[learning notes] Introduction to principal component analysis

One ： brief introduction

Principal component analysis (PCA) is a multivariate statistical analysis method that transforms multiple indicators into several comprehensive indicators by reducing the dimension , Use fewer variables to explain most of the information in the original data .

Two ： Basic principles and procedures

1. The basic principle ： A little .

It should be noted that ：

（1） The results of principal component analysis are influenced by dimensions , Dimensionless treatment is required first , Then the covariance or correlation coefficient matrix is used for analysis .

（2） The principal components selected in the actual study do not exceed 6 individual , Only need to contribute more than 85% that will do .

2. Basic steps ：

3、 ... and ： Example

# Program files Pex11_7.py
import numpy as np
from sklearn.decomposition import PCA
a=np.loadtxt("Pdata11_7.txt")
b=np.r_[a[:,1:4],a[:,-3:]]  # Construct a data matrix 
# #np.r_ It's connecting two matrices in columns , That is to add two matrices up and down , The number of columns is required to be equal .‘
# So the method of constructing the matrix above will be clear .
md=PCA().fit(b)  # Build and train models 
print(" The characteristic value is ：",md.explained_variance_)
print(" Contribution rate of each principal component ：",md.explained_variance_ratio_)
print(" The singular value is ：",md.singular_values_)
print(" Coefficient of each principal component ：\n",md.components_)  # Each line is a principal component 
""" Next, calculate the eigenvalue and eigenvector directly , Compare with library function """
cf=np.cov(b.T)  # Calculate the covariance matrix 
c,d=np.linalg.eig(cf) # Find eigenvalues and eigenvectors 
print(" The characteristic value is ：",c)
print(" The eigenvector is ：\n",d)
print(" The contribution rate of each principal component is ：",c/np.sum(c))

 

  As shown above ： Use library functions PV=CA when , The sign of the principal component is uncontrollable , So we calculate eigenvalues and eigenvectors directly .

  meanwhile ：PCA Principal component analysis using covariance matrix . We can also choose the correlation coefficient matrix . This is equivalent to standardizing the data .

Four ： Application of principal component analysis  

General steps for comprehensive evaluation ：

  example 11.8

 

# Program files Pex11_8.py
import numpy as np
from scipy.stats import zscore
a=np.loadtxt("Pdata11_8.txt")
print(" The correlation coefficient matrix is ：\n",np.corrcoef(a.T))
b=np.delete(a,0,axis=1) # Delete the first 1 Column data 
# This is because r12=r21=1, therefore x1,x2 Completely linear correlation , Consistent selection of indicators , Just keep one 
c=zscore(b); r=np.corrcoef(c.T) # Data standardization and calculation of correlation coefficient matrix 
d,e=np.linalg.eig(r) # Find eigenvalues and eigenvectors 
rate=d/d.sum()  # Calculate the contribution rate of each principal component 
print(" The characteristic value is ：",d)
print(" The eigenvector is ：\n",e)
print(" The contribution rate of each principal component is ：",rate)
k=1; # Put forward the number of principal components 
F=e[:,:k]; score_mat=c.dot(F) # Calculate the principal component score matrix 
score1=score_mat.dot(rate[0:k])  # Calculate the score of each evaluation object 
score2=-score1  # By observation , Adjust the sign of the score 
# This is mainly because the previous eigenvectors do not take the opposite number . If you take this, you don't have to take the opposite 
print(" The score of each evaluation object is ：",score2) 
index=score1.argsort()+1   # The position of each element after sorting in the original array 
print(" The number of each city from high to low is ：",index)

Note here ： First observe the correlation coefficient matrix , If any non diagonal element is 1, It means that the variable selection is unreasonable , The two are completely linear , An indicator should be deleted .

Be careful ： On the principal component score matrix and the score of the evaluation object , As shown in the figure below . Because the title is benefit index value , So the higher the evaluation value, the higher the ranking . Otherwise, the ranking will be reversed .