Ecological Archives A025-078-A2
Katherine J. Willis, Alistair W. R. Seddon, Peter R. Long, Elizabeth S. Jeffers, Neil Caithness, Milo Thurston, Mathijs G. D. Smit, Randi Hagemann, and Marc Macias-Fauria. 2015. Remote assessment of locally important ecological features across landscapes: how representative of reality? Ecological Applications 25:1290–1302. http://dx.doi.org/10.1890/14-1431.1
Appendix B. Sensitivity analysis of the discretization artefact in Fig. 6.
Sensitivity analysis of the discretization artefact in Fig. 6.
Discretization artefacts (or blockiness) are a common feature in discrete wavelet transforms due to the fact that the analysis method is blocky. That is, the decomposition in the discrete wavelet transform is done in discrete steps of 2n pixels (dyadic scales). Whereas a continuous wavelet transform would avoid of these, it would also compromise the independence of the series of maps at successively coarser scales given by the Field Forecast Verification (FFV), and thus the whole multivariate assessment of the agreement between two fields or maps (Briggs and Levine 1997).
The immediate interpretation of a discretization artefact on a given wavelet transform map is that that given area or block has a level of agreement with the wavelet that stands out (positively or negatively) from the other regions. So, in principle, the sharp edge of the area (as depicted in Fig. 6) is not necessarily a bad result. Nevertheless, the presence of an artefact has to be accounted for. In concrete, a sensitivity analysis should be undertaken that evaluates the degree of bias in the agreement statistics introduced by the presence of such feature.
Two approaches were taken in this case:
1. Use of other mother wavelets
The 'best' mother wavelet was selected amongst a library of mother wavelets according to entropy considerations (see Methods). In order to investigate the possibility of the artefact being due to the type of wavelet selected (i.e., algorithm-dependent), we transformed the map that resulted in the presence of a discretization artefact (LEFT β-diversity layer, Fig. 6) using different mother wavelets (increasing coefficients of Daubechies wavelets, along with the Haar wavelet; Daubechies 1992; Press et al. 1992). Results showed a consistent presence of the artefact independently of the type of mother wavelet used (Appendix C). Moreover, r and mse scores did not change with the type of mother wavelet (Appendix D). Hence, the artefact was not due to the type of mother wavelet and an algorithm-dependent bias in the analysis was discarded.
2. Analysis of the data with different row and column offsets
The presence of sea is interpreted as a zone of zeroes in FFV. These can potentially affect the wavelet transform, since the sea is interpreted as a periodic zone in the matrix. Although their effects are in principle thought to be small, they cannot be ignored (Briggs and Levine 1997). Offsetting the initial matrix with increasing numbers of zeroes on their sides can inform on whether the discretization artefact is due to the presence of a large area of zeros corresponding to the sea in the Mahamavo study area. Appendix C shows that the artefact disappears with increasing offset, and thus it confirms that the artefact is due, to a large degree, to the presence of a large area of zeros adjacent to the land where it occurs. This enables testing up to which degree the agreement statistics are distorted by the artefact, since we can compare them with and without offsets (Appendix D). Results show that the artefact has an influence on the agreement statistics, but that the absolute lower boundary of the agreement statistics without artefact is still high. In concrete, r is on average ~0.07–0.08 lower without the artefact (r >0.55), whereas there no noticeable changes are seen in mse (mse ~ 0.012; Appendix D). Moreover, the values remain consistent across scales (Appendix D; Fig. 4). We can thus conclude that there is a minor effect of the artefact on one of the two agreement statistics analyzed, which has been incorporated in the analysis (Fig. 4).
Literature cited
Briggs WM, Levine RA. 1997. Wavelets and Field Forecast Verification. Monthly Weather Review 125:1329–1341.
Daubechies I. 1992. Ten Lectures on Wavelets. SIAM. 357 pp.
Press WH, Teukolsky SA, Vetterling WT, Flannery BP. 1992. Numerical Recipes in C. Second edition. Cambridge University Press. 994 pp.