% This script demonstrates that a data matrix where the data 
% are all of a single sinusoidal frequency produces only two
% nonzero singular values.

% 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 = 500;

% Frequency for the data
w = 2*pi;

% The random amplitude and phase offset for each gene.
phi = 2*pi*rand(n,1);
amp = 10*rand(n,1);

% Generate the sinusoidal data matrix
A = w*del_t*ones(n,1)*[0:N-1] + phi*ones(1,N);
A = diag(amp)*sin(A);

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

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

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

% Plot first 2 "modes"
figure(2);
hold off;
plot(T',V(:,3));
hold on;
plot(T',V(:,20));
hold off;