Another kind of component of the singular value
decomposition is, is known as the variance explained.
And this comes from the singular values that are in the d matrix.
Remember the d matrix, is really a diagonal matrix.
It only has elements that are on the diagonal of that matrix.
And you can think of each singular value as representing the percent of
the total variation in your data
set that's explained by that particular component.
And so, and the components are typically ordered so that the first one explains
the most variation as possible, and the
second one explains the second most, et cetera.
And so you can plot the kind of proportion of
variance explained as we have in the in this plot below.
And so here on the left hand side I've just plotted the raw singular values
and you can see that they kind of decrease in value as you go across the columns.
But of course the raw singular
value doesn't really have much meaning cause it's not on an interpretable scale.
So if I divide by the total sum of
all the singular values, then on the right hand side
I've got the, I can interpret it as the
proportion of the variance explained, and you can see that's
exactly the same plot but the y axis has
changed, and so you can see that the first singular
value if you recall it captures kind of the shift
and the mean in between the rows and the columns.
That,
that captures about 40% of the variation in your data.
Alright, so almost almost half of the variation in the
data is explained by a single kind of singular value.
Or you can think of it as a single dimension.
And then, the remaining variation in the data is explained by the other components.