Can Principal Component Analysis be applied on 2D geometry with X and Y nodes in Python?

Urvesh 0
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Aim of the task: I have sets of coordinates (X and Y) coordinates of the geometry and I want to make my geometry aligned. The coordinate and respective geometry is shown in the picture.

X1_coordinate = [0.0, 0.87, 1.37, 1.87, 2.73, 3.6, 4.46, 4.96, 5.46, 4.6, 3.73, 2.87, 2.0, 1.5, 1.0, 0.5, 2.37, 3.23, 4.1]
Y1_coordinate = [0.0, 0.5, -0.37, -1.23, -0.73, -0.23, 0.27, -0.6, -1.46, -1.96, -2.46, -2.96, -3.46, -2.6, -1.73, -0.87, -2.1, -1.6, -1.1]

enter image description here

Question: Can I apply Principal Component Analysis on 2D geometries to make it aligned such that its principal axis will be parallel to the reference axis (X and Y)?

Expected output: I want my geometry like this. This is just an example. I want my geometry in such as way that, principal axis of geometry lies on the reference axis or be parallel to reference axis.

enter image description here

What I tried: I tried below code to implement PCA and to obtain the geometry aligned.

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
from sklearn.decomposition import PCA
import pandas as pd
from sklearn.preprocessing import StandardScaler
plt.style.use('ggplot')
# Load the data
# iris = datasets.load_iris()
X = X1_coordinate
y = Y1_coordinate
# Z-score the features
scaler = StandardScaler()
scaler.fit(X)
X = scaler.transform(X)
# The PCA model
pca = PCA(n_components=2) # estimate only 2 PCs
X_new = pca.fit_transform(X) # project the original data into the PCA space

However, after running the code, I got error as mentioned below.

enter image description here

Kindly let me know what should I do to make my geometry aligned. Looking forward to get answers.

2d dimensionality-reduction pca geometry python-3.x
1 Answers

Basically, you can apply PCA to this task.

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

circle_pts = sklearn.datasets.make_circles() #get two circles with labels
circle_pts = circle_pts[0][circle_pts[1]==0] #leave only one circle

ang = 63/180*np.pi #radians of rotation
R = np.array([[np.cos(ang), -np.sin(ang)], [np.sin(ang), np.cos(ang)]])

ellipse_pts = circle_pts
ellipse_pts[:,0] *= 4.5
ellipse_rot_pts = ellipse_pts @ R.T

plt.figure()
plt.scatter(ellipse_rot_pts[:,0], ellipse_rot_pts[:,1])
plt.axis("equal")
plt.tight_layout()
plt.show()


scaler = StandardScaler(with_std=False)
scaler.fit(ellipse_rot_pts)
X = scaler.transform(ellipse_rot_pts)
pca = PCA(n_components=2) # estimate only 2 PCs
X_new = pca.fit_transform(X) # project the original data into the PCA space


plt.figure()
plt.scatter(X[:,0],X[:,1])

singular_values = pca.singular_values_
plt.plot([0, singular_values[0]*pca.components_[0,0]], [0, singular_values[0]*pca.components_[0,1]])
plt.plot([0, singular_values[1]*pca.components_[1,0]], [0, singular_values[1]*pca.components_[1,1]])

plt.axis("equal")
plt.show() 


plt.figure()
plt.title("Aligned with axis figure")
plt.scatter(X_new[:,0],X_new[:,1])
plt.axis("equal")
plt.show()

But the problem is that not every geometry is appropriate for this. ellipse has two main axis of symmetry. Your figure for example doesn't. So principal components that are been seeking via maximum variance in data doesn't correspond with your example(expected output) axis alignement.

For example your set of points correspond to this variant of components alignment: Your geometry

And for a modificated little more symmetric object. A little more symmetrical figure

Hope i helped

silug0
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