PCA From Scratch In Python

PCA is a method of dimensional reduction. When a machine learning method has a large number of features, we must choose the features that contribute the most and ignore the datasets that are less significant. This is where PCA comes into play. Consequently, PCA is an unsupervised method for choosing useful datasets from a huge collection of features. It is a statistical technique that converts the observations of correlated features into a set of linearly uncorrelated features via an orthogonal transformation.

PCA on Numerical Dataset

Given the following data use PCA to reduce the dimension from 2 to 1.

Feature	EN.1	EN.2	EN.3	EN.4
x	4	8	13	7
y	11	4	5	14

Following are the steps to obtained the required number of PCA.

Step 1: Dataset

Feature	EN.1	EN.2	EN.3	EN.4
x	4	8	13	7
y	11	4	5	14

No.of feature, n = 2

No.of Sample, N = 4

Step 2: Computation of mean of variables

Step 3: Computation of covariance matrix

ordered paired are,
(x, x), (x, y), (y, x), (y, y)

If we have n variables then we have $$n^2$$ ordered pair.

(i) Covariance of all ordered pairs

(ii) Covariance Matrix

cov(x,x)	cov(x,y)
cov(y,x)	cov(y,y)

14	-11
-11	23

Step 4:Eigen value, Eigen vector, Normalized Eigen vector

(i) Eigen Values

lambda = 30.38, 6.61

(ii) Eigen ectors

Hence,normalized eigen vector u1 of 𝜆1 i.e e1 [-0.83025082, 0.55738997]

for 𝜆2 normalized eigen vector i.e e2 [-0.55738997, -0.83025082]

Step 5: Derive New Dataset

Feature	EN.1	EN.2	EN.3	EN.4
pc1	p11	p12	p13	p14

for pc1:
$$ p11 = e1^T[-4, 2.5]^T = -4.008989537218968 $$
$$ p12 = e1^T[0, 3.5]^T = 3.7361286866113304$$
$$ p13 = e1^T[5, -2.5]^T = 0.1539328264398951$$
$$ p14 = e1^T[-1, 6]^T = 0.1189280241677419$$

Feature	EN.1	EN.2	EN.3	EN.4
pc1	-4.008	3.736	0.153	0.118

Import Necessary Module

import numpy as np
import pandas as pd

DataSet

#Understanding the mathematics behind PCA
A = np.matrix([[4,8,13,7],
               [11,4,5,14]])

A.shape

(2, 4)

dataset = pd.DataFrame(A,columns  = ['EN1','EN2','EN3','EN4'], index = ["x","y"])
dataset

Compute the mean of variable

np.mean(dataset,1)

x    8.0
y    8.5
dtype: float64

Compute co-variance matrix

cov_mat = dataset.T.cov()
cov_mat

Computation of eigenvalues and eigenvectors

w, v = np.linalg.eig(cov_mat)

w

array([ 6.61513568, 30.38486432])

v

array([[-0.83025082,  0.55738997],
       [-0.55738997, -0.83025082]])

v[1:].T
v[1:].shape

(1, 2)

Computation of first principal component

p11 = np.dot(v[1:],  np.array([[4-8],[11-8.5]]))
p11

array([[0.15393283]])

p12 = np.dot(v[1:],  np.array([[8-8],[4-8.5]]))
p12

array([[3.73612869]])

p13 = np.dot(v[1:],  np.array([[13-8],[5-8.5]]))
p13

array([[0.11892802]])

p14 = np.dot(v[1:],  np.array([[7-8],[14-8.5]]))
p14

array([[-4.00898954]])

pc1 = [p11,p12,p13,p14]

pd.Series(pc1)

0    [[0.1539328264398951]]
1    [[3.7361286866113304]]
2    [[0.1189280241677419]]
3    [[-4.008989537218968]]
dtype: object