% This script demonstrates that a data matrix where the data
% are all linear combinations of a two sinusoidal frequencies
% of differing amplitude produces four nonzero singular values
% with principal time series that are each dominated by a single
% frequency sine/cosine wave.


% Interval end points
t_low = 0;
t_high = 3;

% Time step size
del_t = 0.05;

% Time Points
T = [t_low:del_t:t_high];

% Number of time points
N = length(T);

% Number of genes 
n = 100;

% Frequencies for the two sinusoids
w1 = 2*pi;
w2 = pi*pi;

% The random phase offsets for each gene.
phi = 2*pi*rand(n,1);
phi2 = 2*pi*rand(n,1);

% Generate the sinusoidal data matrix
W1 = w1*del_t*ones(n,1)*[1:N] + phi*ones(1,N);
W1 = sin(W1);
W2 = w2*del_t*ones(n,1)*[1:N] + phi2*ones(1,N);
W2 = 10*sin(W2);
A = W1 + W2;

[U,S,V] = svd(A);

S = diag(S);
S(1:10)

% Plot first 2 "modes"
figure(1);
hold off;
plot(T',V(:,1),'k-');
hold on;
plot(T',V(:,2),'k-.');
hold on;

% Plot the second pair of "modes"
figure(2);
hold off;
plot(T',V(:,3),'k-');
hold on;
plot(T',V(:,4),'k-.');
hold on;
hold off;