Displacing points by random distance and direction

Question

I have a shapefile of points representing households in a household survey.

In order to conceal their exact geographic location, I would like to displace these points in a random distance (uniform distribution over 0-2km) and a random direction (uniform distribution over 0-360 degrees).

How can I do this in QGIS?

Although displacing on the x- and y-axis independently creates a random direction, displacing by 2000m on the x and 2000m on the y would lead to more than a 2000m displacement in total. I thought about displacing up to 1414.2135m on both axes, but I wasn't convinced this would generate a uniform distribution over distance (I could be wrong, but I thought it would more likely create smaller distances).

The workaround I came up with was to create a random degree using the field editor rand(0,90) and a random quadrant using rand(1,4). Then I created a random distance using rand(0,2000). Let x* be the displaced x-coordinate of original point; similarly, y* is the displaced y-coordinate of the original point.

In quadrant I : x*=x+distance*sin(degree) and y*=y+distance*sin(90-degree)
In quadrant II : x*=x-distance*sin(90-degree) and y*=y+distance*sin(degree)
In quadrant III : x*=x-distance*sin(degree) and y*=y-distance*sin(90-degree)
In quadrant IV : x*=x+distance*sin(90-degree) and y*=y-distance*sin(degree)

Answer 1

Use "Geometry generator"

Actually, QGIS provides a much easier way to do this: Geometry generator.

No need to create any new layer, nor any new calculated field, just add a Geometry generator marker in the Symbology pane, with soemthing similar to this content:

make_point(
  $x + rand(-2000,2000),
  $y + rand(-2000,2000)
  )

or, to get a true 2km radius:

with_variable('randang', rand(0,360),
with_variable('randdist', rand(0,2000),
make_point(
  $x+@randdist*cos(@randang),
  $y+@randdist*sin(@randang)
  )
  ))

• yellow dots are the original, actual points
• red dots are the randomized ones.

You can of course remove the representation of the actual points, and leave only the randomized one, which addresses your question:

Checking the correctness

You can also check the correctness of this approach by drawing displacement lines instead of displaced points:

This is accomplished selecting LineString / MultiLineString as Geometry type and with the following:

with_variable('randang', rand(0,360),
with_variable('randdist', rand(0,2000),
make_line(
    make_point($x+@randdist*cos(@randang),$y+@randdist*sin(@randang)),
    make_point($x,$y)
  )
))

Answer 2

Alternatively, I can suggest using a "Virtual Layer" through Layer > Add Layer > Add/Edit Virtual Layer...

Let's assume there is a point layer called "points".

With the following query, it is possible to achieve the result where each point will be shifted on a certain delta generated randomly.

SELECT id,
       name,
       make_point(st_x(geometry)+random() % 100,st_y(geometry)+random() % 100) AS geom,
       random() % 100 AS "delta_x",
       random() % 100 AS "delta_y"
FROM "points"

The output Virtual Layer with its corresponding attribute table will look as following

---

If the point layer "points" attached to a polygon layer "polygons" (see image below), basically it means, that a new point has to be generated inside a corresponding polygon.

For this purpose use the following query

SELECT poi.id,
       point_on_surface(p.geometry) AS geom,
       round(st_x(poi.geometry)-st_x(point_on_surface(p.geometry)),4) AS "delta_x",
       round(st_y(poi.geometry)-st_y(point_on_surface(p.geometry)),4) AS "delta_y"
FROM "polygons" AS p
JOIN "points" AS poi ON st_within(poi.geometry, p.geometry)

The query above contains the ST_PointOnSurface() function.

Note: That this solution is working only for 1:1 relation between a point feature and a polygon feature within which the point lays.

---

References:

• Generating only positive random numbers in SQLite
• How is ST_PointOnSurface calculated?

Answer 3

I had never done this before but I think I that you can give my solution a try. What I did was to add two new X and Y fields to my shapefile (my layer projected in UTM18N, which is meters). Then, I used the field calculator to obtain the UTM coordinates of each point but adding to it a random value between 100 and 200. I input $x+(rand(100,200)) and $y+(rand(100,200)).

You export this layer as CSV and then reload it to QGIS as a delimited text file using these new X and Y fields. Your points will be randomly displaced. I tried it out in QGIS and it works; see the image below.

Edit: you can try some thing like $y+(rand(rand(-200,-100),rand(100,200))) if you want more dispersion. I still haven't figure out how to do the 0-360 degrees part.

I hope it helps.

Answer 4

The answer by @1buzz seems complete to me in terms of achieving movement in both random distance and random direction (rotation if you will). Since the x and y random displacements are independent of each other, then this accomplishes a shift in a random direction as well as a random distance from the source point, within the constraints imposed. However, you need to cater for negative displacements as well as positives, so what about $x = $x+randf(-2000,2000) and same for y, which should open up the randomized space to a square centred on the original point (as I understand it, randf returns a floating number rather than an integer). You might decide the integer returned by "rand" is sufficient.