RyotaBannai

import numpy as np
X = np.random.rand(5,10)

RyotaBannai

X -= X.mean(axis=0)
C = np.cov(X,rowvar=False)

RyotaBannai

from numpy import linalg as LA
w, v = LA.eig(C)
idx = w.argsort()[::-1] # In order that bigger Eigenvalue comes first
w, v = w[idx], v[:, idx]

RyotaBannai

n = 2;
n_PC = v[:, 0:n]
#transform matrix X to two dimension matrix 
T = np.dot(X, n_PC)

RyotaBannai

rng = np.random.RandomState(1)
X_raw = np.dot(rng.rand(2, 2), rng.randn(2, 200)).T
X_mean = X_raw.mean(axis=0)
X -= X_mean

RyotaBannai

C = np.cov(X.T) # Now shape is 2 *2 
w, v = LA.eig(C)
inx = w.argsort()[::-1]
w, v = w[inx], v[:, inx]

w_12 = w[:2]
v_12 = v[:, :2]

RyotaBannai

import seaborn as sns; sns.set()

ax = plt.gca()
ax.set_xlabel('Principal component 1')
ax.set_ylabel('Principal component 2')
plt.scatter(X_raw[:, 0], X_raw[:, 1], c='#663399', alpha=0.5)
plt.scatter(X_mean[0], X_mean[1], c='red', s=50)
plt.axis('equal')

for length, vector in zip(w_12, v_12):

RyotaBannai

T = X.dot(v) # Projecting sample data
T = T[:, 0:2]
# Or simply, T = X.dot(v_12)
T_mean = T.mean(axis=0)

C_T = np.cov(T.T) # Shape 2 *2
wT, vT = LA.eig(C_T)
inxT = wT.argsort()[::-1]
wT, v = wT[inxT], vT[:, inxT]

RyotaBannai

ax = plt.gca()
ax.set_xlabel('Principal component 1')
ax.set_ylabel('Principal component 2')
plt.scatter(T[:, 0], T[:, 1], c='#663399', alpha=0.5)
plt.scatter(T_mean[0], T_mean[1], c='red', s=50)

for length, vector in zip(w_12_T, v_12_T):
    dir_ = vector * 3  * np.sqrt(length)
    arrowprops = dict(arrowstyle='->', linewidth=2, shrinkA=0,
                      shrinkB=0, color='red', alpha=0.5)

RyotaBannai

rng = np.random.RandomState(1)
X_raw = np.dot(rng.rand(2, 2), rng.randn(2, 200)).T
X_mean = X_raw.mean(axis=0)
X -= X_mean
U, s, Vt = LA.svd(X, full_matrices=False)
V = Vt.T
S = np.diag(s)
e_values = (s ** 2) / X.shape[0]