Analiza glavnih komponenti¶

In [1]:
import pandas as pd
In [2]:
import numpy as np

Podaci¶

In [3]:
df = pd.read_csv("../data/iris.csv")
In [4]:
df.head()
Out[4]:
Sepal_Length Sepal_Width Petal_Length Petal_Width Species
0 5.1 3.5 1.4 0.2 setosa
1 4.9 3.0 1.4 0.2 setosa
2 4.7 3.2 1.3 0.2 setosa
3 4.6 3.1 1.5 0.2 setosa
4 5.0 3.6 1.4 0.2 setosa
In [5]:
X = df.drop("Species", axis = 1)
y = df["Species"]

Analiza¶

In [6]:
from sklearn.decomposition import PCA
In [7]:
pca = PCA()
# pca = PCA(n_component=0.9)

PCA može da instanciramo tako navodimo koliki deo varijanse želimo da pokriju nove komponente. Ako PCE instanciramo bez parametara, tada se kreiraju sve moguće komponente.

Preprocesiranje¶

Kako bi se izvršila analiza, potrebno je da svi atributi imaju srednju vrednost $0$. U ovu svrhu, podatke možemo da standardizujemo.

In [8]:
from sklearn.preprocessing import StandardScaler
In [9]:
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)
In [10]:
X_scaled = pd.DataFrame(X_scaled, columns=X.columns)
In [11]:
X_scaled
Out[11]:
Sepal_Length Sepal_Width Petal_Length Petal_Width
0 -0.900681 1.019004 -1.340227 -1.315444
1 -1.143017 -0.131979 -1.340227 -1.315444
2 -1.385353 0.328414 -1.397064 -1.315444
3 -1.506521 0.098217 -1.283389 -1.315444
4 -1.021849 1.249201 -1.340227 -1.315444
... ... ... ... ...
145 1.038005 -0.131979 0.819596 1.448832
146 0.553333 -1.282963 0.705921 0.922303
147 0.795669 -0.131979 0.819596 1.053935
148 0.432165 0.788808 0.933271 1.448832
149 0.068662 -0.131979 0.762758 0.790671

150 rows × 4 columns

In [12]:
# predsulvom je da atributi imaju prosek 0
# mozemo da standardizujemo
In [13]:
pca.fit(X_scaled)
x_pca = pca.transform(X_scaled)
In [14]:
pca_names = [f"pca_{i}" for i in range(len(X.columns))]
In [15]:
x_pca = pd.DataFrame(x_pca, columns=pca_names)
In [16]:
x_pca
Out[16]:
pca_0 pca_1 pca_2 pca_3
0 -2.264703 0.480027 -0.127706 -0.024168
1 -2.080961 -0.674134 -0.234609 -0.103007
2 -2.364229 -0.341908 0.044201 -0.028377
3 -2.299384 -0.597395 0.091290 0.065956
4 -2.389842 0.646835 0.015738 0.035923
... ... ... ... ...
145 1.870503 0.386966 0.256274 -0.389257
146 1.564580 -0.896687 -0.026371 -0.220192
147 1.521170 0.269069 0.180178 -0.119171
148 1.372788 1.011254 0.933395 -0.026129
149 0.960656 -0.024332 0.528249 0.163078

150 rows × 4 columns

Analiza¶

In [17]:
from sklearn.decomposition import PCA
In [18]:
pca = PCA()
# pca = PCA(n_component=0.9)

PCA može da instanciramo tako navodimo koliki deo varijanse želimo da pokriju nove komponente. Ako PCE instanciramo bez parametara, tada se kreiraju sve moguće komponente.

In [19]:
pca.fit(X_scaled)
x_pca = pca.transform(X_scaled)
In [ ]:
In [20]:
pca_names = [f"pca_{i}" for i in range(len(X.columns))]
x_pca = pd.DataFrame(x_pca, columns=pca_names)
x_pca
Out[20]:
pca_0 pca_1 pca_2 pca_3
0 -2.264703 0.480027 -0.127706 -0.024168
1 -2.080961 -0.674134 -0.234609 -0.103007
2 -2.364229 -0.341908 0.044201 -0.028377
3 -2.299384 -0.597395 0.091290 0.065956
4 -2.389842 0.646835 0.015738 0.035923
... ... ... ... ...
145 1.870503 0.386966 0.256274 -0.389257
146 1.564580 -0.896687 -0.026371 -0.220192
147 1.521170 0.269069 0.180178 -0.119171
148 1.372788 1.011254 0.933395 -0.026129
149 0.960656 -0.024332 0.528249 0.163078

150 rows × 4 columns

In [ ]:

Možemo da vidimo kako se računaju nove komponente:

In [21]:
pca.components_
Out[21]:
array([[ 0.52106591, -0.26934744,  0.5804131 ,  0.56485654],
       [ 0.37741762,  0.92329566,  0.02449161,  0.06694199],
       [-0.71956635,  0.24438178,  0.14212637,  0.63427274],
       [-0.26128628,  0.12350962,  0.80144925, -0.52359713]])

Dake: $$ PCA_{0} = 0.52 * S_{l} - 0.26 * S_{w} + 0.58 * P_{l} + 0.56 * P_{w} $$ $$ PCA_{1} = 0.38 * S_{l} - 0.92 * S_{w} + 0.02 * P_{l} + 0.06 * P_{w} $$ itd.

Takođe, možemo da vidimo koji procenat varijanse pokriva koja komponenta:

In [22]:
pca.explained_variance_ratio_
Out[22]:
array([0.72962445, 0.22850762, 0.03668922, 0.00517871])

Dakle ${PCA}_{0}$ pokriva $72.9\%$, ${PCA}_{1}$ pokriva $22.8\%$, itd.

Vizualizacija¶

Iz ovih vizualizacija može da se donese zaključak koliko nam komponenti trebaju.

In [23]:
from matplotlib import pyplot as plt
In [24]:
plt.bar(pca_names, pca.explained_variance_ratio_)
Out[24]:
<BarContainer object of 4 artists>
In [25]:
cumulative_variance_ratio = np.cumsum(pca.explained_variance_ratio_)
plt.plot(pca_names, cumulative_variance_ratio, marker = 'o')
Out[25]:
[<matplotlib.lines.Line2D at 0x7fd4863bee60>]

Svođenje atributa na dve dimenzije:

In [26]:
colors = ["red", "green", "blue"]
class_names = y.unique()
for i, class_name in enumerate(class_names):
    class_instances = x_pca.iloc[y[y == class_name].index]
    plt.scatter(class_instances["pca_0"], class_instances["pca_1"], c=colors[i], label=class_name)
    
plt.legend(loc="best")
Out[26]:
<matplotlib.legend.Legend at 0x7fd4864bf0a0>