1 Map overlay and spatial aggregation in sp Edzer Pebesma June 5, 2015 Abstract Numerical map overlay combines spatial features from one map layer with the attribute (numerical) properties of another. This vignette explains the R method over, which provides a consistent way to retrieve indices or attributes from a given spatial object (map layer) at the locations of another. Using this, the R generic aggregate is extended for spatial data, so that any spatial properties can be used to define an aggregation predicate, and any R function can be used as aggregation function. Contents 1 Introduction 1 2 Geometry overlays 2 3 Using over to extract attributes 5 4 Lines, and Polygon-Polygon overlays require rgeos 7 5 Aggregation 10 1 Introduction According to the free e-book by Davidson (2008), An overlay is a clear sheet of plastic or semi-transparent paper. It is used to display supplemental map and tactical information related to military operations. It is often used as a supplement to orders given in the field. Information is plotted on the overlay at the same scale as on the map, aerial photograph, or other graphic being used. Institute for Geoinformatics, University of Muenster, Weseler Strasse 253, Münster, Germany. 1
2 When the overlay is placed over the graphic, the details plotted on the overlay are shown in their true position. This suggests that map overlay is concerned with combining two, or possibly more, map layers by putting them on top of each other. This kind of overlay can be obtained in R e.g. by plotting one map layer, and plotting a second map layer on top of it. If the second one contains polygons, transparent colours can be used to avoid hiding of the first layer. When using the spplot command, the sp.layout argument can be used to combine multiple layers. O Sullivan and Unwin (2003) argue in chapter 10 (Putting maps together: map overlay) that map overlay has to do with the combination of two (or more) maps. They mainly focus on the combination of the selection criteria stemming from several map layers, e.g. finding the deciduous forest area that is less than 5 km from the nearest road. They call this boolean overlays. One could look at this problem as a polygon-polygon overlay, where we are looking for the intersection of the polygons with the deciduous forest with the polygons delineating the area less than 5 km from a road. Other possibilities are to represent one or both coverages as grid maps, and find the grid cells for which both criteria are valid (grid-grid overlay). A third possibility would be that one of the criteria is represented by a polygon, and the other by a grid (polygon-grid overlay, or grid-polygon overlay). In the end, as O Sullivan and Unwin argue, we can overlay any spatial type (points, lines, polygons, pixels/grids) with any other. In addition, we can address spatial attributes (as the case of grid data), or only the geometry (as in the case of the polygon-polygon intersection). This vignette will explain how the over method in package sp can be used to compute map overlays, meaning that instead of overlaying maps visually, the digital information that comes from combining two digital map layers is retrieved. From there, methods to aggregate (compute summary statistics; Heuvelink and Pebesma, 1999) over a spatial domain will be developed and demonstrated. Pebesma (2012) describes overlay and aggregation for spatiotemporal data. 2 Geometry overlays We will use the word geometry to denote the purely spatial characteristics, meaning that attributes (qualities, properties of something at a particular location) are ignored. With location we denote a point, line, polygon or grid cell. Section 3 will discuss how to retrieve and possibly aggregate or summarize attributes found there. Given two geometries, A and B, the following equivalent commands > A %over% B > over(a, B) retrieve the geometry (location) indices of B at the locations of A. More in particular, an integer vector of length length(a) is returned, with NA values for 2
3 locations in A not matching with locations in B (e.g. those points outside a set of polygons). Selecting points of A inside or on some geometry B (e.g. a set of polygons) B is done by > A[B,] which is short for > A[!is.na(over(A,B)),] We will now illustrate this with toy data created by > library(sp) > x = c(0.5, 0.5, 1.2, 1.5) > y = c(1.5, 0.5, 0.5, 0.5) > xy = cbind(x,y) > dimnames(xy)[] = c("a", "b", "c", "d") > pts = SpatialPoints(xy) > xpol = c(0,1,1,0,0) > ypol = c(0,0,1,1,0) > pol = SpatialPolygons(list( + Polygons(list(Polygon(cbind(xpol-1.05,ypol))), ID="x1"), + Polygons(list(Polygon(cbind(xpol,ypol))), ID="x2"), + Polygons(list(Polygon(cbind(xpol,ypol-1.05))), ID="x3"), + Polygons(list(Polygon(cbind(xpol+1.05,ypol))), ID="x4"), + Polygons(list(Polygon(cbind(xpol+.4, ypol+.1))), ID="x5") + )) and shown in figure 1. Now, the polygons pol in which points pts lie are > over(pts, pol) a b c d NA As points b and c fall in two overlapping polygons, we can retrieve the complete information as a list: > over(pts, pol, returnlist = TRUE) $a integer(0) $b  2 5 $c 3
4 x1 x2 x3 a b c d x5 x Figure 1: Toy data: points (a-d), and (overlapping) polygons (x1-x5)  4 5 $d  4 and the appropriate points falling in any of the polygons are selected by > pts[pol] SpatialPoints: x y b c d Coordinate Reference System (CRS) arguments: NA The reverse, identical sequence of commands for selecting polygons pol that have (one or more) points of pts in them is done by > over(pol, pts) 4
5 x1 x2 x3 x4 x5 NA 2 NA 3 2 > over(pol, pts, returnlist = TRUE) $x1 integer(0) $x2  2 $x3 integer(0) $x4  3 4 $x5  2 3 > row.names(pol[pts])  "x2" "x4" "x5" 3 Using over to extract attributes This section shows how over(x,y) is used to extract attribute values of argument y at locations of x. The return value is either an (aggregated) data frame, or a list. We now create an example SpatialPointsDataFrame and a SpatialPolygonsDataFrame using the toy data created earlier: > zdf = data.frame(z1 = 1:4, z2=4:1, f = c("a", "a", "b", "b"), + row.names = c("a", "b", "c", "d")) > zdf z1 z2 f a 1 4 a b 2 3 a c 3 2 b d 4 1 b > ptsdf = SpatialPointsDataFrame(pts, zdf) > zpl = data.frame(z = c(10, 15, 25, 3, 0), zz=1:5, + f = c("z", "q", "r", "z", "q"), row.names = c("x1", "x2", "x3", "x4", "x5")) > zpl 5
6 z zz f x z x q x r x4 3 4 z x5 0 5 q > poldf = SpatialPolygonsDataFrame(pol, zpl) In the simplest example > over(pts, poldf) z zz f a NA NA <NA> b 15 2 q c 3 4 z d 3 4 z a data.frame is created with each row corresponding to the first element of the poldf attributes at locations in pts. As an alternative, we can pass a user-defined function to process the table (selecting those columns to which the function makes sense): > over(pts, poldf[1:2], fn = mean) z zz a NA NA b c d To obtain the complete list of table entries at each point of pts, we use the returnlist argument: > over(pts, poldf, returnlist = TRUE) $a  z zz f <0 rows> (or 0-length row.names) $b z zz f x q x5 0 5 q $c z zz f 6
7 x4 3 x5 0 4 z 5 q $d z zz f x4 3 4 z The same actions can be done when the arguments are reversed: > over(pol, ptsdf) z1 z2 f x1 NA NA <NA> x2 2 3 a x3 NA NA <NA> x4 3 2 b x5 2 3 a > over(pol, ptsdf[1:2], fn = mean) z1 z2 x1 NA NA x x3 NA NA x x Lines, and Polygon-Polygon overlays require rgeos Package sp provides many of the over methods, but not all. Package rgeos can compute geometry intersections, i.e. for any set of (points, lines, polygons) to determine whether they have one ore more points in common. This means that the over methods provided by package sp can be completed by rgeos for any over methods where a SpatialLines object is involved (either as x or y), or where x and y are both of class SpatialPolygons (table 1). For this purpose, objects of class SpatialPixels or SpatialGrid are converted to SpatialPolygons. A toy example combines polygons with lines, created by > l1 = Lines(Line(coordinates(pts)), "L1") > l2 = Lines(Line(rbind(c(1,1.5), c(1.5,1.5))), "L2") > L = SpatialLines(list(l1,l2)) and shown in figure 2. The set of over operations on the polygons, lines and points is shown below (note that lists and vectors are named in this case): 7
8 y: Points y: Lines y: Polygons y: Pixels y: Grid x: Points s r s s s x: Lines r r r r:y r:y x: Polygons s r r s:y s:y x: Pixels s:x r:x s:x s:x s:x x: Grid s:x r:x s:x s:x s:x Table 1: over methods implemented for different x and y arguments. s: provided by sp; r: provided by rgeos. s:x or s:y indicates that the x or y argument is converted to grid cell center points; r:x or r:y indicate grids or pixels are converted to polygons. > library(rgeos) > over(pol, pol) x1 x2 x3 x4 x > over(pol, pol,returnlist = TRUE) $x1 x1 1 $x2 x2 x5 2 5 $x3 x3 3 $x4 x4 x5 4 5 $x5 x2 x4 x > over(pol, L) x1 x2 x3 x4 x5 NA 1 NA 1 1 > over(l, pol) 8
9 x1 x2 x3 x5 L1 x4 L Figure 2: Toy data: two lines and (overlapping) polygons (x1-x5) L1 L2 2 NA > over(l, pol, returnlist = TRUE) $L1 x2 x4 x $L2 named integer(0) > over(l, L) L1 L2 1 2 > over(pts, L) a b c d
10 > over(l, pts) L1 L2 1 NA Another example overlays a line with a grid, shown in figure 3. 5 Aggregation In the following example, the values of a fine grid with 40 m x 40 m cells are aggregated to a course grid with 400 m x 400 m cells. > data(meuse.grid) > gridded(meuse.grid) = ~x+y > off = gridparameters(meuse.grid)$cellcentre.offset + 20 > gt = GridTopology(off, c(400,400), c(8,11)) > SG = SpatialGrid(gt) > agg = aggregate(meuse.grid, SG) Figure 4 shows the result of this aggregation (agg, in colors) and the points (+) of the original grid (meuse.grid). Function aggregate aggregates its first argument over the geometries of the second argument, and returns a geometry with attributes. The default aggregation function (mean) can be overridden. An example of the aggregated values of meuse.grid along (or under) the line shown in Figure?? are > sl.agg = aggregate(meuse.grid[,1:3], sl) > class(sl.agg)  "SpatialLinesDataFrame" attr(,"package")  "sp" > as.data.frame(sl.agg) part.a part.b dist L Function aggregate returns a spatial object of the same class of sl (SpatialLines), and as.data.frame shows the attribute table as a data.frame. References ˆ O Sullivan, D., Unwin, D. (2003) Geographical Information Analysis. Wiley, NJ. ˆ Davidson, R., Reading topographic maps. Free e-book from: http: // 10
11 > data(meuse.grid) > gridded(meuse.grid) = ~x+y > Pt = list(x = c( , ), y = c( , )) > sl = SpatialLines(list(Lines(Line(cbind(Pt$x,Pt$y)), "L1"))) > image(meuse.grid) > xo = over(sl, geometry(meuse.grid), returnlist = TRUE) > image(meuse.grid[xo[], ],col=grey(0.5),add=t) > lines(sl) Figure 3: the line Overlay of line with grid, identifying cells crossed (or touched) by 11
12 Figure 4: aggregation over meuse.grid distance values to a 400 m x 400 m grid ˆ Heuvelink, G.B.M., and E.J. Pebesma, Spatial aggregation and soil process modelling. Geoderma 89, 1-2, ˆ Pebesma, E., Spatio-temporal overlay and aggregation. Package vignette for package spacetime, packages/spacetime/vignettes/sto.pdf 12
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