26
significance. The fact that even red noise can be formally decomposed into
quasi-periodic modes underscores the need for rigorous statistical significance testing in conjunction with the use of these methods.
3.2 Random variability in phase space
The distinctions between red and white noise discussed in section 2.3 are
easily extended to a multi-dimensional phase space. The autocorrelation
between successive points in an individual time series is analogous to the
coherence of the points along a trajectory in phase space. If the correspondence between successive points along the trajectory is so weak that
it is impossible to identify which point follows which, then that trajectory
is more appropriately viewed as 'white noise'. The decrease in temporal
variance with lengthening averaging time is analogous to the collapse of
the "cloud" of time averaged realizations in phase space toward the origin.
The increase in temporal variance with sampling interval is analogous to
the growth of the cloud of data points in phase-space.
The transitional frequency between red and white noise, if it exists, need
not be the same along all axes. If some variables (or PC's) are red out to
lower frequencies than other variables, increasingly strong time averaging
will tend, in addition to collapsing the cloud in all dimensions, to cause
what remains of the cloud to become increasingly elongated along the axes
of the "reddest" variables. In this situation time averaging has the effect
of reducing the dimensionality of the phase space.
4 Concluding Remarks
In view of the complexity of the climate system, with its large number
of degrees of freedom in the space and parameter domains, and its multiplicity of feedbacks, highly ordered structures and modes of evolution
are likely to be the exception, rather than the rule. It follows that the
more conservative or 'coarse' ways of conceptualizing and describing the
climate variability (i.e., those that require calculating less detailed information on the structure and evolution of climate anomalies) are likely to
be the more robust and informative. In those relatively rare instances in
which a more detailed, refined statistical description is warranted, the more
coarse description of the observations or model output can serve as a properly elaborated "null hypothesis" for use in establishing the strength and
significance. The fact that even red noise can be formally decomposed into
quasi-periodic modes underscores the need for rigorous statistical significance testing in conjunction with the use of these methods.
3.2 Random variability in phase space
The distinctions between red and white noise discussed in section 2.3 are
easily extended to a multi-dimensional phase space. The autocorrelation
between successive points in an individual time series is analogous to the
coherence of the points along a trajectory in phase space. If the correspondence between successive points along the trajectory is so weak that
it is impossible to identify which point follows which, then that trajectory
is more appropriately viewed as 'white noise'. The decrease in temporal
variance with lengthening averaging time is analogous to the collapse of
the "cloud" of time averaged realizations in phase space toward the origin.
The increase in temporal variance with sampling interval is analogous to
the growth of the cloud of data points in phase-space.
The transitional frequency between red and white noise, if it exists, need
not be the same along all axes. If some variables (or PC's) are red out to
lower frequencies than other variables, increasingly strong time averaging
will tend, in addition to collapsing the cloud in all dimensions, to cause
what remains of the cloud to become increasingly elongated along the axes
of the "reddest" variables. In this situation time averaging has the effect
of reducing the dimensionality of the phase space.
4 Concluding Remarks
In view of the complexity of the climate system, with its large number
of degrees of freedom in the space and parameter domains, and its multiplicity of feedbacks, highly ordered structures and modes of evolution
are likely to be the exception, rather than the rule. It follows that the
more conservative or 'coarse' ways of conceptualizing and describing the
climate variability (i.e., those that require calculating less detailed information on the structure and evolution of climate anomalies) are likely to
be the more robust and informative. In those relatively rare instances in
which a more detailed, refined statistical description is warranted, the more
coarse description of the observations or model output can serve as a properly elaborated "null hypothesis" for use in establishing the strength and