Connecting Correlated GIS Predictors ina Bayes Network ModelAlix I Gitelman with Kathryn GeorgitisStatistics DepartmentOregon State UniversitySTARMAP, Colorado State UniversityJune, 2005 WNAR/IMS Meeting, Fairbanks AKAcknowledgementThis presentation was developed under STAR (Science to AchieveResults) Research Assistance Agreements CR-829095 and CR-829095awarded by the US Environmental Protection Agency (EPA) toColorado State University and Oregon State University, respectively.The presentation has not been formally reviewed by the EPA. Theviews expressed here are solely those the authors and respectiveprograms under these two agreements. The EPA does not endorse anyproducts or commercial services mentioned in this presentation.Thanks to Nick Danz for providing the data.Talk Outline- Concentric Circles Design- GIS Predictors ad libitum- An Example- Principal Components and Partial Least Squares- Bayes Networks/Graphical Models- Example, RevisitedHabitat AssociationQuestions:1. What landscape characteristics are associated with habitatselection?2. Are there different landscape scales associated with habitatselection?For question 2 Dugan et al. (2002) give a nice discussion of“phenomenon,” “sampling” and “analysis” scales.Concentric Circle Design x Buffer 1 Buffer 2 Buffer 3 Sample point (Bergin et al. 2000; Pearson and Niemi 2000; Hatten and Paradzick2003; Hostetler and Knowles-Yanez 2003; Martinez et al. 2003;Mayer and Cameron 2003; Holland 2004).Predictors ad libitumLandscape variables collected using GIS:• Deciding on classifications• Deciding on spatial scales (extents)• Deciding on pixel size• Deciding on aggregationsAn ExampleA breeding songbird survey conducted in the Western Great Lakesregion of Minnesota and Wisconsin.- Sampling units were forest stands larger than 40 acres ( 16 ha).- 10-minute unlimited radius point count at three subsamples perstand.- 4 circular buffers of different radii (i.e. different spatial extents):100m, 500m, 1000m, and 5000m.- Explanatory variables were derived from a land cover map:aspen-birch; conifer regeneration; hardwood regeneration; lowlandconifer; lowland hardwoods; lowland non-forested; northernhardwoods; pine and oak-pine; spruce-fir; upland non-forested.Predictors ad libitumThese land cover predictors are......correlated within buffers:pXi=1xij≤ 100%where xijis the percent of the ith land cover type within the jthbuffer...correlated across buffers:corr(xij, xik) 6= 0for some land cover types, i and some buffers, j 6= k....numerous (e.g., 10 per buffer).Principal ComponentsFor continuous (Normal) responses, Hwang and Nettleton (2003) givedata-driven (i.e., using the responses) methods for principalcomponents regression (PCR).Schaefer (1985) and others give biased PCR-based estimators in thelogisitic regression setting.Is this the best way to address the questions of interest?Principal ComponentsSome drawbacks:1. Interpretability2. Model selection issues3. Unless you’re lucky, these won’t address questions about scale.Partial Least SquaresDue to Wold (1975), these models seek to combine manifest andlatent variables, where the latent variables are intermediate betweenmanifest explanatory variables and manifest responses.Assumptions:- linear relationships between latent variables- linear relationships between manifest and latent variables- direction of correlations between manifest and latent variablesPartial Least SquaresSome drawbacks:1. Better as a predictive model2. Understanding the latent mechanisms3. Fairly similar to PCRBayes Networks/Graphical Models Land cover variables: Wildlife response Pearl 2000; Shipley 2000; Gitelman and Herlihy (in press)Graphical ModelsSome features:1. Take a holistic approach to modeling the ecological system.2. Reduce or eliminate multicollinearity problems by accounting fordependence among the “explanatory” variables.Some issues:1. Model selection: RJMCMC is a good but computationallyintensive method.2. Model evalutation: how to compare graphical models with moretraditional approaches?Specifying a Graphical ModelLet Xsijdenote the proportion of area in the jth buffer (j = 1, . . . , k)covered by the ith land cover type (i = 1, . . . , p) at sample point s,s = 1, . . . , n.For the first (innermost) buffer, let Zsij= XsijFor each successive buffer, j = 2, . . . , k, takeZsij= Xsij−r2j−1r2jXsi,j−1where rjis the radius of the jth buffer.Model (continued)So the Zsij’s are the proportions of the ith land cover types in the jthdonut around the sample point, s.Z0sj= (Zsij, . . . , Zspj) is a multivariate observation with theconstraint thatpXi=1Zsij≤ 1for all j and all s.Furthermore, as the buffer (donut) sizes increase, we ought not toexpect these proportions to remain constant.Indeed, it might be that the “patchiness” is an important habitatassociation consideration.Model (continued)Let Ysdenote the wildlife response at sample point s.The joint probability distribution of Ysand Zs1, . . . , Zskcan bewritten:f(Ys, Zs1, . . . , Zsk|φ) = f(Ys|Zs1, . . . , Zsk, φY)f(Zs1, . . . , Zsk|φZ),Where φYand φZdenote parameters corresponding to the conditionaldistributions of Ys|Zs1, . . . , Zskand Zs1, . . . , Zsk, respectively, whereφ= (φY,φZ).Using a graphical model approach, we can factor the joint distributionof the Zsj’s and eliminate some of them from the conditionaldistribution of Ys.Example RevisitedBetween buffer correlations (for some land cover types):Land Cover Type r(B1, B2) r(B1, B3) r(B2, B3)Aspen-Birch 0.73 0.62 0.93Conifer (Reg) 0.96 0.79 0.87Lowland Conifer 0.73 0.56 0.87Lowland Non-forest 0.51 0.27 0.33Pine/Oak-Pine 0.81 0.70 0.95Spruce-Fir 0.60 0.50 0.88These estimates are all based on n = 156.Example RevisitedWithin buffer 1 correlations:A-B C (R) LC LNf POP S-FA-B 1.00 0.02 -0.28 0.02 -0.46 -0.05C (R) 1.00 -0.05 0.08 -0.12 0.00LC 1.00 0.27 -0.23 -0.18LNf 1.00 -0.24 -0.03POP 1.00 -0.31S-F 1.00Similar results for buffers 2 and 3.Graphical Model: Part 1 AB2 AB1 AB3 PO3 PO2 LC1 PO1 LC2 LC3 CR1 CR2 CR3SF2 SF3 LN1 LN3 LN2