Hello everyone and welcome to the principal component analysis lecture if you want a more mathematical
background and a deeper dive into principle component analysis.
Go ahead and read Section ten point two of an introduction to statistical learning.
Now let's discuss the basic idea behind principle component analysis or PCA.
PCa is an unsupervised artistical technique used to examine the interrelations among a set of variables
in order to identify the underlying structure of those variables.
It is also known sometimes as a general factor analysis where regression determines a line of best fit
to a data set.
Factor Analysis determines several orthogonal lines of best fit to the data set orthogonal just means
at right angles.
Actually those lines are perpendicular to each other.
In the end dimensional space where the end dimensional space is the variable sample space.
There are as many dimensions as there are variables.
So in a dataset with four variables the sample space is for them.
Let's go ahead and see an example.
Here we have some data plotted along to features X and Y and we went ahead and put a regression line
of best fit what we can do now is add an orthogonal line which is just at a right angle to that first
line.
And now we can begin to understand the components of our data set.
Components are linear transformation that chooses a variable system for the data set such that the greatest
variance of the data set comes to lie on the first axes and then likewise the second greatest variance
on the second axes and so on.
And this process allows us to reduce the number of variables used in an analysis and if we look back
at our figure we can see that that first principle line explains the 70 percent of the variation and
then the line orthogonal to that explains 28 percent of the variation.
And right now we have 2 percent of the variation remaining unexplained.
And note that the components are uncorrelated.
Since in the sample space they are orthogonal to each other.
Or at direkte right angles and we can continue this analysis into higher dimensions.
Here we now have three components.
Component 1 that principal component explaining most of the variance and then component 2 and component
3 and a higher that mention.
As we go into the z axes if we use this technique on a dataset with a large number of variables we can
actually compress the amount of explained variation to just a few components.
However the most challenging part of PCa is interpreting the components for a work with Python or going
to do is walk through an example of how to perform PC-DOS sikat learn and will review this sort of theory
as we go along.
So don't worry if you were 100 percent on the theory right now we're going to review it as we go through
the process of performing PCa with Python.
What we're also want to do is usually standardize our data by some sort of scale.
We're going to cover how to do that as well in the next lecture.
Since this algorithm is usually used for analysis of data and not a fully the playable model there won't
be a portfolio project for this topic.
OK thanks everyone and I'll see you at the next lecture.