New script added to better understand cumulants.
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@ -200,7 +200,7 @@ angle_threshold = 100;
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mean_max_g2_values = zeros(1, N_params);
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skew_max_g2_angle_values = zeros(1, N_params);
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var_max_g2_values = zeros(1, N_params);
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kurt_max_g2_angle_values = zeros(1, N_params);
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fourth_order_cumulant_max_g2_angle_values= zeros(1, N_params);
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fifth_order_cumulant_max_g2_angle_values = zeros(1, N_params);
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sixth_order_cumulant_max_g2_angle_values = zeros(1, N_params);
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@ -265,10 +265,9 @@ for i = 1:N_params
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std_error_g2_values(i) = sqrt(kappa(2)) / sqrt(N_eff);
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skew_max_g2_angle_values(i) = kappa(3);
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kurt_max_g2_angle_values(i) = kappa(4);
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fourth_order_cumulant_max_g2_angle_values(i)= kappa(4);
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fifth_order_cumulant_max_g2_angle_values(i) = kappa(5);
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sixth_order_cumulant_max_g2_angle_values(i) = kappa(6);
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end
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%% Plot PDF of order parameter
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@ -384,10 +383,10 @@ ylabel('$\kappa_3$', 'Interpreter', 'latex', ...
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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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% 4. Binder Cumulant
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subplot(3,2,4);
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plot(scan_vals, kurt_max_g2_angle_values, '^-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Kurtosis of $\mathrm{max}[g^{(2)}_{[50,70]}(\delta\theta)]$', ...
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plot(scan_vals, fourth_order_cumulant_max_g2_angle_values, '^-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Binder Cumulant of $\mathrm{max}[g^{(2)}_{[50,70]}(\delta\theta)]$', ...
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'Interpreter', 'latex', 'FontSize', title_fontsize);
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', ...
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'FontSize', label_fontsize);
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@ -7,16 +7,16 @@ 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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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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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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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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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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@ -78,7 +78,6 @@ for i = 1:N_params
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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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@ -104,8 +103,8 @@ for i = 1:N_params
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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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w1 = 0.5;
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w2 = 0.5;
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mu1 = 1 - 0.3;
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mu2 = 1 + 0.3;
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@ -123,14 +122,14 @@ for i = 1:N_params
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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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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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@ -238,7 +237,7 @@ 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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title('Fourth-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('$\kappa_4$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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@ -0,0 +1,223 @@
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%% Main script: Sweep different parameter pairs
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% Default parameters
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defaults.mu1 = 0.5;
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defaults.mu2 = 1.0;
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defaults.sigma1 = 0.1;
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defaults.sigma2 = 0.1;
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defaults.weight1 = 0.5;
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% Parameter pair definitions
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%{
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param_pairs = {
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'mu1', linspace(0.7, 1.0, 40), ...
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'mu2', linspace(1.0, 1.3, 40);
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'mu1', linspace(0.7, 1.0, 40), ...
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'weight1', linspace(0.2, 0.8, 40);
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'sigma1', linspace(0.05, 0.2, 40), ...
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'sigma2', linspace(0.05, 0.2, 40);
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'mu1', linspace(0.7, 1.0, 40), ...
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'sigma1', linspace(0.05, 0.2, 40);
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'mu2', linspace(1.0, 1.3, 40), ...
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'weight1', linspace(0.2, 0.8, 40);
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};
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%}
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param_pairs = {
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'mu1', linspace(0.1, 1.5, 40), ...
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'mu2', linspace(0.1, 1.5, 40);
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};
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% Cumulant index to visualize (2=variance, 3=skewness, 4=kurtosis)
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cumulant_to_plot = 4;
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% Run sweep for each pair
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for i = 1:size(param_pairs,1)
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param1_name = param_pairs{i,1};
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param1_vals = param_pairs{i,2};
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param2_name = param_pairs{i,3};
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param2_vals = param_pairs{i,4};
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fprintf('Sweeping %s and %s...\n', param1_name, param2_name);
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Z = sweepBimodalCumulants(param1_name, param1_vals, ...
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param2_name, param2_vals, ...
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defaults, cumulant_to_plot);
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end
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%%
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% Parameters
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N_total = 10000;
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mu1 = 0.5;
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sigma1 = 0.1;
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mu2 = 1.0;
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sigma2 = 0.1;
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weight1 = 0.7;
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% Generate data
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data = generateBimodalDistribution(N_total, mu1, mu2, sigma1, sigma2, weight1);
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% Plot histogram
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figure(3);
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clf
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set(gcf, 'Color', 'w', 'Position', [100 100 950 750]);
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histogram(data, 'Normalization', 'pdf', 'EdgeColor', 'none', 'FaceAlpha', 0.5);
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hold on;
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% Overlay smooth density estimate
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[xi, f] = ksdensity(data);
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plot(f, xi, 'r-', 'LineWidth', 2);
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% Labels and title
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xlabel('Value');
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ylabel('Probability Density');
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legend('Histogram', 'Smoothed Density');
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grid on;
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%%
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N = size(Z, 1);
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main_diag_values = diag(Z);
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anti_diag_values = diag(flipud(Z));
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param1_diag_main = param1_vals;
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param2_diag_main = param2_vals;
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param1_diag_anti = param1_vals;
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param2_diag_anti = flip(param2_vals);
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% For example, plot the main diagonal cumulants:
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figure(4);
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set(gcf, 'Color', 'w', 'Position', [100 100 950 750]);
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plot(1:N, main_diag_values, '-o');
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xlabel('Index along diagonal');
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ylabel('$\kappa_4$', 'Interpreter', 'latex');
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title('$\kappa_4$ along anti-diagonal', 'Interpreter', 'latex');
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% Plot anti-diagonal cumulants:
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figure(5);
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set(gcf, 'Color', 'w', 'Position', [100 100 950 750]);
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plot(1:N, anti_diag_values, '-o');
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xlabel('Index along anti-diagonal');
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ylabel('$\kappa_4$', 'Interpreter', 'latex');
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title('$\kappa_4$ along anti-diagonal', 'Interpreter', 'latex');
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%% === Helper: Bimodal Distribution ===
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function data = generateBimodalDistribution(N_total, mu1, mu2, sigma1, sigma2, weight1)
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%GENERATEBIMODALDISTRIBUTION Generates a single bimodal distribution.
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%
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% data = generateBimodalDistribution(N_total, mu1, mu2, sigma1, sigma2, weight1)
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%
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% Inputs:
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% N_total - total number of samples
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% mu1, mu2 - means of the two modes
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% sigma1, sigma2 - standard deviations of the two modes
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% weight1 - fraction of samples from mode 1 (between 0 and 1)
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%
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% Output:
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% data - shuffled samples from the bimodal distribution
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% Validate weight
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weight1 = min(max(weight1, 0), 1);
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weight2 = 1 - weight1;
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% Determine number of samples for each mode
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N1 = round(N_total * weight1);
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N2 = N_total - N1;
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% Generate samples
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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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% Combine and shuffle
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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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%% === 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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centered = data - mu;
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% Preallocate
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c = zeros(1, max_order);
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kappa = zeros(1, max_order);
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% Compute central moments up to max_order
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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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% Assign cumulants based on available order
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if max_order >= 1, kappa(1) = mu; end
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if max_order >= 2, kappa(2) = c(2); end
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if max_order >= 3, kappa(3) = c(3); end
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if max_order >= 4, kappa(4) = c(4) - 3*c(2)^2; end
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if max_order >= 5, kappa(5) = c(5) - 10*c(3)*c(2); end
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if max_order >= 6
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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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end
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%% === Helper: Cumulant Calculation ===
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function Z = sweepBimodalCumulants(param1_name, param1_vals, ...
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param2_name, param2_vals, ...
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fixed_params, ...
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cumulant_index)
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%SWEEPBIMODALCUMULANTS Sweep 2 parameters and return a chosen cumulant.
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%
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% Z = sweepBimodalCumulants(...)
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% Returns a matrix Z of cumulant values at each grid point.
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% Setup grid
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[P1, P2] = meshgrid(param1_vals, param2_vals);
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Z = zeros(size(P1));
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N_samples = 1000;
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maxOrder = max(4, cumulant_index);
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for i = 1:numel(P1)
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% Copy fixed parameters
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params = fixed_params;
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% Override swept parameters
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params.(param1_name) = P1(i);
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params.(param2_name) = P2(i);
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% Generate and compute cumulants
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data = generateBimodalDistribution(N_samples, ...
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params.mu1, params.mu2, ...
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params.sigma1, params.sigma2, ...
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params.weight1);
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kappa = computeCumulants(data, maxOrder);
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Z(i) = kappa(cumulant_index);
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end
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% Plot full heatmap
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figure;
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set(gcf, 'Color', 'w', 'Position', [100 100 950 750]);
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imagesc(param1_vals, param2_vals, Z);
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set(gca, 'YDir', 'normal');
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xlabel(param1_name);
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ylabel(param2_name);
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title(['Cumulant \kappa_', num2str(cumulant_index)]);
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colorbar;
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axis tight;
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% Optional binary colormap (red = ≥0, blue = <0)
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figure;
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set(gcf, 'Color', 'w', 'Position', [100 100 950 750]);
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imagesc(param1_vals, param2_vals, Z);
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set(gca, 'YDir', 'normal');
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xlabel(param1_name);
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ylabel(param2_name);
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title(['Binary color split of \kappa_', num2str(cumulant_index)]);
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clim([-1 1]);
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colormap([0 0 1; 1 0 0]); % Blue (neg), Red (pos & zero)
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colorbar;
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axis tight;
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end
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