Statistical hypothesis testing in wavelet analysis: theoretical developments and applications to Indian rainfall

Abstract

Abstract. 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.

Short communication: On the use of the Benjamini–Hochberg procedure in wavelet analysis and its inferior performance to the cumulative arc‐wise and area‐wise tests

“…Except for the arc‐wise test, all tests reveal a second region of statistical significance at a period of 256 months around 1970. Because this feature is not arc‐wise significant, it may be associated with a relatively transient and abrupt fluctuation rather than a sustained periodicity Schulte ().…”

“…Likewise, the cumulative area‐wise and arc‐wise tests were applied at the 5% arc‐wise ( α arc = .05) and area‐wise ( α aw = .05) significance level, respectively. The implementation of the cumulative arc‐wise and area‐wise involved the selection of a set of point‐wise significance levels (Schulte, ). In this study, the point‐wise significance levels were incrementally varied from .02 to .18 by .02.…”

“…In this study, the point‐wise significance levels were incrementally varied from .02 to .18 by .02. This selection was based on previous work showing how the tests perform well with these specific point‐wise significance levels Schulte (). The reader is referred to Schulte () for more details about the testing procedures.…”

“…Often the application of wavelet analysis involves the estimation of a sample wavelet spectrum and the subsequent comparison of the sample wavelet spectrum to a background noise spectrum (Torrence and Compo, 1998). For this application, it is necessary to implement statistical hypothesis tests, such as the point-wise (Torrence and Compo, 1998), area-wise (Maraun et al, 2007), geometric (Schulte et al, 2015), cumulative area-wise and arc-wise (Schulte, 2016;Schulte, 2019), and topological (Schulte et al, 2015;Schulte, 2019) tests. For the point-wise test, the statistical significance of wavelet quantities is individually assessed without considering the correlation structure among wavelet coefficients arising from the reproducing kernel of the analysing wavelet (Maraun et al, 2007).…”

“…For the point-wise test, the statistical significance of wavelet quantities is individually assessed without considering the correlation structure among wavelet coefficients arising from the reproducing kernel of the analysing wavelet (Maraun et al, 2007). In contrast, all the other tests account for the correlation structure by evaluating the statistical significance of contiguous regions of point-wise significance based on their geometric or topological properties such as area or connectedness (Schulte, 2019). By considering the correlation structure, the more sophisticated tests reduce the number of spurious results arising from the naive point-wise test.…”