RyotaBannai

from sklearn.decomposition import PCA

pca = PCA(n_components=2)
pca.fit(X) # Apply PCA

RyotaBannai

from matplotlib.patches import Ellipse
import matplotlib as mpl

ax = plt.gca(); ax.set_xlabel('X'); ax.set_ylabel('Y')
plt.scatter(X_raw[:, 0], X_raw[:, 1], c='#B8860B', alpha=0.5)
plt.scatter(X_mean[0], X_mean[1], c='red', s=50)

U_, s_, _ = LA.svd(np.cov(X, rowvar=False), full_matrices=False)
angle = np.degrees(np.arctan2(U_[1,0], U_[0,0]))
width, height = 2 * np.sqrt(s_)

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ax = plt.gca()
ax.set_xlabel('X'); ax.set_ylabel('Y')
plt.scatter(X_raw[:, 0], X_raw[:, 1], c='#B8860B', alpha=0.5)
plt.scatter(X_mean[0], X_mean[1], c='red', s=50)
plt.axis('equal')

for length, vector in zip(e_values, V):
    dir_ = -vector * 3  * np.sqrt(length) # Tweak the sign
    start = X_mean; end = start + dir_
    arrowprops = dict(arrowstyle='->',linewidth=2, 

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]

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)

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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

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):

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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

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

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