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R for Macroecology

R for Macroecology. Spatial data continued. Projections. Cylindrical projections. Lambert CEA. Behrmann EA. Latitude of true scale = 30. Choosing a projection. What properties are important? Angles (conformal) Area (equal area) Distance from a point (equidistant)

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R for Macroecology

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  1. R for Macroecology Spatial data continued

  2. Projections • Cylindrical projections Lambert CEA

  3. Behrmann EA • Latitude of true scale = 30

  4. Choosing a projection • What properties are important? • Angles (conformal) • Area (equal area) • Distance from a point (equidistant) • Directions should be strait lines (gnomonic) • Minimize distortion • Cylindrical, conic, azimuthal http://www.geo.hunter.cuny.edu/~jochen/gtech201/lectures/lec6concepts/map%20coordinate%20systems/how%20to%20choose%20a%20projection.htm

  5. Projecting points • project() function in the proj4 package is good • Can also use project() from rgdal • spTransform()(in rgdal)works for SpatialPoints, SpatialLines, SpatialPolygons . . . • Can also handle transformations from one datum to another

  6. Projecting points > lat = rep(seq(-90,90,by = 5),(72+1)) > long = rep(seq(-180,180,by = 5),each = (36+1)) > xy= project(cbind(long,lat),"+proj=cea +datum=WGS84 +lat_ts=30") > par(mfrow= c(1,2)) > plot(long,lat) > plot(xy)

  7. Projecting a grid > mat = raster("MAT.tif") > mat = aggregate(mat,10) > bea= projectExtent(mat,"+proj=cea +datum=WGS84 +lat_ts=30") > mat class : RasterLayer dimensions : 289, 288, 83232 (nrow, ncol, ncell) resolution : 0.04166667, 0.04166667 (x, y) extent : 0, 12, 47.96667, 60.00833 (xmin, xmax, ymin, ymax) projection : +proj=longlat +ellps=WGS84 +datum=WGS84 +no_defs +towgs84=0,0,0 values : in memory min value : -22.88 max value : 113.56 >bea class : RasterLayer dimensions : 289, 288, 83232 (nrow, ncol, ncell) resolution : 4016.896, 3137.077 (x, y) extent : 0, 1156866, 5450663, 6357279 (xmin, xmax, ymin, ymax) projection : +proj=cea +datum=WGS84 +lat_ts=30 +ellps=WGS84 +towgs84=0,0,0 values : none

  8. Projecting a grid > bea = projectExtent(mat,"+proj=cea +datum=WGS84 +lat_ts=30") > res(bea) = xres(bea) > matBEA = projectRaster(mat,bea) > mat class : RasterLayer dimensions : 289, 288, 83232 (nrow, ncol, ncell) resolution : 0.04166667, 0.04166667 (x, y) extent : 0, 12, 47.96667, 60.00833 (xmin, xmax, ymin, ymax) projection : +proj=longlat +ellps=WGS84 +datum=WGS84 +no_defs +towgs84=0,0,0 values : in memory min value : -22.88 max value : 113.56 > matBEA class : RasterLayer dimensions : 169, 288, 48672 (nrow, ncol, ncell) resolution : 4638.312, 4638.312 (x, y) extent : 0, 1335834, 4721690, 5505565 (xmin, xmax, ymin, ymax) projection : +proj=cea +datum=WGS84 +ellps=WGS84 +towgs84=0,0,0 +lat_ts=30 values : in memory min value : -21.65266 max value : 113.3013

  9. How does it look?

  10. What happened? xyLL = project(cbind(x,y), "+proj=cea +datum=WGS84 +latts=30”,inverse = T) plot(xyLL,pch = ".") x = xFromCell(bea,1:ncell(bea)) y = yFromCell(bea,1:ncell(bea)) plot(x,y,pch = ".")

  11. What happened • Grid of points in lat-long (where each point corresponds with a BEA grid cell) • Sample original raster at those points (with interpolation) Different spacing in y direction Identical spacing in x direction

  12. What are the units? > matBEA class : RasterLayer dimensions : 169, 288, 48672 (nrow, ncol, ncell) resolution : 4638.312, 4638.312 (x, y) extent : 0, 1335834, 4721690, 5505565 (xmin, xmax, ymin, ymax) projection : +proj=cea +datum=WGS84 +ellps=WGS84 +towgs84=0,0,0 +lat_ts=30 values : in memory min value : -21.65266 max value : 113.3013 Meters, along the latitude of true scale (30N and 30S)

  13. A break to try things out • Projections

  14. Spatially structured data • Tobler’s first law of geography • “Everything is related to everything else, but near things are more related than distant things.” • Waldo Tobler • Nearby data points often have similar conditions • Understanding these patterns can provide insights into the data, and are critical for statistical tests

  15. Visualizing spatial structure • The correlogram • Often based on Moran’s I, a measure of spatial correlation Positive autocorrelation Negative autocorrelation

  16. Making Correlograms • First goal – get x, y and z vectors x = xFromCell(mat,1:ncell(mat)) y = yFromCell(mat,1:ncell(mat)) z = getValues(mat) assignColors = function(z) { z = (z-min(z,na.rm=T))/(max(z,na.rm=T)-min(z,na.rm=T)) color = rep(NA,length(z)) index = which(!is.na(z)) color[index] = rgb(z[index],0,(1-z[index])) return(color) } plot(x,y,col = assignColors(z),pch = 15, cex = 0.2)

  17. Pairwise distance matrix • Making a correlogram starts with a pairwise distance matrix! • (N data points requires ~ N2 calculations) • Big data sets need to be subsetted down > co = correlog(x,y,z,increment = 10,resamp = 0, latlon = T,na.rm=T) Error in outer(zscal, zscal) : allocMatrix: toomany elements specified

  18. Pairwise distance matrix • Making a correlogram starts with a pairwise distance matrix! • (N data points requires ~ N2 calculations) • Big data sets need to be subsetted down • sample() can help us do this > co = correlog(x,y,z,increment = 100,resamp = 0, latlon = T,na.rm=T) Error in outer(zscal, zscal) : allocMatrix: toomany elements specified > length(x) [1] 83232 > length(x)^2 [1] 6927565824

  19. Quick comments on sample() • sample() takes a vector and draws elements out of it > v = c("a","c","f","g","r") > sample(v,3) [1] "r""f""c" > sample(v,3,replace = T) [1] "c""a""a" > sample(v,6,replace = F) Error in sample(v, 6, replace = F) : cannot take a sample larger than the population when 'replace = FALSE‘ > sample(1:20,20) [1] 12 14 2 8 6 5 10 4 7 9 18 16 1 17 19 15 20 3 13 11

  20. Sampling • What’s wrong with this? • co = correlog( x[sample(1:length(x),1000,replace = F)], y[sample(1:length(y),1000,replace = F)], z[sample(1:length(z),1000,replace = F)], increment = 10, resamp = 0, latlon = T,na.rm=T)

  21. Sampling, the right way > index = sample(1:length(x),1000,replace = F) > co = correlog(x[index],y[index],z[index],increment = 100,resamp = 0, latlon = T,na.rm=T) > plot(co)

  22. Autocorrelation significance • Assessed through random permutation tests • Reassign z values randomly to x and y coordinates • Calculate correlogram • Repeat many times • Does the observed correlogram differ from random? > index = sample(1:length(x),1000,replace = F) > co = correlog(x[index],y[index],z[index],increment = 100,resamp = 100, latlon = T,na.rm=T) > plot(co) Downside – this is slow!

  23. Significant deviation from random

  24. Why is spatial dependence important? • Classic statistical tests assume that data points are independent • Can bias parameter estimates • Leads to incorrect P value estimates • A couple of methods to deal with this (next week!) • Simultaneous autoregressive models • Spatial filters

  25. Just for fun – 3d surfaces • R can render cool 3d surfaces • Demonstrate live • rgl.surface() (package rgl)

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