# Chapter 6 Data Cubes

Data cubes arise naturally when we observe properties of a set of geometries repeatedly over time. Time information may sometimes be considered as an attribute of a feature, e.g. when we register the year of construction of a building, or the date of birth of a person (chapter 5). In other cases it may refer to the time of observing an attribute, or the time for which a prediction of an attribute has been made. In these cases, time is on equal footing with space, and time and space together describe the physical dimensions over which we observe, model, make predictions or make forecasts.

One way of considering our world is that of a four-dimensionsal space, with three space and one time dimension. In that view, events become “things” or “objects” that have as duration their size on the time dimension (Galton 2004). Although such a view does not align well with how we experience and describe the world, from a data analytic perspective, four numbers, along with their reference systems, suffice to describe space and time coordinates of an observation associated with a point location and time instance.

We define data cubes as array data with one or more array dimensions
associated with space and/or time (Lu, Appel, and Pebesma 2018). This
implies that raster data, features with attributes, and time series
data are all special cases of data cubes. Since we do not restrict
to three-dimensional structures, we actually mean *hypercubes*
rather than cubes, and as the cube extent of the different dimensions
does not have to be identical, or have comparable units, the better
term would be *hyperrectangle*. For simplicity, we talk about data
cubes instead.

A canonical form of a data cube is shown in figure 6.1: it shows in a perspective plot a set of raster layers for the same region that were collected (observed, or modelled) at different time steps. The three cube dimensions, longitude, latitude and time are thought of as being orthogonal. Arbitrary two-dimensional cube slices are obtained by fixing one of the dimensions at a particular value, one-dimensional slices are obtained by fixing two of the dimensions at a particular value, and a scalar is obtained by fixing three dimensions at a particular value.

```
# (C) 2019, Edzer Pebesma, CC-BY-SA
set.seed(1331)
suppressPackageStartupMessages(library(stars))
library(colorspace)
tif = system.file("tif/L7_ETMs.tif", package = "stars")
r = read_stars(tif)
nrow = 5
ncol = 8
m = r[[1]][1:nrow,1:ncol,1]
dim(m) = c(x = nrow, y = ncol) # named dim
s = st_as_stars(m)
# s
attr(s, "dimensions")[[1]]$delta = 3
attr(s, "dimensions")[[2]]$delta = -.5
attr(attr(s, "dimensions"), "raster")$affine = c(-1.2, 0.0)
plt = function(x, yoffset = 0, add, li = TRUE) {
attr(x, "dimensions")[[2]]$offset = attr(x, "dimensions")[[2]]$offset + yoffset
l = st_as_sf(x, as_points = FALSE)
pal = sf.colors(10)
if (li)
pal = lighten(pal, 0.3 + rnorm(1, 0, 0.1))
if (! add)
plot(l, axes = FALSE, breaks = "equal", pal = pal, reset = FALSE, border = grey(.75), key.pos = NULL, main = NULL, xlab = "time")
else
plot(l, axes = TRUE, breaks = "equal", pal = pal, add = TRUE, border = grey(.75))
u = st_union(l)
# print(u)
plot(st_geometry(u), add = TRUE, col = NA, border = 'black', lwd = 2.5)
}
pl = function(s, x, y, add = TRUE, randomize = FALSE) {
attr(s, "dimensions")[[1]]$offset = x
attr(s, "dimensions")[[2]]$offset = y
m = r[[1]][y + 1:nrow,x + 1:ncol,1]
if (randomize)
m = m[sample(y + 1:nrow),x + 1:ncol]
dim(m) = c(x = nrow, y = ncol) # named dim
s[[1]] = m
plt(s, 0, add)
plt(s, 1, TRUE)
plt(s, 2, TRUE)
plt(s, 3, TRUE)
plt(s, 4, TRUE)
plt(s, 5, TRUE)
plt(s, 6, TRUE)
plt(s, 7, TRUE)
plt(s, 8, TRUE, FALSE)
}
plot.new()
par(mar = rep(0.5,4))
plot.window(xlim = c(-12,15), ylim = c(-5,10), asp=1)
pl(s, 0, 0)
# box()
text(-10, 0, "time", srt = -90, col = 'black')
text(-5, 6.5, "latitude", srt = 25, col = 'black')
text( 5, 8.5, "longitude", srt = 0, col = 'black')
```

## 6.1 A four-dimensional data cube

```
# (C) 2021, Jonathan Bahlmann, CC-BY-SA
# https://github.com/Open-EO/openeo.org/tree/master/documentation/1.0/datacubes/.scripts
# based on work by Edzer Pebesma, 2019, here: https://gist.github.com/edzer/5f1b0faa3e93073784e01d5a4bb60eca
# plotting runs via a dummy stars object with x, y dimensions (no bands)
# to not be overly dependent on an input image, time steps and bands
# are displayed by replacing the matrix contained in the dummy stars object
# every time something is plotted
# packages, read input ----
set.seed(1331)
library(stars)
suppressPackageStartupMessages(library(colorspace))
suppressPackageStartupMessages(library(scales))
# make color palettes ----
blues <- sequential_hcl(n = 20, h1 = 211, c1 = 80, l1 = 40, l2 = 100, p1 = 2)
greens <- sequential_hcl(n = 20, h1 = 134, c1 = 80, l1 = 40, l2 = 100, p1 = 2)
reds <- sequential_hcl(n = 20, h1 = 360, c1 = 80, l1 = 40, l2 = 100, p1 = 2)
purples <- sequential_hcl(n = 20, h1 = 299, c1 = 80, l1 = 40, l2 = 100, p1 = 2)
greys <- sequential_hcl(n = 20, h1 = 0, c1 = 0, l1 = 40, l2 = 100, p1 = 2)
# matrices from raster ----
# make input matrices from an actual raster image
input <- read_stars("data/iceland_delta_cutout_2.tif") # this raster needs approx 6x7 format
# if the input raster is changed, every image where a pixel value is written as text needs to be checked and corrected accordingly
input <- input[,,,1:4]
warped <- st_warp(input, crs = st_crs(input), cellsize = 200) # warp to approx. 6x7 pixel
# these are only needed for resampling
warped_highres <- st_warp(warped, crs = st_crs(warped), cellsize = 100) # with different input, cellsize must be adapted
# this is a bit of a trick, because 3:4 is different format than 6:7
# when downsampling, the raster of origin isn't so important anyway
warped_lowres <- st_warp(warped_highres[,1:11,,], crs = st_crs(warped), cellsize = 390)
# plot(warped_lowres)
# image(warped[,,,1], text_values = TRUE)
t1 <- floor(matrix(runif(42, -30, 150), ncol = 7)) # create timesteps 2 and 3 randomly
t2 <- floor(matrix(runif(42, -250, 50), ncol = 7))
# create dummy stars object ----
make_dummy_stars <- function(x, y, d1, d2, aff) {
m = warped_highres[[1]][1:x,1:y,1] # underlying raster doesn't matter because it's just dummy construct
dim(m) = c(x = x, y = y) # named dim
dummy = st_as_stars(m)
attr(dummy, "dimensions")[[1]]$delta = d1
attr(dummy, "dimensions")[[2]]$delta = d2
attr(attr(dummy, "dimensions"), "raster")$affine = c(aff, 0.0)
return(dummy)
}
s <- make_dummy_stars(6, 7, 2.5, -.5714286, -1.14) # mainly used, perspective
f <- make_dummy_stars(6, 7, 1, 1, 0) # flat
highres <- make_dummy_stars(12, 14, 1.25, -.2857143, -.57) # for resampling
lowres <- make_dummy_stars(3, 4, 5, -1, -2) # for resampling
# matrices from image ----
make_matrix <- function(image, band, n = 42, ncol = 7, t = 0) {
# this is based on an input image with >= 4 input bands
# n is meant to cut off NAs, ncol is y, t is random matrix for time difference
return(matrix(image[,,,band][[1]][1:n], ncol = ncol) - t)
# before: b3 <- matrix(warped[,,,1][[1]][1:42], ncol = 7) - t2
}
# now use function:
b1 <- make_matrix(warped, 1)
b2 <- make_matrix(warped, 1, t = t1)
b3 <- make_matrix(warped, 1, t = t2)
g1 <- make_matrix(warped, 2)
g2 <- make_matrix(warped, 2, t = t1)
g3 <- make_matrix(warped, 2, t = t2)
r1 <- make_matrix(warped, 3)
r2 <- make_matrix(warped, 3, t = t1)
r3 <- make_matrix(warped, 3, t = t2)
n1 <- make_matrix(warped, 4)
n2 <- make_matrix(warped, 4, t = t1)
n3 <- make_matrix(warped, 4, t = t2)
# plot functions ----
plt = function(x, yoffset = 0, add, li = TRUE, pal, print_geom = TRUE, border = .75, breaks = "equal") {
# pal is color palette
attr(x, "dimensions")[[2]]$offset = attr(x, "dimensions")[[2]]$offset + yoffset
l = st_as_sf(x, as_points = FALSE)
if (li)
pal = lighten(pal, 0.2) # + rnorm(1, 0, 0.1))
if (! add)
plot(l, axes = FALSE, breaks = breaks, pal = pal, reset = FALSE, border = grey(border), key.pos = NULL, main = NULL, xlab = "time")
else
plot(l, axes = TRUE, breaks = breaks, pal = pal, add = TRUE, border = grey(border))
u = st_union(l)
# print(u)
if(print_geom) {
plot(st_geometry(u), add = TRUE, col = NA, border = 'black', lwd = 2.5)
} else {
# not print geometry
}
}
pl_stack = function(s, x, y, add = TRUE, nrM, imgY = 7, inner = 1) {
# nrM is the timestep {1, 2, 3}, cause this function
# prints all 4 bands at once
attr(s, "dimensions")[[1]]$offset = x
attr(s, "dimensions")[[2]]$offset = y
# m = r[[1]][y + 1:nrow,x + 1:ncol,1]
m <- eval(parse(text=paste0("n", nrM)))
s[[1]] = m[,c(imgY:1)] # turn around to have same orientation as flat plot
plt(s, 0, TRUE, pal = purples)
m <- eval(parse(text=paste0("r", nrM)))
s[[1]] = m[,c(imgY:1)]
plt(s, 1*inner, TRUE, pal = reds)
m <- eval(parse(text=paste0("g", nrM)))
s[[1]] = m[,c(imgY:1)]
plt(s, 2*inner, TRUE, pal = greens)
m <- eval(parse(text=paste0("b", nrM)))
s[[1]] = m[,c(imgY:1)]
plt(s, 3*inner, TRUE, pal = blues) # li FALSE deleted
}
# flat plot function
# prints any dummy stars with any single matrix to position
pl = function(s, x, y, add = TRUE, randomize = FALSE, pal, m, print_geom = TRUE, border = .75, breaks = "equal") {
# m is matrix to replace image with
# m <- t(m)
attr(s, "dimensions")[[1]]$offset = x
attr(s, "dimensions")[[2]]$offset = y
# print(m)
s[[1]] = m
plt(s, 0, add = TRUE, pal = pal, print_geom = print_geom, border = border, breaks = breaks)
#plot(s, text_values = TRUE)
}
print_segments <- function(x, y, seg, by = 1, lwd = 4, col = "black") {
seg = seg * by
seg[,1] <- seg[,1] + x
seg[,3] <- seg[,3] + x
seg[,2] <- seg[,2] + y
seg[,4] <- seg[,4] + y
segments(seg[,1], seg[,2], seg[,3], seg[,4], lwd = lwd, col = col)
}
# time series ----
# from: cube1_ts_6x7_bigger.png
offset = 26
plot.new()
#par(mar = c(3, 2, 7, 2))
par(mar = c(0, 0, 0, 0))
#plot.window(xlim = c(10, 50), ylim = c(-3, 10), asp = 1)
plot.window(xlim = c(-15, 75), ylim = c(-3, 10), asp = 1)
pl_stack(s, 0, 0, nrM = 3)
pl_stack(s, offset, 0, nrM = 2)
pl_stack(s, 2 * offset, 0, nrM = 1)
# po <- matrix(c(0,-8,7,0,15,3.5, 0,1,1,5,5,14), ncol = 2)
heads <- matrix(c(3.5, 3.5 + offset, 3.5 + 2*offset, 14,14,14), ncol = 2)
points(heads, pch = 16) # 4 or 16
segments(c(-8, 7, 0, 15), c(-1,-1,3,3), 3.5, 14) # first stack pyramid
segments(c(-8, 7, 0, 15) + offset, c(-1,-1,3,3), 3.5 + offset, 14) # second stack pyramid
segments(c(-8, 7, 0, 15) + 2*offset, c(-1,-1,3,3), 3.5 + 2*offset, 14) # third stack pyramid
arrows(-13, 14, 72, 14, angle = 20, lwd = 2) # timeline
text(7.5, 3.8, "x", col = "black")
text(-10, -2.5, "bands", srt = 90, col = "black")
text(-4.5, 1.8, "y", srt = 27.5, col = "black")
y = 15.8
text(69, y, "time", col = "black")
text(3.5, y, "2020-10-01", col = "black")
text(3.5 + offset, y, "2020-10-13", col = "black")
text(3.5 + 2*offset, y, "2020-10-25", col = "black")
```

Figure 6.2 depicts a four-dimensional raster data cube, where three-dimensional raster data cubes with a spectral dimension (“bands”) are organised along a fourth dimension, a time axis. Color image data always has three bands (Blue, Green, Red), and this example has a fourth band (near infrared, NIR), which is commonly found in spectral remote sensing data.

Figure 6.3 shows exactly the same data, but layed out
flat as a facet plot (or scatterplot matrix), where two dimensions
(\(x\) and \(y\)) are aligned with (or nested within) the dimensions
*bands* and *time*, respectively.

```
# flat ----
xlabels <- seq(attr(warped, "dimensions")[[1]]$offset + attr(warped, "dimensions")[[1]]$delta / 2, length.out = attr(warped, "dimensions")[[1]]$to, by = attr(warped, "dimensions")[[1]]$delta)
ylabels <- seq(attr(warped, "dimensions")[[2]]$offset + attr(warped, "dimensions")[[2]]$delta / 2, length.out = attr(warped, "dimensions")[[2]]$to, by = attr(warped, "dimensions")[[2]]$delta)
print_labels <- function(x, y, off, lab, horizontal, cex = 1) {
if(horizontal) { # x
for(i in 0:(length(lab)-1)) {
text(x + i*off, y, lab[i+1], cex = cex, srt = 90)
}
} else { # y
lab <- lab[length(lab):0]
for(i in 0:(length(lab)-1)) {
text(x, y + i*off, lab[i+1], cex = cex)
}
}
}
# before: width=1000, xlim(-2, 33), date labels x=31
plot.new()
# par(mar = c(0,0,0,0))
par(mar = c(3,0,0,0))
plot.window(xlim = c(-2, 40), ylim = c(0, 25), asp = 1)
pl(f, 7, 0, pal = blues, m = b1)
pl(f, 7, 10, pal = blues, m = b2)
pl(f, 7, 20, pal = blues, m = b3)
pl(f, 14, 0, pal = greens, m = g1)
pl(f, 14, 10, pal = greens, m = g2)
pl(f, 14, 20, pal = greens, m = g3)
pl(f, 21, 0, pal = reds, m = r1)
pl(f, 21, 10, pal = reds, m = r2)
pl(f, 21, 20, pal = reds, m = r3)
pl(f, 28, 0, pal = purples, m = n1)
pl(f, 28, 10, pal = purples, m = n2)
pl(f, 28, 20, pal = purples, m = n3)
print_labels(28.5, -2, 1, xlabels, horizontal = TRUE, cex = 0.7)
print_labels(36, 0.5, 1, ylabels, horizontal = FALSE, cex = 0.7)
# arrows(6, 27, 6, 0, angle = 20, lwd = 2)
# text(5, 14, "time", srt = 90, col = "black")
text(10, 28, "blue", col = "black")
text(17, 28, "green", col = "black")
text(24, 28, "red", col = "black")
text(31, 28, "nir", col = "black")
text(3, 23.5, "2020-10-01", col = "black")
text(3, 13.5, "2020-10-13", col = "black")
text(3, 3.5, "2020-10-25", col = "black")
```

## 6.2 Dimensions, attributes, and support

Phenomena in space and time can be thought of as functions
with domain space and time, and with range one or more observed
attributes. For clearly identifiable discrete events or objects,
the range is typically discrete, and precise delineation involves
describing the precise coordinates where a thing starts or stops,
which is best suited by vector geometries. For continuous phenomena,
variables that take on a value everywhere such as air temperature
or land use type, there are infinitely many values to represent and
a common approach is to discretize space and time *regularly* over
the spatiotemporal domain (extent) of interest. This leads to a
number of familiar data structures:

- time series, depicted as time lines for functions of time
- image or raster data for two-dimensional spatial data
- time sequences of images for dynamic spatial data

The third form of this, where a variable \(Z\) depends on \(x\), \(y\) and \(t\), as in

\[Z = f(x, y, t)\]

is the archetype of a spatiotemporal array or *data cube*: the shape
of the volume where points regularly discretizing the domain forms a
cube. We call the variables that form the range (here: \(x, y, t\)) the
cube *dimensions*. Data cubes may have multiple attributes, as in

\[\{Z_1,Z_2,...,Z_p\} = f(x, y, t)\]

and if \(Z\) is functional, e.g. reflectance values measured over the electromagnetic spectrum, the spectral wavelengths \(\lambda\) may form an additional dimension, as in \(Z = f(x,y,t,\lambda)\); section 6.5 discusses the alternative of representing color bands as attributes.

Multiple time dimensions arise for instance when making forecasts for different times in the future \(t'\) at different times \(t\), or when time is split into multiple dimensions (e.g. year, day-of-year, hour-of-day). The most general definition of a data cube is a functional mapping from \(n\) dimensions to \(p\) attributes:

\[\{Z_1,Z_2,...,Z_p\} = f(D_1,D_2,...,D_n)\]

Here, we will consider any dataset with one or more space dimensions and zero or more time dimensions as data cubes. That includes:

- simple features (section 3.1)
- time series for sets of features
- raster data
- multi-spectral raster data (images)
- time series of multi-spectral raster data (video)

### 6.2.1 Regular dimensions, GDAL’s geotransform

Data cubes are usually stored in multi-dimensional arrays, and the usual relationship between 1-based array index \(i\) and an associated regularly discretized dimension variable \(x\) is

\[x = o_x + (i-1) d_x\]

with \(o_x\) the origin, and \(d_x\) the grid spacing for this dimension.

For more general cases like those in figure 1.5b-c, the relation between \(x\) and \(y\) and array indexes \(i\) and \(j\) is

\[x = o_x + (i-1) d_x + (j-1) a_1\]

\[y = o_y + (i-1) a_2 + (j-1) d_y\]

With two affine parameters \(a_1\) and \(a_2\); this is the so-called
*geotransform* as used in GDAL. When \(a_1=a_2=0\), this
reduces to the regular raster of figure 1.5a
with square cells if \(d_x = d_y\).
For integer indexes, the coordinates are that of the starting *edge*
of a grid cell, and the cell area (pixel) spans a range corresponding to
index values ranging from \(i\) (inclusive) to \(i+1\) (exclusive).
For most common imagery formats, \(d_y\) is negative,
indicating that image row index increases with decreasing \(y\)
values (southward). To get the \(x\)- and \(y\)-coordinate of the grid
cell *center* of the top left grid cell (in case of a negative \(d_y\)),
we use \(i=1.5\) and \(j=1.5\).

For rectilinear rasters, a table that maps array index to dimension values is needed. NetCDF files for instance always stores all values of spatial dimension variables that correspond to the center of spatial grid cells, and may in addition store grid cell boundaries (which is needed to define rectlinear dimensions unambiguously).

For curvilinear rasters an array that maps every combination of \(i,j\) into \(x,y\) pairs is needed, or a parametric function that does this (e.g. an projection or its inverse). NetCDF files often provide both, when available.

### 6.2.2 Support along cube dimensions

Section 5.1 defined *spatial* support of an attribute variable
as the size (length, area, volume) of a geometry a particular
observation or prediction is associated with. The same notion
applies to temporal support. Although time is rarely reported by
explicit time periods having a start- and end-time, in many cases
either the time stamp implies a period (e.g. ISO-8601 indications
like “2021” for a full year, “2021-01” for full month) or the time
period is taken as the period from the time stamp of the current
record up to but not including the time stamp of the next record.

An example is MODIS satellite imagery, where vegetation indexes (NDVI and EVI) are available as 16-day composites, meaning that over 16-day periods all available imagery is aggregated into a single image; such composites have temporal “block support”. Sentinel-2 or Landsat-8 data on the other hand are “snapshot” images and have temporal “point support”. When temporally aggregating data with temporal point support e.g. to monthly values one would select all images falling in the target time interval. When aggregating temporal block support imagery such as the MODIS 16-day composite, one might weigh images, e.g. according to the amount of overlap of the 16-day composite period and the target period, similar to area-weighted interpolation but over the time dimension.

## 6.3 Operations on data cubes

### 6.3.1 Slicing a cube: filter

Data cubes can be sliced into sub-cubes by fixing a dimension at a particular value. Figure 6.4 shows the sub-cubes obtained by doing with each of the dimensions. In this figure, the spatial filtering does not happen by fixing a single spatial dimension at a particular value, but by selecting a particular subregion, which is a more common operation. Fixing \(x\) or \(y\) would give a sub-cube along a transect of constant \(x\) or \(y\), which can be used to show a Hovmöller diagram, where an attribute is plotted (colored) in the space of one space and one time dimension.

```
# filter ----
# mask <- matrix(c(rep(NA, 26), 1,NA,1,NA,1,1,1, rep(NA, 9)), ncol = 7)
mask <- matrix(c(NA,NA,NA,NA,NA,NA,
NA,NA,NA,NA,NA,NA,
NA,NA,NA, 1, 1, 1,
NA,NA, 1, 1, 1,NA,
NA,NA,NA, 1, 1,NA,
NA,NA,NA,NA,NA,NA,
NA,NA,NA,NA,NA,NA), ncol = 7)
print_grid <- function(x, y) {
pl(f, 0+x, 0+y, pal = blues, m = b1)
pl(f, 0+x, 10+y, pal = blues, m = b2)
pl(f, 0+x, 20+y, pal = blues, m = b3)
pl(f, 7+x, 0+y, pal = greens, m = g1)
pl(f, 7+x, 10+y, pal = greens, m = g2)
pl(f, 7+x, 20+y, pal = greens, m = g3)
pl(f, 14+x, 0+y, pal = reds, m = r1)
pl(f, 14+x, 10+y, pal = reds, m = r2)
pl(f, 14+x, 20+y, pal = reds, m = r3)
pl(f, 21+x, 0+y, pal = purples, m = n1)
pl(f, 21+x, 10+y, pal = purples, m = n2)
pl(f, 21+x, 20+y, pal = purples, m = n3)
}
print_alpha_grid <- function(x,y, alp = 0.2, geom = FALSE) {
pl(f, 0+x, 0+y, pal = alpha(blues, alp), print_geom = geom, m = b1, border = 1)
pl(f, 0+x, 10+y, pal = alpha(blues, alp), print_geom = geom, m = b2, border = 1)
pl(f, 0+x, 20+y, pal = alpha(blues, alp), print_geom = geom, m = b3, border = 1)
pl(f, 7+x, 0+y, pal = alpha(greens, alp), print_geom = geom, m = g1, border = 1)
pl(f, 7+x, 10+y, pal = alpha(greens, alp), print_geom = geom, m = g2, border = 1)
pl(f, 7+x, 20+y, pal = alpha(greens, alp), print_geom = geom, m = g3, border = 1)
pl(f, 14+x, 0+y, pal = alpha(reds, alp), print_geom = geom, m = r1, border = 1)
pl(f, 14+x, 10+y, pal = alpha(reds, alp), print_geom = geom, m = r2, border = 1)
pl(f, 14+x, 20+y, pal = alpha(reds, alp), print_geom = geom, m = r3, border = 1)
pl(f, 21+x, 0+y, pal = alpha(purples, alp), print_geom = geom, m = n1, border = 1)
pl(f, 21+x, 10+y, pal = alpha(purples, alp), print_geom = geom, m = n2, border = 1)
invisible(pl(f, 21+x, 20+y, pal = alpha(purples, alp), print_geom = geom, m = n3, border = 1))
}
print_grid_filter <- function(x, y) {
pl(f, 0+x, 0+y, pal = blues, m = matrix(b1[mask == TRUE], ncol = 7))
pl(f, 0+x, 10+y, pal = blues, m = matrix(b2[mask == TRUE], ncol = 7))
pl(f, 0+x, 20+y, pal = blues, m = matrix(b3[mask == TRUE], ncol = 7))
pl(f, 7+x, 0+y, pal = greens, m = matrix(g1[mask == TRUE], ncol = 7))
pl(f, 7+x, 10+y, pal = greens, m = matrix(g2[mask == TRUE], ncol = 7))
pl(f, 7+x, 20+y, pal = greens, m = matrix(g3[mask == TRUE], ncol = 7))
pl(f, 14+x, 0+y, pal = reds, m = matrix(r1[mask == TRUE], ncol = 7))
pl(f, 14+x, 10+y, pal = reds, m = matrix(r2[mask == TRUE], ncol = 7))
pl(f, 14+x, 20+y, pal = reds, m = matrix(r3[mask == TRUE], ncol = 7))
pl(f, 21+x, 0+y, pal = purples, m = matrix(n1[mask == TRUE], ncol = 7))
pl(f, 21+x, 10+y, pal = purples, m = matrix(n2[mask == TRUE], ncol = 7))
pl(f, 21+x, 20+y, pal = purples, m = matrix(n3[mask == TRUE], ncol = 7))
}
print_grid_time_filter <- function(x, y) { # 3x1, 28x7
pl(f, 0+x, 10+y, pal = blues, m = b3)
pl(f, 7+x, 10+y, pal = greens, m = g3)
pl(f, 14+x, 10+y, pal = reds, m = r3)
pl(f, 21+x, 10+y, pal = purples, m = n3)
}
print_grid_bands_filter <- function(x, y, pal = greys) { # 1x3 6x27
pl(f, 0+x, 0+y, pal = pal, m = n1)
pl(f, 0+x, 10+y, pal = pal, m = n2)
pl(f, 0+x, 20+y, pal = pal, m = n3)
}
# build exactly like reduce
plot.new()
par(mar = c(3,3,3,3))
x = 120
y = 100
down = 0
plot.window(xlim = c(0, x), ylim = c(0, y), asp = 1)
print_grid(x/2-28/2,y-27)
print_alpha_grid((x/3-28)/2, 0-down) # alpha grid
print_grid_time_filter((x/3-28)/2, -10-down) # select 3rd
print_alpha_grid(x/3+((x/3-6)/2) -10.5, 0-down) # alpha grid
print_grid_bands_filter(x/3+((x/3-12.4)), 0-down, pal = purples)
print_alpha_grid(2*(x/3)+((x/3-28)/2), 0-down) # alpha grid
print_grid_filter(2*(x/3)+((x/3-28)/2), 0-down)
text(3, 13.5-down, "time", srt = 90, col = "black")
text(43, 13.5-down, "time", srt = 90, col = "black")
text(83, 13.5-down, "time", srt = 90, col = "black")
text(20, 30, "bands", col = "black")
text(60, 30, "bands", col = "black")
text(100, 30, "bands", col = "black")
arrows(x/2-28/2,y-30, x/6,32, angle = 20, lwd = 2)
arrows(x/2,y-30, x/2,32, angle = 20, lwd = 2)
arrows(x/2+28/2,y-30, 100, 32, angle = 20, lwd = 2)
# points(seq(1,120,10), seq(1,120,10))
text(28.5,49, "filter temporally", srt = 55.5, col = "black", cex = 0.8)
text(57,49, "filter bands", srt = 90, col = "black", cex = 0.8)
text(91.5,49, "filter spatially", srt = -55.5, col = "black", cex = 0.8)
print_labels(x = x/2-28/2 + 3, y = y+4, off = 7, lab = c("blue", "green", "red", "nir"),
horizontal = TRUE, cex = 0.6)
print_labels(x = x/2-28/2 - 9, y = y-23, off = 10, lab = c("2020-10-01", "2020-10-13", "2020-10-25"),
horizontal = FALSE, cex = 0.6)
print_labels(x = x/2-28/2 + 21.5, y = y-30, off = 1, lab = xlabels,
horizontal = TRUE, cex = 0.3)
print_labels(x = x/2-28/2 + 30, y = y-26.5, off = 1, lab = ylabels,
horizontal = FALSE, cex = 0.3)
```

### 6.3.2 Applying functions to dimensions

A common analysis involves applying a function over one or more
cube dimensions. Simple cases arise where a function such as `abs`

,
`sin`

or `sqrt`

is applied to all values in the cube, or when a function
takes all values in the cube and returns a single scalar, e.g. when
one computes the mean or maximum value over the entire cube. Other
options include applying the function to selected dimensions,
e.g. applying a temporal low-pass filter to every individual
(pixel/band) time series as shown in figure 6.6,
or applying a *spatial* low-pass filter to every spatial
slice (i.e. every band/time combination), shown in figure
6.5.

```
# apply ----
print_text = function(s, x, y, m) {
# m <- t(m) # transverse for correct order
# print(m)
r <- rep(seq(0.5, 5.5, 1), 7)
r <- r + x # consider offsets
u <- c(rep(0.5, 6), rep(1.5, 6), rep(2.5, 6),
rep(3.5, 6), rep(4.5, 6), rep(5.5, 6), rep(6.5, 6))
u <- u + y # offset
tab <- matrix(c(r,u), nrow = 42, byrow = FALSE) # make point table
for (i in 1:42) {
#text(tab[i, 1], tab[i, 2], labels = paste0("", m[i]), cex = 1.1)
text(tab[i, 1], tab[i, 2], labels = paste0("", m[i]), cex = 0.8)
}
}
abs_brks <- seq(-500,500, 50)
abs_pal <- sequential_hcl(n = 20, h1 = 211, c1 = 80, l1 = 30, l2 = 100, p1 = 1.2)
#png("exp_apply_unary.png", width = 2400, height = 1000, pointsize = 24)
#plot.new()
#par(mar = c(2,2,2,2))
#x = 30
#y = 7.5
#plot.window(xlim = c(0, x), ylim = c(0, y), asp = 1)
#pl(f, 3, 2, pal = abs_pal, m = b3 - 200, breaks = abs_brks)
#pl(f, 1.5, .5, pal = abs_pal, m = b2 - 200, breaks = abs_brks)
#pl(f, 0, -1, pal = abs_pal, m = b1 - 200, breaks = abs_brks)
#print_text(s, 0, -1, m = b1 - 200)
#pl(f, 23, 2, pal = abs_pal, m = abs(b3 - 200), breaks = abs_brks)
#pl(f, 21.5, 0.5, pal = abs_pal, m = abs(b2 - 200), breaks = abs_brks)
#pl(f, 20, -1, pal = abs_pal, m = abs(b1 - 200), breaks = abs_brks)
#print_text(s, 20, -1, m = abs(b1 - 200))
#arrows(11, 4, 17.5, 4, lwd = 3)
#text(14.3, 3.5, "absolute()", cex = 1.4)
#dev.off()
vNeumann_seg <- matrix(c(c(0,0,1,1,2,2,1,1,0,0,-1,-1), c(0,-1,-1,0,0,1,1,2,2,1,1,0),
c(0,1,1,2,2,1,1,0,0,-1,-1,0), c(-1,-1,0,0,1,1,2,2,1,1,0,0)), ncol = 4)
apply_filter <- function(input, pad = TRUE, padValue = 1) {
ras <- raster::focal(raster::raster(input), w = matrix(c(0,0.2,0, 0.2,0.2,0.2, 0,0.2,0), ncol = 3), pad = pad, padValue = padValue)
ras <- raster::as.matrix(ras)
ras[ras == "NaN"] <- -999
return(floor(ras))
}
brks <- seq(0,1000, 50)
plot.new()
par(mar = c(0,2,0,0))
x = 30
y = 7.5
plot.window(xlim = c(0, x), ylim = c(0, y), asp = 1)
pl(f, 3, 2, pal = blues, m = b3, breaks = brks)
pl(f, 1.5, .5, pal = blues, m = b2, breaks = brks)
pl(f, 0, -1, pal = blues, m = b1, breaks = brks)
print_text(s, 0, -1, m = b1) # print text on left first stack
print_segments(2, 3, seg = vNeumann_seg, lwd = 3)
pl(f, 23, 2, pal = blues, m = apply_filter(b3), breaks = brks)
pl(f, 21.5, 0.5, pal = blues, m = apply_filter(b2), breaks = brks)
pl(f, 20, -1, pal = blues, m = apply_filter(b1), breaks = brks)
print_text(s, 20, -1, m = apply_filter(b1)) # set pad = FALSE for -99
print_segments(22, 3, seg = vNeumann_seg, lwd = 3)
arrows(11, 4, 17.5, 4, lwd = 3)
text(14.3, 3.5, "apply_kernel()", cex = 1.4)
print_segments(13.8, 1, seg = vNeumann_seg, lwd = 3)
cex = .8
text(14.3, 1.5, "0.2", cex = cex)
text(13.3, 1.5, "0.2", cex = cex)
text(15.3, 1.5, "0.2", cex = cex)
text(14.3, 2.5, "0.2", cex = cex)
text(14.3, .5, "0.2", cex = cex)
```

```
time_arrow_seg <- matrix(c(c(-1.0, 0.3, 1.5, 2.7, 3.9, 5.1), c(-1.0, 0.3, 1.5, 2.7, 3.9, 5.1),
c(-0.5, 0.7, 1.9, 3.1, 4.3, 5.6), c(-0.5, 0.7, 1.9, 3.1, 4.3, 5.6)), ncol = 4)
time_arrow_flag_seg <- matrix(c(c(-1.0, 1.5, 2.7, 3.9), c(-1.0, 1.5, 2.7, 3.9),
c(0.7, 1.9, 3.1, 5.6), c(0.7, 1.9, 3.1, 5.6)), ncol = 4)
b11 <- b2 - t1
b12 <- b1 - t2 + t1
# png("exp_apply_ts.png", width = 2400, height = 1000, pointsize = 24)
plot.new()
par(mar = c(2,2,2,2))
x = 30
y = 10 # 7.5
plot.window(xlim = c(0, x), ylim = c(0, y), asp = 1)
pl(f, 4.8, 3.8, pal = blues, m = b3, breaks = brks)
print_text(s, 4.8, 3.8, m = b3)
pl(f, 3.6, 2.6, pal = blues, m = b11, breaks = brks)
print_text(s, 3.6, 2.6, m = b11)
pl(f, 2.4, 1.4, pal = blues, m = b12, breaks = brks)
print_text(s, 2.4, 1.4, m = b12)
pl(f, 1.2, .2, pal = blues, m = b2, breaks = brks)
print_text(s, 1.2, .2, m = b2)
pl(f, 0, -1, pal = blues, m = b1, breaks = brks)
print_text(s, 0, -1, m = b1) # print text on left first stack
pl(f, 24.8, 3.8, pal = alpha(greys, 0.1), m = matrix(rep("NA", 42), ncol = 7))
pl(f, 23.6, 2.6, pal = blues, m = (b12 + b11 + b3) / 3, breaks = brks)
print_text(s, 23.6, 2.6, m = floor((b12 + b11 + b3) / 3))
pl(f, 22.4, 1.4, pal = blues, m = (b2 + b12 + b11) / 3, breaks = brks)
print_text(s, 22.4, 1.4, m = floor((b2 + b12 + b11) / 3))
pl(f, 21.2, .2, pal = blues, m = (b1 + b2 + b12) / 3, breaks = brks)
print_text(s, 21.2, .2, m = floor((b1 + b2 + b12) / 3))
pl(f, 20, -1, pal = alpha(greys, 0.1), m = matrix(rep("NA", 42), ncol = 7))
print_segments(5.7, 1.7, seg = time_arrow_seg, col = "forestgreen")
arrows(12.5, 9, 20, 9, lwd = 2)
cex = .9
text(16.3, 8.3, "apply_dimension(dimension = 't')", cex = cex)
print_segments(9.7, 1.7, time_arrow_seg, col = "forestgreen") # draw ma explanation
text(-0.5 + 10, -0.5 + 2, "496", cex = cex)
text(.7 + 10, .7 + 2, "363", cex = cex)
text(1.9 + 10, 1.9 + 2, "658", cex = cex)
text(3.1 + 10, 3.1 + 2, "230", cex = cex)
text(4.3 + 10, 4.3 + 2, "525", cex = cex)
t_formula <- expression("t"[n]*" = (t"[n-1]*" + t"[n]*" + t"[n+1]*") / 3")
# text(13.8, 3, t_formula, srt = 45, cex = 1.2)
text(14.4, 3.6, "calculate moving average", srt = 45, cex = cex)
arrows(15, 5.7, 18, 5.7, lwd = 2)
print_segments(15.4, 1.7, seg = time_arrow_seg, col = "forestgreen") # draw ma explanation
text(-0.5 + 15.7, -0.5 + 2, "NA", cex = cex)
text(.7 + 15.7, .7 + 2, "505", cex = cex)
text(1.9 + 15.7, 1.9 + 2, "417", cex = cex)
text(3.1 + 15.7, 3.1 + 2, "471", cex = cex)
text(4.3 + 15.7, 4.3 + 2, "NA", cex = cex)
print_segments(25.7, 1.7, seg = time_arrow_seg, col = "forestgreen")
```

### 6.3.3 Reducing dimensions

When applying function `mean`

to an entire data cube, all dimensions
vanish: the resulting “data cube” has dimensionality zero. We
can also apply functions to a limited set of dimensions such that
selected dimensions vanish, or are *reduced*. We already saw that
filtering is a special case of this, but more in general we could
for instance compute the maximum of every time series, the mean over
every spatial slice, or a band index such as NDVI that summarizes
different spectral values into a single new “band” with the index
value. Figure 6.7 illustrates these options.

```
# reduce ----
# calc mean over time
timeB <- (b1 + b2 + b3) / 3
timeG <- (g1 + g2 + g3) / 3
timeR <- (r1 + r2 + r3) / 3
timeN <- (n1 + n2 + n3) / 3
print_grid_time <- function(x, y) { # 3x1, 28x7
pl(f, 0+x, 10+y, pal = blues, m = timeB)
pl(f, 7+x, 10+y, pal = greens, m = timeG)
pl(f, 14+x, 10+y, pal = reds, m = timeR)
pl(f, 21+x, 10+y, pal = purples, m = timeN)
}
# calc ndvi
ndvi1 <- (n1 - r1) / (n1 + r1)
ndvi2 <- (n2 - r2) / (n2 + r2)
ndvi3 <- (n3 - r3) / (n3 + r3)
print_grid_bands <- function(x, y, pal = greys) { # 1x3 6x27
pl(f, 0+x, 0+y, pal = pal, m = ndvi1)
pl(f, 0+x, 10+y, pal = pal, m = ndvi2)
pl(f, 0+x, 20+y, pal = pal, m = ndvi3)
}
plte = function(s, x, y, add = TRUE, randomize = FALSE, pal, m) {
attr(s, "dimensions")[[1]]$offset = x
attr(s, "dimensions")[[2]]$offset = y
# m = r[[1]][y + 1:nrow,x + 1:ncol,1]
# dim(m) = c(x = nrow, y = ncol) # named dim
# s[[1]] = m
# me <- floor(mean(s[[1]]))
me <- floor(mean(m))
if (me[1] > 100) { # in case non-artificial grids with very high
me <- m / 10 # numbers are used, make them smaller
me <- floor(mean(me))
}
text(x,y,me,cex = 0.8)
}
print_grid_spat <- function(x, y) {
x = x + 3
y = y + 3.5
plte(s, 0+x, 0+y, pal = blues, m = b1)
plte(s, 0+x, 10+y, pal = blues, m = b2)
plte(s, 0+x, 20+y, pal = blues, m = b3)
plte(s, 7+x, 0+y, pal = greens, m = g1)
plte(s, 7+x, 10+y, pal = greens, m = g2)
plte(s, 7+x, 20+y, pal = greens, m = g3)
plte(s, 14+x, 0+y, pal = reds, m = r1)
plte(s, 14+x, 10+y, pal = reds, m = r2)
plte(s, 14+x, 20+y, pal = reds, m = r3)
plte(s, 21+x, 0+y, pal = purples, m = n1)
plte(s, 21+x, 10+y, pal = purples, m = n2)
plte(s, 21+x, 20+y, pal = purples, m = n3)
}
# png("exp_reduce.png", width = 1200, height = 1000, pointsize = 32)
plot.new()
#par(mar = c(3,3,3,3))
par(mar = c(3,0,2,0))
x = 120
y = 100
plot.window(xlim = c(0, x), ylim = c(0, y), asp = 1)
print_grid(x/2-28/2,y-27)
# print_alpha_grid((x/3-28)/2, 0) # alpha grid
print_grid_time((x/3-28)/2, 0) # off = 5.5
# print_alpha_grid(x/3+((x/3-6)/2) -10.5, 0) # alpha grid
print_grid_bands(x/3+((x/3-6)/2), 0)
print_alpha_grid(2*(x/3)+((x/3-28)/2), 0, alp = 0, geom = TRUE) # alpha grid
print_grid_spat(2*(x/3)+((x/3-28)/2), 0)
text(3, 13.5, "time", srt = 90, col = "black")
#segments(3.6, 8, 3.7, 19, col = "red", lwd=3)
segments(3.4, 8, 3.4, 19, col = "red", lwd = 3)
text(43, 13.5, "time", srt = 90, col = "black")
text(83, 13.5, "time", srt = 90, col = "black")
text(20, 30, "bands", col = "black")
text(60, 30, "bands", col = "black")
segments(53,29.8, 67,29.8, col = "red", lwd = 3)
text(100, 30, "bands", col = "black")
text(30, 7, "x", col = "black")
text(36, 13, "y", col = "black")
text(60, -3, "x", col = "black")
text(66, 3, "y", col = "black")
text(110, -3, "x", col = "black")
text(116, 3, "y", col = "black")
segments(108,-2.4, 112,-3.2, col = "red", lwd = 3)
segments(114,3.2, 118,2.4, col = "red", lwd = 3)
text(60, y+4, "bands", col = "black") # dim names on main
text(43, y-14, "time", srt = 90, col = "black")
text(x/2-28/2 + 24, y-30, "x", col = "black")
text(x/2-28/2 + 30, y-24, "y", col = "black")
arrows(x/2-28/2,y-30, x/6,32, angle = 20, lwd = 2)
arrows(x/2,y-30, x/2,32, angle = 20, lwd = 2)
arrows(x/2+28/2,y-30, 100, 32, angle = 20, lwd = 2)
# points(seq(1,120,10), seq(1,120,10))
text(28.5,49, "reduce temporally", srt = 55.5, col = "black", cex = 0.8)
text(57,49, "reduce bands", srt = 90, col = "black", cex = 0.8)
text(91.5,49, "reduce spatially", srt = -55.5, col = "black", cex = 0.8)
```

## 6.4 Aggregating raster to vector cubes

Figure 6.8 illustrates how a four-dimensional raster
data cube can be aggregated to a three-dimensional *vector data
cube*. Pixels in the raster are grouped by spatial intersection
with a set of vector geometries, and each group is then reduced
to a single value by an aggregation function such as `mean`

or
`max`

. In the example, the *two* spatial dimensions \(x\) and \(y\)
reduce to a single dimension, the one-dimensional sequence of feature
geometries, with geometries that are defined in the space of \(x\)
and \(y\). Grouping geometries can also be `POINT`

geometries, in
which case the aggregation function is obsolete as single values
at the `POINT`

locations are *extracted*, e.g. by querying a pixels
value or by interpolating from the nearest pixels.

```
# aggregate ----
mask_agg <- matrix(c(NA,NA,NA,NA,NA,NA,
NA, 1, 1,NA,NA,NA,
1, 1,NA,NA, 1,NA,
1,NA,NA, 1, 1,NA,
1,NA,NA, 1, 1,NA,
1,NA,NA,NA, 1,NA,
NA,NA,NA,NA,NA,NA), ncol = 7)
pl_stack_agg = function(s, x, y, add = TRUE, nrM, imgY = 7, inner = 1) {
# pl_stack that masks the added matrices
# nrM is the timestep {1, 2, 3}, cause this function
# prints all 4 bands at once
attr(s, "dimensions")[[1]]$offset = x
attr(s, "dimensions")[[2]]$offset = y
# m = r[[1]][y + 1:nrow,x + 1:ncol,1]
m <- eval(parse(text=paste0("n", nrM)))
m <- matrix(m[mask_agg == TRUE], ncol = 7)
s[[1]] = m[,c(imgY:1)] # turn around to have same orientation as flat plot
plt(s, 0, TRUE, pal = purples)
m <- eval(parse(text=paste0("r", nrM)))
m <- matrix(m[mask_agg == TRUE], ncol = 7)
s[[1]] = m[,c(imgY:1)]
plt(s, 1*inner, TRUE, pal = reds)
m <- eval(parse(text=paste0("g", nrM)))
m <- matrix(m[mask_agg == TRUE], ncol = 7)
s[[1]] = m[,c(imgY:1)]
plt(s, 2*inner, TRUE, pal = greens)
m <- eval(parse(text=paste0("b", nrM)))
m <- matrix(m[mask_agg == TRUE], ncol = 7)
s[[1]] = m[,c(imgY:1)]
plt(s, 3*inner, TRUE, pal = blues) # li FALSE
}
polygon_1 <- matrix(c(c(0.0, 5.1, 4.9,-2.3), c(0.0, 2.4, 3.1, 1.8),
c(5.1, 4.9,-2.3, 0.0), c(2.4, 3.1, 1.8, 0.0)), ncol = 4)
a <- make_dummy_stars(6, 7, 5, -1.14, -2.28)
print_vector_content <- function(x, y, cex = 0.8) {
vec <- floor(rnorm(8, 250, 100))
text( 0 + x,12 + y, vec[1], cex = cex)
text( 0 + x, 8 + y, vec[2], cex = cex)
text( 0 + x, 4 + y, vec[3], cex = cex)
text( 0 + x, 0 + y, vec[4], cex = cex)
text( 12 + x,12 + y, vec[5], cex = cex)
text( 12 + x, 8 + y, vec[6], cex = cex)
text( 12 + x, 4 + y, vec[7], cex = cex)
text( 12 + x, 0 + y, vec[8], cex = cex)
}
print_ts <- function(off2, yoff2) {
pl_stack(s, 0 + off2, yoff2, nrM = 3) # input 2
pl_stack(s, off + off2, yoff2, nrM = 2)
pl_stack(s, 2 * off + off2, yoff2, nrM = 1)
arrows(-13 + off2, 14 + yoff2, 72 + off2, 14 + yoff2, angle = 20, lwd = 2) # timeline
heads <- matrix(c(3.5+off2, 3.5 + off + off2, 3.5 + 2*off + off2, 14+yoff2,14+yoff2,14+yoff2), ncol = 2)
points(heads, pch = 16) # 4 or 16
segments(c(-8, 7, 0, 15)+off2, c(-1,-1,3,3)+yoff2, 3.5+off2, 14+yoff2) # first stack pyramid
segments(c(-8, 7, 0, 15) + off + off2, c(-1,-1,3,3)+yoff2, 3.5 + off + off2, 14+yoff2) # second stack pyramid
segments(c(-8, 7, 0, 15) + 2*off + off2, c(-1,-1,3,3)+yoff2, 3.5 + 2*off + off2, 14+yoff2) # third stack pyramid
text(7.5+off2, 4.3+yoff2, "x", col = "black", cex = secText)
text(-9.5+off2, -2.5+yoff2, "bands", srt = 90, col = "black", cex = secText)
text(-4.5+off2, 2+yoff2, "y", srt = 27.5, col = "black", cex = secText)
text(69+off2, 15.5+yoff2+1, "time", col = "black")
text(3.5+off2, 15.5+yoff2, "2020-10-01", col = "black")
text(3.5 + off + off2, 15.5+yoff2, "2020-10-13", col = "black")
text(3.5 + 2*off + off2, 15.5+yoff2, "2020-10-25", col = "black")
}
secText = 0.8 # secondary text size (dimension naming)
off = 26 # image stacks are always 26 apart
x = 72 # png X
y = 48 # png Y
yoff = 30
plot.new()
par(mar = c(5,3,3,3))
plot.window(xlim = c(-1, x), ylim = c(0, y), asp = 1)
print_ts(5, yoff)
col = '#ff5555'
print_segments(10.57, yoff-.43, seg = polygon_1, col = col)
x = 3
segments(c(2, 0, -1.3)+x, c(-.2, 0, 1)+yoff, c(0, -1.3, 1)+x, c(0, 1, 2)+yoff, lwd = 4, col = col)
print_segments(10.57+off, yoff-.43, seg = polygon_1, col = col)
x = 3 + off
segments(c(2, 0, -1.3)+x, c(-.2, 0, 1)+yoff, c(0, -1.3, 1)+x, c(0, 1, 2)+yoff, lwd = 4, col = col)
print_segments(10.57+2*off, yoff-.43, seg = polygon_1, col = col)
x = 3 + 2 * off
segments(c(2, 0, -1.3)+x, c(-.2, 0, 1)+yoff, c(0, -1.3, 1)+x, c(0, 1, 2)+yoff, lwd = 4, col = col)
# old 75 28 poly 86.25, 27.15
pl_stack_agg(a, 5, 5, nrM = 1, inner = 3) # print masked enlargement
print_segments(16.25, 7.15, seg = polygon_1, by = 2, col = col)
x <- 1
y <- 8
segments(c(4, 0, -2.6)+x, c(-.4, 0, 2)+y, c(0, -2.6, 2)+x, c(0, 2, 4)+y, lwd = 4, col = col) # line in large
segments(-3, 25, -7, 9, lwd = 3, col = 'grey')
segments(21, 29, 27.5, 14, lwd = 3, col = 'grey')
text(10, 20, "1. Group by geometry", cex = 1.3)
vecM <- matrix(rep(1,8), ncol = 2)
text(57, 21, "2. Reduce to vector cube", cex = 1.3)
b <- make_dummy_stars(2, 4, 12, 4, 0)
pl(b, 48, -5, m = vecM, pal = alpha("white", 0.9), border = 0)
print_vector_content(54, -3)
pl(b, 46.5, -3.5, m = vecM, pal = alpha("white", 0.9), border = 0)
print_vector_content(52.5, -1.5)
pl(b, 45, -2, m = vecM, pal = alpha("white", 0.9), border = 0)
print_vector_content(51, 0)
text(51.5, 15, "Line_1", col = col)
text(63, 15, "Polygon_1", col = col)
text(57, 17.5, "Geometries", cex = 1.1)
text(42, 12, "blue")
text(42, 8, "green")
text(42, 4, "red")
text(42, 0, "nir")
text(38, 6, "Bands", srt = 90, cex = 1.1)
# arrows(13.5, -2, 13.5, -6, angle = 20, lwd = 3)
text(72, 15.5, "time", srt = 315, cex = 1.1)
arrows(69.5, 15, 72.5, 12, angle = 20, lwd = 2)
```

Further examples of vector data cubes include air quality data, where we could have \(PM_{10}\) measurements over two dimensions:

- a sequence of monitoring stations, and
- a sequence of time intervals

or where we consider time series of demographic or epidemiological data, consisting of (population, disease) counts, with number of persons

- by region, for a sequence of \(n\) regions
- by age class, for \(m\) age classes, and
- by year, for \(p\) years

which forms an array with \(n m p\) elements.

For spatial data science, support of vector and raster data cubes is extremely useful, because many variables are both spatially and temporaly varying, and because we often want to either change dimensions or aggregate them out, but in a fully flexible manner and order. Examples of changing dimensions are:

- interpolating air quality measurements to values on a regular grid (raster; chapter 12)
- estimating density maps from points or lines, e.g. with the number of flights passing by per week within a range of 1 km (chapter 11)
- aggregating climate model predictions to summary indicators for administrative regions
- combining Earth observation data from different sensors, e.g. MODIS (250 m pixels, every 16 days) with Sentinel-2 (10 m, every 5 days)

Examples of aggregating one ore more full dimensions are assessments of:

- which air quality monitoring stations indicate unhealthy conditions (time)
- which region has the highest increase in disease incidence (space, time)
- global warming (e.g. in degrees per year)

## 6.5 Switching dimension with attributes

When we accept that a dimension can also reflect an unordered, categorical variable, then one can easily swap a set of attributes for a single dimension, by replacing

\[\{Z_1,Z_2,...,Z_p\} = f(D_1,D_2,...,D_n)\]

with

\[Z = f(D_1,D_2,...,D_n, D_{n+1})\]

where \(D_{n+1}\) has cardinality \(p\) and has as labels (the names
of) \(Z_1,Z_2,...,Z_p\). Figure 6.9 shows a vector data
cube for air quality stations where one cube dimension reflects
air quality parameters. When the \(Z_i\) have incompatible measurement
units, as in figure 6.9, one would have to take care
when reducing the “parameter” dimension \(D_{n+1}\): numeric functions
like `mean`

or `max`

would be meaningless. Counting the number of
variables that exceed their respective threshold values may however
be meaningful.

```
set.seed(1331)
library(stars)
library(colorspace)
tif = system.file("tif/L7_ETMs.tif", package = "stars")
r = read_stars(tif)
nrow = 5
ncol = 8
#m = matrix(runif(nrow * ncol), nrow = nrow, ncol = ncol)
m = r[[1]][1:nrow,1:ncol,1]
dim(m) = c(x = nrow, y = ncol) # named dim
s = st_as_stars(m)
# s
attr(s, "dimensions")[[1]]$delta = 3
attr(s, "dimensions")[[2]]$delta = -.5
attr(attr(s, "dimensions"), "raster")$affine = c(-1.2, 0.0)
plt = function(x, yoffset = 0, add, li = TRUE) {
attr(x, "dimensions")[[2]]$offset = attr(x, "dimensions")[[2]]$offset + yoffset
l = st_as_sf(x, as_points = FALSE)
pal = sf.colors(10)
if (li)
pal = lighten(pal, 0.3 + rnorm(1, 0, 0.1))
if (! add)
plot(l, axes = FALSE, breaks = "equal", pal = pal, reset = FALSE, border = grey(.75), key.pos = NULL, main = NULL, xlab = "time")
else
plot(l, axes = TRUE, breaks = "equal", pal = pal, add = TRUE, border = grey(.75))
u = st_union(l)
plot(st_geometry(u), add = TRUE, col = NA, border = 'black', lwd = 2.5)
}
pl = function(s, x, y, add = TRUE, randomize = FALSE) {
attr(s, "dimensions")[[1]]$offset = x
attr(s, "dimensions")[[2]]$offset = y
m = r[[1]][y + 1:nrow,x + 1:ncol,1]
if (randomize)
m = m[sample(y + 1:nrow),x + 1:ncol]
dim(m) = c(x = nrow, y = ncol) # named dim
s[[1]] = m
plt(s, 0, add)
plt(s, 1, TRUE)
plt(s, 2, TRUE)
plt(s, 3, TRUE)
plt(s, 4, TRUE)
plt(s, 5, TRUE)
plt(s, 6, TRUE)
plt(s, 7, TRUE)
plt(s, 8, TRUE, FALSE)
}
# point vector data cube:
plot.new()
par(mar = c(5, 0, 5, 0))
plot.window(xlim = c(-10, 16), ylim = c(-2,12), asp = 1)
library(spacetime)
data(air)
de = st_geometry(st_normalize(st_as_sf(DE)))
#
pl(s, 0, 0, TRUE, randomize = TRUE)
de = de * 6 + c(-7, 9)
plot(de, add = TRUE, border = grey(.5))
text(-10, 0, "time", srt = -90, col = 'black')
text(-5, 7.5, "sensor location", srt = 25, col = 'black')
text( 7, 10.5, "air quality parameter", srt = 0, col = 'black')
text( 1.5, 8.5, expression(PM[10]), col = 'black', cex = .75)
text( 4.5, 8.5, expression(NO[x]), col = 'black', cex = .75)
text( 8, 8.5, expression(SO[4]), col = 'black', cex = .75)
text( 11, 8.5, expression(O[3]), col = 'black', cex = .75)
text( 14, 8.5, expression(CO), col = 'black', cex = .75)
# location points:
p = st_coordinates(s[,1])
p[,1] = p[,1]-1.4
p[,2] = p[,2] + 8.2
points(p, col = grey(.7), pch = 16)
# centroids:
set.seed(131)
cent = st_coordinates(st_sample(de, 8))
points(cent, col = grey(.7), pch = 16)
cent = cent[rev(order(cent[,1])),]
seg = cbind(p, cent[1:8,])
segments(seg[,1], seg[,2], seg[,3], seg[,4], col = 'grey')
```

Being able to swap dimensions to attributes flexibly and vice-versa leads to extremely flexible analysis possibilities, as e.g. shown by the array database SciDB (Brown 2010).

## 6.6 Other dynamic spatial data

We have seen several dynamic raster and vector data examples that match the data cube structure well. Other data examples do less so: in particular spatiotemporal point patterns (chapter 11) and trajectories (movement data; for a recent review, see Joo et al. (2020)) are often more straightforward to not handle as a data cube. Spatiotemporal point patterns are the sets of spatiotemporal coordinates of events or objects: accidents, disease cases, traffic jams, lightning strikes, and so on. Trajectory data are time sequences of spatial locations of moving objects (persons, cars, satellites, animals). For such data, the primary information is in the coordinates, and shifting these to a limited set of regularly discretized grid cells covering the space may help some analysis, e.g. to quickly explore patterns in areas of higher densities, but the loss of the exact coordinates also hinders a number of analysis approaches involving distance, direction or speed calculations. Nevertheless, for such data often the first computational steps involves generation of data cube representations by aggregating to a time-fixed spatial and/or space-fixed temporal discretization.

Using sparse array representations of data cubes to represent point pattern or trajectory data, as e.g. offered by SciDB (Brown 2010) or TileDB (Papadopoulos et al. 2016), may strongly limit the loss of coordinate accuracy by choosing dimensions that represent an extremely dense grid, and storing only those grid cell that contain data points. For trajectory data, such representations would need to add a grouping dimension to identify individuals, or individual sequences of consecutive movement observations.

## 6.7 Exercises

Use words to solve the following exercises, if needed or relevant use R code to illustrate the argument(s).

- Why is it difficult to represent trajectories, sequences of \((x,y,t)\) obtained by tracking moving objects, by data cubes as described in this chapter?
- In a socio-economic vector data cube with variables population, life expectancy, and gross domestic product ordered by dimensions country and year, which variables have block support for the spatial dimension, and which have block support for the temporal dimension?
- The Sentinel-2 satellites collect images in 12 spectral bands; list advantages and disadvantages to represent them as (i) different data cubes, (ii) a data cube with 12 attributes, one for each band, and (iii) a single attribute data cube with a spectral dimension.
- Explain why a curvilinear raster as shown in figure 1.5 can be considered a special case of a data cube.
Explain how the following problems can be solved with data cube operations

`filter`

,`apply`

,`reduce`

and/or`aggregate`

, and in which order. Also mention for each which function is applied, and what the dimensionality of the resulting data cube is (if any):- from hourly \(PM_{10}\) measurements for a set of air quality monitoring stations, compute per station the amount of days per year that the average daily \(PM_{10}\) value exceeds 50 \(\mu g/m^3\)
- for a sequence of aerial images of an oil spill, find the time at which the oil spill had its largest extent, and the corresponding extent
- from a 10-year period with global daily sea surface temperature (SST) raster maps, find the area with the 10% largest and 10% smallest temporal trends in SST values.

### References

Brown, Paul G. 2010. “Overview of SciDB: Large Scale Array Storage, Processing and Analysis.” In *Proceedings of the 2010 ACM SIGMOD International Conference on Management of Data*, 963–68. ACM.

Galton, A. 2004. “Fields and Objects in Space, Time and Space-Time.” *Spatial Cognition and Computation* 4.

Joo, Rocío, Matthew E. Boone, Thomas A. Clay, Samantha C. Patrick, Susana Clusella-Trullas, and Mathieu Basille. 2020. “Navigating Through the R Packages for Movement.” *Journal of Animal Ecology* 89 (1): 248–67. https://doi.org/https://doi.org/10.1111/1365-2656.13116.

Lu, Meng, Marius Appel, and Edzer Pebesma. 2018. “Multidimensional Arrays for Analysing Geoscientific Data.” *ISPRS International Journal of Geo-Information* 7 (8): 313.

Papadopoulos, Stavros, Kushal Datta, Samuel Madden, and Timothy Mattson. 2016. “The Tiledb Array Data Storage Manager.” *Proceedings of the VLDB Endowment* 10 (4): 349–60.