Background

## [1] "correlation between X1 and X2: 0.9"

Background

## [1] "correlation between X1 and X2: 0.9"

Which is better?

Goal

working example

(cont'd)

suppose we have two data points, the goal can be written as:

$d_1^2+d_2^2=(a_1\ast e11+b_1\ast e12{)}^2+(a_2\ast e11+b_2\ast e12{)}^2$

$$=e^T.U\Sigma U^T.e=(e^T.U)\Sigma (U^T.e)=n^T\Sigma n$$

Steps:

1. Center the values of each variable in the dataset on 0 (by subtracting the mean of the variable's observed values from each of those values)

2. Compute the covariance matrix of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix

3. Normalize each of the orthogonal eigenvectors to turn them into unit vectors

The scree plot

The profile plot

• shows the correlations between each PC and the original variables
• will become very crowded and hard to read when have many variables

The score plot

projection of data onto the span of PC1 and PC2

The loadings plots

shows the relationship between the PCs and the original variables

The biplot

1. Image compression

## (156, 194, 3)

## (156, 194, 3)

References

[1] https://blogs.sas.com/content/iml/2019/11/06/what-are-biplots.html#:~:text=A%20biplot%20is%20an%20overlay%20of%20a%20score,plots%20and%20overlay%20them%20on%20a%20single%20plot.

[2] https://www.zhihu.com/question/41120789/answer/481966094

[3] https://en.wikipedia.org/wiki/Principal_component_analysis#Intuition