The use of the Mexican Hat and the Morlet wavelets for detection of ecological patterns

In this paper, we compare the relationship between scale and period in ecological pattern analysis and wavelet analysis. We also adapt a commonly used wavelet, the Morlet, to ecological pattern analysis. Using Monte Carlo assessments, we apply methods of statistical significance test to wavelet analysis for pattern analysis. In order to understand the inherent strength and weakness of the Morlet and the Mexican Hat wavelets, we also investigate and compare the properties of two frequently used wavelets by testing with field data and four artificial transects of different typical patterns which is often encountered in ecological research. It is shown that the Mexican Hat provides better detection and localization of patch and gap events over the Morlet, whereas the Morlet offers improved detection and localization of scale over the Mexican Hat. There is always a trade-off between the detection and localization of scale versus patch and gap events. Therefore, the best composite analysis is the combination of their advantages. The properties of wavelet in dealing with ecological data may be affected by characteristics intrinsic to wavelet itself. The peaks of different scales in isograms of wavelet power spectrum from the Mexican Hat may overlap with each other. Alternatively, these peaks of different scales in isograms of wavelet power spectrum may combine with each other unless the size of the analyzed scales is significantly different. These overlapping or combining lead to combining of peaks for different scales, or the masking of trough between peaks of different scales in the scalogram. Ecologists should combine all the information in scalogram and isograms of wavelet coefficient and wavelet power spectrum from different wavelets, which can provide us a broader view and precise pattern information.

[1]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[2]  Bai-lian Li,et al.  Wavelet Analysis of Coherent Structures at the Atmosphere-Forest Interface. , 1993 .

[3]  Sari C. Saunders,et al.  Hierarchical relationships between landscape structure and temperature in a managed forest landscape , 1998, Landscape Ecology.

[4]  C. Torrence,et al.  A Practical Guide to Wavelet Analysis. , 1998 .

[5]  Leslie A. Real,et al.  Monte Carlo assessments of goodness-of-fit for ecological simulation models , 2003 .

[6]  James D. Annan,et al.  Modelling under uncertainty: Monte Carlo methods for temporally varying parameters , 2001 .

[7]  Lonnie H. Hudgins,et al.  Wavelet transforms and atmopsheric turbulence. , 1993, Physical review letters.

[8]  M. Farge.  Wavelet Transforms and their Applications to Turbulence , 1992 .

[9]  M. Evans.  Statistical Distributions , 2000 .

[10]  E. Slezak,et al.  Identification of structures from galaxy counts: use of the wavelet transform , 1990 .

[11]  T. Spies,et al.  Characterizing canopy gap structure in forests using wavelet analysis , 1992 .

[12]  J. O'Brien,et al.  An Introduction to Wavelet Analysis in Oceanography and Meteorology: With Application to the Dispersion of Yanai Waves , 1993 .

[13]  Jessica Gurevitch,et al.  The Consequences of Spatial Structure for the Design and Analysis of Ecological Field Surveys , 2022 .

[14]  Jennifer L. Dungan,et al.  Illustrations and guidelines for selecting statistical methods for quantifying spatial pattern in ecological data , 2002 .

[15]  D. Labat,et al.  Rainfall–runoff relations for karstic springs: multifractal analyses , 2002 .

[16]  Truong Q. Nguyen,et al.  Wavelets and filter banks , 1996 .

[17]  G. A. Bradshaw.  Hierarchical analysis of spatial pattern and processes of Douglas-fir forests using wavelet analysis , 1991 .

[18]  M. B. Usher,et al.  Analysis of Pattern in Real and Artificial Plant Populations , 1975 .

[19]  David R. Williams,et al.  High-frequency oscillations in a solar active region coronal loop , 2001 .

[20]  D. Macisaac,et al.  New methods for the analysis of spatial pattern in vegetation , 1989 .

[21]  K. A. Harper,et al.  Structure and composition of riparian boreal forest: new methods for analyzing edge influence , 2001 .

[22]  S. Minobe.  Spatio-temporal structure of the pentadecadal variability over the North Pacific , 2000 .

[23]  Mark R. T. Dale,et al.  The use of wavelets for spatial pattern analysis in ecology , 1998 .

[24]  Jiquan Chen,et al.  Vegetation responses to landscape structure at multiple scales across a Northern Wisconsin, USA, pine barrens landscape , 1999, Plant Ecology.

[25]  Jay M. Ver Hoef,et al.  Spatial Analysis in Ecology , 2006 .

[26]  David L. Donoho,et al.  Nonlinear Wavelet Methods for Recovery of Signals, Densities, and Spectra from Indirect and Noisy Da , 1993 .

[27]  Paul S. Addison,et al.  LOW-OSCILLATION COMPLEX WAVELETS , 2002 .

[28]  D. Labat,et al.  Rainfall-runoff relations for karstic springs. Part II: Continuous wavelet and discrete orthogonal multiresolution analyses. , 2000 .

[29]  O. Bjørnstad,et al.  Travelling waves and spatial hierarchies in measles epidemics , 2001, Nature.