In his RealClimate post, West Antarctica Still Warming 2, Eric Steig discusses some of the criticisms made by Ryan O’donnell et al. (Improved methods for PCA-based reconstructions: case study using the Steig et al. 2009 Antarctic temperature reconstruction) of his 2009 Nature paper, Warming of the Antarctic ice-sheet surface since the 1957 International Geophysical Year.
Second, that in doing the analysis, we retain too few (just 3) EOF patterns. These are decompositions of the satellite field into its linearly independent spatial patterns. In general, the problem with retaining too many EOFs in this sort of calculation is that one’s ability to reconstruct high order spatial patterns is limited with a sparse data set, and in general it does not makes sense to retain more than the first few EOFs. O’Donnell et al. show, however, that we could safely have retained at least 5 (and perhaps more) EOFs, and that this is likely to give a more complete picture.
Some background may be required for the non-statistically oriented reader. A singular value decomposition allows one to take a set of numerical data sequences (usually arranged into a matrix array) and to decompose it into a new set of sequences (called Principal Components or Empirical Orthogonal Functions), a set of weights indicating the amount of variability of the original set accounted for by each of the PCs (called eigenvalues or singular values) and a set of coefficients which relate each PC to each of the original sequences. To reconstruct the original sequences, one can take each PC, multiply it by its eigenvalue and then use the coefficients to reproduce the original data.
This decomposition has certain properties which can be very useful in understanding and analyzing the original data. In particular, when there are strong relationships between the data sequences, several of the eigenvalues may be much larger than the rest. Using only the PCs which belong to those eigenvalues can create a very good replica of the data, but with fewer “moving parts”.
In the Steig paper, the authors divided the Antarctic into 5509 grid cells. They took a huge amount of satellite data and from it formed a monthly temperature sequence ( from January 1982 to December 2006 – 300 months) for each of the grid cells. The problem was to estimate the behavior of various regions of Antarctica during the longer period from January 1957 to the end of 2006 (a total of 600 months). Since the data before the satellite era was sparse both geographically and temporally, it was decided to try to “extend” the satellite data to the earlier period by first relating it to the ground temperatures that were available and then using that relationship to guess what the satellite temperatures might have been prior to 1982.
This is a good idea, but, as always, the devil is in the details. Using the totality of the available satellite sequences was unwieldy, both from a mathematical and statistical standpoint. This is where the decision to use a PC approach came in handy. The satellite temperature sequences have a great deal of relationship within their structure. For example, one would expect that geographically adjacent grid cells would have very similar behavior so it was apparent that this approach could reasonably produce something useful.
How many PCs should one use? This is the specific disagreement mentioned in the above quote from the RC post. Too many PCs mean a larger number of values to be estimated from the earlier station data (300 for each PC wherein the “overfitting” claim arises). On the other hand, too few PCs mean that the reconstruction will be unable to properly separate the spatial and temporal temperature patterns. The temperatures from the peninsula can be “smeared” to West Antarctica, there could be no ability to make separate examinations of the temperatures during the various seasons or any combination of these items.
One could argue that the graph from Odonnell et al. displayed in the RC post illustrates this difference:
Too few PCs will produce a monotone colored plot: With a single PC, every grid cell will have exactly the same characteristics since only a single multiplier and a single PC are available to reconstruct it. With two PCs, only two coefficients are available to differentiate the entire cell sequence from all of the others, etc. As more PCs are added, more variation in the coloring becomes possible. Whether this represents a greater reality or not becomes an issue.
What method should be used to “establish the relationship” between the satellites and the ground and to extend the sequences to the pre-1982 era? There are several of these that are available. They have different properties and it is important to understand what the drawbacks of each can be. The two mentioned in the RC post are TTLS (truncated total least squares – advocated by Mann and Steig) and ‘iridge’ (individual ridge regression – part of the methodology used by OD 2010). Discussion of these is a complicated matter which is beyond the discussion here.
The point of this post is to look at the specific 3 PCs which were used by the Steig paper. One can download the Antarctic reconstruction from the papers web site (warning –it is a VERY large text file) and using the R svd function decompose it using a singular value decomposition. Since the means of the sequences are not all zero, the “PCs” are not orthogonal, but it does not impact the point being made here. A plot of the 3 PCs used by Steig et al (oriented so that all trends are positive) produce the following:
The third PC looks somewhat different from the other two. The (extended) portion prior to 1982 is pretty close to being identically zero. Any reconstruction of the satellite data using these PCS becomes essentially a two PC reconstruction prior to 1982 and three PCs afterward. The end effect of the third PC on the overall results is to put a bend in the trends at that point (upward if the coefficient is positive and down if negative).
However, can this reconstruction differentiate well between the Antarctic regions in the early temperature record? I sincerely doubt it unless someone believes that the record is sufficiently homogeneous both spatially and temporally to justify that possibility. Perhaps, the authors of Steig et al. could explain this in more detail – I presume that they would have seen the same graph when they were writing the paper.
Why did this occur? My best guess is that it might be a result of using the total least squares function in the procedure.
I will give three more plots. Each of these is a plot of the relative size of the grid cell coefficient for each of the PCs when combining them to create the three PC satellite series. NOTE: These are NOT the values of the trends.
PC1 produces a general increasing trend throughout the continent. This trend is somewhat more pronounced in the central area.
PC2 is the main driver for the peninsula – West Antarctica relationship.
The main effects of this are felt after 1982 – it adds to the cooling lower right and the warming upper left.