Monday, December 21, 2009

Hockey Stick, Karhunen-Loeve, and If and If and If

The above is the now well quoted and often questioned hockey stick from Mann et al from Nature in 1998. Simply stated Mann et al made the following calculations:

1. They took data from a variety of sources that they contend reflects the temperature at some time.

2. These data sets then had an inferred T at some time t and at some location x.

3. Now each of these data sets had some variability due to several factors:

a. The inference of temperature from say tree ring size may not be one to one
b. the actual measurement may have an error
c. there may be other factors driving tree ring size
d. and on and on

4. They then used what is called the Karhunen-Loeve approach to obtain a single curve by weighting each of the multiple inferred curves. For example if they had ice cores, tree rings, and water height, then they could infer the temperature in say 1492 as:

T(1492)= a Tree (1492) + b Ice Core (1492) + c Water Height (1492)

This is quite simplified but one can get the idea. A good reference is the book by Fukunaga on Pattern Recognition.

5. Then they selectively smoothed the data and as with the Karhunen-Loeve approach they used the series with the largest "eigen values" or the series which would give the smallest statistical error.

So far so good. Yet there are many ifs here. For example take tree rings. As with the old Karhunen-Loeve approach which I know and have used extensively over the years, I also know a bit about growth in plants having spent the last twenty five years measuring and modelling that effect. You see there is not a one to one relationship between temperature and growth. Rain, soil conditions, insects, fires, drought, and the like play an equal if not more dominant effect. I have tried to find the underlying physiological models but they are still not well worked out yet. Thus if you rely upon a dendrologist to give you reasonable accuracy on tree ring width and annual temperature you are "shingling the roof in the fog" as I am fond to say.

The along comes McKitrick and McIntyre. They start the critiques by actually looking at the data. They state:

The database used by MBH98 contains the errors and defects listed below. We detail each of these points in this section, then in Section 3 we show how correcting these errors and defects affects the calculation of the Northern Hemisphere average temperature index using MBH98 methodology.

(a) unjustified truncation of 3 series;

(b) copying 1980 values from one series onto other series, resulting in incorrect values in at least 13 series;

(c) displacement of 18 series to one year earlier than apparently intended;

(d) unjustified extrapolations or interpolations to cover missing entries in 19 series;

(e) geographical mislocations and missing identifiers of location;

(f) inconsistent use of seasonal temperature data where annual data are available;

(g) obsolete data in at least 24 series, some of which may have been already obsolete at the time of the MBH98 calculations;

(h) listing of unused proxies;

(i) incorrect calculation of all 28 tree ring principal components.

And they conclude:

"The MBH98 hockey stick-shaped NH temperature reconstruction discussed here has been extremely influential in discussions of 20th century global warming. Together with a pre-1400 extension derived in Mann et. al. (1999) and a spliced instrumental temperature series, this index figured prominently in the IPCC Third Assessment Report (IPCC 2001) and numerous other publications. However, the dataset used to make this construction contained collation errors, unjustified truncation or extrapolation of source data, obsolete data, incorrect principal component calculations, geographical mislocations and other serious defects. These errors and defects substantially affect the temperature index."

Their work upon review has substantial merit. The work of Mann becomes highly questionable Mann makes the types of assumptions as:

If a tree ring can be accurately measured
If the ring width is a one to one mapping to the annual temperature in a year
If the other data can be similarly represented
If one can then select the sole single largest eigen value for the series

Then.....a hockey stick.

McKitrick then continues, and clearly starts to dismantle the Mann report as follows:

"The hockey stick debate is about two things. At a technical level it concerns a well-known study that characterized the state of the Earth’s climate over the past thousand years and seemed to prove a recent and unprecedented global warming. I will explain how the study got the results it did, examine some key flaws in the methodology and explain why the conclusions are unsupported by the data. At the political level the emerging debate is about whether the enormous international trust that has been placed in the IPCC was betrayed. The hockey stick story reveals that the IPCC allowed a deeply flawed study to dominate the Third Assessment Report, which suggests the possibility of bias in the Report-writing process. In view of the massive global influence of IPCC Reports, there is an urgent need to bias-proof future assessments in order to put climate policy onto a new foundation that will better serve the public interest."

Thus the whole basis of the Copenhagen meeting and the need for the Cap and Trade becomes problematic at best. So as my snow piles up before Christmas and my little squirrel friends hunker down from the cold, one may still question the Mann data and wonder what all the fuss is about.

I really can't wait till Spring!