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9 Statistical Background
Pages 83-97

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From page 83...
... · The variability of proxy reconstructed temperatures will be less than the variability of the actual temperatures and may not reproduce the actual temperature pattern at particular timescales. · Examining the prediction of the reconstruction in a validation period is important, but the length of this period sets limits on a statistical appraisal of the uncertainty in the reconstruction.
From page 84...
... 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.
From page 85...
... The solid black line is the least-squares fitted line and the blue lines indicate 95 percent prediction intervals for temperature using this linear relationship. The dashed line and the red line indicate possible departures from a linear relationship between the proxy data and the temperature data.
From page 86...
... However, as in the single-proxy case, the prediction errors using several proxies will increase as the values deviate from those observed in the calibration period. Variability in the Regression Predictions Strictly speaking, assumption 1 posits a straight-line relationship between the average value of the climate variable, given the proxy, and the value of the proxy.
From page 87...
... For example, the error bounds in Figure 9-1 are based on statistical assumptions on how the temperature departs from an exact linear relationship. These assumptions can also be checked using the training and calibration periods, and often more complicated regression methods can be used to adjust for particular features in the data that violate the assumptions.
From page 88...
... However, geophysical data are often autocorrelated, which has the effect of reducing the effective sample size of the data. This reduction in sample size decreases the accuracy of the estimated regression coefficients and causes the standard error to be underestimated during the calibration period.
From page 89...
... The prediction mean squared error is the square of the standard error and is the sum of two terms. One is the variance of the errors in the regression equation, which is estimated from calibration data, and may be modified in the light of differences between the calibration errors and the validation errors.
From page 90...
... demonstrated that under some conditions the leading principal component can exhibit a spurious trendlike appearance, which could then lead to a spurious trend in the proxy-based reconstruction. To see how this can happen, suppose that instead of proxy climate data, one simply used a random sample of autocorrelated time series that did not contain a coherent signal.
From page 91...
... In this case, an argument can be made for using the variables without further normalization. However, the highervariance variables tend to make correspondingly higher contributions to the principal components, so the decision whether to equalize variances or not should be based on the scientific considerations of the climate information represented in each of the proxies.
From page 92...
... An inherent difficulty in validating a climate reconstruction is that the validation period is limited to the historical instrumental record, so it is not possible to obtain a direct estimate of the reconstruction skill at earlier periods. Because of the autocorrelation in most geophysical time series, a validation period adjacent to the calibration period cannot be truly independent; if the autocorrelation is short term, the lack of independence does not seriously bias the validation results.
From page 93...
... is the mean squared error of using the sample average temperature over the calibration period (a constant, yc ) to predict temperatures during the period of interest: MSE(yc )
From page 94...
... Distinguishing Between RE and CE and the Validation Period The combination of a high RE and a low CE or r2 means that the reconstruction identified the change in mean levels between the calibration and validation periods reasonably well but failed to track the variations within the validation period. One way that this discrepancy can occur is for the proxies and the temperatures to be related by a common trend in the calibration period.
From page 95...
... For example, the MSE calculated for the validation period provides a useful measure of the accuracy of the reconstruction; the square root of MSE can be used as an estimate of the reconstruction standard error. Reconstructions that have poor validation statistics (i.e., low CE)
From page 96...
... . This calculation will also provide a theoretical MSE for the validation period, which can be compared to the actual mean squared validation error as a check on the method.
From page 97...
... As an alternative, statistical methods exist for generating an ensemble of temperature reconstructions that can be interpreted in the more traditional way as a random sample. Although this requires additional statistical assumptions on the joint distribution of the proxies and temperatures, it simplifies the interpretation of the reconstruction.


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