Not exact matches
Don't forget cycles in
autocorrelation do not necessarily
result from such nice sine wave looking cycles in the data, but do show repetitions of pattern.
Ammann and Wahl (2007) has argued that using
autocorrelation coefficients estimated from actual proxies
results in
The more linear (lower variance) the average output in two models, the higher the r2, purely as a
result of increased
autocorrelation.
Monthly, or even weekly or daily, data will get you more degrees of freedom for calculating statistical significance for short time periods, but unless you have an assured method of compensating for the
autocorrelations you could be obtaining misleading
results.
As per Papoulis (for me, 2nd edition, page 233), if a Gaussian process with
autocorrelation function R (t) is squared, the
resulting autocorrelation function, which I will call Q (t), is
These
results are why I question the use of monthly data with its
autocorrelations (that have to be corrected with methods such as Cochrane - Orcutt) when the annual data does not require corrections (that could be of uncertain validity — see Steve M remarks on C - O CIs versus those CIs derived using maximum liklihood approach).
Looking at these
results, that are admittedly anecdotal at this point, I see generally better fits to a normal distribution and lower
autocorrelation (AR1) in the residuals as one goes from monthly to individual months to annual data series and as one goes to sub periods of a long term temperature anomaly series.
However, we find the estimation of statistical significance ascribed to these
results to be in error: MS00 based this calculation on 12 - month smoothed data, from a calculation of the effective sample size (taking into account
autocorrelation effects).
Additionally, climatological field variables exhibit high spatial
autocorrelation,
resulting in an increased probability of committing a Type 1 error, or falsely rejecting the global null hypothesis (Wilks 2006b).
It's not due to the
autocorrelation structure being more complicated than the e-AR1 series, because the
results show interpolation skill improves as the noise becomes more structured.
Honestly, the p - values should be generated by constructing a Monte Carlo ensemble of model
results, per model, and looking at the actual distribution of (and variance of,
autocorrelation of, etc) the ensemble of outcomes where the outcomes ARE iid samples drawn from a distribution of model
results, and then use a correctly generated mean / sd to determinea p - value on the null hypothesis.
-LSB-...] In the end, the
autocorrelation issue turned out to be the least of the original paper's problems: Ryan O'Donnell, Micholas Lewis, Steve McIntyre and Jeff Condon (J. Climate April 2011, 24:2099 - 2115) have shown that the main
results of the paper are dependent on oversmoothing that
results from retaining too few principal components of the satellite covariance matrix.
Using a sixty - three year average on your data puts the
autocorrelation through the roof, making your
results statistically insignificant.
Using the SIC, which leads to no «lags» being used,
results in remaining
autocorrelation in the errors of the test equation.
In fact, a great many time series in geophysics exhibit
autocorrelation, which makes the
results of trend analysis less precise, sometimes greatly so.»
Results: The global justice model with
autocorrelations had the most satisfactory goodness - of - fit indices.