1.18 Machine learning lecture 19  (Page 12/15)

As an extremely simple – as a, sort of, an extremely simplified model of what the dynamics of, like, a plane or object moving in 2-D may look like. So just imagine that you have simulation and you have a radar and you’re tracking blips on your radar and you want to estimate the position, or the state, of the helicopter as just as its XY position and its XY velocity and you have a very simple dynamical model of what the helicopter may do. So this matrix, this just says that XT plus one equals XT plus X star T plus noise, so that’s this first equation. The second equation says that X star T plus one equals 0.9 times X star T plus nine. Yes, this is an amazingly simplified model of what a flying vehicle may look like.

Here’s the more interesting part, which is that with – if you’re tracking a helicopter with some sensor you won’t get to observe the full state explicitly. But just for this cartoon example, let’s say that we get to observe YT, which is CST plus VT where the VT is a random variables – Gaussian random variables with, say, zero mean and a Gaussian noise with covariance given by sigma V. Okay? So in this example let’s say that C is that and – so CST is equal to XY, right? Take this state vector and multiply it by Z, you just get XY. So let’s see what the sensor, maybe a radar, maybe a vision system, I don’t know, something that only gets to observe the position of the helicopter that you’re trying to track.

So here’s the cartoon. So a helicopter may fly through some sequence of states, let’s say it flies through some smooth trajectory, whatever. It makes a slow turn. So the true state is four-dimensional, but I’m just drawing two dimensions, right? So maybe you have a camera sensor down here, or a radar or whatever, and for this cartoon example, let’s say the noise in your observations is larger in the vertical axis than the horizontal axis. So what you get is actually one sample from the sequence of five Gaussians. So you may observe the helicopter there at times step one, observe it there at time step two, observe it there at time three, time four, time five. Okay? All right. So that’s what your – there’s a sequence of positions that your camera estimate gives you. And given these sorts of observations, can you estimate the actual state of the system? Okay? So these orange things, I guess, right? Okay?

These orange things are your observations YT. And test for the state of helicopter every time. Just for it, so the position of the helicopter at every time. Clearly you don’t want to just rely on the orange crosses because that’s too noisy and they also don’t give you velocities, right? So you only observe the subset of the state of variables. So what can you do? So concretely – well, you don’t actually ever get to observe the true positions, right? All you get to do is observe those orange crosses. I guess I should erase the ellipses if I can. Right. You get the idea. The question is given – yeah. You know what I’m trying to do. Given just the orange crosses can you get a good estimate of the state of the system at every time step?