數據預處理之PCA主成分分析

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
# decomposition  降解  
# 經過運算屬性變了   原來的數據變了
# pca 將數據屬性變少,少量得數據可以代表原來比較多的數據

from sklearn.decomposition import PCA

from sklearn.linear_model import LogisticRegression
from sklearn import datasets
from sklearn.model_selection import train_test_split

import warnings 
warnings.filterwarnings('ignore')

X,y=datasets.load_iris(True)

# 降維
pca=PCA(n_components=0.98,whiten=False)  # n_components 保留多少屬性 小數 重要性,當一個屬性的重要性不夠時,找到可以加和 第二大的依次類推
                                                                        # 整數位個數  whiten 爲是否標準化   

X_pca=pca.fit_transform(X)

X_pca.head()   #數據有正有負 則無原來的物理意義

array([[-2.68412563,  0.31939725, -0.02791483],
       [-2.71414169, -0.17700123, -0.21046427],
       [-2.88899057, -0.14494943,  0.01790026],
       [-2.74534286, -0.31829898,  0.03155937],
       [-2.72871654,  0.32675451,  0.09007924],])

PCA原理

# 1.去中心化
B=X-X.mean(axis=0)
B[:5]

array([[-0.74333333,  0.44266667, -2.358     , -0.99933333],
       [-0.94333333, -0.05733333, -2.358     , -0.99933333],
       [-1.14333333,  0.14266667, -2.458     , -0.99933333],
       [-1.24333333,  0.04266667, -2.258     , -0.99933333],
       [-0.84333333,  0.54266667, -2.358     , -0.99933333]])

 # 2計算協方差  矩陣
V=np.cov(B,rowvar=False)
V

array([[ 0.68569351, -0.042434  ,  1.27431544,  0.51627069],
       [-0.042434  ,  0.18997942, -0.32965638, -0.12163937],
       [ 1.27431544, -0.32965638,  3.11627785,  1.2956094 ],
       [ 0.51627069, -0.12163937,  1.2956094 ,  0.58100626]])

# 3,協方差矩陣的特徵值和特徵向量
eigen,ev=np.linalg.eigh(V)
display(eigen,ev)  # 特徵值 和特徵向量

array([0.02383509, 0.0782095 , 0.24267075, 4.22824171])



array([[ 0.31548719,  0.58202985,  0.65658877, -0.36138659],
       [-0.3197231 , -0.59791083,  0.73016143,  0.08452251],
       [-0.47983899, -0.07623608, -0.17337266, -0.85667061],
       [ 0.75365743, -0.54583143, -0.07548102, -0.3582892 ]])

eigen/eigen.sum() # 重要性

array([0.00521218, 0.01710261, 0.05306648, 0.92461872])

# 4 降維標準 2個特徵  選取最大的兩個特徵值所對應的的特徵的特徵向量
# 百分比   計算各特徵值 佔權重 累加可以
vector=ev[:,2:]

# 5,進行矩陣運算
PCA_result=B.dot(vector)
PCA_result[:10]

array([[ 0.31939725,  2.68412563],
       [-0.17700123,  2.71414169],
       [-0.14494943,  2.88899057],
       [-0.31829898,  2.74534286],
       [ 0.32675451,  2.72871654],
       [ 0.74133045,  2.28085963],
       [-0.08946138,  2.82053775],
       [ 0.16338496,  2.62614497],
       [-0.57831175,  2.88638273],
       [-0.11377425,  2.6727558 ]])

驗證降維數據,準確

# 降維數據  爲0.87

lr=LogisticRegression()
X_train,X_test,y_train,y_test=train_test_split(X_pca,y,test_size=0.2,random_state=102)
lr.fit(X_train,y_train)
lr.score(X_test,y_test)

0.9

# 原數據

lr=LogisticRegression()
X_train,X_test,y_train,y_test=train_test_split(X,y,test_size=0.2,random_state=102)
lr.fit(X_train,y_train)
lr.score(X_test,y_test)

0.9666666666666667