Integrate polar histogram values about a certain value

I have a set of directional (circular) data, and I also have the mean value of this data. I need to find out how to integrate the normlaized polar histogram to clockwise and counter-clockwise away from the mean. How could this be done, given that the data is some value X? Additionally, how could find the angle away from the mean such that I have 0.25 on eitherside of the mean? To put it in math terms, I need to determine what and are the following

Here the PMF is the normalized polar histogram for X.

Answer 1

[Ayush]

Here is the function to integrate normalized polar histogram to clockwise & counterclockwise away from mean:

function [clockwise_cumulative_sum, counterclockwise_cumulative_sum] = integrate_polar_histogram(data, mean_value) 
% Step 1: Normalize the circular data 
normalized_data = deg2rad(data - mean_value); 

% Step 2: Create a polar histogram 
num_bins = 36;  % Number of bins for the histogram 
[histogram, ~] = histcounts(normalized_data, num_bins, 'BinLimits', [-pi pi]); 

% Step 3: Normalize the histogram 
normalized_histogram = histogram / numel(normalized_data); 

% Step 4: Integrate the normalized histogram 
clockwise_cumulative_sum = cumsum(normalized_histogram); 
counterclockwise_cumulative_sum = flip(cumsum(flip(normalized_histogram))); 

end 

Refer to following documentation for further information:

1. https://in.mathworks.com/help/matlab/ref/histcounts.html
2. https://www.mathworks.com/help/matlab/ref/cumsum.html
3. https://www.mathworks.com/help/matlab/ref/flip.html
4. https://www.mathworks.com/help/matlab/ref/deg2rad.html

Here is the function to get the angle on either side of the mean to have:

function angle_with_0_25 = find_angle_with_0_25_on_either_side(clockwise_cumulative_sum) 
% Find the bin where the cumulative sum reaches 0.25 
bin_index = find(clockwise_cumulative_sum >= 0.25, 1); 

% Calculate the angle at the midpoint of the bin 
angle_with_0_25 = (bin_index + 0.5) * (2 * pi / numel(clockwise_cumulative_sum)); 

angle_with_0_25 = rad2deg(angle_with_0_25); 

end 

An example to run the above workflow:

% Example usage 
data = [10, 20, 30, 40, 350, 355, 5];  % Example circular data 
mean_value = 0;  % Example mean value 

[clockwise_cumulative_sum, counterclockwise_cumulative_sum] = integrate_polar_histogram(data, mean_value); 
angle_with_0_25 = find_angle_with_0_25_on_either_side(clockwise_cumulative_sum); 
disp(['Angle away from the mean with 0.25 on either side: ' num2str(angle_with_0_25)]);