Section 9.4: Serial Correlation
171
superior for increasing density. Moreover, the test environment here is un~
representative in that the covariance structure of the data is known precisely.
In practice this is rarely the case, the covariance matrix must be estimated
with the practical result that the exact Hotelling-T 2 test is either ineffective or cannot be used at all here except for m = 2. 3 This implies that the
permutation procedure is a competitive alternative to the exact test for all
situations examined here.
Third, note the diminishing returns with increasing density of the network.
This illustrates the point made by Hasselmann (1979) that with large-scale
signals denser networks often add little non-redundant information ab out the
signal but a considerable amount of noise.
The powers of three exact scalar tests are also compared to the permutation
tests. These scalar tests look for signals in three pre-specified directions
respectively: the first EOF, the known signal J1.o and the maximum signal-tanoise ratio v. All utilize a test statistic that is normally distributed and all
require the use of the known covariance structure. They are related to several
of the tests covered in Chapter 8 which are alternatives in situations with
small n and spatial correlation. For further details refer to Zwiers (1987).
It is dear from the left half of Table 9.2 that all three scalar tests are at
least as powerful as the permutations tests in all situations examined, with
the exception of the first EOF test for the larger geographically expanded
networks. For these same cases analogous to the larger spatial domains the J1.o
and v tests are considerably superior to the permutation procedure. However,
as emphasized above, in practice covariance matrices must be estimated,
thereby requiring the use of different test statistics and less powerful tests.
Nor are signals known perfectly apriori in practice. For both ofthese reasons
the powers of the scalar tests shown in Table 9.2 are likely to be overestimates,
especially for the J1.o and v tests.
Overall, the results in Table 9.2 and the stated caveats suggest that for
~ less than about five (the range studied here) permutation methods are
viable test alternatives in situations with spatial correlation. Further , in the
absence of apriori knowledge of the signal to be detected they are practically
the only currently available choice for ~ less than about one.
9.4 Serial Correlation
The discussion ofserial correlation has been deferred to the end ofthis chapter
despite the fact that it is perhaps the most serious difficulty the analyst
faces in the application of permutation or Monte Carlo techniques. Without
attention to its effects it is completely debilitating to the effectiveness of
the procedures. If two sampies to be compared each have serial correlation
the process of resampling destroys this property of the data and leads to
3 With an estimated covariance matrix the Hotelling T2 test requires that n -1 = 9 > m
and is ineffective unless n :> m (see Chapter 8).
-
171
superior for increasing density. Moreover, the test environment here is un~
representative in that the covariance structure of the data is known precisely.
In practice this is rarely the case, the covariance matrix must be estimated
with the practical result that the exact Hotelling-T 2 test is either ineffective or cannot be used at all here except for m = 2. 3 This implies that the
permutation procedure is a competitive alternative to the exact test for all
situations examined here.
Third, note the diminishing returns with increasing density of the network.
This illustrates the point made by Hasselmann (1979) that with large-scale
signals denser networks often add little non-redundant information ab out the
signal but a considerable amount of noise.
The powers of three exact scalar tests are also compared to the permutation
tests. These scalar tests look for signals in three pre-specified directions
respectively: the first EOF, the known signal J1.o and the maximum signal-tanoise ratio v. All utilize a test statistic that is normally distributed and all
require the use of the known covariance structure. They are related to several
of the tests covered in Chapter 8 which are alternatives in situations with
small n and spatial correlation. For further details refer to Zwiers (1987).
It is dear from the left half of Table 9.2 that all three scalar tests are at
least as powerful as the permutations tests in all situations examined, with
the exception of the first EOF test for the larger geographically expanded
networks. For these same cases analogous to the larger spatial domains the J1.o
and v tests are considerably superior to the permutation procedure. However,
as emphasized above, in practice covariance matrices must be estimated,
thereby requiring the use of different test statistics and less powerful tests.
Nor are signals known perfectly apriori in practice. For both ofthese reasons
the powers of the scalar tests shown in Table 9.2 are likely to be overestimates,
especially for the J1.o and v tests.
Overall, the results in Table 9.2 and the stated caveats suggest that for
~ less than about five (the range studied here) permutation methods are
viable test alternatives in situations with spatial correlation. Further , in the
absence of apriori knowledge of the signal to be detected they are practically
the only currently available choice for ~ less than about one.
9.4 Serial Correlation
The discussion ofserial correlation has been deferred to the end ofthis chapter
despite the fact that it is perhaps the most serious difficulty the analyst
faces in the application of permutation or Monte Carlo techniques. Without
attention to its effects it is completely debilitating to the effectiveness of
the procedures. If two sampies to be compared each have serial correlation
the process of resampling destroys this property of the data and leads to
3 With an estimated covariance matrix the Hotelling T2 test requires that n -1 = 9 > m
and is ineffective unless n :> m (see Chapter 8).
-