In this video, we're going to investigate the Jokowi matrixes and actually define what they are.
So the Jacobean matrix is I am by and matrix for the function f x, whose elements are the partial derivatives
of the M outputs of the function with respect to the end inputs of the function.
So what that means is if we have a function here, takes a input vector and it produces an output vector,
the input vector X is going to have an inputs.
So we're going to have a vector with one, two, three all the way up to any inputs and the function
F is going to output M outputs.
So I can say we have these F1, F2 or up to M outputs now that you come in of this function can be written
like this.
So if we look here we can see that the size of this matrix here, we're going to have M outputs down
here.
So we're going to have F1, F2 all the way down to form a partial derivatives.
And then across the top here, we're going to have the end inputs so we can see we have x1 X to adapt
X and so we have the partial derivative of x1 x2 all the way up to X.
And so we can pretty much say that each of the elements inside this matrix follows this relationship
here.
So the IJA element of this matrix here is going to be the partial derivative of the output.
If I with respect to the input state x j.
So now linking this to the process model state Jacobean, the previous model gave you an F is simply
the curbing of the process model given by this equation here with respect to the state vector X and
it's evaluated about the current best estimate, state X.
So the Matrix is going to be attained by a matrix where the N is a number of states inside this function
here.
So we can say the F matrix is going to be the Jacobean matrix and it's just going to be the partial
derivative of the state transition matrix f with respect to different states.
And we're going to evaluate this around the current best estimate X.
We can also say the same for the process model noise, Cobian, so the process model noise you Cobian
in for the ill is simply the Jakobsson of the process model shown here with respect to noise.
Victus with respect to this factor here, and again, it's going to be evaluated about the current best
estimate of the state.
So X, K minus one.
So we're going to evaluate it around this position here.
So the process model you Cobian is going to be in biometrics where N is a number of states and L is
number of noise components.
So looking at the definition here, we can say that L is the process model Noge Cobian.
So it's the Jakobsson of the process model f respect to the noise input w so it is a partial derivative
of f respect to w evaluated at where X is equal to the best current state estimate.
So now let's actually have a look at more of an example.
So if we have an example for the process model state Jacobean, we have the process model that we've
been looking at showing here and now we want to work out the Jacobean of this process model with respect
to the state vector X.
So we're saying that we can form the state Eucumbene by differentiating the output with respect to different
input states.
So we can say in this matrix here we have the different outputs.
So we have V.K. Sike p.
S p y.
So that's the different states here.
And then we have the partial derivatives with respect to the states across the top as well.
So we have the V K minus one side, minus one minus one, P Wykeham minus one.
So these are going to be the input states.
So that's going to be this time here.
So basically we can evaluate each of these individual terms.
So we go back to our process model equations so we can actually write this out as expanded set of equations.
So we get these equations here.
So now if we want to work out the different terms, the equations, we can actually differentiate these
equations, define these terms here so we can say that this is the output.
V.K., with respect to the input, V.K. minus one.
So that's basically saying we want to take this equation here for V.K. and differentiate it with respect
to our V.K. minus one.
And if we do that, we can see that derivative or this is simply just going to be one we can do the
same thing with this time here.
So we differentiate the same equation with respect to six minus one.
So I differentiate this work with respect to CYB minus one.
We can see that we don't have any of that time there.
So it's going to be zero.
And the same thing here is going to be zero and going to be zero here because these terms do not apply
this equation here.
So if we do this for all the different elements inside this Jacobean here, we end up with this Jacobean
matrix here.
And you can see that the when we actually evaluate this Jacobean, we're evaluating it around a current
state estimate so we can see that we have these nonlinear timesaver cosmic minus one.
So this is what it means when we evaluate it around our ICSC minus one every time that we change a current
state estimate.
So this that here we're gonna have to re update the Giacobbe matrix because these elements inside the
describing are going to change.
And we can do the similar prices for the noisy Caribbean as well.
And again, we have exactly the same price, this model.
But now we want to work out that you came in with respect to the noise components shown here.
So the president's model noised Caribbean is can be written like this.
We have the different outputs so we can see CapEx KPK.
So these are the so these are the outputs of the process model.
And then across the top here, we have the different input derivatives that we want to differentiate
respect to.
So we have the different noise vector components.
So we have all the noise for I've lost a noise for our heading, a noise for our position on noise for
our Y position.
So this vector here.
So this Jakobsson here is going to be pretty simple to calculate.
You can see that on these equations here we only have additive noise, so the noise is only added onto
the end.
So obviously if we do all that differentiation, we're just going to end up with an identity matrix
for the noise Jacobean.
So this is fairly simple.
This is just showing us that we have additive noise.
We can add the noise covariance directly onto the previous model covariance.