Integrate polar histogram values about a certain value

2023 8 14
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更新时间：2023 8 22
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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.

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Ayush
Ayush 2023-8-22

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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)]);

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