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PRINCIPAL COMPONENT
ANALYSIS
Soham Patra
CSE-B
13000120121
PEC-IT501B

It is an unsupervised learning algorithm that is used for
the dimensionality reduction in machine learning
Unsupervised learning – discovering patterns in the data
set without human help
example - clustering
Dimensionality Reduction –
Types:
1. Feature Elimination – eliminating some of the features to reduce
feature space
2. Feature Extraction – create new features where each new feature
is a combination of the old features
What is Principal Component Analysis ?

PCA converts correlated features into a set of
uncorrelated features with the help of orthogonal
transformation.
These new features are called Principal Components.
It is a technique to draw strong patterns from the given
dataset by reducing the no of variables.
PCA Algorithm is based on mainly two mathematical
concepts:
1. Variance and Covariance
2. Eigenvalues and Eigenvector
What is Principal Component Analysis ?

Lets consider a dataset in which we compare 6 algorithms to solve a problem on the
basis of Time complexity, Space Complexity and Lines of Code
Algo 1 Algo 2 Algo 3 Algo 4 Algo 5 Algo 6
Time
Complexity 9 8 6 3 3 1
Space
Complexity 7 8 9 5 3 4
Lines of
Code 7 8 7 2 4 2
1
2
5
4
6
3
Example

Algo 1 Algo 2 Algo 3 Algo 4 Algo 5 Algo 6
Time
Complexity 4 3 1 -2 -2 -4
Space
Complexity 1 2 3 -1 -3 -2
Lines of
Code 2 3 2 -3 -1 -3
So in order to standardize the data we need to find the mean vector and subtract it from
each point
Mean for Time complexity = (9+8+6+3+3+1) / 6 = 5
Mean for Space Complexity = (7+8+9+5+3+4) / 6 = 6
Mean for Lines of Code = (7+8+7+2+4+2) / 6 = 5
Mean vector = (5,6,5)
So the new data set looks like this:
1
2
5
4
6
3
Standardize the Data

So now we need to form a n * m matrix from the data given, let it be matrix A
(n = no of dimensions variables
A = m = no of points )
Now the covariance matrix C = A * AT
C = * =
4 3 1 -2 -2 -4
1 2 3 -1 -3 -2
2 3 2 -3 -1 -3
4 1 2
3 2 3
1 3 2
-2 -1 -3
-2 -3 -1
-4 -2 -3
50 29 39
29 28 26
39 26 36
Finding the Covariance Matrix
4 3 1 -2 -2 -4
1 2 3 -1 -3 -2
2 3 2 -3 -1 -3

To find the eigenvalues of a n*n matrix we need to solve the following equation:
det(C – 𝝀I) =0
– = =
= – 𝝀3 + 114 𝝀 2 – 1170 𝝀 + 2548 = 0
Solving the above equation gives us the roots as
𝝀 1 = 102.86
𝝀 2 = 8.06
𝝀 3 = 3.07
Finding the Eigenvalues
50 29 39
29 28 26
39 26 36
50 - 𝝀 29 39
29 28 - 𝝀 26
39 26 36 - 𝝀
(50 – 𝝀) [ (28- 𝝀)(36- 𝝀) – 26*26 ]
+ 29 [ 26*39 – 29*(36- 𝝀) ]
+ 39 [ 29*26 – (28- 𝝀)*39 ]
Since the eigenvalue 𝝀 3 is very less it can be left out and we can carry
out the operations with 𝝀 1 and 𝝀 2
𝝀 0 0
0 𝝀 0
0 0 𝝀

Lets find the first eigenvector by solving the equation:
Cx = 𝝀1x
x = 102.86
Let z = 1,
eq1 ⇒ – 52.86 x + 29 y + 39 = 0………...eq4
eq2 ⇒ 29 x – 74.86 y + 26 = 0…………..eq5
solving these we get
y = 0.682 x………eq6
Replacing eq6 in eq4 :
– 52.86 x + 29 * 0.682 x + 39 = 0 ⇒ x = 1.178
Finding the Eigenvectors
50 29 39
29 28 26
39 26 36
x
y
z
x
y
z
Solving this gives us three equations:
– 52.86 x + 29 y + 39 z = 0………..eq1
29 x – 74.86 y + 26 z = 0………….eq2
39 x + 26 y – 66.86 z = 0………….eq3
Therefore the eigenvector for the eigenvalue 102.86 is
(1.178 , 0.803 , 1)
This is also known as Principal Component 1 (PC1)

50 29 39
29 28 26
39 26 36
Let’s find the second eigenvector by solving the equation:
Cx = 𝝀2x
x = 8.06
Let z = 1,
eq1 ⇒ 41.94 x + 29 y + 39 = 0……………eq4
eq2 ⇒ 29 x + 19.94 y + 26 = 0……………eq5
solving these we get
y = – 1.714 x………eq6
Replacing eq6 in eq4 :
41.94 x + 29 * (– 1.714) x + 39 = 0 ⇒ x = 5.021
x
y
z
x
y
z
Solving this gives us three equations:
41.94 x + 29 y + 39 z = 0………….eq1
29 x + 19.94 y + 26 z = 0………….eq2
39 x + 26 y + 27.94 z = 0………….eq3
Therefore the eigenvector for the eigenvalue 8.06 is
(5.021 , – 8.605 , 1)
This is also known as Principal Component 2 (PC2)
50 29 39
29 28 26
39 26 36
Finding the Eigenvectors

So to get the axes of the new 2D plot we need to find the unit vector along the principal components:
Unit vector along PC1 =
(1.178,0.803,1)
(1.178)2 + (0.803)2 + (1)2
= ( 0.676, 0.461, 0.574 )
Unit vector along PC2 =
(5.021,−8.605,1)
(5.021)2 + (−8.605)2 + (1)2
= ( 0.501, – 0.859, 0.099)
Getting the new axes

Plotting the Data on the new graph
TC SC LOC
PC1
(0.676,0.471,
0.574)
PC2
(0.501, -0.859,
0.099)
Algo 1 9 7 7 13.399 -0.811
Algo 2 8 8 8 13.768 -2.072
Algo 3 6 9 7 12.313 -4.032
Algo 4 3 5 2 5.531 -2.594
Algo 5 3 3 4 5.737 -0.678
Algo 6 1 4 2 3.708 -2.737
1
2
3
5
4
6
PC2
PC1

Variation in PCA for each PC is determined by dividing the eigenvalue by sample size -1
Variation for PC1 = 102.86/5 = 20.572
Variation for PC2 = 8.06/5 = 1.612
Variation for PC3 = 3.07/5 = 0.614
Total variation = 22.798
Percentage Variation:
PC1 = 20.572 / 22.798 % = 90.2 %
PC2 = 1.612 / 22.798 % = 7.07 %
PC3 = 0.614 / 22.798 % = 2.73 %
Variation in PCA
90.2
7.07
2.73
0
10
20
30
40
50
60
70
80
90
100
PC 1 PC 2 PC 3
SCREE PLOT

Application
The principal component analysis is a widely used unsupervised learning method to
perform dimensionality reduction.
1. used for finding hidden patterns if data has high dimensions like in finance, data
mining, Psychology
2. Used in image compression
3. Used in noise cancellation

Advantages and Disadvantages
Advantages:
1. Less misleading data means model accuracy improves.
2. Fewer dimensions mean less computing. Less data means that algorithms train faster.
3. Less data means less storage space required.
4. Removes redundant features and noise.
5. Dimensionality Reduction helps us to visualize the data that is present in higher
dimensions in 2D or 3D
Disadvantages:
1. While doing dimensionality reduction, we lost some of the information, which can
possibly affect the performance of subsequent training algorithms.
2. It can be computationally intensive.
3. Transformed features are often hard to interpret.
4. It makes the independent variables less interpretable.

References
1. https://towardsdatascience.com/a-one-stop-shop-for-principal-component-analysis-5582fb7e0a9c
2. https://www.youtube.com/watch?v=FgakZw6K1QQ
3. https://www.youtube.com/watch?v=FgakZw6K1QQ
4. https://www.javatpoint.com/principal-component-analysis
5. https://www.simplilearn.com/tutorials/machine-learning-tutorial/principal-component-analysis
6. https://www.analyticsvidhya.com/blog/2021/05/20-questions-to-test-your-skills-on-dimensionality-
reduction-pca/
7. https://www.wolframalpha.com/
8. https://c3d.libretexts.org/CalcPlot3D/index.html
9. https://research.google.com/colaboratory/

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