5  What are transforms?

5.1 Motivation

In Part I of the book, we learned that our GeoTable representation of geospatial data provides the data access pattern of the DataFrame, a feature that is very convenient for data science. To recap, let’s consider the following geotable with four random variables:

N = 10000
a = [2randn(N÷2) .+ 6; randn(N÷2)]
b = [3randn(N÷2); 2randn(N÷2)]
c = randn(N)
d = c .+ 0.6randn(N)

table = (; a, b, c, d)

gtb = georef(table, CartesianGrid(100, 100))

9990 rows omitted

10000×5 GeoTable over 100×100 CartesianGrid
a	b	c	d	geometry
Continuous	Continuous	Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	0.0486913	0.213044	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	-0.953847	-1.9141	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	0.279604	1.93854	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	0.23392	0.373733	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	-2.22837	-2.20047	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	0.673263	1.04732	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	0.577041	0.542995	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	0.165784	0.439732	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	-0.61076	-0.14848	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	-0.796178	-0.407364	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮	⋮	⋮

We can easily retrieve the “a” column of the geotable as a vector, and plot its histogram:

Mke.hist(gtb.a, color = "gray80")

We can compute the cross-correlation between columns “a” and “b”:

cor(gtb.a, gtb.b)

0.013963135945675801

And inspect bivariate distributions of the values of the geotable with PairPlots.jl by Thompson (2023):

using PairPlots

pairplot(values(gtb))

This pattern is useful to answer geoscientific questions via marginal analysis (i.e. entire columns treated as measurements of a single random variable). However, the answers to many questions in geosciences depend on where the measurements were made.

Attempting to answer geoscientific questions with basic access to rows and columns can be very frustrating. In particular, this approach is prone to unintentional removal of geospatial information:

gtb.a

10000-element Vector{Float64}:
  6.858805187538858
  6.708506186608441
  6.498515181388323
  7.351731309799403
  5.732813587949375
  2.8024934108195043
  7.368501190403044
  5.946786137471679
  5.679291258127283
  7.706094697145358
  ⋮
 -0.8099899630294426
  0.9952759820943204
  0.0851955286875299
  0.5021485240218768
  1.6307225844094215
  1.6632900041283243
  0.03767354162417888
  0.36596494524883627
  0.6105847738837629

Note

Any script that is written in terms of direct column access has the potential to discard the special geometry column, and become unreadable very quickly with the use of auxiliary indices for rows.

We propose a new approach to geospatial data science with the concept of transforms, which we introduce in three classes with practical examples:

1. Feature transforms
2. Geometric transforms
3. Geospatial transforms

5.2 Feature transforms

A feature transform is a function that takes the values of the geotable and produces a new set of values over the same geospatial domain. The framework provides over 30 such transforms, ranging from basic selection of columns, to data cleaning, to advanced multivariate statistical transforms.

5.2.1 Basic

Let’s start with two basic and important transforms, Select and Reject. The Select transform can be used to select columns of interest from a geotable:

gtb |> Select("a", "b") # select columns "a" and "b"

9990 rows omitted

10000×3 GeoTable over 100×100 CartesianGrid
a	b	geometry
Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮

In the example above, we selected the columns “a” and “b” explicitly, but Select has various methods for more flexible column selection:

gtb |> Select(1:3) # select columns 1 to 3

9990 rows omitted

10000×4 GeoTable over 100×100 CartesianGrid
a	b	c	geometry
Continuous	Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	0.0486913	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	-0.953847	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	0.279604	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	0.23392	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	-2.22837	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	0.673263	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	0.577041	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	0.165784	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	-0.61076	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	-0.796178	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮	⋮

gtb |> Select(r"[bcd]") # columns matching regular expression

9990 rows omitted

10000×4 GeoTable over 100×100 CartesianGrid
b	c	d	geometry
Continuous	Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
-0.384438	0.0486913	0.213044	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
-2.51621	-0.953847	-1.9141	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
0.69291	0.279604	1.93854	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
-3.10133	0.23392	0.373733	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
0.199764	-2.22837	-2.20047	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
1.72443	0.673263	1.04732	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
1.12664	0.577041	0.542995	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
3.12004	0.165784	0.439732	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
3.67148	-0.61076	-0.14848	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
-2.00082	-0.796178	-0.407364	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮	⋮

A convenient method is also provided to select and rename columns:

gtb |> Select("a" => "A", "b" => "B")

9990 rows omitted

10000×3 GeoTable over 100×100 CartesianGrid
A	B	geometry
Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮

The Reject transform can be used to reject columns from a geotable that are not relevant for a given analysis. It supports the same column specification of Select:

gtb |> Reject("b") # reject column "b"

9990 rows omitted

10000×4 GeoTable over 100×100 CartesianGrid
a	c	d	geometry
Continuous	Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	0.0486913	0.213044	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-0.953847	-1.9141	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.279604	1.93854	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	0.23392	0.373733	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	-2.22837	-2.20047	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	0.673263	1.04732	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	0.577041	0.542995	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	0.165784	0.439732	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	-0.61076	-0.14848	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-0.796178	-0.407364	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮	⋮

Note

Unlike direct column access, the Select and Reject transforms preserve geospatial information.

NoteTip for all users

The Select transform can be used in conjunction with the viewer to quickly visualize a specific variable:

gtb |> Select("a") |> viewer

The Rename transform can be used to rename specific columns of a geotable. It preserves all other columns that are not part of the column specification:

gtb |> Rename("a" => "A", "b" => "B")

9990 rows omitted

10000×5 GeoTable over 100×100 CartesianGrid
A	B	c	d	geometry
Continuous	Continuous	Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	0.0486913	0.213044	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	-0.953847	-1.9141	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	0.279604	1.93854	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	0.23392	0.373733	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	-2.22837	-2.20047	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	0.673263	1.04732	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	0.577041	0.542995	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	0.165784	0.439732	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	-0.61076	-0.14848	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	-0.796178	-0.407364	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮	⋮	⋮

The Identity transform can be used as a placeholder to forward the geotable without modifications to the next transform:

gtb |> Identity()

9990 rows omitted

10000×5 GeoTable over 100×100 CartesianGrid
a	b	c	d	geometry
Continuous	Continuous	Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	0.0486913	0.213044	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	-0.953847	-1.9141	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	0.279604	1.93854	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	0.23392	0.373733	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	-2.22837	-2.20047	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	0.673263	1.04732	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	0.577041	0.542995	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	0.165784	0.439732	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	-0.61076	-0.14848	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	-0.796178	-0.407364	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮	⋮	⋮

The RowTable and ColTable transforms change the underlying table representation of the values of the geotable as discussed in the first chapter of the book:

rtb = gtb |> RowTable()

9990 rows omitted

10000×5 GeoTable over 100×100 CartesianGrid
a	b	c	d	geometry
Continuous	Continuous	Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	0.0486913	0.213044	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	-0.953847	-1.9141	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	0.279604	1.93854	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	0.23392	0.373733	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	-2.22837	-2.20047	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	0.673263	1.04732	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	0.577041	0.542995	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	0.165784	0.439732	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	-0.61076	-0.14848	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	-0.796178	-0.407364	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮	⋮	⋮

rtb |> values |> typeof

Vector{@NamedTuple{a::Float64, b::Float64, c::Float64, d::Float64}} (alias for Array{@NamedTuple{a::Float64, b::Float64, c::Float64, d::Float64}, 1})

The Functional transform can be used to apply a function to columns of a geotable in place:

gtb |> Functional(cos) |> values |> pairplot

gtb |> Functional("a" => cos, "b" => sin) |> values |> pairplot

The Map transform can be used to create new columns from existing columns in the geotable. It takes a column specification, calls a function on the selected columns row-by-row, and returns the result as a new column:

gtb |> Map("a" => sin, "b" => cos => "cos(b)")

9990 rows omitted

10000×3 GeoTable over 100×100 CartesianGrid
sin_a	cos(b)	geometry
Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
0.544355	0.927009	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
0.412613	-0.810739	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
0.21367	0.76939	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
0.876501	-0.99919	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
-0.523004	0.980113	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
0.332638	-0.153027	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
0.884451	0.429698	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
-0.33009	-0.999768	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
-0.567852	-0.862862	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
0.989085	-0.41689	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮

gtb |> Map([2, 3] => ((b, c) -> 2b + c) => "f(b, c)")

9990 rows omitted

10000×2 GeoTable over 100×100 CartesianGrid
f(b, c)	geometry
Continuous	Quadrangle
[NoUnits]	🖈 Cartesian{NoDatum}
-0.720186	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
-5.98627	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
1.66543	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
-5.96874	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
-1.82885	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
4.12212	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
2.83032	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
6.40586	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
6.73221	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
-4.79781	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮

The name of the resulting column can be provided or omitted. If the name is omitted like in the example above with the column “a”, it is created by concatenation of column and function names.

To filter rows in the geotable based on a given predicate (i.e., a function that returns true or false), we can use the Filter transform:

gtb |> Filter(row -> row.a < 0 && row.b > 0)

1240 rows omitted

1250×5 GeoTable over 1250 view(::CartesianGrid, [545, 1657, 4125, 5002, ..., 9983, 9984, 9989, 9990])
a	b	c	d	geometry
Continuous	Continuous	Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
-0.448396	5.57693	-0.385385	0.9321	Quadrangle((x: 44.0 m, y: 5.0 m), ..., (x: 44.0 m, y: 6.0 m))
-1.28025	0.837619	1.92832	1.8777	Quadrangle((x: 56.0 m, y: 16.0 m), ..., (x: 56.0 m, y: 17.0 m))
-0.90238	2.72992	0.800659	1.15751	Quadrangle((x: 24.0 m, y: 41.0 m), ..., (x: 24.0 m, y: 42.0 m))
-0.00457009	0.0328381	-0.116339	-0.762298	Quadrangle((x: 1.0 m, y: 50.0 m), ..., (x: 1.0 m, y: 51.0 m))
-0.702477	1.36537	0.144648	0.284003	Quadrangle((x: 3.0 m, y: 50.0 m), ..., (x: 3.0 m, y: 51.0 m))
-0.315433	2.84356	0.47901	0.0857618	Quadrangle((x: 15.0 m, y: 50.0 m), ..., (x: 15.0 m, y: 51.0 m))
-2.18453	0.247497	-0.342363	-1.83445	Quadrangle((x: 24.0 m, y: 50.0 m), ..., (x: 24.0 m, y: 51.0 m))
-1.13553	1.93667	0.23416	0.513091	Quadrangle((x: 26.0 m, y: 50.0 m), ..., (x: 26.0 m, y: 51.0 m))
-1.70005	1.56619	-0.279746	-0.290534	Quadrangle((x: 44.0 m, y: 50.0 m), ..., (x: 44.0 m, y: 51.0 m))
-0.912214	2.16239	-0.318865	0.330739	Quadrangle((x: 48.0 m, y: 50.0 m), ..., (x: 48.0 m, y: 51.0 m))
⋮	⋮	⋮	⋮	⋮

To sort rows based on specific columns we can use the Sort transform:

gtb |> Sort("a", "b")

9990 rows omitted

10000×5 GeoTable over 10000 view(::CartesianGrid, [9001, 6043, 7533, 7747, ..., 3288, 1599, 2836, 24])
a	b	c	d	geometry
Continuous	Continuous	Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
-3.7765	0.00749113	1.79548	0.355031	Quadrangle((x: 0.0 m, y: 90.0 m), ..., (x: 0.0 m, y: 91.0 m))
-3.73386	-1.86234	-0.516951	-1.04768	Quadrangle((x: 42.0 m, y: 60.0 m), ..., (x: 42.0 m, y: 61.0 m))
-2.98602	1.90583	-0.39574	0.334386	Quadrangle((x: 32.0 m, y: 75.0 m), ..., (x: 32.0 m, y: 76.0 m))
-2.95241	0.705734	1.07332	1.2065	Quadrangle((x: 46.0 m, y: 77.0 m), ..., (x: 46.0 m, y: 78.0 m))
-2.94824	2.11828	-0.627909	-0.725522	Quadrangle((x: 36.0 m, y: 76.0 m), ..., (x: 36.0 m, y: 77.0 m))
-2.87803	0.676606	0.0405132	-0.745099	Quadrangle((x: 66.0 m, y: 81.0 m), ..., (x: 66.0 m, y: 82.0 m))
-2.86149	1.36436	1.14843	2.36141	Quadrangle((x: 80.0 m, y: 90.0 m), ..., (x: 80.0 m, y: 91.0 m))
-2.81949	4.04475	0.181401	-0.376488	Quadrangle((x: 92.0 m, y: 84.0 m), ..., (x: 92.0 m, y: 85.0 m))
-2.81306	0.810338	-0.509473	-0.189381	Quadrangle((x: 26.0 m, y: 69.0 m), ..., (x: 26.0 m, y: 70.0 m))
-2.79825	1.72454	-0.731735	-0.912083	Quadrangle((x: 29.0 m, y: 89.0 m), ..., (x: 29.0 m, y: 90.0 m))
⋮	⋮	⋮	⋮	⋮

This transform accepts all options of the sortperm function in Julia, including the option to sort in reverse order:

gtb |> Sort("a", "b", rev=true)

9990 rows omitted

10000×5 GeoTable over 10000 view(::CartesianGrid, [24, 2836, 1599, 3288, ..., 7747, 7533, 6043, 9001])
a	b	c	d	geometry
Continuous	Continuous	Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
13.2762	2.70473	-0.623466	-0.541657	Quadrangle((x: 23.0 m, y: 0.0 m), ..., (x: 23.0 m, y: 1.0 m))
13.1167	-4.4244	0.0620008	0.479773	Quadrangle((x: 35.0 m, y: 28.0 m), ..., (x: 35.0 m, y: 29.0 m))
12.9939	-3.29937	1.17894	0.225023	Quadrangle((x: 98.0 m, y: 15.0 m), ..., (x: 98.0 m, y: 16.0 m))
12.9193	0.823505	-0.703004	-0.32756	Quadrangle((x: 87.0 m, y: 32.0 m), ..., (x: 87.0 m, y: 33.0 m))
12.4635	1.45564	1.84552	1.76056	Quadrangle((x: 6.0 m, y: 18.0 m), ..., (x: 6.0 m, y: 19.0 m))
12.4067	-1.27658	1.50589	1.15777	Quadrangle((x: 47.0 m, y: 10.0 m), ..., (x: 47.0 m, y: 11.0 m))
12.3768	1.55913	-0.562865	0.251249	Quadrangle((x: 84.0 m, y: 3.0 m), ..., (x: 84.0 m, y: 4.0 m))
12.2327	0.293555	-0.15182	-0.825646	Quadrangle((x: 37.0 m, y: 11.0 m), ..., (x: 37.0 m, y: 12.0 m))
12.0943	-3.91462	2.10218	2.4034	Quadrangle((x: 14.0 m, y: 49.0 m), ..., (x: 14.0 m, y: 50.0 m))
12.0817	-2.20415	0.806313	0.214543	Quadrangle((x: 70.0 m, y: 8.0 m), ..., (x: 70.0 m, y: 9.0 m))
⋮	⋮	⋮	⋮	⋮

5.2.2 Cleaning

Some feature transforms are used to clean the data before geostatistical analysis. For example, the StdNames transform can be used to standardize variable names that are not very readable due to file format limitations. To illustrate this transform, let’s create a geotable with unreadable variable names:

utb = gtb |> Select("a" => "aBc De-F", "b" => "b_2 (1)")

9990 rows omitted

10000×3 GeoTable over 100×100 CartesianGrid
aBc De-F	b_2 (1)	geometry
Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮

We can standardize the names with:

utb |> StdNames()

9990 rows omitted

10000×3 GeoTable over 100×100 CartesianGrid
ABC_DE_F	B_2_1	geometry
Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮

By default the transform, uses the :uppersnake naming convention. Other conventions can be specified depending on personal preference:

utb |> StdNames(:uppercamel)

9990 rows omitted

10000×3 GeoTable over 100×100 CartesianGrid
AbcDeF	B21	geometry
Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮

utb |> StdNames(:upperflat)

9990 rows omitted

10000×3 GeoTable over 100×100 CartesianGrid
ABCDEF	B21	geometry
Continuous	Continuous	Quadrangle
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
6.85881	-0.384438	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 1.0 m))
6.70851	-2.51621	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 1.0 m))
6.49852	0.69291	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 1.0 m))
7.35173	-3.10133	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 1.0 m))
5.73281	0.199764	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 1.0 m))
2.80249	1.72443	Quadrangle((x: 5.0 m, y: 0.0 m), ..., (x: 5.0 m, y: 1.0 m))
7.3685	1.12664	Quadrangle((x: 6.0 m, y: 0.0 m), ..., (x: 6.0 m, y: 1.0 m))
5.94679	3.12004	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 1.0 m))
5.67929	3.67148	Quadrangle((x: 8.0 m, y: 0.0 m), ..., (x: 8.0 m, y: 1.0 m))
7.70609	-2.00082	Quadrangle((x: 9.0 m, y: 0.0 m), ..., (x: 9.0 m, y: 1.0 m))
⋮	⋮	⋮

The Replace transform can be used to replace specific values in the geotable by new values that are meaningful to the analysis. For example, we can replace the values -999 and NaN that are used to represent missing values in some file formats:

rtb = georef((a=[1,-999,3], b=[NaN,5,6]))

3×3 GeoTable over 3 CartesianGrid
a	b	geometry
Categorical	Continuous	Segment
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
1	NaN	Segment((x: 0.0 m), (x: 1.0 m))
-999	5.0	Segment((x: 1.0 m), (x: 2.0 m))
3	6.0	Segment((x: 2.0 m), (x: 3.0 m))

rtb |> Replace(-999 => missing, NaN => missing)

3×3 GeoTable over 3 CartesianGrid
a	b	geometry
Categorical	Continuous	Segment
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
1	missing	Segment((x: 0.0 m), (x: 1.0 m))
missing	5.0	Segment((x: 1.0 m), (x: 2.0 m))
3	6.0	Segment((x: 2.0 m), (x: 3.0 m))

or replace all negative values using a predicate function:

rtb |> Replace(<(0) => missing)

3×3 GeoTable over 3 CartesianGrid
a	b	geometry
Categorical	Continuous	Segment
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
1	NaN	Segment((x: 0.0 m), (x: 1.0 m))
missing	5.0	Segment((x: 1.0 m), (x: 2.0 m))
3	6.0	Segment((x: 2.0 m), (x: 3.0 m))

Note

In Julia, the expression <(0) is equivalent to the predicate function x -> x < 0.

Although Replace could be used to replace missing values by new values, there is a specific transform for this purpose named Coalesce:

ctb = georef((a=[1,missing,3], b=[4,5,6])) |> Coalesce(value=2)

3×3 GeoTable over 3 CartesianGrid
a	b	geometry
Categorical	Categorical	Segment
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
1	4	Segment((x: 0.0 m), (x: 1.0 m))
2	5	Segment((x: 1.0 m), (x: 2.0 m))
3	6	Segment((x: 2.0 m), (x: 3.0 m))

Note

Unlike Replace, the Coalesce transform also changes the column type to make sure that no missing values can be stored in the future:

typeof(ctb.a)

Vector{Int64} (alias for Array{Int64, 1})

In many applications, it is enough to simply drop all rows for which the selected column values are missing. This is the purpose of the DropMissing transform:

georef((a=[1,missing,3], b=[4,5,6])) |> DropMissing()

2×3 GeoTable over 2 view(::CartesianGrid, [1, 3])
a	b	geometry
Categorical	Categorical	Segment
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
1	4	Segment((x: 0.0 m), (x: 1.0 m))
3	6	Segment((x: 2.0 m), (x: 3.0 m))

The DropNaN is an alternative to drop all rows for which the selected column values are NaN.

5.2.3 Statistical

The framework provides various feature transforms for statistical analysis. We will cover some of these transforms in more detail in Part V of the book with real data. In the following examples we illustrate the most basic statistical transforms with synthetic data.

The Sample transform can be used to sample rows of the geotable at random, with or without replacement depending on the replace option. Other options are available such as rng to set the random number generator and ordered to preserve the order of rows in the original geotable:

gtb |> Sample(1000, replace=false) |> viewer

Note

Similar to Filter and Sort, the Sample transform is lazy. It simply stores the indices of sampled rows for future construction of the new geotable.

The Center and Scale transforms can be used to standardize the range of values in a geotable. Aliases are provided for specific types of Scale such as MinMax and Interquartile. We can use the describe function to visualize basic statistics before and after the transforms:

gtb |> describe

Table with 6 columns and 4 rows:
     variable  mean         minimum   median       maximum  nmissing
   ┌────────────────────────────────────────────────────────────────
 1 │ a         3.01902      -3.7765   1.94986      13.2762  0
 2 │ b         0.0475736    -10.9491  0.0532241    10.452   0
 3 │ c         0.000573241  -3.89229  -0.00935594  3.60953  0
 4 │ d         0.000545703  -4.48257  -0.0155962   4.42181  0

gtb |> Center("a") |> describe

Table with 6 columns and 4 rows:
     variable  mean          minimum   median       maximum  nmissing
   ┌─────────────────────────────────────────────────────────────────
 1 │ a         -3.63798e-16  -6.79552  -1.06916     10.2571  0
 2 │ b         0.0475736     -10.9491  0.0532241    10.452   0
 3 │ c         0.000573241   -3.89229  -0.00935594  3.60953  0
 4 │ d         0.000545703   -4.48257  -0.0155962   4.42181  0

gtb |> MinMax() |> describe

Table with 6 columns and 4 rows:
     variable  mean      minimum  median    maximum  nmissing
   ┌─────────────────────────────────────────────────────────
 1 │ a         0.398502  0.0      0.335804  1.0      0
 2 │ b         0.513838  0.0      0.514102  1.0      0
 3 │ c         0.518922  0.0      0.517599  1.0      0
 4 │ d         0.503473  0.0      0.501661  1.0      0

The ZScore transform is similar to the Scale transform, but it uses the mean and the standard deviation to standardize the range:

gtb |> ZScore() |> describe

Table with 6 columns and 4 rows:
     variable  mean          minimum   median       maximum  nmissing
   ┌─────────────────────────────────────────────────────────────────
 1 │ a         0.0           -1.99223  -0.313444    3.00706  0
 2 │ b         -1.84741e-17  -4.3683   0.00224463   4.13301  0
 3 │ c         -1.56319e-17  -3.88728  -0.00991493  3.60378  0
 4 │ d         -2.41585e-17  -3.86172  -0.0139045   3.80844  0

Another important univariate transform is the Quantile transform, which can be used to convert empirical distribution in a column of the geotable to any given distribution from Distributions.jl by Lin et al. (2023). Selected columns are converted to a Normal distribution by default, but more than 60 distributions are available:

gtb |> Quantile() |> values |> pairplot

In data science, scientific traits are used to link data types to adequate statistical algorithms. The most popular scientific traits encountered in geoscientific applications are the Continuous and the Categorical scientific traits. To convert (or coerce) the scientific traits of columns in a geotable, we can use the Coerce transform:

stb = georef((a=[1,2,2,2,3,3], b=[1,2,3,4,5,6])) |> Coerce("b" => Continuous)

6×3 GeoTable over 6 CartesianGrid
a	b	geometry
Categorical	Continuous	Segment
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
1	1.0	Segment((x: 0.0 m), (x: 1.0 m))
2	2.0	Segment((x: 1.0 m), (x: 2.0 m))
2	3.0	Segment((x: 2.0 m), (x: 3.0 m))
2	4.0	Segment((x: 3.0 m), (x: 4.0 m))
3	5.0	Segment((x: 4.0 m), (x: 5.0 m))
3	6.0	Segment((x: 5.0 m), (x: 6.0 m))

eltype(stb.b)

Float64

Note

All scientific traits are documented in the DataScienceTraits.jl module, and can be used to select variables:

stb |> Only(Continuous)

6×2 GeoTable over 6 CartesianGrid
b	geometry
Continuous	Segment
[NoUnits]	🖈 Cartesian{NoDatum}
1.0	Segment((x: 0.0 m), (x: 1.0 m))
2.0	Segment((x: 1.0 m), (x: 2.0 m))
3.0	Segment((x: 2.0 m), (x: 3.0 m))
4.0	Segment((x: 3.0 m), (x: 4.0 m))
5.0	Segment((x: 4.0 m), (x: 5.0 m))
6.0	Segment((x: 5.0 m), (x: 6.0 m))

The Levels transform can be used to adjust the categories (or levels) of Categorical columns in case the sampling process does not include all possible values:

stb = stb |> Levels("a" => [1,2,3,4])

6×3 GeoTable over 6 CartesianGrid
a	b	geometry
Categorical	Continuous	Segment
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
1	1.0	Segment((x: 0.0 m), (x: 1.0 m))
2	2.0	Segment((x: 1.0 m), (x: 2.0 m))
2	3.0	Segment((x: 2.0 m), (x: 3.0 m))
2	4.0	Segment((x: 3.0 m), (x: 4.0 m))
3	5.0	Segment((x: 4.0 m), (x: 5.0 m))
3	6.0	Segment((x: 5.0 m), (x: 6.0 m))

levels(stb.a)

4-element CategoricalArrays.CategoricalArray{Int64,1,UInt32}:
 1
 2
 3
 4

Another popular transform in statistical learning is the OneHot transform. It converts a Categorical column into multiple columns of true/false values, one column for each level:

stb |> OneHot("a")

6×6 GeoTable over 6 CartesianGrid
a_1	a_2	a_3	a_4	b	geometry
Categorical	Categorical	Categorical	Categorical	Continuous	Segment
[NoUnits]	[NoUnits]	[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
true	false	false	false	1.0	Segment((x: 0.0 m), (x: 1.0 m))
false	true	false	false	2.0	Segment((x: 1.0 m), (x: 2.0 m))
false	true	false	false	3.0	Segment((x: 2.0 m), (x: 3.0 m))
false	true	false	false	4.0	Segment((x: 3.0 m), (x: 4.0 m))
false	false	true	false	5.0	Segment((x: 4.0 m), (x: 5.0 m))
false	false	true	false	6.0	Segment((x: 5.0 m), (x: 6.0 m))

A similar transform for Continuous columns is the Indicator transform. It converts the column into multiple columns based on threshold values on the support of the data. By default, the threshold values are computed on a quantile scale:

stb |> Indicator("b", k=3, scale=:quantile)

6×5 GeoTable over 6 CartesianGrid
a	b_1	b_2	b_3	geometry
Categorical	Categorical	Categorical	Categorical	Segment
[NoUnits]	[NoUnits]	[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
1	true	true	true	Segment((x: 0.0 m), (x: 1.0 m))
2	true	true	true	Segment((x: 1.0 m), (x: 2.0 m))
2	false	true	true	Segment((x: 2.0 m), (x: 3.0 m))
2	false	true	true	Segment((x: 3.0 m), (x: 4.0 m))
3	false	false	true	Segment((x: 4.0 m), (x: 5.0 m))
3	false	false	true	Segment((x: 5.0 m), (x: 6.0 m))

More advanced statistical transforms such as EigenAnalysis, PCA, DRS, SDS, ProjectionPursuit for multivariate data analysis and Remainder, Closure, LogRatio, ALR, CLR, ILR for compositional data analysis will be covered in future chapters.

5.3 Geometric transforms

While feature transforms operate on the values of the geotable, geometric transforms operate on the geospatial domain. The framework provides various geometric transforms for 2D and 3D space.

5.3.1 Coordinate

A coordinate transform is a geometric transform that modifies the coordinates of all points in the domain without any advanced topological modification (i.e., connectivities are preserved). The most prominent examples of coordinate transforms are Translate, Rotate and Scale.

Let’s load an additional geotable to see these transforms in action:

using GeoIO

btb = GeoIO.load("data/beethoven.ply")

viz(btb.geometry)

The Beethoven domain has been saved in the .ply file in a position that is not ideal for visualization. We can rotate this domain with any active rotation specification from Rotations.jl by Koolen et al. (2023) to improve the visualization. For example, we can specify that we want to rotate all points in the mesh by analogy with a rotation between coordinates (0, 1, 0) and coordinates (0, 0, 1):

rtb = btb |> Rotate((0, 1, 0), (0, 0, 1))

viz(rtb.geometry)

Beethoven is now standing up, but still facing the wall. Let’s rotate it once again by analogy between coordinates (1, 0, 0) and (-1, 1, 0):

rtb = rtb |> Rotate((1, 0, 0), (-1, 1, 0))

viz(rtb.geometry)

Rotation specifications are also available in 2D space. As an example, we can rotate the 2D grid of our synthetic geotable by the counter clockwise angle π/4:

gtb |> Rotate(Angle2d(π/4)) |> viewer

In GIS, this new geotable would be called a rotated “raster”. As another example, let’s translate the geotable to the origin of the coordinate system with the Translate transform:

c = centroid(gtb.geometry)

gtb |> Translate(-to(c)...) |> viewer

and scale it with a positive factor for each dimension:

gtb |> Scale(0.1, 0.2) |> viewer

The StdCoords transform combines Translate and Scale to standardize the coordinates of the domain to the interval [-0.5, 0.5]:

gtb |> StdCoords() |> viewer

In GIS, another very important coordinate transform is the Proj transform. We will cover this transform in the next chapter because it depends on the concept of map projection, which deserves more attention.

Note

In our framework, the Proj transform is just another coordinate transform. It is implemented with the same code optimizations, and can be used in conjunction with many other transforms that are not available elsewhere.

5.3.2 Advanced

Advanced geometric transforms are provided that change the topology of the domain besides the coordinates of points. Some of these transforms can be useful to repair problematic geometries acquired from sensors in the real world.

The Repair transform is parameterized by an integer K that identifies the repair to be performed. For example, Repair{0}() is a transform that removes duplicated vertices and faces in a domain represented by a mesh. The Repair{9}() on the other hand fixes the orientation of rings in polygonal areas so that the external boundary is oriented counter clockwise and the inner boundaries are oriented clockwise. The list of available repairs will continue to grow with the implementation of new geometric algorithms in the framework.

To understand why geometric transforms are more general than coordinate transforms, let’s consider the following polygonal area with holes:

outer = [(0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0)]
hole1 = [(0.2, 0.2), (0.2, 0.4), (0.4, 0.4), (0.4, 0.2)]
hole2 = [(0.6, 0.2), (0.6, 0.4), (0.8, 0.4), (0.8, 0.2)]
poly  = PolyArea([outer, hole1, hole2])

viz(poly)

We can connect the holes with the external boundary (or ring) using the Bridge transform:

poly |> Bridge(0.01) |> viz

By looking at the visualization, we observe that the number of vertices changed to accommodate the so called “bridges” between the rings. The topology also changed as there are no holes in the resulting geometry.

As a final example of advanced geometric transform, we illustrate the TaubinSmoothing transform, which gradually removes sharp boundaries of a manifold mesh:

stb = btb |> TaubinSmoothing(30)

fig = Mke.Figure()
viz(fig[1,1], btb.geometry)
viz(fig[1,2], stb.geometry)
fig

For more advanced geometric transforms, please consult the official documentation.

5.4 Geospatial transforms

Geospatial transforms are those transforms that operate on both the values and the domain of the geotable. They are common in geostatistical workflows that need to remove geospatial “trends” or workflows that need to extract geometries from domains.

As an example, let’s consider the following geotable with a variable z that made of a trend component μ and a noise component ϵ:

# quadratic + noise
r = range(-1, stop=1, length=100)
μ = [x^2 + y^2 for x in r, y in r]
ϵ = 0.1rand(100, 100)
t = georef((z=μ+ϵ,))

viewer(t)

We can use the Detrend transform to remove a trend of polynomial degree 2:

t |> Detrend(degree=2) |> viewer

The remaining component can then be modeled with geostatistical models of geospatial correlation, which will be covered in Part IV of the book.

Models of geospatial correlation such as variograms (Hoffimann and Zadrozny 2019) require unique coordinates in the geotable and that is the purpose of the UniqueCoords transform. It removes duplicate points in the geotable and aggregates the values with custom aggregation functions.

Let’s consider the following geotable stored in a .png file to illustrate another geospatial transform:

letters = GeoIO.load("data/letters.png")

44245 rows omitted

44255×2 GeoTable over 265×167 TransformedGrid
color	geometry
Colorful	Quadrangle
[NoUnits]	🖈 Cartesian{NoDatum}
Gray{N0f8}(1.0)	Quadrangle((x: -1.62266e-14 m, y: 265.0 m), ..., (x: 1.0 m, y: 265.0 m))
Gray{N0f8}(1.0)	Quadrangle((x: -1.61653e-14 m, y: 264.0 m), ..., (x: 1.0 m, y: 264.0 m))
Gray{N0f8}(1.0)	Quadrangle((x: -1.61041e-14 m, y: 263.0 m), ..., (x: 1.0 m, y: 263.0 m))
Gray{N0f8}(1.0)	Quadrangle((x: -1.60429e-14 m, y: 262.0 m), ..., (x: 1.0 m, y: 262.0 m))
Gray{N0f8}(1.0)	Quadrangle((x: -1.59816e-14 m, y: 261.0 m), ..., (x: 1.0 m, y: 261.0 m))
Gray{N0f8}(1.0)	Quadrangle((x: -1.59204e-14 m, y: 260.0 m), ..., (x: 1.0 m, y: 260.0 m))
Gray{N0f8}(1.0)	Quadrangle((x: -1.58592e-14 m, y: 259.0 m), ..., (x: 1.0 m, y: 259.0 m))
Gray{N0f8}(1.0)	Quadrangle((x: -1.57979e-14 m, y: 258.0 m), ..., (x: 1.0 m, y: 258.0 m))
Gray{N0f8}(1.0)	Quadrangle((x: -1.57367e-14 m, y: 257.0 m), ..., (x: 1.0 m, y: 257.0 m))
Gray{N0f8}(1.0)	Quadrangle((x: -1.56755e-14 m, y: 256.0 m), ..., (x: 1.0 m, y: 256.0 m))
⋮	⋮

The Potrace transform can be used to extract complex geometries from a geotable over a 2D Grid. It transforms the Grid domain into a GeometrySet based on any column that contains a discrete set of marker values. In this example, we use the color as the column with markers:

Ab = letters |> Potrace("color", ϵ=0.8)

2×2 GeoTable over 2 GeometrySet
color	geometry
Colorful	MultiPolygon
[NoUnits]	🖈 Cartesian{NoDatum}
Gray{N0f8}(1.0)	Multi(4×PolyArea)
Gray{N0f8}(0.0)	Multi(2×PolyArea)

The option ϵ controls the deviation tolerance used to simplify the boundaries of the geometries. The higher is the tolerance, the less is the number of segments in the boundary:

viz(Ab.geometry[2], color = "black")

In the reverse direction, we have the Rasterize transform, which takes a geotable over a GeometrySet and assigns the geometries to a Grid. In this transform, we can either provide an external grid for the the assignments, or request a grid size to discretize the boundingbox of all geometries:

A = [1, 2, 3, 4, 5]
B = [1.1, 2.2, 3.3, 4.4, 5.5]
p1 = Triangle((2, 0), (6, 2), (2, 2))
p2 = Triangle((0, 6), (3, 8), (0, 10))
p3 = Quadrangle((3, 6), (9, 6), (9, 9), (6, 9))
p4 = Quadrangle((7, 0), (10, 0), (10, 4), (7, 4))
p5 = Pentagon((1, 3), (5, 3), (6, 6), (3, 8), (0, 6))
gtb = georef((; A, B), [p1, p2, p3, p4, p5])

5×3 GeoTable over 5 GeometrySet
A	B	geometry
Categorical	Continuous	Ngon
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
1	1.1	Triangle((x: 2.0 m, y: 0.0 m), (x: 6.0 m, y: 2.0 m), (x: 2.0 m, y: 2.0 m))
2	2.2	Triangle((x: 0.0 m, y: 6.0 m), (x: 3.0 m, y: 8.0 m), (x: 0.0 m, y: 10.0 m))
3	3.3	Quadrangle((x: 3.0 m, y: 6.0 m), ..., (x: 6.0 m, y: 9.0 m))
4	4.4	Quadrangle((x: 7.0 m, y: 0.0 m), ..., (x: 7.0 m, y: 4.0 m))
5	5.5	Pentagon((x: 1.0 m, y: 3.0 m), ..., (x: 0.0 m, y: 6.0 m))

gtb |> viewer

ntb = gtb |> Rasterize(20, 20)

390 rows omitted

400×3 GeoTable over 20×20 CartesianGrid
A	B	geometry
Categorical	Continuous	Quadrangle
[NoUnits]	[NoUnits]	🖈 Cartesian{NoDatum}
missing	missing	Quadrangle((x: 0.0 m, y: 0.0 m), ..., (x: 0.0 m, y: 0.5 m))
missing	missing	Quadrangle((x: 0.5 m, y: 0.0 m), ..., (x: 0.5 m, y: 0.5 m))
missing	missing	Quadrangle((x: 1.0 m, y: 0.0 m), ..., (x: 1.0 m, y: 0.5 m))
1	1.1	Quadrangle((x: 1.5 m, y: 0.0 m), ..., (x: 1.5 m, y: 0.5 m))
1	1.1	Quadrangle((x: 2.0 m, y: 0.0 m), ..., (x: 2.0 m, y: 0.5 m))
missing	missing	Quadrangle((x: 2.5 m, y: 0.0 m), ..., (x: 2.5 m, y: 0.5 m))
missing	missing	Quadrangle((x: 3.0 m, y: 0.0 m), ..., (x: 3.0 m, y: 0.5 m))
missing	missing	Quadrangle((x: 3.5 m, y: 0.0 m), ..., (x: 3.5 m, y: 0.5 m))
missing	missing	Quadrangle((x: 4.0 m, y: 0.0 m), ..., (x: 4.0 m, y: 0.5 m))
missing	missing	Quadrangle((x: 4.5 m, y: 0.0 m), ..., (x: 4.5 m, y: 0.5 m))
⋮	⋮	⋮

ntb |> viewer

The values of the variables are aggregated at geometric intersections using a default aggregation function, which can be overwritten with an option. Once the geotable is defined over a Grid, it is possible to refine or coarsen the grid with the Downscale and Upscale transforms.

The Transfer of values to a new geospatial domain is another very useful geospatial transform. The Aggregate transform is related, but aggregates the values with given aggregation functions.

5.5 Remarks

In this chapter we learned the important concept of transforms, and saw examples of the concept in action with synthetic data. In order to leverage the large number of transforms implemented in the framework, all we need to do is load our geospatial data as a geotable using georef or GeoIO.jl.

Some additional remarks:

• One of the major advantages of transforms compared to traditional row/column access in data science is that they preserve geospatial information. There is no need to keep track of indices in arrays to repeatedly reattach values to geometries.
• Transforms can be organized into three classes—feature, geometric and geospatial—depending on how they operate with the values and the domain of the geotable:
  • Feature transforms operate on the values. They include column selection, data cleaning, statistical analysis and any transform designed for traditional Tables.jl.
  • Geometric transforms operate on the domain. They include coordinate transforms that simply modify the coordinates of points as well as more advanced transforms that can change the topology of the domain.
  • Geospatial transforms operate on both the values and domain. They include geostatistical transforms and transforms that use other columns besides the geometry column to produce new columns and geometries.

In the next chapters, we will review map projections with the Proj coordinate transform, and will introduce one of the greatest features of the framework known as transform pipelines.

Hoffimann, Júlio, and Bianca Zadrozny. 2019. “Efficient Variography with Partition Variograms.” Computers & Geosciences 131: 52–59. https://doi.org/https://doi.org/10.1016/j.cageo.2019.06.013.

Koolen, Twan, Yuto Horikawa, Andy Ferris, Claire Foster, awbsmith, ryanelandt, Jan Weidner, et al. 2023. “JuliaGeometry/Rotations.jl: V1.6.0.” Zenodo. https://doi.org/10.5281/zenodo.8366010.

Lin, Dahua, David Widmann, Simon Byrne, John Myles White, Andreas Noack, Mathieu Besançon, Douglas Bates, et al. 2023. “JuliaStats/Distributions.jl: V0.25.100.” Zenodo. https://doi.org/10.5281/zenodo.8224988.

Thompson, William. 2023. “PairPlots.jl Beautiful and Flexible Visualizations of High Dimensional Data.” https://sefffal.github.io/PairPlots.jl/dev.