# Introduction

A method of dimensional reduction is PCA. 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.

#### Step1: 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

#### Step2: Computation of mean of variables

$$\bar{x} = \frac{4 + 8 + 13 + 7}{4}= 8$$ $$\bar{y} = \frac{11 + 4 + 5+ 14}{4}= 8.5$$

#### Step3: 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

$cov(x,x) = \frac{1}{N-1} \sum_{k = 1}^{N}(xik - \bar{xi}) (xij - \bar{xj})$ $=\frac{1}{4-1}[(4- 8)^2 + (8- 8)^2 + (13- 8)^2 + (7- 8)^2]= 14$ $cov(x,y) = \frac{1}{4-1}[(4- 8)(11- 8.5) + (8- 8)(4- 8.5) + (13- 8)(5- 8.5) + (7- 8)(14- 8.5)] = -11$ $cov(x,y) = cov(y, x) = -11$ $cov(y, y) = \frac{1}{4-1}[(11- 8.5)^2 + (4- 8.5)^2 + (5- 8.5)^2 + (14- 8.5)^2]= 23$

(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

$det(S- \lambda I) = (14 - \lambda)(23 - \lambda) - (-11 \times -11)$

lambda = 30.38, 6.61

(ii) Eigen Vectors $$(S- \lambda 1 I)U1 = 0$$ $$(14- \lambda 1)U1 - 11 U2 = 0$$ $$-11 U1 - (23- \lambda 1)U2 = 0$$

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]

Step5: 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

EN1 EN2 EN3 EN4
x 4 8 13 7
y 11 4 5 14

# Compute the mean of variable

np.mean(dataset,1)

x    8.0
y    8.5
dtype: float64


# Compute covariance matrix

cov_mat = dataset.T.cov()
cov_mat

x y
x 14.0 -11.0
y -11.0 23.0

# 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




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