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  1. CHAPTER 3 - UNSUPERVISED LEARNING
  2. Principal Component Analysis

Interpretation of PCA result

PreviousHow Does It Work?NextApplications in Transportation

Last updated 1 year ago

We will give an interpretation of the PCA result in this section. Let’s take the Iris database as an example again. Because there are four features for each Iris sample, each principal direction is a 4-dimensional reduction. By taking a Singular vector decomposition method on the standardized Iris dataset, the principal direction and the corresponding weights can be shown as below:

principal directions:

[ 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]

corresponding variance:

[2.93808505, 0.9201649 , 0.14774182, 0.02085386]

corresponding variance ratio:

[0.72962445, 0.22850762, 0.03668922, 0.00517871]

As can be seen from the variance ratio, the first two principal components can capture 95.813207% of the original dataset. If we multiply the dataset with the first principal direction, we will get a value for each sample. Multiplying the dataset with the second principal direction, we will get a value for each sample. These values and values are the corresponding principal component for each principal direction. By plotting each sample with related value and value, we can the visualization of the Iris dataset as in Figure 1.

The principal components are linearly related to the elements of principal direction . The feature which is corresponding to the higher absolute value of has more influence on this direction. For example, in the Iris dataset, the weights of feature 1, feature 2, and feature 4 in the first principal direction are higher than feature 2. Therefore, these three features are more important in that direction. For the second direction, feature 2 is predominant than other features. By plotting the weights of different principal direction, we can plot the result for each feature as shown by the red lines in Figure 1.