Sampling random directional lines within a polygon

Had another idea which is now implemented...

Consider any segmented path of segments of lengths L_i at angles A_i. Its
endpoint will be the vector sum of those segments, ie at (x,y) = (sum(L_i
cos(A_i)), sum(L_i sin(A_i)).

To create a segmented path to a given (x,y), solve that expression for the
angles A_i. In R you can treat this as an optimisation problem - find a set
of angles A_i that minimise the distance of the end of the segmented path
from the target end point.

Here's some code that does that for a path from 0,0 to 0,1:

pathit <- function(segments){
    obj = function(angles){
        dxdy = dxdy(segments, angles)
        xerr = dxdy$dx-1
        yerr = dxdy$dy
        err = sqrt(xerr^2 + yerr^2)
        err
    }
    angles = runif(length(segments), 0 , 2*pi)
    optim(angles, obj)
}

dxdy = function(segments, angles){
    dx = sum(segments * cos(angles))
    dy = sum(segments * sin(angles))
    list(dx=dx, dy=dy)
}

plotsegs <- function(segments, angles){
    x = rep(NA, length(segments) +1)
    y = x
    x[1] = 0
    y[1] = 0
    for(i in 2:(length(segments)+1)){
        x[i] = x[i-1] + segments[i-1]*cos(angles[i-1])
        y[i] = y[i-1] + segments[i-1]*sin(angles[i-1])
    }
    cbind(x,y)
}

This is deliberately written naively for clarity.

To use, set the segment sizes, optimise, and then plot:

 s1 = c(.1,.3,.2,.1,.3,.3,.1)
 a1 = pathit(s1)
 plot(plotsegs(s1,a1$par),type="l")

which should show a path of seven segments from 0,0 to 0,1 - since the
initial starting values are random the model can find different solutions.
Run again for a different path.

To see what the space of paths looks like, do lots and overplot them:

  lots = lapply(1:1000, function(i)plotsegs(s1,pathit(s1)$par))
  plot(c(-.1,1.1),c(-1,1))
  p = lapply(lots, function(xy){lines(xy)})

this should show 1000 paths, and illustrates the "ellipse" of path
possibles that I mentioned in the previous email.

Sometimes the optimiser struggles to find a solution and so you should
probably test the output from optim for convergence and to make sure the
target function is close enough to zero for your purposes.  For the example
above most of the time the end point is about 1e-5  from (1,0) but for
harder problems such as s = rep(.1, 11) which only has 0.1 of extra "slack"
length, the error can be 0.02 and failed convergence. Possibly longer optim
runs would help or constraining the angles.

Anyway, interesting problem....

Barry







On Wed, Feb 6, 2019 at 8:23 PM Barry Rowlingson <b.rowlingson at gmail.com>
wrote:

Do you want to generate these for input into some statistical process, or
to generate some test data that looks a bit like real data? I think
generating test data isn't too difficult, but anything that you might want
to put into a statistical test (eg testing some hypothesis about the birds
maximum deviation from the straight line A-B) needs a lot more care in
formulating the path generating process.

Here's some thoughts - if you consider one of the red segments in your
map as a piece of string (rather than 10 segments) anchored at the points,
then you can stretch it taut with a pencil and draw an ellipse with A and B
as the foci. Any path created with that string - taut or slack as in your
map - has to strictly lie within the ellipse. Now you can wiggle that
string inside that ellipse and create an infinity of paths from A to B of
the same length. I'm not sure how you can sample uniformly from that
infinity such that any path has an equal sampling probability. Your problem
is similar but has the additional rigid segment constraint.

Any two adjacent segments of a chain, eg 1----2-------3, as long as it
isn't taut (ie straight) can be perturbed by holding 1 and 3 still and
moving 2 to the "mirror image" point over the straight line from 1 to 3.
You can also take three segments 1--2--3--4 and hold 1 and 4 still and
perturb 2 and 3 fairly easily. In this way you could set up an initial
chain and then run multiple perturbations on the chain to get a "random"
chain, but quite what set of all chains it would be a sample from is not
clear. It could at least generate reasonable looking paths, but I wouldn't
want to test a hypothesis against it.

I'm going to generate a path from my office to my home now.

Barry






On Wed, Feb 6, 2019 at 7:50 PM Hannah Justen <justen at tamu.edu> wrote:

Hello everyone,

Thank you very much for the suggestions.

Regarding more details (please also see attached figure):
I would like to simulate a bird's migration between breeding (starting
polygon - blue in the figure) and wintering grounds (end polygon - green in
the figure). The lines can start from anywhere within the starting polygon
and end anywhere in the end polygon. The lines shall consist of 10
connected segments with variable length between 300 and 1000 km (see red
lines in figure; two examples of possible lines with 10 segments between
the polygons).

Thank you very much for you help,
Hannah

---
PhD Student |Ecology and Evolutionary Biology
Texas A&M University

On Wed, Feb 6, 2019 at 5:12 AM Barry Rowlingson <
b.rowlingson at lancaster.ac.uk> wrote:

Interesting, but I think we need more details...

Do the lines have to start and finish at specific locations in the
polygons - like the centroid, or anywhere?

So one line might be 3 segments of 10km each connecting two polygon
centroids that are 15km apart? Imagining three rigid rods of length 10
connected at their ends and with the first and last also connected to two
fixed points tells me there's an infinite number of possible solutions.
There's probably also a number of ways of sampling from those solutions.
Its going to get very complicated with a larger number of segments.

Hmmmmm.....

Barry




On Tue, Feb 5, 2019 at 11:30 PM Hannah Justen <justen at tamu.edu> wrote:

Hi everyone,

I am studying migratory tracks of birds for my dissertation and I would
like to model possible pathways between two polygons. Therefore, I
would
like to sample random lines between the polygons. These lines can
differ in
total length but should consist of x - number of fragments of equal
length.
Each fragment can have slightly different orientation but overall the
lines
should connect the two polygons.

I fail to find the appropriate R package that will allow me to do this
type
of analysis. Does anyone have a suggestion how to approach analysis?

Thank you,
Hannah

---
PhD Student |Ecology and Evolutionary Biology
Texas A&M University

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