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In this video, we're going to investigate the Jokowi matrixes and actually define what they are.

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So the Jacobean matrix is I am by and matrix for the function f x, whose elements are the partial derivatives

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of the M outputs of the function with respect to the end inputs of the function.

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So what that means is if we have a function here, takes a input vector and it produces an output vector,

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the input vector X is going to have an inputs.

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So we're going to have a vector with one, two, three all the way up to any inputs and the function

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F is going to output M outputs.

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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

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like this.

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So if we look here we can see that the size of this matrix here, we're going to have M outputs down

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here.

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So we're going to have F1, F2 all the way down to form a partial derivatives.

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And then across the top here, we're going to have the end inputs so we can see we have x1 X to adapt

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X and so we have the partial derivative of x1 x2 all the way up to X.

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And so we can pretty much say that each of the elements inside this matrix follows this relationship

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here.

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So the IJA element of this matrix here is going to be the partial derivative of the output.

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If I with respect to the input state x j.

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So now linking this to the process model state Jacobean, the previous model gave you an F is simply

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the curbing of the process model given by this equation here with respect to the state vector X and

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it's evaluated about the current best estimate, state X.

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So the Matrix is going to be attained by a matrix where the N is a number of states inside this function

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here.

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So we can say the F matrix is going to be the Jacobean matrix and it's just going to be the partial

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derivative of the state transition matrix f with respect to different states.

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And we're going to evaluate this around the current best estimate X.

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We can also say the same for the process model noise, Cobian, so the process model noise you Cobian

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in for the ill is simply the Jakobsson of the process model shown here with respect to noise.

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Victus with respect to this factor here, and again, it's going to be evaluated about the current best

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estimate of the state.

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So X, K minus one.

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So we're going to evaluate it around this position here.

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So the process model you Cobian is going to be in biometrics where N is a number of states and L is

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number of noise components.

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So looking at the definition here, we can say that L is the process model Noge Cobian.

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So it's the Jakobsson of the process model f respect to the noise input w so it is a partial derivative

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of f respect to w evaluated at where X is equal to the best current state estimate.

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So now let's actually have a look at more of an example.

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So if we have an example for the process model state Jacobean, we have the process model that we've

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been looking at showing here and now we want to work out the Jacobean of this process model with respect

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to the state vector X.

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So we're saying that we can form the state Eucumbene by differentiating the output with respect to different

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input states.

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So we can say in this matrix here we have the different outputs.

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So we have V.K. Sike p.

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S p y.

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So that's the different states here.

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And then we have the partial derivatives with respect to the states across the top as well.

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So we have the V K minus one side, minus one minus one, P Wykeham minus one.

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So these are going to be the input states.

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So that's going to be this time here.

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So basically we can evaluate each of these individual terms.

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So we go back to our process model equations so we can actually write this out as expanded set of equations.

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So we get these equations here.

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So now if we want to work out the different terms, the equations, we can actually differentiate these

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equations, define these terms here so we can say that this is the output.

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V.K., with respect to the input, V.K. minus one.

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So that's basically saying we want to take this equation here for V.K. and differentiate it with respect

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to our V.K. minus one.

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And if we do that, we can see that derivative or this is simply just going to be one we can do the

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same thing with this time here.

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So we differentiate the same equation with respect to six minus one.

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So I differentiate this work with respect to CYB minus one.

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We can see that we don't have any of that time there.

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So it's going to be zero.

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And the same thing here is going to be zero and going to be zero here because these terms do not apply

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this equation here.

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So if we do this for all the different elements inside this Jacobean here, we end up with this Jacobean

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matrix here.

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And you can see that the when we actually evaluate this Jacobean, we're evaluating it around a current

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state estimate so we can see that we have these nonlinear timesaver cosmic minus one.

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So this is what it means when we evaluate it around our ICSC minus one every time that we change a current

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state estimate.

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So this that here we're gonna have to re update the Giacobbe matrix because these elements inside the

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describing are going to change.

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And we can do the similar prices for the noisy Caribbean as well.

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And again, we have exactly the same price, this model.

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But now we want to work out that you came in with respect to the noise components shown here.

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So the president's model noised Caribbean is can be written like this.

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We have the different outputs so we can see CapEx KPK.

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So these are the so these are the outputs of the process model.

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And then across the top here, we have the different input derivatives that we want to differentiate

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respect to.

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So we have the different noise vector components.

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So we have all the noise for I've lost a noise for our heading, a noise for our position on noise for

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our Y position.

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So this vector here.

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So this Jakobsson here is going to be pretty simple to calculate.

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You can see that on these equations here we only have additive noise, so the noise is only added onto

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the end.

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So obviously if we do all that differentiation, we're just going to end up with an identity matrix

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for the noise Jacobean.

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So this is fairly simple.

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This is just showing us that we have additive noise.

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We can add the noise covariance directly onto the previous model covariance.