around 1965.

Contribution from first Eigenmode

Contribution from first Eigenmode

The second pane shows the residual between the first mode and the data. I’ve eyeballed two linear trends on the top pane fit to the apparent residual trends over these intervals. Note the residual trend since 1970 is steeper than the earlier trend and is going against the mode 1 trend whose slope is decreasing during this period. Maybe the warmists are right to ring the alarm bells.

But watch what happens when we add in the second mode (k=1,2) This mode is very sinusoidal and has the 60 year period that others have found in many climate records.

Contribution from modes 1 and 2

Contribution from modes 1 and 2

Both trends disappear in the residual and the residual itself shows zero overall trend.

Discussion
One can argue about the physical source of these modes. Both eigenpairs are in rough quadrature which indicates significance and there is a clear step-wise separation from the noise floor which also is indicative of a noise free measurement(note the mode amplitudes in the bottom panes are in dB). It is also more than passing strange that the two modes are harmonically related as noted in my last post. But in reality, their source, whether internal to the climate system or an external forcing is irrelevant. They are what’s left of the climate signature once we’ve removed the extraneous noise.

These two modes account for about .7 degrees of warming and all of the trend of the past century. But suppose only the second mode is significant (and as I’ve mentioned, there is mounting peer-reviewed evidence that it is). The lack of stationarity in the first mode would only impact our ability to forecast, not the measured trend for the past century. As can be seen from the residual which contains no sign of an AGW signature, any such signature must be contained within the extracted mode(s), because their sum plus the residual recreates the data exactly. But the primary trend has been losing steam since 1965 or so while CO2 has risen exponentially during this interval. The mode 2 trend is nearly sinusoidal over the entire record. There is no anthropomorphic explanation for this characteristic. Therefore those who argue that some natural variation (a fancy term for “we don’t know”) has masked the AGW effect will have find it in the residual left from mode 1+2, because that’s the only garden left for them to dig.

On to part 3…

Advertisements