Some Effects of Finite Sample Size and Persistence on Meteorological Statistics.Part II: Potential Predictability

Kevin E. Trenberth National Center for Atmospheric Research, Boulder, CO 80307

Search for other papers by Kevin E. Trenberth in
Current site
Google Scholar
PubMed
Close
Full access

Abstract

Potential predictability of a meteorological time series can be estimated from the ratio of the actual interannual variability to the natural variability associated with climatic noise. The extent to which this ratio is larger than one is taken as a measure of the climatic signal-to-noise variance ratio. However, there are major problems in separating out the signal from the noise which are compounded by persistence in the time series, the presence of an annual cycle and the effects of finite sample size.

An F test may be used to deduce that a signal is present by rejecting the null hypothesis of no signal. However, it is shown that, generally, the null hypothesis should not be accepted just because it cannot be rejected. Confidence limits can be very large when a signal is present in a finite sample since the signal will be fictitiously correlated with the noise due to sampling in a manner that is unknown. Although the correlation coefficient is statistically not significant, there is a large impact on the confidence limits of the F ratio. A case is presented using many artificially generated, and thus known, time series that include a signalcombined with a first order autoregressive process (red noise). With a signal-to-noise ratio of 1 (<F) = 2) only about half of the time could the null hypothesis be rejected at the 5% level and 5% of the time the sample F ratio was found to be less than I.

The problem of assessing potential predictability is analyzed in detail in the time domain. The presence of any low frequency signal can lead to overestimates of the level of climatic noise since the signal will add to the persistence of the time series. A method is devised to adjust the statistics and remove the effects of the signal and thus obtain a more accurate signal-to-noise ratio.

The results are compared with the alternative approach of Jones and of Madden in the frequency domain. The latter uses a low frequency white noise extension of the power spectrum to estimate the climatic noise. The method is shown to work quite well, with minimal impact of a low frequency signal on results, but the confidence limits were very large. It is necessary to greatly increase the degrees of freedom of the spectral estimates before the results can be considered reliable; even then they are still subject to the inherent uncertainty associated with the unknown correlation between signal and noise.

Abstract

Potential predictability of a meteorological time series can be estimated from the ratio of the actual interannual variability to the natural variability associated with climatic noise. The extent to which this ratio is larger than one is taken as a measure of the climatic signal-to-noise variance ratio. However, there are major problems in separating out the signal from the noise which are compounded by persistence in the time series, the presence of an annual cycle and the effects of finite sample size.

An F test may be used to deduce that a signal is present by rejecting the null hypothesis of no signal. However, it is shown that, generally, the null hypothesis should not be accepted just because it cannot be rejected. Confidence limits can be very large when a signal is present in a finite sample since the signal will be fictitiously correlated with the noise due to sampling in a manner that is unknown. Although the correlation coefficient is statistically not significant, there is a large impact on the confidence limits of the F ratio. A case is presented using many artificially generated, and thus known, time series that include a signalcombined with a first order autoregressive process (red noise). With a signal-to-noise ratio of 1 (<F) = 2) only about half of the time could the null hypothesis be rejected at the 5% level and 5% of the time the sample F ratio was found to be less than I.

The problem of assessing potential predictability is analyzed in detail in the time domain. The presence of any low frequency signal can lead to overestimates of the level of climatic noise since the signal will add to the persistence of the time series. A method is devised to adjust the statistics and remove the effects of the signal and thus obtain a more accurate signal-to-noise ratio.

The results are compared with the alternative approach of Jones and of Madden in the frequency domain. The latter uses a low frequency white noise extension of the power spectrum to estimate the climatic noise. The method is shown to work quite well, with minimal impact of a low frequency signal on results, but the confidence limits were very large. It is necessary to greatly increase the degrees of freedom of the spectral estimates before the results can be considered reliable; even then they are still subject to the inherent uncertainty associated with the unknown correlation between signal and noise.

Save