A number of techniques have been developed within different scientific sub-disciplines, to help distinguish and isolate specific non-linear signals from either system noise or other valid, but unwanted, signals. One example is the technique of Cepstral analysis, used in the fields of speech analysis and in the performance testing of audio equipment (loud speakers).

The following diagram compares the same time series put through different analyses. The input data is the CET (Central England Temperature) series, a composite air temperature record dating from the mid 1600s. From Thomson*(2000) the solid black line shows the results of a Cepstral analysis, the dashed black line a multi-period trigonometric fit of the Cepstral analysis, and the red solid line the output from a MONACLE filter.

Thomson used the trigonometric fit (using basic solar cycles) to verify that the non-stationary in the CET records was not random but was “reasonably” systematic. If this is the case it implies that these interactions have specific causes and hence may be predicted.

Although many sophisticated analysis techniques exist their results are often hard to interpret, and even harder to relate to the environment. Like many of those techniques, MONACLE is also based on auto-correlation.

However, its advantages are that it is potentially

• simple to apply
• easier to interpret in terms of (system) persistence
• easier to relate to potential physical processes
• capable of tracking abrupt (system) changes

SIMULATIONS
In order to test how any filtering/analysis technique works, a standard test is to put known signals, such as sine waves, through the analysis. Two sine waves of wavelength 15 and 33 years were analysed with MONACLE using an 11-year auto-correlation window. The small inset plot shows the output from such a 1000 year analysis applied to the sum of two sine waves [green] and to the non-linear series [red]; and in the main figure, the non-linear series with 20% random noise added [blue]. Note that once noise has been added to this simulation the output or feedback can be negative. Also note that with the addition of random noise, no simulation can be reproduced identically.

MONACLE applied to -linear -non-linear -non-linear + noise

Experimenting with these simple simulations shows us that, for a given set of sine waves and correlation window, the greater the level of non-linearity the smaller the level of noise required to give negative feedback (vice-versa). So this type of filtering [MONACLE] can emphasise weak linearity in a system. The high and low outputs could also be viewed as periods of strong/weak environmental persistence.

Abrupt transitions to decade-long negative feedback states can also be simulated; for example, see below, with wavelengths of 11 and 57 years, 70% noise and a 9 year correlation window. Here negative feedback persists for up to 14 years and positive for typically 30-60 years.

MONACLE applied to: a + b + 0.9*ab + 0.7*noise

The correlation window, number of years over which the autocorrelation is performed, is effectively a third cycle in the simulation. A very short correlation window can produce noisy output. So, this type of filtering approach is best suited for marine systems that respond over several years (eg: fish) rather than over months (eg: zooplankton) or days (eg: phytoplankton).