Statistical hypothesis testing in wavelet analysis: theoretical developments and applications to Indian rainfall
Statistical hypothesis tests in wavelet analysis are methods that
assess the degree to which a wavelet quantity (e.g., power and coherence)
exceeds background noise. Commonly, a point-wise approach is adopted in
which a wavelet quantity at every point in a wavelet spectrum is
individually compared to the critical level of the point-wise test. However,
because adjacent wavelet coefficients are correlated and wavelet spectra
often contain many wavelet quantities, the point-wise test can produce many
false positive results that occur in clusters or patches. To circumvent the
point-wise test drawbacks, it is necessary to implement the recently
developed area-wise, geometric, cumulative area-wise, and topological
significance tests, which are reviewed and developed in this paper. To
improve the computational efficiency of the cumulative area-wise test, a
simplified version of the testing procedure is created based on the idea
that its output is the mean of individual estimates of statistical
significance calculated from the geometric test applied at a set of
point-wise significance levels. Ideal examples are used to show that the
geometric and cumulative area-wise tests are unable to differentiate wavelet
spectral features arising from singularity-like structures from those
associated with periodicities. A cumulative arc-wise test is therefore
developed to strictly test for periodicities by using normalized arclength, which is defined as the number of points composing a cross section of a patch
divided by the wavelet scale in question. A previously proposed topological
significance test is formalized using persistent homology profiles (PHPs)
measuring the number of patches and holes corresponding to the set of all
point-wise significance values. Ideal examples show that the PHPs can be
used to distinguish time series containing signal components from those that
are purely noise. To demonstrate the practical uses of the existing and
newly developed statistical methodologies, a first comprehensive wavelet
analysis of Indian rainfall is also provided. An R software package has been
written by the author to implement the various testing procedures.
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