# 1 – Kalman Filter Land RENDER 1 V1

To explain how this works, I have to talk about high-dimensional Gaussians. These are often called Multivariate Gaussians. The mean is now a vector with one element for each of the dimensions. The variance square is replaced by what’s called a covariance and it’s a matrix with D rows and D columns if the dimensionality of the estimator is D, and the formula is something you have to get used to. I’m writing it out for you, but you’ll never get to see this again. Pay the tooth even I have to look up the formula for this class, so I don’t have it in my head and please don’t get confused. Let me explain it to you more intuitively. Here’s a two-dimensional space. A two-dimensional Gaussian is defined over that space and it is possible to draw the contour lines of the Gaussian that might look like this. The mean of this Gaussian is this x_0, y_0 pair and the covariance now defines the spread of the Gaussian as indicated by these contour lines. A Gaussian with small amounts of uncertainty might look like this. It may be possible to have a fairly small uncertainty in one dimension but a huge uncertainty in the other. The uncertainty in the x dimension is small and the y dimension is large, and when the Gaussian is tilted as shown over here, then the uncertainty of x and y is correlated which means if I get information about x that actually sits over here, they would make me believe that y probably sits somewhere over here. That’s called correlation. I can explain to you the entire effect of estimating velocity and using it in filtering using Gaussians like this and it becomes really simple. The problem I’m going to choose is a one-dimensional motion example. Let’s assume at t equals 1, we see our options over here, at t equals 2 right over here, at t equals 3 over here, then you would assume that at t equals 4, the object sits over here. And the reason why you would assume this is even though we just seen these different discrete locations, you can infer from where there’s actually velocity that drives the object to the right side to the point over here. Now, how does the Kalman Filter addresses? This is the true beauty of the Kalman filter