Measures of Reconstruction Accuracy
Let yt denote a temperature at time t and ŷt the prediction based on a proxy reconstruction.
Mean squared error (MSE)
where the sum on the right-hand side of the equation is over times of interest (either the calibration or validation period) and N is the number of time points.
Reduction of error statistic (RE)
where is the mean squared error of using the sample average temperature over the calibration period (a constant, ) to predict temperatures during the period of interest:
Coefficient of efficiency (CE)
where is the mean squared error of using the sample average over the period of interest as a predictor of temperatures during the period of interest:
Squared correlation (r2 )
If ŷt are the predictions from a linear regression of yt on the proxies, and the period of interest is the calibration period, then RE, CE, and r2 are all equal. Otherwise, CE is less than both RE and r2.
value, however, will always have a high r2, and this is another justification for considering the CE.
Figure 9-3 gives some examples of a hypothetical temperature series and several reconstruction series, where the black line is the actual temperatures and the colored lines are various reconstructions. The red line has an r2 of 1 but a CE of –18.9 and is