## 1. Introduction

[2] The term “hockey stick” is often used to describe the shape of the Northern Hemisphere (NH) mean temperature index introduced in *Mann et al.* [1998] (hereinafter referred to as MBH98). For convenience, we define the “hockey stick index” of a series as the difference between the mean of the closing sub-segment (here 1902–1980) and the mean of the entire series (typically 1400–1980 in this discussion) in units of the long-term standard deviation (σ), and a “hockey stick shaped” series is defined as one having a hockey stick index of at least 1 σ. Such series may be either upside-up (i.e., the “blade” trends upwards) or upside-down. Our focus here is on the 1400–1450 step (“AD1400 step”) of MBH98, because of controversy over early 15th century temperature reconstructions [*McIntyre and McKitrick*, 2003; M. E. Mann et al., Note on paper by McIntyre and McKitrick in *Energy and Environment*, unpublished manuscript, 2003, available at ftp://holocene.evsc.virginia.edu/pub/mann/EandEPaperProblem.pdf, hereinafter referred to as Mann et al., unpublished manuscript, 2003]. Our particular interest in the performance of the Reduction of Error (RE) statistic arises out of that controversy. We also focus on the North American tree ring network (“NOAMER”), because the first principal component (“PC1”) of this network has been identified as essential for controversial periods of the MBH98 temperature reconstruction [*Mann et al.*, 1999, unpublished manuscript, 2003]. MBH98 has recently been criticized on other grounds in *von Storch et al.* [2004].

[3] MBH98 used principal components (PCs) to reduce the dimensionality of tree ring networks and stated that they used “conventional” PC analysis. A conventional PC algorithm centers the data by subtracting the column means of the underlying series. For the AD1400 step highlighted here, this would be the full 1400–1980 interval. Instead, MBH98 Fortran code (ftp://holocene.evsc.virginia.edu/pub/MBH98/TREE/ITRDB/NOAMER/pca-noamer) contains an unusual data transformation prior to PC calculation that has never been reported in print. Each tree ring series was transformed by subtracting the 1902–1980 mean, then dividing by the 1902–1980 standard deviation and dividing again by the standard deviation of the residuals from fitting a linear trend in the 1902–1980 period. The PCs were then computed using singular value decomposition on the transformed data. (The effects reported here would have been partly mitigated if PCs had been calculated using the covariance or correlation matrix.) This previously unreported transformation was recently acknowledged in the Supplementary Information to a Corrigendum to MBH98 [*Mann et al.*, 2004], where they asserted that it has no effect on the results, a claim we refute herein.

[4] PCs can be strongly affected by linear transformations of the raw data. Under the MBH98 method, for those series in which the 1902–1980 mean is close to the 1400–1980 mean, subtraction of the 1902–1980 mean has little impact on weightings for the PC1. But if the 1902–1980 mean is different than the 1400–1980 mean (i.e., a hockey stick shape), the transformation translates the “shaft” off a zero mean; the magnitude of the residuals along the shaft is increased, and the series variance, which grows with the square of each residual, gets inflated. Since PC algorithms choose weights that maximize variance, the method re-allocates variance so that hockey stick shaped series get overweighted. In effect, the MBH98 data transformation results in the PC algorithm mining the data for hockey stick patterns.

[5] In a network of persistent red noise, there will be some series that randomly “trend” up or down during the ending sub-segment of the series (as well as other sub-segments). In the next section, we discuss a Monte Carlo experiment to show that these spurious “trends” in a closing segment are sufficient for the MBH98 method, when applied to a network of red noise, to yield hockey stick PC1s, even though the underlying data generating process has no trend component. We then examine the effect of this procedure on actual MBH98 weights for the North American PC1. Finally we use the simulated PC1s to establish benchmarks for the Reduction of Error (RE) verification statistic used by MBH98, and we discuss R^{2} and other verification statistics for the MBH98 reconstruction.