PCA on sklearn - how to interpret pca.components_

问题:

I ran PCA on a data frame with 10 features using this simple code:

pca = PCA()
fit = pca.fit(dfPca)

The result of pca.explained_variance_ratio_ shows:

array([  5.01173322e-01,   2.98421951e-01,   1.00968655e-01,
         4.28813755e-02,   2.46887288e-02,   1.40976609e-02,
         1.24905823e-02,   3.43255532e-03,   1.84516942e-03,
         4.50314168e-16])

I believe that means that the first PC explains 52% of the variance, the second component explains 29% and so on...

What I dont undestand is the output of pca.components_. If I do the following:

df = pd.DataFrame(pca.components_, columns=list(dfPca.columns))

I get the data frame bellow where each line is a principal component. What I'd like to understand is how to interpret that table. I know that if I square all the features on each component and sum them I get 1, but what does the -0.56 on PC1 mean? Dos it tell something about "Feature E" since it is the highest magnitude on a component that explains 52% of the variance?

Thanks

回答1:

Terminology: First of all, the results of a PCA are usually discussed in terms of component scores, sometimes called factor scores (the transformed variable values corresponding to a particular data point), and loadings (the weight by which each standardized original variable should be multiplied to get the component score).

PART1: I explain how to check the importance of the features and how to plot a biplot.

PART2: I explain how to check the importance of the features and how to save them into a pandas dataframe using the feature names.

---

PART 1:

In your case, the value -0.56 for Feature E is the score of this feature on the PC1. This value tells us 'how much' the feature influences the PC (in our case the PC1).

So the higher the value in absolute value, the higher the influence on the principal component.

After performing the PCA analysis, people usually plot the known 'biplot' to see the transformed features in the N dimensions (2 in our case) and the original variables (features).

I wrote a function to plot this.

---

Example using iris data:

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
import pandas as pd
from sklearn.preprocessing import StandardScaler
from sklearn.decomposition import PCA

iris = datasets.load_iris()
X = iris.data
y = iris.target

#In general it is a good idea to scale the data
scaler = StandardScaler()
scaler.fit(X)
X=scaler.transform(X)

pca = PCA()
pca.fit(X,y)
x_new = pca.transform(X)   

def myplot(score,coeff,labels=None):
    xs = score[:,0]
    ys = score[:,1]
    n = coeff.shape[0]

    plt.scatter(xs ,ys, c = y) #without scaling
    for i in range(n):
        plt.arrow(0, 0, coeff[i,0], coeff[i,1],color = 'r',alpha = 0.5)
        if labels is None:
            plt.text(coeff[i,0]* 1.15, coeff[i,1] * 1.15, "Var"+str(i+1), color = 'g', ha = 'center', va = 'center')
        else:
            plt.text(coeff[i,0]* 1.15, coeff[i,1] * 1.15, labels[i], color = 'g', ha = 'center', va = 'center')

plt.xlabel("PC{}".format(1))
plt.ylabel("PC{}".format(2))
plt.grid()

#Call the function. 
myplot(x_new[:,0:2], pca. components_) 
plt.show()

---

Results

PART 2:

The important features are the ones that influence more the components and thus, have a large absolute value on the component.

TO get the most important features on the PCs with names and save them into a pandas dataframe use this:

from sklearn.decomposition import PCA
import pandas as pd
import numpy as np
np.random.seed(0)

# 10 samples with 5 features
train_features = np.random.rand(10,5)

model = PCA(n_components=2).fit(train_features)
X_pc = model.transform(train_features)

# number of components
n_pcs= model.components_.shape[0]

# get the index of the most important feature on EACH component
# LIST COMPREHENSION HERE
most_important = [np.abs(model.components_[i]).argmax() for i in range(n_pcs)]

initial_feature_names = ['a','b','c','d','e']
# get the names
most_important_names = [initial_feature_names[most_important[i]] for i in range(n_pcs)]

# LIST COMPREHENSION HERE AGAIN
dic = {'PC{}'.format(i): most_important_names[i] for i in range(n_pcs)}

# build the dataframe
df = pd.DataFrame(dic.items())

This prints:

     0  1
 0  PC0  e
 1  PC1  d

So on the PC1 the feature named e is the most important and on PC2 the d.

回答2:

Basic Idea

The Principle Component breakdown by features that you have there basically tells you the "direction" each principle component points to in terms of the direction of the features.

In each principle component, features that have a greater absolute weight "pull" the principle component more to that feature's direction.

For example, we can say that in PC1, since Feature A, Feature B, Feature I, and Feature J have relatively low weights (in absolute value), PC1 is not as much pointing in the direction of these features in the feature space. PC1 will be pointing most to the direction of Feature E relative to other directions.

Visualization in Lower Dimensions

For a visualization of this, look at the following figures taken from here and here:

The following shows an example of running PCA on correlated data.

We can visually see that both eigenvectors derived from PCA are being "pulled" in both the Feature 1 and Feature 2 directions. Thus, if we were to make a principle component breakdown table like you made, we would expect to see some weightage from both Feature 1 and Feature 2 explaining PC1 and PC2.

Next, we have an example with uncorrelated data.

Let us call the green principle component as PC1 and the pink one as PC2. It's clear that PC1 is not pulled in the direction of feature x', and as isn't PC2 in the direction of feature y'. Thus, in our table, we must have a weightage of 0 for feature x' in PC1 and a weightage of 0 for feature y' in PC2.

I hope this gives an idea of what you're seeing in your table.