proxies require statistical analyses. The first step is typically to separate the period of instrumental measurements into two segments: a calibration period and a validation period. The statistical relationship between proxy data (e.g., tree ring width or a multiproxy assemblage) and instrumental measurements of a climate variable (e.g., surface temperature) is determined over the calibration period. Past variations in the climate variable, including those during the validation period, are then reconstructed by using this statistical relationship to predict the variable from the proxy data. Before the proxy reconstruction is accepted as valid, the relationship between the reconstruction and the instrumental measurements during the validation period is examined to test the accuracy of the reconstruction. In a complete statistical analysis, the validation step should also include the calculation of measures of uncertainty, which gives an idea of the confidence one should place in the reconstructed record.

This chapter outlines and discusses some key elements of the statistical process described in the preceding paragraph and alluded to in other chapters of this report. Viewing the statistical analysis from a more fundamental level will help to clarify some of the methodologies used in surface temperature reconstruction and highlight the different types of uncertainties associated with these various methods. Resolving the numerous methodological differences and criticisms of proxy reconstruction is beyond the scope of this chapter, but we will address some key issues related to temporal correlation, the use of principal components, and the interpretation of validation statistics. As a concrete example, the chapter focuses on the Northern Hemisphere annual mean surface temperature reconstructed from annually resolved proxies such as tree rings. However, the basic principles can be generalized to other climate proxies and other meteorological variables. Spatially resolved reconstructions can also be reproduced using these methods, but a discussion of this application is not possible within the length of this chapter.

The most common form of proxy reconstruction depends on the use of a multivariate linear regression. This methodology requires two key assumptions:

1. Linearity: There is a linear statistical relationship between the proxies and the expected value of the climate variable.

2. Stationarity: The statistical relationship between the proxies and the climate variable is the same throughout the calibration period, validation period, and reconstruction period. Note that the stationarity of the relationship does not require stationarity of the series themselves, which would imply constant means, constant variances, and time-homogeneous correlations.

These two assumptions have precise mathematical formulations and address the two key questions concerning climate reconstructions: (1) How is the proxy related to the climate variable? (2) Is this relationship consistent across both the instrumental period and at earlier times? In statistical terminology, these assumptions comprise a statistical model because they define a statistical relationship among the data.