1 00:00:00,000 --> 00:00:08,000 To explain how this works, I have to talk about high dimesional Gaussians. 2 00:00:08,000 --> 00:00:13,000 These are often called multivariate Gaussians. 3 00:00:13,000 --> 00:00:20,000 The mean is now a vector with 1 element for each of the dimensions. 4 00:00:20,000 --> 00:00:23,000 The variance here is replaced by what's called a co-variance, 5 00:00:23,000 --> 00:00:27,000 and it's a matrix with D rows and D columns, 6 00:00:27,000 --> 00:00:30,000 if the dimensionality of the estimate is D. 7 00:00:30,000 --> 00:00:36,000 The formula is something you have to get used to. 8 00:00:36,000 --> 00:00:41,000 I'm writing it out for you, but you never get to see this again. 9 00:00:41,000 --> 00:00:44,000 To tell you the truth, even I have to look up the formula for this class, 10 00:00:44,000 --> 00:00:48,000 so I don't have it in my head, and please, don't get confused. 11 00:00:48,000 --> 00:00:52,000 Let me explain it to you more intuitively. 12 00:00:52,000 --> 00:00:55,000 Here's a 2-dimensional space. 13 00:00:55,000 --> 00:01:00,000 A 2-dimensional Gaussian is defined over that space, 14 00:01:00,000 --> 00:01:05,000 and it's possible to draw the contour lines of the Gaussian. It might look like this. 15 00:01:05,000 --> 00:01:10,000 The mean of this Gaussian is this x0, y0 pair, 16 00:01:10,000 --> 00:01:14,000 and the co-variance now defines the spread of the Gaussian 17 00:01:14,000 --> 00:01:17,000 as indicated by these contour lines. 18 00:01:17,000 --> 00:01:21,000 A Gaussian with a small amount of uncertainty might look like this. 19 00:01:21,000 --> 00:01:25,000 It might be possible to have a fairly small uncertainty in 1 dimension, 20 00:01:25,000 --> 00:01:28,000 but a huge uncertainty in the other. 21 00:01:28,000 --> 00:01:32,000 Huge uncertainty in the x-dimension is small, and the y- dimension is large. 22 00:01:32,000 --> 00:01:36,000 When the Gaussian is tilted as showed over here, 23 00:01:36,000 --> 00:01:41,000 then the uncertainty of x and y is correlated, which means if I get information about x-- 24 00:01:41,000 --> 00:01:46,000 it actually sits over here--that would make me believe that y probably sits 25 00:01:46,000 --> 00:01:50,000 somewhere over here. That's called correlation. 26 00:01:50,000 --> 00:01:57,000 I can explain to you the entire effect of estimating velocity and using it in filtering 27 00:01:57,000 --> 00:01:59,000 using Gaussians like these, 28 00:01:59,000 --> 00:02:01,000 and it becomes really simple. 29 00:02:01,000 --> 00:02:05,000 The problem I'm going to choose is a 1-dimensional motion example. 30 00:02:05,000 --> 00:02:09,000 Let's assume at t = 1, we see our object over here. 31 00:02:09,000 --> 00:02:11,000 A t = 2 right over here. 32 00:02:11,000 --> 00:02:14,000 A t = 3 over here. 33 00:02:14,000 --> 00:02:20,000 Then you would assume that at t = 4, the object sits over here, 34 00:02:20,000 --> 00:02:24,000 and the reason why you would assume this is--even though it's just seen these different 35 00:02:24,000 --> 00:02:29,000 discrete locations, you can infer from it there is actually velocity that drives the object 36 00:02:29,000 --> 00:02:32,000 to the right side to the point over here. 37 00:02:32,000 --> 00:02:35,000 Now how does the Kalman filter address this? 38 00:02:35,000 --> 00:02:37,838 This is the true beauty of the Kalman filter.