That was the end of quite a hectic session! Here’s a summary of what you’ve learnt so far.
First, you came to know about PCA and how it is essentially a dimensionality reduction technique. You saw the necessity for doing PCA in a couple of situations like
- A predictive model setup where there are a lot of features to eliminate
- A dataset where you needed to perform EDA and Data Visualisation
You understood how PCA not only helps in resolving the above two issues but has applications in several other areas like noise reduction, finding latent Themes and so on. Then you got a brief understanding of its definition:
It is a statistical procedure that finds principal components or directions that are:
- Linear combination of the original variables
- Are uncorrelated
- Capture Maximum information in the dataset.
After that, you went ahead and learnt some essential linear algebra concepts like vectors and their properties along with their associated operations. Then you studied another tool called matrix multiplication and matrix inverse, both of which proved invaluable in understanding the first fundamental building block of PCA: Basis
Basis is essentially the fundamental units in which you express your data. As you saw in the lecture videos, it is similar to how we use units for physical objects to measure things like height, weight, temperature, etc.
In vectors and vector spaces, we use basis vectors to represent the points in space. You understood how every observation in the space can be represented by scaling and adding the scaled basis vectors. This process is also called a linear combination.
Then you learnt one of the key ideas that helped you connect basis vectors and the idea of dimensionality reduction: using different basis vectors to represent the same points.
From there, you learnt how to change from one basis space to another using matrices. Here’s a list of rules to help you revise the same.
1.) If you’re moving from a basis space B to the standard basis, then the change of basis matrix M is the same as the basis vectors of B written as its column vectors. Therefore, if there is a vector v represented in B and you want to find its representation in the standard basis, then you’d have to perfrom Mv.
2.) If you want to go the other way around, where you have v represented in the standard basis and want to find its representation in B you multiply it by its inverse – M−1v
3.) Finally, if you want to find the change of basis matrix M where you move from two non-standard basis vectors – say from B1 to B2 then you can get that by calculating this value –
B−12B1. Note that in all the above cases, the basis vectors should be represented in the same units.
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