Problem with identifying the entropy stabilization point in SSA for accurate signal decomposition.

In Singular Spectrum Analysis (SSA), the goal is to decompose a signal into three main components:
Trend: The overall trend of the signal, which represents long-term changes.
Oscillations: Parts of the signal that change periodically, typically corresponding to medium-term variations.
Noise: Parts of the signal that change randomly, usually representing noise or disturbances in the signal.
In this process, the signal is first transformed into a trajectory matrix, and then its components are extracted via Singular Value Decomposition (SVD).
Main Challenge:
After performing SVD and calculating the entropy of each component, we are looking for a point where the entropy stabilizes. This point is considered where additional components no longer contribute meaningful information to the model.
The problem is that the point of entropy stabilization is not being correctly identified, which results in an improper decomposition of the signal into the three components (trend, oscillations, and noise).
Question: Is there a method in MATLAB that can more accurately identify the point of entropy stabilization and correctly decompose the signal into trend, oscillations, and noise?

clc;
clear;
close all;

% Load the CSV file
[file, path] = uigetfile(‘*.csv’, ‘Please upload your CSV file:’);
if isequal(file, 0)
error(‘No file selected.’);
end
data = readtable(fullfile(path, file));

% Display available columns for the user to choose from
disp(‘Available columns in your dataset:’);
disp(data.Properties.VariableNames’);
column_name = input(‘Enter the column name you want to analyze: ‘, ‘s’);

% Extract the signal (time series) from the chosen column
if ~any(strcmp(data.Properties.VariableNames, column_name))
error([‘Column "’, column_name, ‘" not found in the dataset.’]);
end
signal = data.(column_name);
signal = signal(~isnan(signal)); % Drop NaN values

% SSA Functions
function X = trajectory_matrix(signal, window_length)
N = length(signal);
K = N – window_length + 1;
X = zeros(window_length, K);
for i = 1:K
X(:, i) = signal(i:i + window_length – 1);
end
end

function entropy = singular_entropy(S)
P = S / sum(S);
entropy = -sum(P .* log(P));
end

function reconstructed_signal = reconstruct_signal(U, S, V, indices, original_length)
reconstructed_trajectory = zeros(size(U, 1), size(V, 1));
for i = indices
reconstructed_trajectory = reconstructed_trajectory + S(i) * (U(:, i) * V(i, :));
end

[rows, cols] = size(reconstructed_trajectory);
signal_length = rows + cols – 1;
reconstructed_signal = zeros(signal_length, 1);
counts = zeros(signal_length, 1);

% Extracting anti-diagonals
for k = 1:(rows + cols – 1)
anti_diag_elements = [];
for i = 1:rows
j = k – i + 1;
if j >= 1 && j <= cols
anti_diag_elements = [anti_diag_elements; reconstructed_trajectory(i, j)];
end
end
reconstructed_signal(k) = mean(anti_diag_elements);
counts(k) = length(anti_diag_elements);
end

reconstructed_signal = reconstructed_signal ./ counts; % Normalize the signal
reconstructed_signal = reconstructed_signal(1:original_length);
end

% Perform SSA
window_length = floor(length(signal) / 3); % Adjust window length
X = trajectory_matrix(signal, window_length);

% Perform SVD decomposition
[U, S_mat, V] = svd(X, ‘econ’);
S = diag(S_mat);

% Calculate Singular Entropy for each component cumulatively
cumulative_entropy = zeros(length(S), 1);
for i = 1:length(S)
cumulative_entropy(i) = singular_entropy(S(1:i));
end

% Plot Singular Entropy to observe stabilization point
figure;
plot(1:length(S), cumulative_entropy, ‘-o’, ‘LineWidth’, 1.5);
xlabel(‘Component Index’);
ylabel(‘Cumulative Singular Entropy’);
title(‘Cumulative Singular Entropy vs. Component Index’);
grid on;

% Calculate the point where entropy stabilizes
entropy_diff = diff(cumulative_entropy); % Difference between consecutive entropy values
threshold = 0.01; % Threshold to detect stabilization
stable_point = find(entropy_diff < threshold, 1);

% Mark the stabilization point
hold on;
xline(stable_point + 1, ‘–r’, [‘Stabilization Point: ‘, num2str(stable_point + 1)]);
legend(‘Cumulative Singular Entropy’, ‘Stabilization Point’);

% Number of components based on entropy stabilization
num_components = stable_point + 1;
disp([‘Number of components based on entropy stabilization: ‘, num2str(num_components)]);

% Reconstruct the signal by categorizing components
trend_indices = 1; % Trend corresponds to the first component
oscillation_indices = 2:num_components; % Oscillations from second component to stabilization point
noise_indices = (num_components + 1):length(S); % Noise corresponds to components after stabilization

% Reconstruct signals based on selected indices
trend_signal = reconstruct_signal(U, S, V’, trend_indices, length(signal));
oscillation_signal = reconstruct_signal(U, S, V’, oscillation_indices, length(signal));
noise_signal = reconstruct_signal(U, S, V’, noise_indices, length(signal));

% Plot the original signal, Trend, Oscillations, and Noise
figure;
plot(signal, ‘b’, ‘DisplayName’, ‘Original Signal’, ‘LineWidth’, 1.5);
hold on;
plot(trend_signal, ‘g’, ‘DisplayName’, ‘Trend’, ‘LineWidth’, 1.5);
plot(oscillation_signal, ‘orange’, ‘DisplayName’, ‘Oscillations’, ‘LineWidth’, 1.5);
plot(noise_signal, ‘r’, ‘DisplayName’, ‘Noise’, ‘LineWidth’, 1.5);
title(‘Signal Decomposition’);
xlabel(‘Time’);
ylabel(‘Amplitude’);
legend;
grid on;In Singular Spectrum Analysis (SSA), the goal is to decompose a signal into three main components:
Trend: The overall trend of the signal, which represents long-term changes.
Oscillations: Parts of the signal that change periodically, typically corresponding to medium-term variations.
Noise: Parts of the signal that change randomly, usually representing noise or disturbances in the signal.
In this process, the signal is first transformed into a trajectory matrix, and then its components are extracted via Singular Value Decomposition (SVD).
Main Challenge:
After performing SVD and calculating the entropy of each component, we are looking for a point where the entropy stabilizes. This point is considered where additional components no longer contribute meaningful information to the model.
The problem is that the point of entropy stabilization is not being correctly identified, which results in an improper decomposition of the signal into the three components (trend, oscillations, and noise).
Question: Is there a method in MATLAB that can more accurately identify the point of entropy stabilization and correctly decompose the signal into trend, oscillations, and noise?