Scikit-learn principal component analysis (PCA) for dimension reduction

Question

I want to perform principal component analysis for dimension reduction and data integration.

I have 3 features(variables) and 5 samples like below. I want to integrate them into 1-dimensional(1 feature) output by transforming them(computing 1st PC). I want to use transformed data for further statistical analysis, because I believe that it displays the 'main' characteristics of 3 input features.

I first wrote a test code with python using scikit-learn like below. It is the simple case that the values of 3 features are all equivalent. In other word, I applied PCA for three same vector, [0, 1, 2, 1, 0].

Code

import numpy as np
from sklearn.decomposition import PCA
pca = PCA(n_components=1)
samples = np.array([[0,0,0],[1,1,1],[2,2,2],[1,1,1],[0,0,0]])
pc1 = pca.fit_transform(samples)
print (pc1)

Output

[[-1.38564065]
[ 0.34641016]
[ 2.07846097]
[ 0.34641016]
[-1.38564065]]

1. Is taking 1st PCA after dimension reduction proper approach for data integration?

1-2. For example, if features are like [power rank, speed rank], and power have roughly negative correlation with speed, when it is a 2-feature case. I want to know the sample which have both 'high power' and 'high speed'. It is easy to decide that [power 1, speed 1] is better than [power 2, speed 2], but difficult for the case like [power 4, speed 2] vs [power 3, speed 3]. So I want to apply PCA to 2-dimensional 'power and speed' dataset, and take 1st PC, then use the rank of '1st PC'. Is this kind of approach still proper?

2. In this case, I think the output should also be [0, 1, 2, 1, 0] which is the same as the input. But output was [-1.38564065, 0.34641016, 2.07846097, 0.34641016, -1.38564065]. Are there any problem with the code, or is it the right answer?

Answer 1

1. Yes. It is also called data projection (to the lower dimension).
2. The resulting output is centered and normalized according to the train data. The result is correct.

In case of only 5 samples I don't think it is wise to run any statistical methods. And if you believe that your features are the same, just check that correlation between dimensions is close to 1, and then you can just disregard other dimensions.