Random walk (based in random angles). Math help....!!

I created two diferent movement rules for animal path simulation in Matlab, based on some field data. I used iterated steps (iterated algorithm), ramdomizing the angle of direction.

The first path rule is a complete (I suppose) random movement, where the direction of the next step could vary ramdomly from 0 to 360 (or 180 to -180) degrees.

And in the second path rule the steps direction are more correlated with the previous steps, as the angle of the next step could vary only 30 degrees (from -15 to 15 degrees).

The randomized variable is always the angle of direction. And for these these two path patterns the path lenght does not change during the displacement.

I am trying to deal with statistical theory, but so far Iam a little confuse about the simulations, so I came ask for some help:

- I created a “random walk” and a “correlated random walk” correct ? These are non stationary process right?

- I need to symbolize mathematically those rules (for scientific writing). For this I first should specify the media, the variace, and the distribuition correct ? But particulary at this point Iam stucked. Could you help me ? I really tried the bibliography but until now I havent had sucess understanding.

Thank you in advance.

[Walter Roberson]

This appears to be essentially a correlated random walk from a random starting point on the circle one unit in radius from the origin.

[Image Analyst]

I moved Paulo's "Answer" moved here since it's a comment to Walter, not reall an answer in itself: Paulo, see my answer below.

Thank you for the insight! I look for some references with circular random walk and it really seens to be the case. But I am still stucked with the measures (media, variance, distributions)

As for giving more informations:

In fact I created two movements paths. One of them I hope is a random walk and the another a correlated random walk.

The random walk I allowed to random the angle direction in every steps, so it sorted an angle from 360 options, so: media zero, and uniform distribution ? and how about the variance ?

Answer 1

[Image Analyst]

Try this:

maxNumberOfSteps = 10; % or whatever
radius = 1;
x = zeros(1, maxNumberOfSteps)
y = zeros(1, maxNumberOfSteps)
for stepNumber = 2 : maxNumberOfSteps
  % Pick one of the two angle strategies.
%   angle(stepNumber) = 360*rand(1); % Independent.
  angle(stepNumber) = angle(stepNumber-1) + 30 * rand(1) - 15; % Correlated.
  x(stepNumber) = x(stepNumber - 1) + radius * cosd(angle(stepNumber));
  y(stepNumber) = y(stepNumber - 1) + radius * sind(angle(stepNumber));
  cla;
  plot(x, y, 'bo-', 'LineWidth', 3, 'MarkerSize', 15);
  grid on;
end
% Enlarge figure to full screen.
set(gcf, 'units','normalized','outerposition',[0 0 1 1]);

[Image Analyst]

Here's an earlier random walk I wrote that calculates distance and labels the points. See attached file in blue below.

[Paulo]

Thank you Image Analyst!

The code I used before is this one:

% start position

p1x=-100+(100--100)*rand;
p1y=-100+(100--100)*rand;
pr1xi(1,:)=[p1x];pr1yi(1,:)=[p1y];

% k is the step lenght (constant) and n is total steps

k=1;
n=100;

%movement iteration

azimute=360;
thp1i(1,:)=-azimute + (azimute--azimute)*rand ;

   for i=2:n+1

   phi1=180;

   thp1i(i:i,1)=thp1i(i-1,1)+((-1+(1--1)*rand)*phi1);   

   %changing for cartesian coordenates
   pr1xi(i:i,1)=pr1xi(i-1,1)+(k*cos((thp1i(i-1,1))*pi/180));
   pr1yi(i:i,1)=pr1yi(i-1,1)+(k*sin((thp1i(i-1,1))*pi/180));

   plot(pr1xi,pr1yi)

end

Now Iam strugling to deal with the writing formalization.

What is the all assumptions behind this randow walk ? I supposed that I had a uniform distribution, as at a new step the chances for each angle to be sorted is the same (from the rand function), correct ?

But if I had a sorted angle from -180 to 180, how do I formalize the mean (the mean is zero?) and the variance ?

Also, if I want to calculate the chances of two particles meet each other, under this rule, is possible ? From where should I begin...? Any sugestions abou this ? Thank you!!