n = 10
p = 2
N = 15
print(f'Example 1: {N} points on Gr({p}, {m}) embedded in Gr({p}, {n})\n')
sig = 0.1
gr_low = Grassmann(m, p, N)
gr = Grassmann(n, p, N)
gr_map = Grassmann(n, m)
X_low = gr_low.rand() # N x m x p
A = gr_map.rand() # n x m
#B = np.random.normal(0, 0.1, (n, p)) # n x p
B = np.zeros((n,p))
AAT = np.matmul(A, A.T)
IAATB = np.matmul(np.eye(n) - AAT, B)
X = np.array([np.linalg.qr(np.matmul(A, X_low[i]) + IAATB)[0] for i in range(N)]) # N x n x p
X = gr.exp(X, sig * gr.randvec(X)) # perturb the emdedded X
scores = PNG(X, log = True)
print('\n')
print('PCA of the scores\n')
n_c = 5
pca = PCA(n_components = n_c)
pca.fit(scores)
print(f'The ratios of expressed variance first {n_c} PCs.')
print(pca.explained_variance_ratio_.round(2))
print(f'The cumulative ratios of expressed variance first {n_c} PCs.')
print(np.cumsum(pca.explained_variance_ratio_).round(2))
