﻿[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:00.00,0:00:08.00,Default,,0000,0000,0000,,To explain how this works, I have to talk about high dimesional Gaussians. Dialogue: 0,0:00:08.00,0:00:13.00,Default,,0000,0000,0000,,These are often called multivariate Gaussians. Dialogue: 0,0:00:13.00,0:00:20.00,Default,,0000,0000,0000,,The mean is now a vector with 1 element for each of the dimensions. Dialogue: 0,0:00:20.00,0:00:23.00,Default,,0000,0000,0000,,The variance here is replaced by what's called a co-variance, Dialogue: 0,0:00:23.00,0:00:27.00,Default,,0000,0000,0000,,and it's a matrix with D rows and D columns, Dialogue: 0,0:00:27.00,0:00:30.00,Default,,0000,0000,0000,,if the dimensionality of the estimate is D. Dialogue: 0,0:00:30.00,0:00:36.00,Default,,0000,0000,0000,,The formula is something you have to get used to. Dialogue: 0,0:00:36.00,0:00:41.00,Default,,0000,0000,0000,,I'm writing it out for you, but you never get to see this again. Dialogue: 0,0:00:41.00,0:00:44.00,Default,,0000,0000,0000,,To tell you the truth, even I have to look up the formula for this class, Dialogue: 0,0:00:44.00,0:00:48.00,Default,,0000,0000,0000,,so I don't have it in my head, and please, don't get confused. Dialogue: 0,0:00:48.00,0:00:52.00,Default,,0000,0000,0000,,Let me explain it to you more intuitively. Dialogue: 0,0:00:52.00,0:00:55.00,Default,,0000,0000,0000,,Here's a 2-dimensional space. Dialogue: 0,0:00:55.00,0:01:00.00,Default,,0000,0000,0000,,A 2-dimensional Gaussian is defined over that space, Dialogue: 0,0:01:00.00,0:01:05.00,Default,,0000,0000,0000,,and it's possible to draw the contour lines of the Gaussian. It might look like this. Dialogue: 0,0:01:05.00,0:01:10.00,Default,,0000,0000,0000,,The mean of this Gaussian is this x0, y0 pair, Dialogue: 0,0:01:10.00,0:01:14.00,Default,,0000,0000,0000,,and the co-variance now defines the spread of the Gaussian Dialogue: 0,0:01:14.00,0:01:17.00,Default,,0000,0000,0000,,as indicated by these contour lines. Dialogue: 0,0:01:17.00,0:01:21.00,Default,,0000,0000,0000,,A Gaussian with a small amount of uncertainty might look like this. Dialogue: 0,0:01:21.00,0:01:25.00,Default,,0000,0000,0000,,It might be possible to have a fairly small uncertainty in 1 dimension, Dialogue: 0,0:01:25.00,0:01:28.00,Default,,0000,0000,0000,,but a huge uncertainty in the other. Dialogue: 0,0:01:28.00,0:01:32.00,Default,,0000,0000,0000,,Huge uncertainty in the x-dimension is small, and the y- dimension is large. Dialogue: 0,0:01:32.00,0:01:36.00,Default,,0000,0000,0000,,When the Gaussian is tilted as showed over here, Dialogue: 0,0:01:36.00,0:01:41.00,Default,,0000,0000,0000,,then the uncertainty of x and y is correlated, which means if I get information about x-- Dialogue: 0,0:01:41.00,0:01:46.00,Default,,0000,0000,0000,,it actually sits over here--that would make me believe that y probably sits Dialogue: 0,0:01:46.00,0:01:50.00,Default,,0000,0000,0000,,somewhere over here. That's called correlation. Dialogue: 0,0:01:50.00,0:01:57.00,Default,,0000,0000,0000,,I can explain to you the entire effect of estimating velocity and using it in filtering Dialogue: 0,0:01:57.00,0:01:59.00,Default,,0000,0000,0000,,using Gaussians like these, Dialogue: 0,0:01:59.00,0:02:01.00,Default,,0000,0000,0000,,and it becomes really simple. Dialogue: 0,0:02:01.00,0:02:05.00,Default,,0000,0000,0000,,The problem I'm going to choose is a 1-dimensional motion example. Dialogue: 0,0:02:05.00,0:02:09.00,Default,,0000,0000,0000,,Let's assume at t = 1, we see our object over here. Dialogue: 0,0:02:09.00,0:02:11.00,Default,,0000,0000,0000,,A t = 2 right over here. Dialogue: 0,0:02:11.00,0:02:14.00,Default,,0000,0000,0000,,A t = 3 over here. Dialogue: 0,0:02:14.00,0:02:20.00,Default,,0000,0000,0000,,Then you would assume that at t = 4, the object sits over here, Dialogue: 0,0:02:20.00,0:02:24.00,Default,,0000,0000,0000,,and the reason why you would assume this is--even though it's just seen these different Dialogue: 0,0:02:24.00,0:02:29.00,Default,,0000,0000,0000,,discrete locations, you can infer from it there is actually velocity that drives the object Dialogue: 0,0:02:29.00,0:02:32.00,Default,,0000,0000,0000,,to the right side to the point over here. Dialogue: 0,0:02:32.00,0:02:35.00,Default,,0000,0000,0000,,Now how does the Kalman filter address this? Dialogue: 0,0:02:35.00,0:02:37.84,Default,,0000,0000,0000,,This is the true beauty of the Kalman filter.