[Behind the ML] Principal Component Analysis II.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import tensorflow as tf
CSV_COLUMN_NAMES = [‘SepalLength’, ‘SepalWidth’, ‘PetalLength’ , ‘PetalWidth’, ‘Species’]SPECIES = [‘Setosa’, ‘Versicolor’, ‘Virginica’]
train_path = tf.keras.utils.get_file("iris_training.csv", "https://storage.googleapis.com/download.tensorflow.org/data/iris_training.csv")test_path = tf.keras.utils.get_file("iris_test.csv", "https://storage.googleapis.com/download.tensorflow.org/data/iris_test.csv")train = pd.read_csv(train_path, names=CSV_COLUMN_NAMES, header=0)
test = pd.read_csv(test_path, names=CSV_COLUMN_NAMES, header=0)
train.head()train_y = train.pop('Species')
test_y = test.pop('Species')
# The label column has now been removed from the features.train.head()

0. Exploratory Data Analysis with simple data visualizations

import seaborn as snsimport matplotlib.pyplot as plt%matplotlib inlinesns.pairplot(train,hue=’Species’,size=2.6)
train_y = train.pop(‘Species’)test_y = test.pop(‘Species’)# The label column has now been removed from the features.train.head()

1. Normalize

from sklearn.preprocessing import StandardScaler 
X_std = StandardScaler().fit_transform(x)
x_scaled = (x-x.mean(axis = 0))/x.std(axis = 0)

2. Eigendecomposition

  • 1) Mean value of the scaled data
  • 2) The dot product between the transpose of (x_scaled — mean)and (x_scaled — mean)
  • 3) Dividing the value into (n-1), and ’n’ is the number of units(rows) in x_scaled, which can be reached with python code “x_scaled.shape[0]”.
mean_vec = np.mean(x_scaled, axis = 0) # 1
dotP = (x_scaled — mean_vec).T.dot(x_scaled-mean_vec) # 2
n = x_scaled.shape[0] # 3
cov_mat = dotP / (n-1)
eig_vals, eig_vecs = np.linalg.eig(cov_mat)
u,s,v = np.linalg.svd(x_scaled.T)u

3. Sorting out principal components

  • 1) (eigenvalue, eigenvector) pair, let’s name this as ‘eigen_pair’.
  • 2) Sorting out the ‘eigen_pair’ depending on the eigenvalue which each can be reached with the code ‘eigen_pair[0]’.
  • 3) A total sum of eigenvalues
  • 4) Percentage calculation to show how much variance has been explained with the eigenvector
eig_pairs = [(np.abs(eig_vals[i]), eig_vecs[:,i]) for i in range(len(eig_vals))] # 1eig_pairs.sort(key=lambda x: x[0], reverse=True) # 2total_sum = sum(eig_vals) # 3variance_explained = [(i/total_sum)*100 for i in sorted(eig_vals, reverse = True)] # 4

4. Projection Matrix

  • 1) The first eigenvector with the highest eigenvalue
  • 2) The second-highest eigenvalue’s eigenvector
  • 3) Concatenating the two eigenvectors column-wise, more powerful one comes first.
the_highest = eig_pairs[0][1] # 1second_highest = eig_pairs[1][1] # 2matrix_w = np.hstack((the_highest.reshape(4,1),second_highest.reshape(4,1)))# 3

5. Migration from the existing data space into the newly constructed feature space using the projection matrix

y = x_scaled.dot(matrix_w)

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