The next topic is Variance Covariance
Matrices.
[COUGH]
So this is tool that you use to, if you're
building an, sorry, if you're
building, portfolio of assets, you sort of
want to know how risky they are.
You have a measure of risk for each asset.
But what also ends up being useful is the
measure of risk between assets.
So sometimes you'll see if you have assets
in
the same sector, so if you have two
technologies stocks.
So, for instance, like Oracle or
Microsoft.
If technology stocks are doing good, they
both go up together.
Not exactly the same amount, but when one
of them goes up,
the other one is much more likely to go up
than go down.
And and vice versa for when they go down.
sometimes you have things that are
negatively correlated, so I'm trying to
think.
There were all sorts of good ex examples
of this one.
You know, in 2007,
2008, you had things like McDonald's and
Spam going up.
And more expensive types of food going
down.
So when you had a shock that made.
Say, I don't know, what's an expensive
type of food?
[LAUGH]
Lobster.
Yeah, lobster maybe.
Maybe lobster goes down.
that would be an indication that maybe you
should buy McDonalds.
The stock, not the food.
[LAUGH]
[COUGH]
So I'll start off with some concepts from
statistics something
called the sample mean, I hope everybody's
aware of.
So if I have a vector x and it's got m
elements, m
components in it, and I want to just find
the average value of those.
What I'm going to do is just, so this
notation
means, sum, so I'm going to sum all of
those up.
So add all m of these values together and
divide by m, I'm
going to call that x bar and that's going
to be called the sample mean.
The sample variance of the elements in x,
I'm going to get by, so
I need my sample mean from step one, or
from part one up here.
I subtract that from each element of my
vector, x.
I square that, that difference.
And then I add up all of the squared
differences.
And this time I have to divide by m minus
1.
And that I'm going
to call the sample variance of my vector
x.
And then suppose y is also a vector of
length m.
That I can define something called the
sample covariance of x and y.
So looks kind of like the variance except
instead of having
this thing squared, I just have x oops x
minus its mean.
Times y minus its mean.
I form that product and then I sum up over
all so over each of the m components.
And then again for that I have to divide
by m minus 1.
So these things are sample means, sample
variants, sample covariants.
And so now let's look.
So let's let e be a column vector of m1,
so this is just, you
know, 11111 down a column, so what if I do
e transpose x?
This is the same thing as e.x, that would
be one times
x1 plus one times x2 plus one times x3 and
so on.
So that's another way of writing the sum
of all
of the elements, all of the components in
this vector x.
And then m is a scaler, so putting this
over m is,
this is just going to give me another way
to write x bar.
So this is sort of my way of writing dark
product that is 1 times x1,
1 times x2 and so on, and then 1 over m,
so I divide by m.
And that was just the definition of the
sample mean I had on the previous slide.
So I can also write the sample mean as e
transpose x divided by m.
And x bar is going to be a scaler.
So when I do this a dot product gives me
scaler and then I'm dividing that by m,
which is scaler.
So this is just going to be oops, a single
real number.
And then I can make a column vector with x
bar in each position just by multiplying e
by x bar.
Normally, I would want to write this with
the, the scalar value times
the vector, but I'm going to write it as
the vector times the scalar value.
I mean the same thing, but it's just
going to be a little bit nicer
when I try and use some other calculations
later if it comes in this order.
So this will be a column vector.
Where each element is x bar.
So now I want to make a new vector that
I'm going to call x tilda, and
that's going to be equal to x, so that was
my original vector, minus,
now remember this thing.
This fraction here, that's just x bar.
And then I'm going to multiply that times
this column vector of 1's e.
So that's going to give me x minus ex bar,
so if I look at that, that vector
difference now, the first element is
going to be x1 minus x bar, the second
element is going to be x2 minus x bar.
And so on.
And so the i'th element is x tilde i, is
xi minus x bar.
And so now if I go back and look at my
sample variance formula,
I have xi minus x bar.
Well, that's just x tilde i.
I can't highlight.
So I can replace that, this square
difference
with just xi squared or x tilde i squared.
But another way of writing that is x tilda
transpose x
tilda and now I'm dividing that still by m
minus 1.
And again, this is just a dot product, so
this is, what I do x transpose x where x
is a column vector, that's going to be a
scalar
value, so that's just the matrix notation
for dot product.
And then, I'm dividing that by scalar n
minus 1 and
that's going to be the number that's the
sample variance of x.
And another way I could write that, is
this is the length squared of x divided
by m minus 1.
So really what I'm kind of trying to do
here is, show
that there are lots of different ways to
approach the same problem.
And then what we'll do later on is we're
going to pick the
way that makes it easiest to get to the
answer that we want.
So I can do the same thing now for the,
the sample covariance.
So I'll make, oops.
e transpose y is just going to be the sum
of all of the elements in my vector y.
Divided by m, that's going to give me the
mean value,
the sample mean of the elements of my
vector y.
So, y tilde is just going to be y, minus
its mean.
And the sample covariance then.
So I have x minus its mean, y minus its
mean.
That's just x tilde I, y tilda I, and I
can write that
as x tilde transpose y tilda divided by m
minus 1.
So this again is a dot product or an inner
product divided by a scalar.
So it's a scalar divided by a scalar, so
that's the
number that gives me the sample covariance
of x and y.
And so I started out in, in this section,
I don't know if
everybody remembered from the, from the
outline
I called this a variance covariance
matrix.
and that's basically because the variance
of x
goes, ends up being down the, the
diagonal.
So if I have x and y, if y is equal to x,
I
get a variance and if y is different than
x, I get a covariance.
I'm just going to think now of the
variance
of x as being the covariance of x with
itself.
And then only call this thing a
co-variance matrix from now
on, because it gets really difficult to
pronounce variance, co-variance matrix.
Also it makes the slides wider than they
need to be.
So the Cov x,x that's going to give the,
the
variance but from now on I'm just going to
think of
this whole structure as a covariance
structure, and sometimes I
just end up calculating covariances of x
with itself, okay.
So now let's suppose that x and y, I could
put those in the columns of a matrix r.
So here r, I chose r just because it's a
rectangular matrix.
So we have m rows and 2 columns.
The first row is my vector x, second row
is my vector y.
I can now make a matrix called r tilde.
So I know how to subtract off the sample
mean from each element of
my vector now, and that gives me these
vectors x tilda and y tilda.
So why don't I put those in a matrix and
call that r tilda?
And sample variance covariance matrix.
So that the notation, remember I'm just
going to
call this a covariance matrix, kind of,
after this set of slides.
So I'll write Cov of r, where r is this
matrix here, is going to be the, on
the diagonal so I have, I think of this
column, the first column as being the x
column.
The second column as being the y column,
and the first row
as being the x row and second row as being
the y row.
And then where each
of those intersect that tells me what I'm
going to put in the.
In that element of the matrix, so here, I
have the x
row intersecting the x column, so I put
covariance of x, x.
Here, I have the x row intersecting the y
column, so I put covariance of x, y.
Here, I have the y row intersecting the x
column, so I get Co of y, x.
And then Cov y, y for the, for the bottom
right slot.
And another way I can write this now, I've
already said that Cov x, x,
that was just x tilda transpose x tilde
divided by 1 over m minus 1.
And so I'm just going to fill in these.
I'll replace these Covs with these.
products divided by 1 over m minus 1.
But then I can factor that 1 over m minus
1 out, and just stick that in front of my
matrix.
And so it turns out that this matrix is
also just the matrix product.
r tilde transpose, r tilda.
So if I take this matrix r tilde, if I
take the transpose of it, then I'm
going to get one row that's equal to the
vector x, and the second row equal to the.
Err, sorry, first row equal to x tilde,
second row
equal to y tilde, and then when I do r
transpose
r, these are the elements of that 2 by 2
matrix I'll end up with.
And then because I have to divide by this
m minus
1, I'll have r transpose r divided by m
minus 1.
And so this is a sort of more
matrixy expression for the covariance
structure of this matrix.
[INAUDIBLE]
r that has column x and column y.
And I'm going to say that this matrix is
symmetric.
So that would imply that x tilda transpose
y
tilda is equal to y tilda transpose x
tilda.