Is There a Wavelength on This Transect? Two dimensions suffice, one does not, and counting bands needs a window condition

Whether a spatial pattern has a selected characteristic wavelength is usually decided by looking for a peak in the power spectrum at a finite wavenumber. The operating characteristics of that step are rarely calibrated, and a string of downstream conclusions takes its output as an independent variable. This paper puts the commonly used statistics on one starting line within a single simulation. Six are compared in one dimension: the autocorrelation trough in two versions (search window told where the wavelength is, and search window not told), the unsmoothed periodogram peak ratio, the multitaper peak ratio, Fisher’s g, and the Whittle nested likelihood ratio of background against background-plus-peak. Three are compared in two dimensions: the radially averaged peak ratio, the two-dimensional spectral peak ratio, and the nested likelihood ratio on the radial average. Every threshold is the 95th percentile under a reference null of the same family; nominal level 0.05. There are five results. First, dimension matters far more than the choice of statistic. At a structure fraction of 0.10, the best statistic on a transect covering 20 wavelengths gives 0.436, whereas a two-dimensional window only 4 wavelengths a side gives 1.000. The same window still gives 0.996 when the fraction drops to 0.05. Second, cutting one transect out of a two-dimensional pattern makes the power collapse with the orientation of the stripes, and the transect itself carries no information about where on that curve it sits. With a window 8 wavelengths a side and a structure fraction of 0.20, the two-dimensional reading on the same fields is 1.000 at every angle, while the cut transect falls from 0.384 to 0.020 as the angle turns. Third, the ranking of statistics reverses between the two dimensions, and the dividing line is the test level. In one dimension the closest to nominal is the nested likelihood ratio, whose upper bound over 30 cross-family cells is 0.090, still 1.8 times nominal, while the blind version of the autocorrelation trough reaches 0.516. In two dimensions the closest is the two-dimensional spectral peak ratio at 0.170, while the radially averaged peak ratio reaches 0.736. Nominal level is 0.05 throughout, so no statistic truly holds its level when the background shape is unknown. Fourth, the number of bands cannot be counted in one dimension, and can be counted in two dimensions only under a window condition that can be written in closed form. Sixteen cells in one dimension top out at 0.240. In two dimensions the ratio of the window side to the wavelengths of the two bands must satisfy two conditions at once: the ratio of the two wavelengths must be near 3, and the longer band must lie inside the analysis band. A window 8 wavelengths a side fails the second; the values of 0.988 to 1.000 read off its ratio-3 column are an edge artefact, the fitted lower peak landing on the lower edge of the analysis band in 100 of 100 realisations. Enlarging the window to 16 wavelengths a side gives 0.997 to 1.000 in the same cells, with both fitted peaks within one bin of their true positions in 100 of 100 realisations; that reading is genuine. Fifth, two places where the author’s earlier registration does not agree with what is reported here; the source of each has been identified. See Section 4, reading two, and Section 9. The gap in the latter comes from how the random fields are generated: a generator that fixes the amplitude spectrum and randomises only the phase systematically overstates detection. From this follows a decision table indexed by four entry points, and one recommendation for downstream work: conclusions that take “whether a characteristic scale exists” as an independent variable should be executed on two-dimensional material, and conclusions that take “how many characteristic scales there are” as an independent variable cannot at present be executed at all.