Plot ECDF and histogramm of distribution

I try to plot custom empirical CDF in matlab for cauchy distribution. Below I present my code:

clear
clc
format long

% Parameters of the Cauchy distribution
location = 17;  % Location parameter (median)
scale = 2;     % Scale parameter (interquartile range)

% Number of data points to generate
num_samples = 20;

% Generate data from a Cauchy distribution
y = location + scale * trnd(1, num_samples, 1);

stats = bootstrp(20, @(x)[mean(x), median(x)], y);

% Extract the first column of stats
first_column = stats(:, 2);

% Compute the histogram without plotting it
[counts, binEdges] = histcounts(first_column, 10);

% Normalize the histogram counts to have a maximum value of 1
max_count = max(counts);
normalized_counts = counts / max_count;

% Compute bin centers from edges
binCenters = (binEdges(1:end-1) + binEdges(2:end)) / 2;

% Create a figure for plotting
figure;

% Plot the normalized histogram with heights scaled to 1
bar(binCenters, normalized_counts, 'hist');

hold on; % This will allow you to plot the ECDF on the same graph

grid on
grid minor

% Compute the ECDF values for the first column
ecdf_values = sort(first_column);
ecdf = (1:numel(ecdf_values)) / numel(ecdf_values);

% Normalize the ECDF values to have a maximum value of 1
ecdf = ecdf / max(ecdf);

% Plot the ECDF as a stairs plot
stairs(ecdf_values, ecdf, 'r-', 'LineWidth', 2);

% Add a horizontal line at y=0 before the minimum data value
line([min(ecdf_values)-10, min(ecdf_values)], [0, 0], 'Color', 'r', 'LineStyle', '-', 'LineWidth', 2);

% Find the first minimal value of custom ECDF
first_minimal_value = min(ecdf(ecdf > 0));

% Find the corresponding x-value
x_of_first_minimal = ecdf_values(find(ecdf == first_minimal_value, 1));

% Add a vertical line at the point of growth where height equals the first minimal value
line([x_of_first_minimal, x_of_first_minimal], [0, first_minimal_value], 'Color', 'r', 'LineStyle', '-', 'LineWidth', 2);

% Add horizontal lines at y=1 after the maximum data value
plot([ecdf_values(end), ecdf_values(end) + 10], [1, 1], 'r-', 'LineWidth', 2);

% Set y-axis limits
ylim([-0.05 1.05]);

% Set x-axis limits (for example, from -2 to 22)
% xlim([-2 22]);

% Add labels and a legend
xlabel('Value');
ylabel('Normalized Probability');
% legend('Normalized Histogram', 'Normalized ECDF');
title('Normalized Histogram and ECDF for First Column of stats');

hold off;

My problem: CDF also have values y = 0 and y = 1 before start my custom stairs and after end of my custom stairs. How to plot this values in thius case?

I understand, that this solution not correct and optimal, but work -- how to improve ecdf visualization in this case?

Answer 1

[the cyclist]

Do you mean you want to avoid drawing the additional horizontal red lines?

I think you'll need to do something "manually", because MATLAB does not know you have data there. The following seems a bit more natural than what you are doing:

clear
clc
format long

% Parameters of the Cauchy distribution
location = 17;  % Location parameter (median)
scale = 2;     % Scale parameter (interquartile range)

% Number of data points to generate
num_samples = 20;

% Generate data from a Cauchy distribution
y = location + scale * trnd(1, num_samples, 1);

stats = bootstrp(20, @(x)[mean(x), median(x)], y);

% Extract the first column of stats
first_column = stats(:, 2);

% Compute the histogram without plotting it
[counts, binEdges] = histcounts(first_column, 10);

% Normalize the histogram counts to have a maximum value of 1
max_count = max(counts);
normalized_counts = counts / max_count;

% Compute bin centers from edges
binCenters = (binEdges(1:end-1) + binEdges(2:end)) / 2;

% Create a figure for plotting
figure;

% Plot the normalized histogram with heights scaled to 1
bar(binCenters, normalized_counts, 'hist');

hold on; % This will allow you to plot the ECDF on the same graph

grid on
grid minor

% Compute the ECDF values for the first column
ecdf_values = sort(first_column);
ecdf = (1:numel(ecdf_values)) / numel(ecdf_values);

% Normalize the ECDF values to have a maximum value of 1
ecdf = ecdf / max(ecdf);

% Append values to extend ECDF range
ecdf_values = [min(ecdf_values)-10; ecdf_values; max(ecdf_values)+10];
ecdf = [0 ecdf 1];

% Plot the ECDF as a stairs plot
stairs(ecdf_values, ecdf, 'r-', 'LineWidth', 2);

% Find the first minimal value of custom ECDF
first_minimal_value = min(ecdf(ecdf > 0));

% Find the corresponding x-value
x_of_first_minimal = ecdf_values(find(ecdf == first_minimal_value, 1));

% Add a vertical line at the point of growth where height equals the first minimal value
line([x_of_first_minimal, x_of_first_minimal], [0, first_minimal_value], 'Color', 'r', 'LineStyle', '-', 'LineWidth', 2);

% Set y-axis limits
ylim([-0.05 1.05]);

% Set x-axis limits (for example, from -2 to 22)
% xlim([-2 22]);

% Add labels and a legend
xlabel('Value');
ylabel('Normalized Probability');
% legend('Normalized Histogram', 'Normalized ECDF');
title('Normalized Histogram and ECDF for First Column of stats');

hold off;

Note that I added the lines

% Append values to extend ECDF range
ecdf_values = [min(ecdf_values)-10; ecdf_values; max(ecdf_values)+10];
ecdf = [0 ecdf 1];

after you calculate the ECDF.

[Alex Milns]

Thanks a lot, exactly what is needed!