1 00:00:03,900 --> 00:00:08,700 So in the previous video, we showed you the equations that you have to use for the answer to come a 2 00:00:08,700 --> 00:00:14,240 field update step, however, we want to go into a bit more detail about how we actually get those equations. 3 00:00:15,000 --> 00:00:18,960 So we have seen that the state vector and coverage metrics are updated. 4 00:00:18,960 --> 00:00:25,320 Using these following equations, we calculate a common gain based on this cross covariance term. 5 00:00:25,320 --> 00:00:30,960 Here we use the predictor corrective form to update the current estimates based on how much error we 6 00:00:30,960 --> 00:00:32,100 have in the current system. 7 00:00:32,280 --> 00:00:37,230 And then we can update the uncertainty inside the system that using this equation here. 8 00:00:38,280 --> 00:00:41,560 So the question becomes, where do these equations actually come from? 9 00:00:41,580 --> 00:00:43,890 So this is what we're going to cover inside this video. 10 00:00:45,680 --> 00:00:51,410 So we can start off by expanding the estimated error equation with the innovation correction step. 11 00:00:51,890 --> 00:00:54,030 So this is the estimation error. 12 00:00:54,050 --> 00:00:58,640 So this is how much error there is between the true state of the system and our predicted state of the 13 00:00:58,640 --> 00:00:59,070 system. 14 00:00:59,720 --> 00:01:05,690 So this time here can be broken down into the state estimates before the update as shown here. 15 00:01:06,380 --> 00:01:12,380 Now we can regroup these two terms over here so these two terms can get regrouped to form this equation 16 00:01:12,380 --> 00:01:12,650 here. 17 00:01:13,040 --> 00:01:18,860 And then you should be pretty easy to notice that this is just the estimation error so they can simplify 18 00:01:18,860 --> 00:01:21,340 this down into our exterior. 19 00:01:21,770 --> 00:01:25,640 But this is before we apply the update so I can say this is after the update. 20 00:01:25,650 --> 00:01:27,850 This is how much error we have once we do the correction step. 21 00:01:28,160 --> 00:01:30,750 This is how much error we have before the correction step. 22 00:01:31,280 --> 00:01:34,010 So we basically end up with this recursive relationship here. 23 00:01:35,630 --> 00:01:42,590 Continuing on from that, we can use the definition of the posterior error covariance matrix, so that's 24 00:01:42,590 --> 00:01:43,800 this equation shown here. 25 00:01:43,820 --> 00:01:46,100 So this is the error covariance matrix. 26 00:01:46,100 --> 00:01:52,280 So this is the estimation error squared up for all the measurements up to timestep, including times 27 00:01:52,280 --> 00:01:52,390 up. 28 00:01:52,390 --> 00:02:00,080 Okay, so substituting in our recursive equation that we had on the previous slide, we can form this 29 00:02:00,080 --> 00:02:01,100 equation here. 30 00:02:01,130 --> 00:02:08,390 So we substitute in for Alex Tilde and then we expanded out and we can form this long equation here 31 00:02:08,390 --> 00:02:12,260 with the expectation of right now investigating this equation. 32 00:02:12,260 --> 00:02:15,380 You should be able to see that the few things that we can simplify. 33 00:02:15,890 --> 00:02:21,110 The first thing is that this term over here, this is just equal to the covariance matrix before we 34 00:02:21,110 --> 00:02:21,770 do the update. 35 00:02:21,770 --> 00:02:23,810 So the apriority covariance matrix. 36 00:02:25,360 --> 00:02:28,630 This expectation of right down here is also very common. 37 00:02:28,720 --> 00:02:32,320 This is the innovation conference, so we can just replace this by S.. 38 00:02:32,830 --> 00:02:37,630 So once we substitute in these two values here, we end up with this equation here. 39 00:02:37,870 --> 00:02:44,200 So now this is a recursive equation that links the covariance before the update to the covariance, 40 00:02:44,200 --> 00:02:45,100 after the update. 41 00:02:45,280 --> 00:02:48,730 And it's only a function of these key metrics here. 42 00:02:50,110 --> 00:02:52,660 Now, the common filter is a optimal filter. 43 00:02:52,720 --> 00:02:59,230 So we want to find the solution for K that minimizes the trace of the updated covariance matrix. 44 00:02:59,560 --> 00:03:04,540 Then this will give us the optimum solution, which minimizes the conditional mean squared error. 45 00:03:04,630 --> 00:03:07,840 So this is the original problem of the common filter. 46 00:03:08,560 --> 00:03:10,590 And to do this is fairly straightforward. 47 00:03:10,930 --> 00:03:13,960 We want to minimize the trace of this covariance matrix. 48 00:03:14,080 --> 00:03:19,580 We define the trace of the covariance matrix to be this intermediate value L.. 49 00:03:20,200 --> 00:03:26,770 And then we want to minimize the value l and to do that, we take the derivative of L with respect to 50 00:03:26,770 --> 00:03:29,800 our common game and we want to set it to zero. 51 00:03:29,950 --> 00:03:31,420 So this is what this is doing here. 52 00:03:31,420 --> 00:03:36,760 Derivative of L with respect to the come again we end up with this is the derivative here. 53 00:03:37,240 --> 00:03:41,120 We want to set it to zero because we want to find the point that minimizes solution. 54 00:03:41,140 --> 00:03:43,150 So it's going to be the minimum minimum point. 55 00:03:43,960 --> 00:03:49,480 And then once we have this equation, here is a fairly simple operation just to rearrange it to find 56 00:03:49,480 --> 00:03:49,900 K. 57 00:03:49,900 --> 00:03:52,430 So that's what this equation here gives us. 58 00:03:53,200 --> 00:03:59,860 So the common game matrix for the uncynical to filter is just the covariance matrix for the cross covariance 59 00:03:59,860 --> 00:04:04,000 between the state and the measurement multiplied by the inverse of the measurement. 60 00:04:04,000 --> 00:04:05,080 Uncertainty is. 61 00:04:07,680 --> 00:04:13,260 So now that we've calculate the common gain, we can use that to update the main for the U.S. economy 62 00:04:13,300 --> 00:04:13,670 filter. 63 00:04:14,160 --> 00:04:18,490 So the solution for the update step is simply the predictive corrective arm. 64 00:04:18,510 --> 00:04:20,510 So we have the measurement innovation here. 65 00:04:20,640 --> 00:04:25,890 We multiply it by the common gain to work out how much we need to update the current state by to get 66 00:04:25,890 --> 00:04:28,510 a better estimate using this current information. 67 00:04:29,250 --> 00:04:32,710 So this equation here is exactly the same as all the other common filters. 68 00:04:33,240 --> 00:04:37,850 However, right now, to calculate the common gain, we're going to use this modified equation here. 69 00:04:38,220 --> 00:04:43,620 So to work out the common gain, we're going to have to calculate this across covariance time as part 70 00:04:43,620 --> 00:04:45,900 of the unscented filter update step. 71 00:04:48,190 --> 00:04:53,950 Now to finalize the update, we want to look at how we can update the covariance, as we've already 72 00:04:53,950 --> 00:04:59,360 seen, that the solution for the covariance update is sharing in this recursive relationship here. 73 00:04:59,380 --> 00:05:01,840 So this is what we use to calculate the common gain. 74 00:05:02,380 --> 00:05:06,960 However, now that we know what the common gain is, we can back into this relationship. 75 00:05:07,570 --> 00:05:12,830 And then this gives us this equation here is basically a simplification. 76 00:05:12,850 --> 00:05:20,440 So now we have the covariance matrix before the update, minus K times as times K transpose. 77 00:05:20,470 --> 00:05:25,900 So this is how much we've minimized the uncertainty inside the system by using in the measurement and 78 00:05:25,900 --> 00:05:29,730 then doing this operation here will give us the updated covariance matrix. 79 00:05:30,040 --> 00:05:34,510 And of course, this updated matrix here is going to be smaller than this matrix here because we've 80 00:05:34,510 --> 00:05:36,690 reduced the uncertainty inside the system. 81 00:05:38,380 --> 00:05:44,500 So the derivation of the uncertainty coming up next, it should be fairly straightforward, is very 82 00:05:44,500 --> 00:05:49,320 similar to the steps we use for the linnear and extending common fields. 83 00:05:49,630 --> 00:05:54,520 However, we have to make a few approximations at a higher level, so we end up with a few more general 84 00:05:54,520 --> 00:05:55,210 equations. 85 00:05:55,720 --> 00:06:02,390 Sorry, the common thought estate update equation is just shown here and the common field covariance 86 00:06:02,410 --> 00:06:04,430 update equation is just shown here. 87 00:06:05,320 --> 00:06:10,990 So these equations are very straightforward compared to the extent the computer is just about how we 88 00:06:10,990 --> 00:06:16,270 go about calculating these terms here and we're going to be using the unscented transform to do that.