We have seen in the past that the linear transform of a one dimensional Gaussian distribution is just
another Gaussian distribution, with the mean and variance transformed.
So if we have a random variable X that belongs to a Gaussian distribution with I mean, of X bar and
a variance of sigma squared X, if we multiply this Gaussian distribution and transform it by a linear
transform of A, X plus B, we end up with a new Gaussian distribution with the main shifted based on
the transformation.
And the variance is just going to be scaled by the square of a scaling factor.
Now we can extend this concept to a multidimensional Gaussian as well.
So let's suppose we have a random vector X and this random vector is pulled from a multidimensional
Gaussian distribution with Amane of our exposure and covariance of C of X.
Let's suppose that we want to transform this Gaussian distribution using the linear transform of X plus
B, so A here is going to be a transformation matrix multiplied by our random vector plus an offset
vector.
So this function here, we're going to call it G.
This is the transformation that we want to apply to this Gaussian distribution up here.
So the first step is to work out our inversed mapping function, so g inverse, and that's just going
to be our inverse, a Thom's Y minus our inverse A time is B.
So this gives us our inverse mapping function and then we can differentiate this mapping function with
respect to Y just to end up without inverse A..
Now we can use the relationship that we've covered in the past, which allows us to perform a mathematical
transformation of a probability density function from one to the other.
So it allows us to apply a function to the density function.
So filling in the terms here, we know the probability density function is going to be our equation
for our Gaussian multidimensional distribution.
We have our derivative of our investment function.
So if we fill all this in, we end up with this equation here.
So we find out this is a probability of the multidimensional Gaussian distribution after is undergone
our transformation of X plus B, so now inspecting these terms here, we can have a look.
You can say now this is a function of why we know our Y bar is now just going to be X bar plus B, we
also know that the covariance of this relationship, of this multidimensional Gaussian distribution
is now going to be this term here.
So this new covariance of the covariance y is just going to be a times variance matrix times our transpose.
So by looking at these terms here, we can see that the linear transform of a multidimensional Gaussian
distribution is just again a linear transform of the main and covariance parameters.
So the linear transform of Gaussian PDF is just another Gaussian UPDF with the mean and variance transformed.
So if we have this random variable vector X here and it comes from a multidimensional Gaussian distribution
described by these parameters here, and we want to apply a linear transformation of X plus B, we end
up with this Gaussian distribution described by these parameters here.
So.
So we just transform the main and we just transform the covariance using this relationship.
So this is pretty important.
It tells us if C X represents the uncertainty covariance, then it can be transformed into a number
of frame using the linear transformation Y is equal to X..
So if we have the covariance in frame X and we want to transform it to frame Y, and we know the transformation
matrix, we can transform the covariance using this relationship here.
So covariance and why is this going to be eight times covariance in X times a transpose?
So we're going to use this transformation relationship quite a lot later on inside our common fiodor.
So this is an important relationship and an important outcome of this process here.
So if we have a random variable vector X and it undergoes a transformation and we know the original
covariance before the transformation, we can work out the uncertainty after the transformation so we
can transform the uncertainty from one frame together.
Just like that, we can transform a vector from one frame to another.