estimate densities from point patterns

content/tutorials/spatial_point_pattern/index.Rmd

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+---
+title: Estimating densities from a point pattern
+maintainer: Thierry Onkelinx
+date: '2017-06-28'
+output: 
+  html_document: default
+  md_document:
+    preserve_yaml: true
+    variant: gfm+footnotes
+params:
+  cellsize: 20
+---
+
+In this example we focus on a set of 10450 coordinates in a small area.
+The goal is to estimate the local density of points, expressed as the number of points per unit area.
+The raw coordinates are given in [WGS84 (EPSG:4326)](https://epsg.io/4326), which is a geodetic coordinate reference system.
+That is not suited for calculating distances, so we need to re-project the points into a local projected coordinate reference system.
+In this case we use [Belgian Lambert72 (EPSG:3170)](https://epsg.io/31370). 
+Next we calculate the density.
+To visualise the density, we have to transform the results back to WGS84. 
+
+The data used in this example is real data but centred to a different location for privacy reasons. The dataset is available on [GitHub](https://github.com/ThierryO/my_blog/tree/master/data/20170628).
+
+First we must read the data into R. Plotting the raw data helps to check errors in the data.
+
+```{r read-data, fig.cap = "Raw data"}
+points <- read.delim("point-pattern/points.txt", sep = " ")
+library(ggmap)
+map <- get_map(
+  location = c(lon = mean(points$lon), lat = mean(points$lat)),
+  zoom = 17,
+  maptype = "satellite",
+  source = "google"
+)
+ggmap(map) +
+  geom_point(data = points, alpha = 0.1, colour = "blue", shape = 4)
+```
+
+The next step is to convert the dataset to a `SpatialPoints` object with WGS84 projection and re-project it to Belgian Lambert72.
+`sp::CRS()` defines the coordinate reference systems (CRS).
+`sp::coordinates()<-` is an easy way to convert a `data.frame` into a `SpatialPointsDataFrame`, but without specifying a CRS.
+Therefore we need to override the CRS with the correct one.
+
+`sp` will derive a 'WKT2 string' representation from an EPSG-code [^epsg] that we provide, in order to represent the CRS.
+The WKT2 string (well known text) is a recent open standard by the Open Geospatial Consortium to represent a CRS, and it replaces the older (deprecated) 'PROJ.4 string'.
+Currently, the function to set the CRS is still called `proj4string()`.
+`sp::spTransform()` converts the spatial object from the current CRS to another CRS.
+
+[^epsg]: Most coordinate reference systems have an [EPSG](https://en.wikipedia.org/wiki/International_Association_of_Oil_%26_Gas_Producers#European_Petroleum_Survey_Group) code which you can find at http://epsg.io/.
+
+```{r crs-objects}
+library(sp)
+crs_wgs84 <- CRS(SRS_string = "EPSG:4326")
+crs_lambert <- CRS(SRS_string = "EPSG:31370")
+```
+
+The warning above once again demonstrates that some PROJ.4 information is not supported anymore.
+
+```{r reproject}
+coordinates(points) <- ~lon + lat
+# proj4string() - still the only available function to set the CRS - may in the
+# future get a more general name:
+proj4string(points) <- crs_wgs84
+points_lambert <- spTransform(points, crs_lambert)
+```
+
+Once we have the points into a projected coordinate system, we can calculate the densities. We start by defining a grid. `cellsize` is the dimension of the square grid cell in the units of the projected coordinate system. Meters in case of Lambert72. The boundaries of the grid are defined using `pretty()`, which turns a vector of numbers into a "pretty" vector with rounded numbers. The combination of the boundaries and the cell size determine the number of grid cells `n` in each dimension. `diff()` calculates the difference between to adjacent numbers of a vector. The density is calculated with `MASS::kde2d()` based on the vectors with the longitude and latitude, the number of grid cells in each dimension and the boundaries of the grid. This returns the grid as a list with elements `x` (a vector of longitude coordinates of the centroids),  `y` (a vector of latitude coordinates of the centroids) and `z` (a matrix with densities). The values in `z` are densities for the 'average' point per unit area. When we multiply the value `z` with the area of the grid cell and sum all of them we get 1. So if we multiple `z` with the number of points we get the density of the points per unit area.
+
+We use [`dplyr::mutate()`](http://dplyr.tidyverse.org/) to convert it into a `data.frame`. The last two steps convert the centroids into a set of coordinates for square polygons.
+
+```{r density}
+library(MASS)
+library(dplyr)
+xlim <- range(pretty(points_lambert$lon)) + c(-100, 100)
+ylim <- range(pretty(points_lambert$lat)) + c(-100, 100)
+n <- c(
+  diff(xlim),
+  diff(ylim)
+) / params$cellsize + 1
+dens <- kde2d(
+  x = points_lambert$lon,
+  y = points_lambert$lat,
+  n = n,
+  lims = c(xlim, ylim)
+)
+dx <- diff(dens$x[1:2])
+dy <- diff(dens$y[1:2])
+sum(dens$z * dx * dy)
+dens <- expand.grid(
+  lon = dens$x,
+  lat = dens$y
+) %>%
+  mutate(
+    density = as.vector(dens$z) * length(points_lambert),
+    id = seq_along(density)
+  ) %>%
+  merge(
+    data.frame(
+      x = dx * (c(0, 0, 1, 1, 0) - 0.5),
+      y = dy * (c(0, 1, 1, 0, 0) - 0.5)
+    )
+  ) %>%
+  mutate(
+    lon = lon + x,
+    lat = lat + y
+  )
+```
+
+In order to visualise the result, we have to re-project the coordinates back to WGS84. Then we can display the raster with a web based background image.
+
+```{r ggmap, fig.cap = "Static image of density"}
+coordinates(dens) <- ~lon + lat
+proj4string(dens) <- crs_lambert
+dens_wgs <- spTransform(dens, crs_wgs84) %>%
+  as.data.frame()
+ggmap(map) +
+  geom_polygon(data = dens_wgs, aes(group = id, fill = density), alpha = 0.5) +
+  scale_fill_gradientn(
+    "density\n(#/m²)",
+    colours = rev(rainbow(100, start = 0, end = .7)),
+    limits = c(0, NA)
+  )
+```
+
+Using `leaflet` to generate a map was a bit more laborious.
+Using the `data.frame dens_wgs`directly failed.
+So we converted the `data.frame` to a `SpatialPolygonsDataframe`, which is a combination of a `SpatialPolygons` and a `data.frame`.
+The `SpatialPolygons` consists of a list of `Polygons`, one for each row of the `data.frame`.
+A `Polygons` object consists of a list of one or more `Polygon` objects.
+In this example it is a single polygon which represents the grid cell.
+
+```{r convert-to-Spatial-Polygons}
+dens_sp <- lapply(
+  unique(dens_wgs$id),
+  function(i){
+    filter(dens_wgs, id == i) %>%
+      select(lon, lat) %>%
+      Polygon() %>%
+      list() %>%
+      Polygons(ID = i)
+  }
+) %>%
+  SpatialPolygons() %>%
+  SpatialPolygonsDataFrame(
+    data = dens_wgs %>%
+      distinct(id, density),
+    match.ID = FALSE
+  )
+```
+
+`leaflet` requires a predefined function with a colour palette.
+We use `leaflet::colorNumeric()` to get a continuous palette.
+Setting `stroke = FALSE` removes the borders of the polygon.
+`fillOpacity` sets the transparency of the polygons.
+
+```{r leaflet, fig.cap = "Dynamic map of density"}
+library(leaflet)
+pal <- colorNumeric(
+  palette = rev(rainbow(100, start = 0, end = .7)),
+  domain = c(0, dens_sp$density)
+)
+leaflet(dens_sp) %>%
+  addTiles() %>%
+  addPolygons(color = ~pal(density), stroke = FALSE, fillOpacity = 0.5) %>%
+  addLegend(pal = pal, values = ~density)
+```
+