Briffa et al [Holocene 2002] describes a procedure for estimating confidence intervals that is applied in Jones et al [2001] as well. Bloomfield (pers. comm.) is cited as a source, although it’s inconceivable that he was aware of the Briffa truncation. I don’t see how you can claim precise confidence intervals when your post-1960 completely falls apart. But here is Briffa’s argument (any comments welcome; Im going to post up a puzzling excerpt from MBH99 as well):
Briffa et al [Holocene 2002]
Appendix: computing uncertainty ranges for reconstructions
The estimates of regional growing-season temperature constructed by simple linear regression (shown in Figures 11 and 12) include uncertainty ranges in terms of +- 1 and +- 2 standard errors about each estimated value. Each error range is based upon the uncertainty in the regression coefficients (the intercept, a, and the gradient, b) and the residual temperature variance (s^2) that is not captured by the regression-based reconstruction. The latter is typically the largest source of error in the reconstructions presented here, though the other terms can be non-negligible. The standard errors of the regression coefficients (SEa and SEb) are readily obtained (see, e.g., equations 1.4.5 and 1.4.2 of Draper and Smith, 1981). Given a time series of the predictor, x(t), then the predicted temperature is simply
yË’€ (t) = a + x(t) * b (A1)
where the Ë’€ indicates that this is a prediction of the actual temperature, y. When the residuals from the regression are not autocorrelated, then the standard error of this prediction, SEy, is given by:
[SEy(t)]^2 = s^2 + SEa^2+ [x(t) – mean(x)]^2 * SEb^2 (A2)
where mean(x) is the mean of the predictors over the calibration period.
The uncertainty ranges are in fact timescale-dependent, and therefore need to be computed for each timescale at which the results are presented. Equation A2 is valid for the raw reconstruction, but in general a reconstruction might be presented after applying a smoothing function, replacing the value at time t by àŽ⡨i) w[i]*y^[t-i] where the weights, w[i], sum to one. In Figures 11 and 12, we use a Gaussian-shaped smoothing operator. For the smoothed reconstruction, the standard error must be modified (Peter Bloomfield, personal communication) to give
[SEy(t)]^2 = s^2 * àŽ⡠w[i] ^2 + SEb^2 * { àŽ⡠ w[i] * (x[t-i]-mean(x)) }^2 (A3)
(comparing equations A2 and A3, note that the predictor anomaly is replaced by its smoothed value in the final term, but the more important change is that the first term is now scaled by the sum of squared weights, which is always 1 and the uncertainty range is increased. For the case of a smoothed reconstruction, the first term on the right-hand side of equation A3 is multiplied by the same factor.
Sir Humphrey and the Boehlert Questions
Yes, Minister was a wonderfully funny British comedy of a generation ago, (some info here and here or google yes minister humphrey) which featured Sir Humphrey Appleby as a senior civil servant (deputy minister) "managing" the Minister, his nominal boss, but who was always outwitted by Sir Humphrey. Each episode would start off with the Minister having some bright idea of instituting change, often involving a cut to civil service budgets. Sir Humphrey would resist and no budget would ever be cut. In a way, Sir Humphrey is descended from PG Wodehouse’s Jeeves. Here are some typical bon mots:
Keep that thought in mind as we see how Sir Humphrey (Cicerone) dealt with the Boehlert questions in classic Yes, Minister style. Continue reading →