296 lines
9.7 KiB
Matlab
296 lines
9.7 KiB
Matlab
%% Evolve to skewed to single gaussian with lower mean
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clear; clc;
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N_params = 50;
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N_reps = 500;
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alpha_values = linspace(0, 45, N_params);
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all_data = cell(1, N_params);
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% Transition control
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alpha_start = 5; % where transition begins
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alpha_end = 40; % where transition ends
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mu_start = 1.2; % high initial mean
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mu_end = 0.8; % low final mean
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sigma_start = 0.2; % wide initial std
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sigma_end = 0.07; % narrow final std
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max_skew = 5; % peak skew strength
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% Loop through alpha
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for i = 1:N_params
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alpha = alpha_values(i);
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% Normalized transition variable t ∈ [0, 1]
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t = min(max((alpha - alpha_start) / (alpha_end - alpha_start), 0), 1);
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% Use cosine-based smooth interpolation
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smooth_t = (1 - cos(pi * t)) / 2; % ease-in-out
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% Mean and sigma interpolation (smoothly decrease)
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mu = mu_start * (1 - smooth_t) + mu_end * smooth_t;
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sigma = sigma_start * (1 - smooth_t) + sigma_end * smooth_t;
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% Skewness: sinusoidal profile, max at middle
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skew_strength = max_skew * sin(t * pi);
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% Generate data
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if abs(skew_strength) < 1e-2 % near-zero skew, use normal
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data = normrnd(mu, sigma, [N_reps, 1]);
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else
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data = skewnormrnd(mu, sigma, skew_strength, N_reps);
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end
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all_data{i} = data;
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% Cumulants
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kappa = computeCumulants(data, 6);
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mean_vals(i) = kappa(1);
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var_vals(i) = kappa(2);
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skew_vals(i) = kappa(3);
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kurt_vals(i) = kappa(4);
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kappa5_vals(i) = kappa(5);
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kappa6_vals(i) = kappa(6);
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end
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%% Evolve to bimodal
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clear; clc;
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N_params = 50;
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N_reps = 500;
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alpha_values = linspace(0, 45, N_params);
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all_data = cell(1, N_params);
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bimodal_start = 20; % start earlier
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transition_width = 5; % wider window for smoothness
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for i = 1:N_params
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alpha = alpha_values(i);
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if alpha < (bimodal_start - transition_width)
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% Pure skewed unimodal, smaller max skewness for subtlety
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skew_strength = 3 * (alpha / bimodal_start);
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data = skewnormrnd(1, 0.1, skew_strength, N_reps);
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elseif alpha <= (bimodal_start + transition_width)
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% Smooth transition window
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t = (alpha - (bimodal_start - transition_width)) / (2 * transition_width);
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% Weights transition 1 -> 0.7
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w1 = 1 - 0.3 * t;
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w2 = 1 - w1;
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% Peak separation smaller (max delta = 0.3)
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delta_max = 0.3;
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delta = delta_max * t;
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mu1 = 1 - delta;
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mu2 = 1 + delta;
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sigma1 = 0.1;
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sigma2 = 0.1;
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N1 = round(N_reps * w1);
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N2 = N_reps - N1;
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mode1_samples = normrnd(mu1, sigma1, [N1, 1]);
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mode2_samples = normrnd(mu2, sigma2, [N2, 1]);
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data = [mode1_samples; mode2_samples];
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data = data(randperm(length(data)));
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else
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% After transition: bimodal, but not strongly balanced
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w1 = 0.7;
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w2 = 0.3;
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mu1 = 1 - 0.3;
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mu2 = 1 + 0.3;
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sigma1 = 0.1;
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sigma2 = 0.1;
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N1 = round(N_reps * w1);
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N2 = N_reps - N1;
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mode1_samples = normrnd(mu1, sigma1, [N1, 1]);
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mode2_samples = normrnd(mu2, sigma2, [N2, 1]);
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data = [mode1_samples; mode2_samples];
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data = data(randperm(length(data)));
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end
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all_data{i} = data;
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kappa = computeCumulants(data,6);
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mean_vals(i) = kappa(1);
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var_vals(i) = kappa(2);
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skew_vals(i) = kappa(3);
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kurt_vals(i) = kappa(4);
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kappa5_vals(i) = kappa(5);
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kappa6_vals(i) = kappa(6);
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end
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%% Animate evolving distribution and cumulant value
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figure(1); clf;
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set(gcf, 'Color', 'w', 'Position',[100 100 1300 750])
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for i = 1:N_params
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clf;
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% PDF
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subplot(1,2,1); cla; hold on;
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data = all_data{i};
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% Plot histogram with normalized PDF
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histogram(data, 'Normalization', 'pdf', 'BinWidth', 0.03, ...
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'FaceColor', [0.3 0.5 0.8], 'EdgeColor', 'k', 'FaceAlpha', 0.6);
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title(sprintf('Histogram at $\\alpha = %.1f^\\circ$', alpha_values(i)), ...
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'Interpreter', 'latex', 'FontSize', 16);
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xlabel('$\mathrm{max}[g^{(2)}]$', 'Interpreter', 'latex', 'FontSize', 14);
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ylabel('PDF', 'FontSize', 14);
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set(gca, 'FontSize', 12); grid on;
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xlim([0.5, 1.8]);
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% Cumulant evolution (e.g., Variance)
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subplot(1,2,2); hold on;
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plot(alpha_values(1:i), var_vals(1:i), 'bo-', 'LineWidth', 2);
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title('Cumulant Tracking', 'Interpreter', 'latex', 'FontSize', 16);
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', 14);
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ylabel('$\kappa_2$', 'Interpreter', 'latex', 'FontSize', 14);
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xlim([0, 45]); grid on;
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set(gca, 'FontSize', 12);
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pause(0.3);
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end
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%% === Compute 2D PDF heatmap: f(x, alpha) ===
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x_grid = linspace(0.5, 1.8, 200); % max[g²] values on y-axis
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pdf_matrix = zeros(numel(x_grid), N_params); % Now: rows = y, columns = alpha
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for i = 1:N_params
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data = all_data{i};
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f = ksdensity(data, x_grid, 'Bandwidth', 0.025);
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pdf_matrix(:, i) = f; % Transpose for y-axis to be vertical
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end
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% === Plot PDF vs. alpha heatmap ===
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figure(2); clf;
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set(gcf, 'Color', 'w', 'Position',[100 100 950 750])
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imagesc(alpha_values, x_grid, pdf_matrix);
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set(gca, 'YDir', 'normal'); % Flip y-axis to normal orientation
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', 14);
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ylabel('$\mathrm{max}[g^{(2)}]$', 'Interpreter', 'latex', 'FontSize', 14);
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title('Evolving PDF of $\mathrm{max}[g^{(2)}]$', ...
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'Interpreter', 'latex', 'FontSize', 16);
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colormap(Colormaps.coolwarm()); % More aesthetic than default
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colorbar;
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c = colorbar;
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ylabel(c, 'PDF', 'FontSize', 14, 'Interpreter', 'latex');
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set(gca, 'FontSize', 14);
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%% === Plotting ===
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figure(1)
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set(gcf, 'Color', 'w', 'Position', [100 100 950 750])
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scan_vals = alpha_values; % your parameter sweep values
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% Define font style for consistency
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axis_fontsize = 14;
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label_fontsize = 16;
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title_fontsize = 16;
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% 1. Mean with error bars (if you have error data, else just plot)
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% If no error, replace errorbar with plot or omit error data
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% For now, no error bars assumed
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subplot(3,2,1);
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plot(scan_vals, mean_vals, 'o-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Mean of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_1$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% 2. Variance
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subplot(3,2,2);
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plot(scan_vals, var_vals, 's-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Variance of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_2$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% 3. Skewness
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subplot(3,2,3);
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plot(scan_vals, skew_vals, 'd-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Skewness of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_3$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% 4. Kurtosis
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subplot(3,2,4);
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plot(scan_vals, kurt_vals, '^-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Kurtosis of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_4$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% 5. 5th-order cumulant
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subplot(3,2,5);
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plot(scan_vals, kappa5_vals, 'v-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Fifth-order Cumulant of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_5$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% 6. 6th-order cumulant
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subplot(3,2,6);
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plot(scan_vals, kappa6_vals, '>-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Sixth-order Cumulant of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_6$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% Super title (you can customize the string)
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sgtitle('Cumulants of a simulated evolving distribution', ...
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'FontWeight', 'bold', 'FontSize', 18, 'Interpreter', 'latex');
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%% === Helper: Cumulant Calculation ===
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function kappa = computeCumulants(data, max_order)
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data = data(:);
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mu = mean(data);
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c = zeros(1, max_order);
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centered = data - mu;
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for n = 1:max_order
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c(n) = mean(centered.^n);
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end
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kappa = zeros(1, max_order);
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kappa(1) = mu;
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kappa(2) = c(2);
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kappa(3) = c(3);
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kappa(4) = c(4) - 3*c(2)^2;
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kappa(5) = c(5) - 10*c(3)*c(2);
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kappa(6) = c(6) - 15*c(4)*c(2) - 10*c(3)^2 + 30*c(2)^3;
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end
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%% === Helper: Skewed Normal Distribution ===
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function x = skewnormrnd(mu, sigma, alpha, n)
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% Skew-normal using Azzalini's method
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delta = alpha / sqrt(1 + alpha^2);
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u0 = randn(n,1);
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v = randn(n,1);
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u1 = delta * u0 + sqrt(1 - delta^2) * v;
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x = mu + sigma * u1 .* sign(u0);
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end
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