As argued above, applying sliding window approaches or repeating analyses for varying analysis parameters may lead to patches of false positive significant points within the analysis results due to intrinsic correlations. To correct for this, an areawise significance test can be applied on top of the pointwise significance test. In the case of the areawise test for the wavelet spectrogram introduced in Maraun et al. (2007), a pointwise significant point is areawise significant if it lies within a patch of pointwise significant points that is larger than the reproducing kernel of the wavelet at the corresponding scale. Thus, patches that are smaller than the reproducing kernel of
4.2. Areawise significance tests the wavelet are identified as resulting from random fluctuations and are sorted out by the areawise test.
This idea of the areawise test can be generalised straightforwardly by replacing the concept of the wavelet reproducing kernel with the more universal concept of a decorrelation length of the intrinsic correlations. The main problem is to quantify the scale of decay of the intrinsic correlations, i. e., to estimate the decorrelation length.
We here put forward a numerical approach to do so consisting of the following steps:
• First, the domains in which the analysis and the pointwise significance test are repeatedly applied are identified (e. g., the time domain for windowed analyses).
• Second, for each domain, the parameters on which the intrinsic correlations depend, are identified.
• Third, a null model against which the areawise significance test is applied is chosen.
• Fourth, the scale of decay of the intrinsic correlations is numerically estimated by calculating the decorrelation lengths in each domain as a function of the parameters identified in the second step by using surrogate data sets created according to the null model specified in the previous step.
• Finally, the areawise significance test is performed as follows: For each pointwise significant point, get the decorrelation lengths corresponding to the analysis parameters at that point and check whether all neighbouring points within the range of the decorrelation lengths are also pointwise significant. The point is areawise significant if this is the case, otherwise it is not areawise significant.
To be precise, we denote the time series that we want to analyse as ⃗x(t) and identify the domains di, i = 1, . . . , n as the domains in which the analysis is repeatedly applied (e. g., the time domain for windowed analyses). In the following, we only consider the case n = 2 but a generalisation to more domains is straightforward.
The analysis results of the time series ⃗x(t) are stored in the matrix Q = (Qk1,k2) with k1 ∈ [1, N1] and k2 ∈ [1, N2] where N1,2 is the number of parameter values analysed in the corresponding domain. The vectors ⃗P(j)=(P1(j), . . . , PN(j)
j
)(j = 1, 2) contain the corresponding values of the analysis parameters. The results of the pointwise significance test of the analysis results are given in a binary matrix Spw with Skpw1,k2 = 1 if the result Qk1,k2 is pointwise significant and Skpw1,k2 = 0 if it is not pointwise significant.
Then, we identify the parameters pj, j = 1, . . . , Np, on which the intrinsic corre-lations depend. We here restrict ourselves to the case Np = 1 and denote p1 = p. Again, the test can easily be generalised to higher values of Np.
Before numerically assessing the intrinsic correlations, we need to choose a null model. From this null model, Ns sets of surrogate data are created and analysed over the range of analysis parameters ⃗P(1) and ⃗P(2). The decorrelation length can then be determined for each domain d by calculating correlations between the analysis
results for different analysis parameters and choosing the value of the parameter at which the correlations fall below 1/ e as a preliminary estimate of the decorrelation length ldi. By repeating this procedure for different values of the parameter p, we obtain the decorrelation lengths as a function of this parameter. Fitting the mean of the different realisations with respect to a linear model
ldi = mdip + ndi, (4.1)
gives the final estimate of the decorrelation length as a function of p in the corre-sponding domain di where the mdi denote the slope and the ndi the intercept of the linear fits. We stress that even though there is no particular reason to prefer a linear model over a more general fitting model, we here only consider this linear model for the sake of simplicity and, because, in many cases, it seems to capture the behaviour of the decorrelation lengths reasonably well (compare section 4.3).
A pointwise significant point in Q corresponding to Skpw1,k2 = 1is said to be areawise significant, i. e., has the entry Skaw1,k2 = 1 in the areawise significance matrix Saw, if
holds. That is, a point is areawise significant if all points within the rectangle of side lengths ld1− 1and ld2− 1centred at the point Qk1,k2 are pointwise significant.
We use this rectangular shape because it has the least implicit assumptions about the decay of correlations in the different domains. In fact, the rectangular shape corresponds to the case where the decays of correlations in d1 and d2 are mutually independent. If an analytical treatment of the decay of correlations is available, the information can in principle be used for determining an alternative shape to define the environment of neighbouring points that have to be pointwise significant for the considered point to be areawise significant.
For this type of areawise significance test, it is possible to derive confidence levels of the test by using surrogate data. For the confidence levels, we use the opposite point of view as Maraun et al. who first choose a confidence level 1 − αaw and then scale the reproducing kernel of the wavelet such that the number of pointwise significant points Apw and the number of areawise significant points Aaw are related by Aaw = αawApw (Maraun et al., 2007). Here, we estimate the decorrelation lengths as specified above for the chosen null model and then apply the areawise significance test to analysis results obtained from data corresponding to the null model. The average fraction of areawise and pointwise significant points in the surrogate results is then used to determine the areawise confidence level saw as
saw= 1 −
⟨Aaw Apw
⟩
. (4.3)