2.1.1. A first example: from 2D to 1D

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2.1.1. A first example: from 2D to 1D

 

To explain how to use the PCA technique to do « dimentionality reduction », we will start with a small example. We have here many points in 2D (2 dimensions) and we want to reduce the dimension to 1D:

 

STARDU~1_img25

 

 
(For the census-income database, we have points in 7D and we want to reduce the coordinates in 3D).  

 

 
The output of the PCA analysis is a set of different perpendicular directions represented in red on the chart below:

 

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Let’s project all the blue points of the database on the first PCA axis (PCA1): the projection is illustrated in this chart in green:

 

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We obtain a new dataset that is represented here in blue:

 

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This new dataset can be “seen” in one dimension along the PCA1 direction:

 

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We just reduced the dimension: the 2D points are now in a one-dimensional-space. During the dimensionality reduction, we didn’t lose too much information because the distance between the (same) point “before” and “after” projection is small. This is thus a good dimensionality reduction.