In [1]:
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
Data¶
In [2]:
points = [[0.71, 0.18, 2.50, 0.45, 0.03,
0.12, 0.30, 2.65, 0.90, 0.46],
[0.14, 0.23, 2.30, 0.17, 0.44,
0.24, 0.03, 2.10, 0.92, 0.33]]
In [3]:
data = np.array(points).T
data.shape
Out[3]:
(10, 2)
In [4]:
plt.plot(data[:, 0], data[:, 1], 'o')
plt.grid()
Move data using mean¶
In [5]:
x = data[:, 0]
y = data[:, 1]
In [6]:
x_bar = round(x.mean(), 2)
x_bar
Out[6]:
0.83
In [7]:
y_bar = round(y.mean(), 2)
y_bar
Out[7]:
0.69
In [8]:
scaled_x = x - x_bar
scaled_x
Out[8]:
array([-0.12, -0.65, 1.67, -0.38, -0.8 , -0.71, -0.53, 1.82, 0.07,
-0.37])
In [9]:
scaled_y = y - y_bar
scaled_y
Out[9]:
array([-0.55, -0.46, 1.61, -0.52, -0.25, -0.45, -0.66, 1.41, 0.23,
-0.36])
In [10]:
plt.plot(scaled_x, scaled_y, 'o')
plt.grid()
Covariance matrix¶
$cov(x, y)=cov(y,x)=\Sigma_{1}^{10}\frac{(x_i-\bar{x})(y_i-\bar{y})}{10-1}$
In [11]:
cov_xy = round(np.sum(scaled_x * scaled_y) / 9, 5)
cov_xy
Out[11]:
0.75957
$cov(x, x)=var(x)=\Sigma_{1}^{10}\frac{(x_i-\bar{x})(x_i-\bar{x})}{10-1}$
In [12]:
var_x = round(np.sum(scaled_x * scaled_x) / 9, 5)
var_x
Out[12]:
0.9166
$cov(y, y)=var(y)=\Sigma_{1}^{10}\frac{(y_i-\bar{y})(y_i-\bar{y})}{10-1}$
In [13]:
var_y = round(np.sum(scaled_y * scaled_y) / 9, 5)
var_y
Out[13]:
0.6942
In [14]:
np.cov(x, y, ddof=1)
Out[14]:
array([[0.9166 , 0.75956667],
[0.75956667, 0.6942 ]])
Eigenvectors of covariance matrix¶
In [15]:
e_vals, e_vecs = np.linalg.eig(np.cov(x, y, ddof=1))
In [16]:
e_vals
Out[16]:
array([1.57306331, 0.03773669])
In [17]:
e_vecs
Out[17]:
array([[ 0.75658944, -0.65389022],
[ 0.65389022, 0.75658944]])
첫 번째 eigenvect를 선택한다.
In [18]:
e_vecs[:, 0]
Out[18]:
array([0.75658944, 0.65389022])
Points on principal component¶
In [19]:
scaled_data = np.c_[scaled_x, scaled_y]
scaled_data
Out[19]:
array([[-0.12, -0.55],
[-0.65, -0.46],
[ 1.67, 1.61],
[-0.38, -0.52],
[-0.8 , -0.25],
[-0.71, -0.45],
[-0.53, -0.66],
[ 1.82, 1.41],
[ 0.07, 0.23],
[-0.37, -0.36]])
Scaled data를 principal components에 투영시키면 새로운 points가 생성된다.
In [20]:
new_points = np.dot(scaled_data, e_vecs[:, 0])
new_points
Out[20]:
array([-0.45043035, -0.79257264, 2.31626762, -0.6275269 , -0.76874411,
-0.8314291 , -0.83255995, 2.29897799, 0.20335601, -0.51533857])
Visualization
In [21]:
origin = [0], [0]
plt.plot(scaled_x, scaled_y, 'o')
plt.quiver(*origin, e_vecs[:, 0], e_vecs[:, 1], color=['r','b','g'], scale=3)
plt.grid()
In [22]:
# 두 eigenvector의 내적값은 0이다. 즉 90도 이다.
np.dot(e_vecs[:, 0], e_vecs[:, 1])
Out[22]:
0.0
Sklearn을 사용하여 결과를 비교해보자.¶
In [23]:
from sklearn.decomposition import PCA
In [24]:
pca = PCA(n_components=2)
pca.fit(data)
Out[24]:
PCA(copy=True, iterated_power='auto', n_components=2, random_state=None, svd_solver='auto', tol=0.0, whiten=False)
Covariance matrix
In [25]:
pca.get_covariance()
Out[25]:
array([[0.9166 , 0.75956667],
[0.75956667, 0.6942 ]])
Pricipal components위로 projection된 데이터
In [26]:
pca.transform(data)
Out[26]:
array([[-0.45043035, 0.33765737],
[-0.79257264, -0.0769975 ],
[ 2.31626762, -0.12611234],
[-0.6275269 , 0.14494823],
[-0.76874411, -0.33396481],
[-0.8314291 , -0.1237968 ],
[-0.83255995, 0.15278722],
[ 2.29897799, 0.12328908],
[ 0.20335601, -0.12824326],
[-0.51533857, 0.03043282]])