Section 2.4: Misleading Names
19
• The definition (2.7) of a decorrelation time makes sense when dealing with
the problem 0/ the mean 0/ n-consecutive serially correlated observations.
However, its abritrariness in defining a characteristic time scale becomes
obvious when we reformulate our problem by replacing the mean in (2.6)
by, for instance, the variance. Then, the characteristic time scale is
(Trenberth, 1984):
Thus characteristic time scales T depends markedly on the statistical
problem under consideration. These numbers are, in general, not physically defined numbers.
• For an AR(l)-process we have to distinguish between the physically
meaningful processes with positive memory (0' > 0) and the physically
meaningless processes with negative memory (0' < 0). If 0' > 0 then
formula (2.8) gives a time TD > Llt representative of the decay of the
auto-correlation function. Thus, in this case, TD may be seen as a physically useful time scale, namely a "persistence time scale" (but see the
dependency on the time step discussed below). If 0' < 0 then (2.8) returns times TD < Llt, even though prob ability statements for any two
states with an even time lag are identical to probabilities of an AR(p)
process with an AR-coefficient 10' 1.
Thus the number TD makes sense as a characteristic time scale when
dealing with red noise processes. But for many high er order AR(p)processes the number TD does not reflect a useful physical information.
• The Decorrelation Time depends on the time increment D.t: To demonstrate this dependency we consider again the AR(l)-process (2.3) with
a time increment of Llt = 1 and a 2: O. Then we may construct other
AR(l) processes with time increments k by noting that
(2.9)
with some noise term N~ which is a function of Nt . .. N t-k+l' The
decorrelation times TD ofthe two processes (2.3,2.9) are because of 0' < 1:
1+0'
TD 1 = -- . 1 > 1 and
I
1-0'
-
1 + O'k
TD k = -l-- k • k > k
I
_
0'
-
(2.10)
so that
(2.11)
19
• The definition (2.7) of a decorrelation time makes sense when dealing with
the problem 0/ the mean 0/ n-consecutive serially correlated observations.
However, its abritrariness in defining a characteristic time scale becomes
obvious when we reformulate our problem by replacing the mean in (2.6)
by, for instance, the variance. Then, the characteristic time scale is
(Trenberth, 1984):
Thus characteristic time scales T depends markedly on the statistical
problem under consideration. These numbers are, in general, not physically defined numbers.
• For an AR(l)-process we have to distinguish between the physically
meaningful processes with positive memory (0' > 0) and the physically
meaningless processes with negative memory (0' < 0). If 0' > 0 then
formula (2.8) gives a time TD > Llt representative of the decay of the
auto-correlation function. Thus, in this case, TD may be seen as a physically useful time scale, namely a "persistence time scale" (but see the
dependency on the time step discussed below). If 0' < 0 then (2.8) returns times TD < Llt, even though prob ability statements for any two
states with an even time lag are identical to probabilities of an AR(p)
process with an AR-coefficient 10' 1.
Thus the number TD makes sense as a characteristic time scale when
dealing with red noise processes. But for many high er order AR(p)processes the number TD does not reflect a useful physical information.
• The Decorrelation Time depends on the time increment D.t: To demonstrate this dependency we consider again the AR(l)-process (2.3) with
a time increment of Llt = 1 and a 2: O. Then we may construct other
AR(l) processes with time increments k by noting that
(2.9)
with some noise term N~ which is a function of Nt . .. N t-k+l' The
decorrelation times TD ofthe two processes (2.3,2.9) are because of 0' < 1:
1+0'
TD 1 = -- . 1 > 1 and
I
1-0'
-
1 + O'k
TD k = -l-- k • k > k
I
_
0'
-
(2.10)
so that
(2.11)