Modeling Bird Species Richness at Multiple Spatial Scales Using Two-Dimensional Wavelet Analysis

For. Sci. 61(1):1–16 http://dx.doi.org/10.5849/forsci.11-041 Copyright © 2015 Society of American Foresters FUNDAMENTAL RESEARCH biometrics Modeling Bird Species Richness at Multiple Spatial Scales Using Two-Dimensional Wavelet Analysis Spatial scale is one of the important issues for analyzing and modeling spatial patterns of species richness. In this study, two-dimensional discrete wavelet transforms were used to explore the effect of spatial scales on the relationship between bird species richness and a number of explanatory variables using the New York State Breeding Bird Atlas. The results indicated that the spatial distributions of species richness, as well as the relationships between species richness and environmental variables (i.e., climate and landscape variables), were significantly scale-dependent and nonconstant across different spatial scales. The results of two-dimensional wavelet regression showed that the predictive power of explanatory variables varied and shifted across different spatial scales. The climatic variables were more important than landscape variables for bird species richness, especially at large spatial scales. Further, summer average temperature was more important than summer average precipitation, and the variations in the climate variables (precipitation and temperature) were more important than their averages to account for the spatial distribution and variations of bird species richness. Our study demonstrated that two-dimensional wavelet transform and wavelet regression are promising tools for exploring the issue of spatial scales in forestry and ecological applications. Keywords: two-dimensional discrete wavelet transform, spatial scale, wavelet regression, wavelet variance, species distribution model, multiresolution analysis G eographical scale is one of the key issues in spatial analysis and modeling of the relationships between species richness and a suite of explanatory variables (Myers 1997). Studies have showed that the spatial patterns and processes of species richness are scale dependent (Godfray and Lawton 2001, Blackburn and Gaston 2002, Bond and Chase 2002), and the impact of environmental variables on the spatial processes and relationships also varies as the spatial scale changes (Rahbek and Graves 2001, Whittaker et al. 2001, Nogués-Bravo and Rahbek 2011). However, no single scale is absolutely right or wrong for analyzing and modeling the patterns of species richness (Blackburn and Gaston 2002). Shifting scales may lead to changes in the degree of and tests for spatial autocorrelation and heterogeneity (Dutilleul and Legendre 1993, Li and Reynolds 1995). Thus, multiresolution analysis has become more and more popular in recent years, and many methods have been developed and used to investigate the effects of varying spatial scales. For example, Garrigues et al. (2006) used intrablock and interblock spatial variability to quantify spatial heterogeneity at different sizes of blocks or spatial scales. Zhang et al. (2008, 2009) used spatial correlograms to explore the changes in Moran’s I of model residuals across different lag distances. Foody (2004) applied geo- graphically weighted regression (GWR) to investigate the scale dependence of the relationships between species richness and some explanatory variables. Although many methods have been explored, it is still a challenging problem to directly evaluate the relationships between variables at a specific spatial scale. In recent years, the wavelet technique has been shown to be an effective means to study spatial structure at different spatial scales and positions. Wavelets are mathematical functions with “small waves” and are scale and location specific. Two relevant characteristics of wavelets are oscillation and rapid decay to zero (hence “small”) (Nason 2008). Thus, wavelets can be used to decompose spatial data of both dependent and independent variables into orthogonal components operating at different spatial scales and then to directly detect scale-specific variations as well as explore scale-specific relationships in the wavelet transformed data. Wavelet transforms have been applied to identify gaps and patches (Bradshaw and Spies 1992), remove spatial autocorrelation (Carl and Kühn 2008), and analyze the scale- and location-dependent variations of soils (Lark and Webster 1999). Most of these ecological studies focus on using wavelets for time series analysis (e.g., Cazelles et al. 2008), and few use wavelet analysis for regularly Manuscript received April 18, 2011; accepted April 1, 2014; published online May 8, 2014. Affiliations: Zhihai Ma (), State University of New York College of Environmental Science and Forestry. Lianjun Zhang (), State University of New York College of Environmental Science and Forestry, Department of Forest and Natural Resource Management, Syracuse, NY. Acknowledgments: We thank the thousands of volunteers and regional coordinators who collected data for the 1980 and 2000 New York State Breeding Bird Atlases and Dr. Benjamin Zuckerberg of the Cornell Laboratory of Ornithology for providing the data and insights on the projects. We acknowledge the financial support from the McIntire-Stennis Cooperative Forestry Research Program (Project 1046505, SUNY-ESF). The authors appreciate the associate editor and two anonymous reviewers for their valuable suggestions and comments on the previous versions of the manuscript. Forest Science • February 2015 1 Zhihai Ma and Lianjun Zhang Father (left) and mother (right) wavelets: S8 is the Daubechies wavelet with a width of 8 and Haar wavelet. spaced spatial data (e.g., Bradshaw and Spies 1992, Mi et al. 2005, He et al. 2007). Keitt and Urban (2005) introduced one-dimensional (1D) wavelet coefficient regression to describe scale-specific patterns. They demonstrated that a linear regression between wavelet coefficients, extracted separately by spatial scales, can identify scale-specific relationships between a response variable and several predictor variables. They found that different environmental variables showed up to substantially different predictive powers at different scales. Carl and Kühn (2008) analyzed spatial ecological data using wavelet regression, but their objective was to remove the effects of autocorrelation in an attempt to predict the habitat of a plant species, rather than to use it to extract scale-specific relationships between the plant occurrence and environmental predictors. In practice, many data sets in ecology and wildlife studies are collected in two-dimensional (2D) space, whereas most applications of wavelet analysis are for 1D data. To our best knowledge to date, however, no study has been done to explore spatial variations and model species distributions at multiple spatial scales using 2D wavelet transforms. The major objective of this study was to explore the issue of spatial scale in modeling the spatial pattern of bird species richness in Ne (...truncated)