7. Statistical modelling with stars objects

stars objects as data.frames

The analogy of stars objects to data.frame is this:

• each attribute (array) becomes a single column
• dimensions become added (index) columns

To see how this works with the 6-band example dataset, consider this:

library(stars)
l7 = system.file("tif/L7_ETMs.tif", package = "stars") %>%
  read_stars()
l7
## stars object with 3 dimensions and 1 attribute
## attribute(s):
##              Min. 1st Qu. Median     Mean 3rd Qu. Max.
## L7_ETMs.tif     1      54     69 68.91242      86  255
## dimension(s):
##      from  to  offset delta                     refsys point x/y
## x       1 349  288776  28.5 SIRGAS 2000 / UTM zone 25S FALSE [x]
## y       1 352 9120761 -28.5 SIRGAS 2000 / UTM zone 25S FALSE [y]
## band    1   6      NA    NA                         NA    NA
as.data.frame(l7) %>% head()
##          x       y band L7_ETMs.tif
## 1 288790.5 9120747    1          69
## 2 288819.0 9120747    1          69
## 3 288847.5 9120747    1          63
## 4 288876.0 9120747    1          60
## 5 288904.5 9120747    1          61
## 6 288933.0 9120747    1          61

We see that we get one single variable with the object (array) name, and added columns with the dimension values (x, y, band). In a typical case, we would like to have the six bands distributed over six variables, and have a single observation (row) for each x/y pair. For this, we could use e.g. utils::unstack or dplyr::pivot_wider on this data.frame, but a more efficient way is to use the dedicated split method for stars objects, which resolves a dimension and splits it over attributes, one for each dimension value:

l7 %>% split("band") %>%
  as.data.frame() %>% 
  head()
##          x       y X1 X2 X3 X4 X5 X6
## 1 288790.5 9120747 69 56 46 79 86 46
## 2 288819.0 9120747 69 57 49 75 88 49
## 3 288847.5 9120747 63 52 45 66 75 41
## 4 288876.0 9120747 60 45 35 66 69 38
## 5 288904.5 9120747 61 52 44 76 92 60
## 6 288933.0 9120747 61 50 37 78 74 38

The reason that split is more efficient than the mentioned alternatives is that (i) split does not have to match records based on dimensions (x/y), and (ii) it works for out-of-memory (stars_proxy) arrays, in the chunked process/write loop of write_stars(). ### Predict for stars objects

The pattern to obtain predictions for all pixels of a stars objects is:

• use the full dataset or a sample of it to train the model, using as.data.frame() (possibly after a split)
• use predict(star_object, model) to predict for all pixels of stars_object, using the stars-wrapper of the predict method for model.
• if there is no predict method for model, provide one (see the kmeans example below)

This works both for stars objects (in-memory) as stars_proxy objects (out-of memory). For plotting stars_proxy objects, downsampling is done before prediction (predicting only the pixels that are shown), full rasters can be written to disk with write_stars(), which will carry out predictions on chunks being read and written.

Linear regression on pixel time series

We can read in the avhrr dataset, containing only 9 days:

library(stars)
x = c("avhrr-only-v2.19810901.nc",
"avhrr-only-v2.19810902.nc",
"avhrr-only-v2.19810903.nc",
"avhrr-only-v2.19810904.nc",
"avhrr-only-v2.19810905.nc",
"avhrr-only-v2.19810906.nc",
"avhrr-only-v2.19810907.nc",
"avhrr-only-v2.19810908.nc",
"avhrr-only-v2.19810909.nc")
file_list = system.file(paste0("netcdf/", x), package = "starsdata")
y = read_stars(file_list, sub = "sst", quiet = TRUE, proxy = TRUE)
(t = st_get_dimension_values(y, 4))

We will use a function that computes the slope of the regression line for temperature with time. We get temperatures as a vector in the first argument of the function supplied to st_apply, and have t already defined. The function could look like

slope = function(x) {
  if (any(is.na(x)))
    NA_real_
  else
    coeffients(lm(x~t))[2]
}

but we will optimize this a bit, using anyNA and lm.fit rather than lm:

slope = function(x) {
  if (anyNA(x))
    NA_real_
  else
    lm.fit(cbind(1, t), x)$coefficients[2]
}

The result is lazily defined by (adrop drops the singular dimension)

out = st_apply(adrop(y), c(1,2), slope)

but only computed by the following command, where the computations are restricted to the pixels plotted:

plot(out, breaks = "equal", main = "9-day time trend (slope)")

An interesting pattern appears (despite the very short time series!): where SST reveals a main signal of colder when getting further from the equator, changes in SST show much more fine grained structures of areas going up, and others going down. A diverging color ramp would be a better choice here, to distinguish positive from negative trends.

Principal components

In the first example, we build principal components on the entire dataset, because it is rather small.

tif = system.file("tif/L7_ETMs.tif", package = "stars")
r = split(read_stars(tif))
pc = prcomp(as.data.frame(r)[,-(1:2)]) # based on all data
out = predict(r, pc)
plot(merge(out), breaks = "equal", join_zlim = FALSE)

We see, amongst others, that PC1 picks up the difference between sea (dark) and land, and PC2 and 3 structures in sea and coastal waters.

In the second example, we build principal components from a sample of the entire dataset, because the entire dataset is rather large. We apply it, using predict, to pixels shown in the plot (i.e. at reduced rather than full resolution)

granule = system.file("sentinel/S2A_MSIL1C_20180220T105051_N0206_R051_T32ULE_20180221T134037.zip", 
   package = "starsdata")
s2 = paste0("SENTINEL2_L1C:/vsizip/", granule, 
"/S2A_MSIL1C_20180220T105051_N0206_R051_T32ULE_20180221T134037.SAFE/MTD_MSIL1C.xml:10m:EPSG_32632")
p = read_stars(s2, proxy = TRUE, NA_value = 0) %>%
    split()
r = st_sample(p, 1000)
pc = prcomp(na.omit(as.data.frame(r))[,-(1:2)]) # based on all data
out = predict(p, pc)

Before plotting this, we’ll add country borders that delineate sea, obtained from the mapdata package:

bb = st_bbox(p) %>% 
  st_as_sfc() %>%
  st_transform(4326) %>%
  st_bbox()
library(maps)
library(mapdata)
m = map("worldHires", xlim = bb[c(1,3)], ylim = bb[c(2,4)], plot=F,fill=TRUE) %>%
  st_as_sfc() %>%
  st_transform(st_crs(r))

We plot the results with independent color ranges, so every PC is stretched over the entire grey scale.

plt_boundary = function() plot(m, border = 'orange', add = TRUE)
plot(merge(out), hook = plt_boundary, join_zlim = FALSE)

This suggests that PC1 picks up the difference cloud signal (difference between clouds and non-clouds), PC2 the difference between sea and land areas, and PC4 some sensor artefacts (striping in swath direction).

To compute full resolution (10000 x 10000 pixels) results and write them to a file, use

write_stars(merge(out), "out.tif")

K-means clustering

library(clue)
predict.kmeans = function(object, newdata, ...) {
    unclass(clue::cl_predict(object, newdata[, -c(1:2)], ...))
}

For a small dataset:

tif = system.file("tif/L7_ETMs.tif", package = "stars")
i = read_stars(tif, proxy = TRUE) %>%
    split()
nclus = 5

sam = st_sample(i, 1000)
k = kmeans(na.omit(as.data.frame(sam)[, -c(1:2)]), nclus)
out = predict(i, k)
plot(out, col = sf.colors(nclus, categorical=TRUE))

This seems to pick up a fair number of land cover classes: water (5), rural (3), and densely populated (1, 2).

For the large(r) dataset:

i = read_stars(s2, proxy = TRUE, NA_value = 0) %>%
    split()
sam = st_sample(i, 1000)
k = kmeans(na.omit(as.data.frame(sam)[, -c(1:2)]), nclus)
out = predict(i, k)
plot(out, col = sf.colors(nclus, categorical=TRUE), reset = FALSE)
plot(m, add = TRUE)

we see that class 1 and 3 identify with the unclouded area, 3 with land, the other classes seem to mainly catch aspects of the cloud signal.

Random Forest land use classification

The following example is purely for educational purposes; the classified “land use” is just a rough approximation from what seems to be easily visible on the image: sea, land, and areas with both but partially covered by clouds. We opted therefore for four classes: sea, land, clouds over sea, clouds over land.

We have polygon areas where the land use was classified, residing in a GeoPackage file. (This file was created using QGIS, using the instructions found here.)

# for all, multi-resolution, use:
bands = c("B04", "B03", "B02", "B08", "B01", "B05", "B06", "B07", "B8A", "B09", "B10", "B11", "B12")
# bands = c("B04", "B03", "B02", "B08")
s2 = paste0("/vsizip/", granule, 
"/S2A_MSIL1C_20180220T105051_N0206_R051_T32ULE_20180221T134037.SAFE/GRANULE/L1C_T32ULE_A013919_20180220T105539/IMG_DATA/T32ULE_20180220T105051_", bands, ".jp2")
r = read_stars(s2, proxy = TRUE, NA_value = 0) %>%
    setNames(bands) 
cl = read_sf(system.file("gpkg/s2.gpkg", package = "stars")) %>%
  st_transform(st_crs(r))
plot(r, reset = FALSE)
plot(cl, add = TRUE)
plot(m, add = TRUE, border = 'orange')

Next, we need points, sampled inside these polygons, for which we need to extract the satellite spectral data

pts = st_sample(cl, 1000, "regular") %>%
    st_as_sf() %>%
    st_intersection(cl)
train = st_extract(r, pts)
train$use = as.factor(pts$use) # no need for join, since the order did not change
train

library(randomForest)
train = as.data.frame(train)
train$x = NULL # remove geometry
rf = randomForest(use ~ ., train) # ~ . : use all other attributes
pr = predict(r, rf)
plot(pr, reset = FALSE, key.pos = 1)
# add country outline:
plot(m, add = TRUE)

This comes with the rather trivial finding that land and sea can be well predicted when there are no clouds, and the less trivial finding that they can be reasonably distinguished through patchy clouds of this kind. Note that predictions of this kind are pure pixel-based: for each prediction only the spectral bands for this pixel are considered, not for instance of any neighboring pixels.